TY - JOUR
AU - Troquard,, Nicolas
AB - Abstract To introduce agent-based technologies in real-world systems, one needs to acknowledge that the agents often have limited access to resources. They have to seek after resource objectives and compete for those resources. We introduce a class of resource games where resources and preferences are specified with the language of a resource-sensitive logic. The agents are endowed with a bag of resources and try to achieve a resource objective. For each agent, an action consists in making available a part of their endowed resources. All the resources made available can be used towards the agents’ objectives. We study three decision problems, the first of which is deciding whether an action profile is a Nash equilibrium: when all the agents have chosen an action, it is a Nash Equilibrium if no agent has an incentive to change their action unilaterally. When dealing with resources, interesting questions arise as to whether some equilibria can be eliminated or constructed by a central authority by redistributing the available resources among the agents. In our economies, division of property in divorce law exemplifies how a central authority can redistribute the resources of individuals and why they would desire to do so. We thus study two related decision problems: (i) rational elimination: given an action profile’s outcome, can the endowed resources be redistributed so that it is not the outcome of a Nash equilibrium? (ii) Rational construction: given an action profile’s outcome, can the endowed resources be redistributed so that it is the outcome of a Nash equilibrium? Among other results, we prove that all three problems are |$\mathsf{PSPACE}$|-complete when the resources are described in the very expressive language of the propositional multiplicative and additive linear logic. We also identify a new modest fragment of linear logic that we call MULT, suitable to represent multisets and reason about the inclusion and equality of bags of resources. We show that when the resources are described in MULT, the problem of deciding whether a profile is a Nash equilibrium is in |$\textsf{PTIME}$|⁠. 1 Introduction Agents, or players, are entities capable of action and trying to reach their goals. In the physical or cyber world, these agents have limited access to resources. They have to seek after resource objectives and compete for those resources. This paper makes use of resource-sensitive logics, linear logic [16] specifically, to model and solve problems of rational agents interacting in a resource-aware environment. We use linear logic to define and reason about a new class of non-cooperative games [30]. Every linear logic formula represents a resource. In these games, each agent is endowed with a bag of resources and has an objective to achieve by transforming the resources made available from the agents’ endowed resources. Can we decide whether the resources made available by the agents constitute a Nash equilibrium, i.e. whether it is locally optimal under individual strategic considerations? If a local optimal is not desirable, could an arbitrator redistribute the resources in the endowments among the agent so that it is not a Nash equilibrium anymore, thus eliminating it? To the contrary, if an outcome is desirable, could an arbitrator redistribute the resources so that it becomes the outcome of a Nash equilibrium, thus constructing it? In this paper, we are going to address the computational complexity of the decision problems corresponding to these questions. As we study the computational complexity of answering these questions about resource-sensitive game theoretical interactions, we will be particularly interested in a few varying parameters: What kind of preferences the agents have? 1. Do they only care about reaching their resource objectives? (dichotomous) 2. Do they also care about how much resource the consume? (parsimonious) What is the exact language for talking about resources, and what is the complexity of reasoning about resources in this language? Which resource-sensitive logic exactly is used to reason about the resources? – Can resources be disposed of freely during reasoning? (affine reasoning) – Must all resources be accounted for during reasoning? (linear reasoning) This paper is putting together: 1. Linear logic, which enables the specification of resources and the reasoning about them. 2. Game theory and Nash equilibria, which give us a guideline to characterize normatively good outcome in games whose actions and preferences are defined in terms of the resources expressed in linear logic. 3. Computational complexity, which helps us towards an algorithmic treatment of our resource games. It is intimately affected by the precise linear logic used to represent the resources. The models and the algorithms presented here can be used as analytical tools at the disposition of actors and policy makers, for instance in interconnected economies [2, 7]. They can serve at gauging the possible strategic behaviours of the actors and of their competitors and at identifying possible issues of resource scarcity in a commons. Our games are reminiscent of notable models existing in the literature. They share the logic-based approach of Boolean games [4, 19]. In Boolean games, each player controls a set of Boolean variables and produces truth values which can be used without restriction towards the Boolean goals. As such, resources proper are absent from Boolean games. Our games also share the resource-sensitiveness of congestion games [39]. In congestion games, the players choose a set of resources (e.g. edges to travel in a graph), and their utility depends on the cost (e.g. delays) of using the shared resources, which depends on the number of players travelling them. Despite some apparent similarities, they are rather superficial. One thing should be obvious: the resources in congestion games are limited to basic resources and lack a rich specification language of resources like the one of resource-sensitive logics. Using resource-sensitive logic languages to represent goods that are transformed and exchanged between agents owes to previous work, e.g. [17, 33, 34], in multiagent systems and computational social choice [6, 12]. A short version of this paper appeared as [42]. Logic: exploiting resource-sensitive logics. In this paper, we study games of resources that are aimed at representing the strategic interactions between rational agents where some combinations of resources replace the abstract notions of action and preferences. In these games, players are endowed with some resources and have preferences upon some resources to be available after the game is played. Players’ actions also consist in making available some of the resources they are endowed with. We propose a class of games of resources that exploits the formalisms and reasoning methods coming from the literature in knowledge representation and computational logics, namely resource-sensitive logics: e.g. linear logic, separation logic and BI logic [16, 29, 38]. The languages of these logics allow a fine-grained description of resources, processes and their harmonious combinations. In computer science, they have been quite successful at modelling systems for multi-party access and modification of shared structures, by allocation and deallocation of resources. The resources used in this paper are not based on a trivial and naïve set theory. Instead, they are based on rich logical languages, supported by elaborate reasoning features. A resource is represented by one formula of a resource-sensitive logic LOG. More specifically, we assume here that LOG is some propositional variant of linear logic. We provide an informal presentation of the resource interpretation of linear logic in Section 2 so that the conceptual aspects of the paper can be grasped without a great understanding of linear logic. Game theory: individual resource games. We will consider individual resource games defined formally in Section 3. Each player |$i$| of a game will be endowed with a multiset of resources |$\epsilon _i$|⁠. An action for Player |$i$| will be to contribute a subset of |$\epsilon _i$|⁠. An (action) profile specifies a contribution for every player. An outcome will be a context consisting of a multiset of resources resulting from a profile. Then, each Player |$i$| has a goal |$\gamma _i$|⁠, which is a resource, represented by one formula of LOG. An outcome |$X$| satisfies the goal of Player |$i$| if there is a proof of |$X \vdash \gamma _i$| in the logic LOG. This will mean that the resources in |$X$|can be consumed so as to produce |$\gamma _i$|⁠.1 Intuitively, we can imagine a game taking place around a table. Each player has an objective to create some resource. Each player has also a bag of resources. To play, each player chooses to take some resources (possibly none) from their respective bags and put them on the table in front of them. The outcome is the collection of resources on the table after every player has chosen. A player is satisfied if we can transform the resources on the table so as to produce her goal. It is a Nash equilibrium when no player has an incentive to take back any resources she put on the table, or to add more resources from her bag. What should be an incentive to take back or to add resources? We will study these games of resources with two kinds of preferences. We will first consider, in Section 4, preferences over outcomes that are dichotomous. We can thus initially say that Player |$i$| prefers an outcome |$X$| over an outcome |$Y$| iff |$X \vdash \gamma _i$| and |$Y \not \vdash \gamma _i$|⁠. Some formal results will lead us to define in Section 5, parsimonious preferences, a finer notion of preference where |$i$| may be qualitatively indifferent between |$X$| and |$Y$|⁠, but still prefer |$X$| over |$Y$| because |$i$|’s contribution is strictly less in |$X$| than in |$Y$|⁠. Algorithms and complexity: solving problems. We will study three decision problems defined also in Section 3, the first of which is deciding whether an action profile is a Nash equilibrium. A Nash equilibrium is, under strategic considerations, a local optimal. A situation in which every agent has picked an action is a Nash equilibrium when no agent has an incentive to change their mind. A variant of this example, with one additional player, will be formalized later in Section 6.3. Example 1.1 In a local telecom industry, anti-trust laws forbid a priori cooperation, and regulations oblige the companies to accept traffic from each other. (These telecom companies operate in an interconnected economy [2, 7].) Consider two competing telecommunication companies. Company |$A$| manages a 3G network of comprised capacity |$3$| (bundled as capacities |$1$| and |$2$|⁠). Company |$B$| manages a 4G network of capacity |$3$| (bundled as capacities |$1$| and |$2$|⁠). Company |$A$| needs to offer their customers 3G at capacity |$2$| and 4G at capacity |$1$|⁠. Company |$B$| needs to offer their customers 3G at capacity |$2$| and 4G at capacity |$2$|⁠. Activating a network at some capacity has a cost. Companies can privately activate and deactivate networks on the fly. What are the possible equilibria? There are two Nash equilibria. First, there is the one where Company |$A$| provides a bundle of two 3G antennas and Company |$B$| provides a bundle of two 4G antennas. Both companies can achieve their goal, and none has an incentive to reduce their contribution as they would not satisfy their goal anymore. Second, there is the one where both Company |$A$| and Company |$B$| contribute nothing. None of them has an incentive to change their contribution since they would not be able to achieve their goal on their own. When dealing with resources, interesting questions arise as to whether some equilibria can be eliminated or constructed by a central authority by redistributing the available resources among the players [18]. In the tradition of social mechanism design, redistribution schemes can be used by a central authority to enforce some behavior, either by disincentivizing a behavior or incentivizing a behavior. Formal frameworks dealing with redistribution schemes and economic policies have been studied [11, 25, 28]. Some profiles that are not equilibria can have desirable outcomes. Some equilibria can have outcomes that are undesirable. Desirability must here be understood from the point of view of a system designer. A system designer can redistribute the resources of the players in a game so as to steer the interaction to or away from a particular outcome. A redistribution consists in reallocating the resources endowed to the players. To every redistribution corresponds a new game where the players maintain their objectives, but their possible actions have changed. If |$G^\epsilon $| is the original game, and |$\epsilon ^{\prime}$| is a redistribution of the endowment function |$\epsilon $|⁠, then |$G^{\epsilon ^{\prime}}$| is a new game. We will investigate how resource distribution schemes can contribute to eliminate undesirable game equilibria and construct desirable game equilibria. They are a form of redistribution of wealth, which consists in wealth being transferred from some individuals to others. In our economies, it exists in the form of social mechanisms such as taxation, public services and confiscation. Division of property and division of debt in divorce law are good imagery of what a designer can do in the mechanisms we propose in this paper. This example will be formalized later in Section 6.2. Example 1.2 Ann and Bernard, a couple of bakers, have filed for divorce. Ann is officially the tenant of the business premises of the bakery. Bernard is the owner of the baking equipment. He also owns enough flour to make bread for two years. Ann would like to be able to keep the means of production and being able to make bread for one year. Bernard wants to keep the shop. In this context, if Ann and Bernard are parsimonious, the outcome is very likely to be the one where Ann does not use the shop and Bernard does not use the breadmaking equipment and the flour. It is the only equilibrium. Neither of them satisfy their objective. However, an arbitrator can redistribute their endowments. He can give the equipment and half the flour to Ann and give the shop to Bernard. Doing so, the outcome where Ann and Bernard do not use any of their endowment can be eliminated. Moreover, a new outcome equilibrium can be constructed where both satisfy their objectives. We will thus look at two decision problems related to Nash equilibria: rational elimination and rational construction of Nash equilibria. In a game |$G^{\epsilon }$|⁠, a profile can be rationally eliminated from a game if there exists a redistribution |$\epsilon ^{\prime}$| of |$\epsilon $| such that there is no profile with the same outcome which is a Nash equilibrium in |$G^{\epsilon ^{\prime}}$|⁠. A profile can be rationally constructed if there exists a redistribution |$\epsilon ^{\prime}$| such that there is a profile in |$G^{\epsilon ^{\prime}}$| with the same outcome, which a Nash equilibrium. Outline. We make a brief presentation of linear logic in Section 2. We explain how the language can be used to capture a variety of resources which we will put to use in the remainder of the paper. We present individual resource games formally in Section 3. We also introduce precisely the decision problems NASH EQUILIBRIUM, RATIONAL ELIMINATION and RATIONAL CONSTRUCTION. We will use and study two kinds of preferences over action profiles. We define dichotomous preferences in Section 4. We study all three decision problems. We propose general algorithms and general complexity results depending on the complexity of sequent provability in LOG and on whether LOG admits the weakening rule or not (i.e. whether LOG is linear or affine). We do the same for parsimonious preferences in Section 5. We also illustrate the decision problems with a few small examples. We present more thorough examples in Section 6. In particular, we formalize Example 1.2 in Section 6.2, and a variant Example 1.1 in Section 6.3, and we illustrate the findings of this paper on them. Some concluding remarks are offered in Section 7. We provide a technical appendix. Specifically, Appendix A presents the sequent rules of the biggest fragment of linear logic used in the paper. Appendix B briefly summarizes some elements of computational complexity that can be useful to the reader. 2 Resources and linear logic One contribution of this paper is to show that resource-sensitive logics are a useful tool for studying the formal aspects of resources in game theoretical settings. Another contribution is to demonstrate that it is possible to obtain rather general results for a large class of games of resources depending on the formal properties of the logic LOG we start with. This offers the opportunity to tailor a game to the needs of a certain application without changing the framework. We can indeed choose any sensible fragment of a resource-sensitive logic. We will work with some fragments of linear logic [16]. The conceptual aspects of the paper can be grasped without a great understanding of linear logic, but the technical results will draw upon the proof theory and its rules presented in the Appendix A. A basic understanding of logic is thus necessary to follow the proofs in general, and some intuitions about the resource interpretation of linear logic can hopefully contribute to make reading through the remainder of this paper less dull. 2.1 Formulas and sequents A good introduction to linear logic and its variants is [41]. We will use logics defined on the language of propositional linear logic. The classical tautology splits into the additive |$\top $| and the multiplicative |$\textbf{1}$|⁠. The classical falsum splits into the additive |$\textbf{0}$| and the multiplicative |$\bot $|⁠. The additive conjunction and disjunction are, respectively, & and |$\oplus $|⁠. The multiplicative conjunction and disjunction are, respectively, and |$\otimes $|⁠. The linear implication is |$A \multimap B$| and combines with the multiplicative conjunction such that |$(A \otimes (A \multimap B)) \multimap B$| is a valid principle. The linear negation is |$\mathop \sim A$|⁠. MLL is the multiplicative fragment, whose language is formalized by the grammar where |$p$| is an atomic formula. It only contains the multiplicative connectives. MALL is the fragment with both additive and multiplicative operators . We now introduce some terminology and notations. A sequent is a statement |$ \varGamma \vdash \varDelta $| where |$\varGamma $| and |$\varDelta $| are finite multisets of occurrences of formulas of |${\textsf{LOG}}$|⁠. Often, we can conveniently write a multiset |$\{A_1, \ldots , A_n\}$| as the list of formulas |$A_1, \ldots , A_n$|⁠. Also, we use the notation |$\varGamma ^* = \bigotimes _{A \in \varGamma } A$| and |$\emptyset ^* = \textbf{1}$|⁠. An intuitionistic sequent is a sequent |$\varGamma \vdash A$| with only one formula to the right. Sequent provability will play an important part in the technical work of the paper. A sequent |$\varGamma \vdash \varDelta $| is provable in LOG if there exists a linear proof using the rules of the logic LOG. Intuitively, |$\varGamma \vdash \varDelta $| being provable means that the resources in |$\varGamma $| can be transformed into either of the resources in |$\varDelta $|⁠. If a sequent |$\varGamma \vdash \varDelta $| is not provable, we can write |$\varGamma \not \vdash \varDelta $|⁠, although we will also often simply write ‘not |$\varGamma \vdash \varDelta $|’. Section 2.4 summarizes the computational complexity characterizations of a few fragments of linear logic in terms sequent provability. In the individual resource games introduced in this paper, the action of a player |$i$| consists in making available a multiset |$C_i$| of formulas/resources. The outcome of an action is the multiset union of all the individual actions: |$\varGamma = \biguplus _i C_i$|⁠.2 The goal of a player is a formula/resource |$\gamma $|⁠. To decide whether the profile with outcome |$\varGamma $| satisfies the goal |$\gamma $| of a player, we will evaluate the provability of the (intuitionistic) sequent |$\varGamma \vdash \gamma $|⁠. The logic captured by all the rules in the Appendix A is Affine MALL. A rule that is not part of the calculus is the structural rule of contraction. One rule of contraction (left contraction) says that if something can be proved with two occurrences of |$A$|⁠, then it can be proved with only one occurrence. Symbolically, |$ $| This is prohibited in every resource-sensitive logic. Integrating it into linear logic, one consequence would be that |$A \vdash A \otimes A$|⁠. If we interpret formulas as resources—as we do—contraction would be a license to duplicate resources at will. (See [37] for a detailed account of logics without contraction.) We must concede that some of the connectives of MLL and MALL do not have an intuitive interpretation in terms of resources, in and of themselves. This is the case of the multiplicative and the additive falsums (⁠|$\bot $|⁠, |$\textbf{0}$|⁠) and of the somehow infamous multiplicative disjunction . Fortunately, we do not need them to enjoy the full expressivity of linear logic. To see that, Table 1 shows how the connectives interact. From it, it is clear that we can as well make without some language redundancy. The resource-interpretable language of MLL is $$\begin{equation*}A::= \textbf{1} \mid p \mid \mathop \sim A \mid A \otimes A \mid A \multimap A \enspace,\end{equation*}$$ Table 1 Remarkable relationship between the linear logic connectives. The symbol |$\dashv \vdash $| indicates provability in both directions. |$\mathop \sim \mathop \sim A$| . |$\dashv \vdash $| . |$A$| . ∼(A & B) |$\dashv \vdash $| |$(\mathop \sim A) \oplus (\mathop \sim B)$| AB |$\dashv \vdash $| |$(\mathop \sim A) \multimap B$| |$\mathop \sim (A \otimes B)$| |$\dashv \vdash $| |$(\mathop \sim A)$||$(\mathop \sim B)$| |$A$||$\bot$| |$\dashv \vdash $| |$A$| |$A \otimes \textbf{1}$| |$\dashv \vdash $| |$A$| |$A$| & |$\top $| |$\dashv \vdash $| |$A$| |$A \oplus \textbf{0}$| |$\dashv \vdash $| |$A$| |$\textbf{0}$| |$\dashv \vdash $| |$\mathop \sim \top $| |$\boldsymbol{\bot }$| |$\dashv \vdash $| |$\mathop \sim \textbf{1}$| |$\mathop \sim \mathop \sim A$| . |$\dashv \vdash $| . |$A$| . ∼(A & B) |$\dashv \vdash $| |$(\mathop \sim A) \oplus (\mathop \sim B)$| AB |$\dashv \vdash $| |$(\mathop \sim A) \multimap B$| |$\mathop \sim (A \otimes B)$| |$\dashv \vdash $| |$(\mathop \sim A)$||$(\mathop \sim B)$| |$A$||$\bot$| |$\dashv \vdash $| |$A$| |$A \otimes \textbf{1}$| |$\dashv \vdash $| |$A$| |$A$| & |$\top $| |$\dashv \vdash $| |$A$| |$A \oplus \textbf{0}$| |$\dashv \vdash $| |$A$| |$\textbf{0}$| |$\dashv \vdash $| |$\mathop \sim \top $| |$\boldsymbol{\bot }$| |$\dashv \vdash $| |$\mathop \sim \textbf{1}$| Open in new tab Table 1 Remarkable relationship between the linear logic connectives. The symbol |$\dashv \vdash $| indicates provability in both directions. |$\mathop \sim \mathop \sim A$| . |$\dashv \vdash $| . |$A$| . ∼(A & B) |$\dashv \vdash $| |$(\mathop \sim A) \oplus (\mathop \sim B)$| AB |$\dashv \vdash $| |$(\mathop \sim A) \multimap B$| |$\mathop \sim (A \otimes B)$| |$\dashv \vdash $| |$(\mathop \sim A)$||$(\mathop \sim B)$| |$A$||$\bot$| |$\dashv \vdash $| |$A$| |$A \otimes \textbf{1}$| |$\dashv \vdash $| |$A$| |$A$| & |$\top $| |$\dashv \vdash $| |$A$| |$A \oplus \textbf{0}$| |$\dashv \vdash $| |$A$| |$\textbf{0}$| |$\dashv \vdash $| |$\mathop \sim \top $| |$\boldsymbol{\bot }$| |$\dashv \vdash $| |$\mathop \sim \textbf{1}$| |$\mathop \sim \mathop \sim A$| . |$\dashv \vdash $| . |$A$| . ∼(A & B) |$\dashv \vdash $| |$(\mathop \sim A) \oplus (\mathop \sim B)$| AB |$\dashv \vdash $| |$(\mathop \sim A) \multimap B$| |$\mathop \sim (A \otimes B)$| |$\dashv \vdash $| |$(\mathop \sim A)$||$(\mathop \sim B)$| |$A$||$\bot$| |$\dashv \vdash $| |$A$| |$A \otimes \textbf{1}$| |$\dashv \vdash $| |$A$| |$A$| & |$\top $| |$\dashv \vdash $| |$A$| |$A \oplus \textbf{0}$| |$\dashv \vdash $| |$A$| |$\textbf{0}$| |$\dashv \vdash $| |$\mathop \sim \top $| |$\boldsymbol{\bot }$| |$\dashv \vdash $| |$\mathop \sim \textbf{1}$| Open in new tab and the resource-interpretable language of MALL is |$ $| It suffices to see the other connectives as definitions, following the equivalences of Table 1. We define |$\bot = \mathop \sim \textbf{1}$|⁠, |$\textbf{0} = \mathop \sim \top $| and |$A$||$B = (\mathop \sim A) \multimap B$|⁠. 2.2 Resources as propositions A resource captured by a proposition of linear logic can be atomic like one mole of hydrogen |$\mathsf{H}$| or one mole of oxygen |$\mathsf{O}$|⁠. It can be a simultaneous combination of resources, e.g. |$\mathsf{O} \otimes \mathsf{O}$| being two moles of oxygen. A resource can be a process transforming resources, e.g. |$\mathsf{H_2O} \otimes \mathsf{H_2O} \multimap \mathsf{H_2} \otimes \mathsf{H_2} \otimes O_2$| would be the well-known chemical reaction of electrolysis. It consumes two moles of water to produce two moles of dihydrogen and one mole of dioxygen. Working harmoniously with resources and resource transformation processes with this meticulous control over their combination is made possible using resource-sensitive logics. In a game where a player is endowed with |$2q$| moles of water and a player is endowed with |$q$| processes of electrolysis, it is possible to consume these resources and produce |$2q$| moles of hydrogen gas and |$q$| of oxygen gas. But not more! In Section 6.1, we will illustrate our games with an example using chemical reactions. But for the time being, we explain in more details how the refined operators of linear logic can be used to formalize and grasp a variety of resources. Table 2 reports possible readings of the connectives. Table 2 Possible resource interpretations of formulas. |$A \otimes B$| . |$A$| and |$B$| simultaneously . A & B A deterministic choice between |$A$| and |$B$|⁠; not both |$A \oplus B$| |$A$| or |$B$| non-deterministically; not both |$A \multimap B$| |$A$| is sufficient to produce |$B$| (losing |$A$| in the process) |$\textbf{1}$| Vacuous resource |$\top $| Some resource |$A \otimes B$| . |$A$| and |$B$| simultaneously . A & B A deterministic choice between |$A$| and |$B$|⁠; not both |$A \oplus B$| |$A$| or |$B$| non-deterministically; not both |$A \multimap B$| |$A$| is sufficient to produce |$B$| (losing |$A$| in the process) |$\textbf{1}$| Vacuous resource |$\top $| Some resource Open in new tab Table 2 Possible resource interpretations of formulas. |$A \otimes B$| . |$A$| and |$B$| simultaneously . A & B A deterministic choice between |$A$| and |$B$|⁠; not both |$A \oplus B$| |$A$| or |$B$| non-deterministically; not both |$A \multimap B$| |$A$| is sufficient to produce |$B$| (losing |$A$| in the process) |$\textbf{1}$| Vacuous resource |$\top $| Some resource |$A \otimes B$| . |$A$| and |$B$| simultaneously . A & B A deterministic choice between |$A$| and |$B$|⁠; not both |$A \oplus B$| |$A$| or |$B$| non-deterministically; not both |$A \multimap B$| |$A$| is sufficient to produce |$B$| (losing |$A$| in the process) |$\textbf{1}$| Vacuous resource |$\top $| Some resource Open in new tab Now, whether the occurrence of a resource indicates a consumption or a production of the resources depends on where a formula appears in the sequent. The sequent of linear logic $$\begin{equation*}A \vdash B\end{equation*}$$ can be read as $$\begin{equation*}\text{`if you give {$A$} you can receive {$B^{\prime}$}} \ .\end{equation*}$$ Hence, as it should be expected, we give the resources at the left of the sequent and receive the resources at the right of the sequent. Table 3 reports possible readings of the sequents. Table 3 Possible resource interpretations of sequents. |$\varGamma \vdash A \otimes B$| . Receive |$A$| and |$B$| simultaneously . |$\varGamma \vdash A$| & |$B$| Choose whether to receive |$A$| or |$B$|⁠; you can’t receive both |$\varGamma \vdash A \oplus B$| Receive |$A$| or |$B$|⁠; you don’t choose; you won’t receive both |$\varGamma \vdash A \multimap B$| Receive a resource that can be used in such a way that, if you give |$A$|⁠, you receive |$B$| (losing |$A$| in the process) |$A \otimes B \vdash \varDelta $| Give |$A$| and |$B$| simultaneously |$A$| & |$B \vdash \varDelta $| Choose whether to give |$A$| or |$B$|⁠; you don’t give both |$A \oplus B \vdash \varDelta $| Give |$A$| or |$B$|⁠; you don’t choose; you don’t give both |$A \multimap B \vdash \varDelta $| Give a resource that can be used in such a way that, if you give |$A$|⁠, you receive |$B$| (losing |$A$| in the process) |$\varGamma \vdash A \otimes B$| . Receive |$A$| and |$B$| simultaneously . |$\varGamma \vdash A$| & |$B$| Choose whether to receive |$A$| or |$B$|⁠; you can’t receive both |$\varGamma \vdash A \oplus B$| Receive |$A$| or |$B$|⁠; you don’t choose; you won’t receive both |$\varGamma \vdash A \multimap B$| Receive a resource that can be used in such a way that, if you give |$A$|⁠, you receive |$B$| (losing |$A$| in the process) |$A \otimes B \vdash \varDelta $| Give |$A$| and |$B$| simultaneously |$A$| & |$B \vdash \varDelta $| Choose whether to give |$A$| or |$B$|⁠; you don’t give both |$A \oplus B \vdash \varDelta $| Give |$A$| or |$B$|⁠; you don’t choose; you don’t give both |$A \multimap B \vdash \varDelta $| Give a resource that can be used in such a way that, if you give |$A$|⁠, you receive |$B$| (losing |$A$| in the process) Open in new tab Table 3 Possible resource interpretations of sequents. |$\varGamma \vdash A \otimes B$| . Receive |$A$| and |$B$| simultaneously . |$\varGamma \vdash A$| & |$B$| Choose whether to receive |$A$| or |$B$|⁠; you can’t receive both |$\varGamma \vdash A \oplus B$| Receive |$A$| or |$B$|⁠; you don’t choose; you won’t receive both |$\varGamma \vdash A \multimap B$| Receive a resource that can be used in such a way that, if you give |$A$|⁠, you receive |$B$| (losing |$A$| in the process) |$A \otimes B \vdash \varDelta $| Give |$A$| and |$B$| simultaneously |$A$| & |$B \vdash \varDelta $| Choose whether to give |$A$| or |$B$|⁠; you don’t give both |$A \oplus B \vdash \varDelta $| Give |$A$| or |$B$|⁠; you don’t choose; you don’t give both |$A \multimap B \vdash \varDelta $| Give a resource that can be used in such a way that, if you give |$A$|⁠, you receive |$B$| (losing |$A$| in the process) |$\varGamma \vdash A \otimes B$| . Receive |$A$| and |$B$| simultaneously . |$\varGamma \vdash A$| & |$B$| Choose whether to receive |$A$| or |$B$|⁠; you can’t receive both |$\varGamma \vdash A \oplus B$| Receive |$A$| or |$B$|⁠; you don’t choose; you won’t receive both |$\varGamma \vdash A \multimap B$| Receive a resource that can be used in such a way that, if you give |$A$|⁠, you receive |$B$| (losing |$A$| in the process) |$A \otimes B \vdash \varDelta $| Give |$A$| and |$B$| simultaneously |$A$| & |$B \vdash \varDelta $| Choose whether to give |$A$| or |$B$|⁠; you don’t give both |$A \oplus B \vdash \varDelta $| Give |$A$| or |$B$|⁠; you don’t choose; you don’t give both |$A \multimap B \vdash \varDelta $| Give a resource that can be used in such a way that, if you give |$A$|⁠, you receive |$B$| (losing |$A$| in the process) Open in new tab The linear negation allows one to switch the give/receive mode. The sequent |$A \vdash \mathop \sim B$| represents ‘give |$A$| and |$B$|⁠, and receive nothing’. The sequent |$A, \mathop \sim B \vdash \bot $| represents ‘give |$A$| and receive |$B$|’. Example 2.1 A few items can be obtained from vending machine in exchange of money. For instance, giving |$\mathsf{\$1}$| you can choose to receive a chocolate bar or a soft-drink. This is captured by |$ $| Also, giving |$\mathsf{\$0.8}$| you can receive |$2$| packs of gum. This is captured by $$\begin{equation*}\$0.8 \vdash \mathsf{gum} \otimes \mathsf{gum}\enspace.\end{equation*}$$ In the previous example, the formula |$\mathsf{chocobar}$| & |$\mathsf{drink}$| denotes a deliberative choice between |$\mathsf{chocobar}$| and |$\mathsf{drink}$|⁠. One and the other can be obtained from |$\mathsf{\$1}$|⁠, but not both. This is significantly different from |$\mathsf{\$1} \vdash \mathsf{chocobar} \oplus \mathsf{drink}$| which denotes something more akin to the classical disjunction: |$\mathsf{chocobar}$| or |$\mathsf{drink}$| can be obtained from |$\mathsf{\$1}$|⁠. But for all we know, it might be impossible to actually get one or to get the other, and we don’t get to decide. Example 2.2 We can represent a simple act of gambling. The sequent $$\begin{equation*} \mathsf{\$1} \vdash (\mathsf{\$1}\otimes \mathsf{\$1}) \oplus \textbf{1} \end{equation*}$$ captures the fact that you can give |$\mathsf{\$1}$| to receive |$\$2$| or nothing (the vacuous resource), but you don’t choose what you get. The next example uses most of the resource-interpretable connectives. Example 2.3 We can capture the fact that |$\mathsf{\$17}$| get you a menu: $$\begin{equation*}\mathsf{\$17} \vdash \mathsf{menu} \enspace.\end{equation*}$$ The menu consists of a main dish, a side dish and a dessert: $$\begin{equation*}\mathsf{menu} \vdash \mathsf{dish} \otimes \mathsf{side} \otimes \mathsf{dessert} \enspace.\end{equation*}$$ As main dish, you can choose between fish and meat: $$\begin{equation*} \mathsf{dish} \vdash \mathsf{fish} \ \& \ \mathsf{meat} \enspace. \end{equation*}$$ The side dish depends on the season; you don’t choose; it is either aubergine, or parsnip with leek, or asparagus: $$\begin{equation*}\mathsf{side} \vdash \mathsf{aubergine} \oplus (\mathsf{parsnip} \otimes \mathsf{leek}) \oplus \mathsf{asparagus} \enspace.\end{equation*}$$ Finally, as dessert, you choose between the strudel and the chocolate tart. Moreover, you choose whether to have ice cream for |$\mathsf{\$1}$| extra or to have no extra (the vacuous resource). $$\begin{equation*}\mathsf{dessert} \vdash (\mathsf{strudel} \ \& \ \mathsf{chocotart}) \otimes ((\mathsf{\$1} \multimap \mathsf{icecream}) \ \& \ \textbf{1}) \enspace. \end{equation*}$$ We have not illustrated the additive unit |$\top $| yet. The next example hints at the upcoming formalization of Example 1.2 in Section 6.2. Example 2.4 We can formalize the function of the whole baking equipment (mixer, oven, etc.) as the resource transformation process |$\mathsf{flour} \multimap \mathsf{bread}$|⁠. That is, the equipment transforms flour into bread. (Arguably ignoring that we would also need water and electricity. For simplicity, water and electricity could here be considered resources that are provably equivalent to the vacuous resource |$\textbf{1}$|⁠.) The sequent $$\begin{equation*}\mathsf{flour}, \mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread} \vdash \mathsf{bread} \otimes \top\end{equation*}$$ indicates that with two ‘tokens’ of flour and the breadmaking equipment, one can make bread, and some resources will remain in excess, viz., |$\mathsf{flour}$|⁠. The additive unit |$\top $| has some connection with the relationship between linear and affine reasoning that we now discuss briefly. 2.3 Linear vs. affine reasoning and preferences Weakening (rules |$(W)$| in the Appendix A) in the logic LOG can play a crucial role in the satisfaction of the goals of the players. It will also have striking consequences for the algorithmic solutions of the decision problems that we study in this paper. In the context of resource-sensitive logic, one rule of weakening (left weakening) says that if something can be obtained from a set of resources then it can also be obtained from more resources. Symbolically, |$ $| Weakening gives a monotonic flavour to the process of deduction in the logic. Following the terminology in linear logic, a logic LOG admitting weakening will be referred to as affine and a logic LOG without weakening will just be referred to as linear. Despite the fact the affine logic admits more inference rules than linear logic, the unit |$\top $| allows one to simulate the reasoning in affine logic with the provability of linear logic. Indeed, the sequent |$\varGamma \vdash A$| is provable in a logic LOG with the rule of weakening iff the sequent |$\varGamma \vdash A \otimes \top $| is provable the logic LOG without using weakening. In the affine case, |$A, B \vdash A$| is a provable sequent. If a player has a goal |$\gamma = A$|⁠, then she will find her objective satisfied with an outcome |$\{A,B\}$|⁠. In the linear case, we have in general |$A, B \not \vdash A$| (unless |$B$| is a vacuous resource equivalent to |$\textbf{1}$|⁠). A player with a goal |$\gamma = A$| will not be satisfied with an outcome |$\{A,B\}$| as she wants |$A$| and nothing more. If she is indeed indifferent to leftover resources, her goal can be expressed as |$\gamma = A \otimes \top $|⁠, when LOG is linear. Affine logic should be used when extra resources can be disposed of freely. That is, when we can assume that a player satisfied with an outcome would be satisfied with a more sizeable outcome. As we will see in Section 5, this does not prevent players to behave more parsimoniously when they can. 2.4 Sequent provability and some complexity characterizations Given a sequent in a fragment LOG of linear logic, the problem of sequent provability (or provability for short) asks whether the sequent is provable from the sequent rules for LOG. When convenient, we write ‘LOG is in |$\mathsf{C}$|’ when the problem of sequent provability in the logic LOG is in the complexity class |$\mathsf{C}$|⁠. Before moving to the technical part of this paper, we quickly summarize the complexity of sequent provability in some fragments and variants of linear logic that could be used as the LOG parameter in our analysis resource games.3 The results of this paper will be applicable to every fragment mentioned here. MALL is |$\mathsf{PSPACE}$|-complete; MLL is |$\mathsf{NP}$|-complete; affine MLL is |$\mathsf{NP}$|-complete; affine MALL is |$\mathsf{PSPACE}$|-complete; intuitionistic MALL is |$\mathsf{PSPACE}$|-complete; intuitionistic MLL is |$\mathsf{NP}$|-complete. Remarkably, and unlike classical logic, these fragments of linear logic behave well computationally also in the first-order case. First-order MLL is |$\mathsf{NP}$|-complete and first-order MALL is |$\mathsf{NEXPTIME}$|-complete. See [20, 26]. We will also consider the weaker fragment that we call MULT: $$\begin{equation*}A::= \textbf{1} \mid p \mid A \otimes A \enspace.\end{equation*}$$ Proposition 2.5 Sequent provability in intuitionistic affine and intuitionistic linear MULT is in |$\mathsf{PTIME}$|⁠. Proof. Linear MULT is captured by the rules (ax), (cut), (E), (⁠|$\otimes $|R), (⁠|$\otimes $|L), (⁠|$\textbf{1}$|L) and (⁠|$\textbf{1}$|R). Affine MULT also requires |$(W)$|⁠. To check whether the intuitionistic sequent |$\varGamma \vdash A$| is provable, it suffices to check whether |$\flat ^{\bullet }(\varGamma ) \supseteq \flat (A)$| in the case of affine MULT or |$\flat ^{\bullet }(\varGamma ) = \flat (A)$| in the case of linear MULT, where the flattening functions |$\flat $| and |$\flat ^{\bullet }$| are defined as follows: • |$\flat (\textbf{1}) = \emptyset $| • |$\flat (p) = \{p\}$| • |$\flat (A\otimes B) = \flat (A) \uplus \flat (B)$| • |$\flat ^{\bullet }(\emptyset ) = \emptyset $| • |$\flat ^{\bullet }(\{A\} \uplus \varDelta ) = \flat (A) \uplus \flat ^{\bullet }(\varDelta )$| Both multiset inclusion and multiset equality can be performed in linear time in the number of elements in the sets. 3 Individual resource games and decision problems We formally define our models of individual resource games.4 Definition 3.1 An individual resource game (IRG) is a tuple |$G = (N,\gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| where • |$N = \{1, \ldots , n\}$| is a finite set of players; • |$\gamma _i$| is a formula of LOG (⁠|$i$|’s goal, or objective); • |$\epsilon _i$| is a finite multiset of formulas of LOG (⁠|$i$|’s endowment). Let |$G = (N,\gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$|⁠; we define: the set of possible actions of |$i$| as the set of multisets |$\mathsf{ch}_i(G) = \{C \mid C \subseteq \epsilon _i \}$|⁠, and the set of profiles in |$G$| as |$\mathsf{ch}(G) = \prod _{i \in N} \mathsf{ch}_i(G)$|⁠. When |$P= (C_1, \ldots , C_k) \in \mathsf{ch}(G)$| and |$1 \leq i \leq k$|⁠, then |$P_{-i} = (C_1, \ldots , C_{i-1},C_{i+1}, \ldots , C_k)$|⁠. That is, |$P_{-i}$| denotes |$P$| without player |$i$|’s contribution. The outcome of a profile |$P = (C_1, \ldots , C_n)$| is given by the multiset of resources |$\mathsf{out}(P) = \biguplus _{1\leq i \leq n} C_i$|⁠. We will define ‘|$i$| strongly prefers |$P$| over |$P^{\prime}$|’ in due time, reflecting dichotomous preferences first (Section 4) and parsimonious preferences second (Section 5). Definition 3.2 Let |$G = (N,\gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$|⁠. A profile |$P \in \mathsf{ch}(G)$| is a Nash equilibrium iff for all |$i \in N$| and for all |$C_i \in \mathsf{ch}_i(G)$|⁠, we have that |$i$| does not strongly prefer |$(P_{-i}, C_i)$| over |$P$|⁠. Let us note |$NE(G)$| the set of Nash equilibria in |$\mathsf{ch}(G)$|⁠. A basic decision problem is the one of determining whether a choice profile is a Nash equilibrium. NASH EQUILIBRIUM (NE) (in) An individual resource game |$G$| and |$P \in \mathsf{ch}(G)$|⁠. (out)|$P \in NE(G)$|? Some profiles that are not equilibria can have desirable outcomes. Some equilibria can have outcomes that are undesirable. Hence, it is interesting to investigate how resource distribution schemes influence how undesirable game equilibria can be eliminated and how desirable game equilibria can be constructed. In the tradition of social mechanism design, redistribution schemes can be used by a central authority to enforce some behavior, either by disincentivizing a behavior or incentivizing a behavior. We will study redistribution schemes in individual resource games. Let |$\epsilon $| be an endowment function such that for every player |$i$| we have |$\epsilon (i) = \epsilon _i$|⁠, a multiset of formulas of LOG. A redistribution scheme of |$\epsilon $| is an endowment function |$\epsilon ^{\prime}$| such that $$\begin{equation*}\biguplus_{i \in N} \epsilon(i) = \biguplus_{i \in N} \epsilon^{\prime}(i) \enspace.\end{equation*}$$ We note |$\mathsf{redis}(\epsilon )$| the set of redistributions of the endowment function |$\epsilon $|⁠. Given the individual resource game |$G^{\epsilon } = (N, \gamma _1, \ldots , \gamma _n, \epsilon (1), \ldots , \epsilon (n))$|⁠, we can apply a redistribution scheme where we modify the endowment function |$\epsilon $| into |$\epsilon ^{\prime}$|⁠. We thus obtain the individual resource game |$G^{\epsilon ^{\prime}} = (N, \gamma _1, \ldots , \gamma _n, \epsilon ^{\prime}(1), \ldots , \epsilon ^{\prime}(n))$|⁠. We will investigate two decision problems inspired by [18], which are related to resource redistributions. We will look at whether the outcome of a resource game can be rationally eliminated. That is whether there is a resource redistribution such that no Nash equilibrium of the new resource game yields this outcome. RATIONAL ELIMINATION (RE) (in) An individual resource game |$G^\epsilon $| and |$P \in \mathsf{ch}(G^\epsilon )$|⁠. (out) Is there a redistribution |$\epsilon ^{\prime}$| of |$\epsilon $| such that for all |$P^{\prime} \in \mathsf{ch}(G^{\epsilon ^{\prime}})$|⁠, if |$\mathsf{out}(P^{\prime}) = \mathsf{out}(P)$| then |$P^{\prime} \not \in NE(G^{\epsilon ^{\prime}})$|? Conversely, we will look at whether the outcome of a resource game can be rationally constructed. That is whether there is a resource redistribution such that the outcome is the outcome of some Nash equilibrium in the new resource game. RATIONAL CONSTRUCTION (RC) (in) An individual resource game |$G^\epsilon $| and |$P \in \mathsf{ch}(G^\epsilon )$|⁠. (out) Is there a redistribution |$\epsilon ^{\prime}$| of |$\epsilon $| such that there is |$P^{\prime} \in \mathsf{ch}(G^{\epsilon ^{\prime}})$| where |$\mathsf{out}(P^{\prime}) = \mathsf{out}(P)$| and |$P^{\prime} \in NE(G^{\epsilon ^{\prime}})$|? Note that being a game equilibrium is without ambiguity a property of profile. However, after a redistribution of resources in an individual resource game, the space of actions and the space of profiles change. Thus, elimination and construction are more about the outcomes of profiles. Sections 4.2 and 5.1 will illustrate these decision problems in due time. 4 Dichotomous preferences Let |$G = (N, \gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game. Player |$i$|⁠, whose goal is |$\gamma _i$|⁠, realizes her objectives in a profile |$P$| when |$\mathsf{out}(P) \vdash \gamma _i$|⁠. That is, the resources in |$\mathsf{out}(P)$| can be transformed into a shareable resource |$\gamma _i$|⁠. For |$P \in \mathsf{ch}(G)$| and |$Q \in \mathsf{ch}(G)$|⁠, we say that player |$i \in N$| (dichotomously) strongly prefers |$P$| over |$Q$| (noted |$Q \prec _i P$|⁠) iff |$\mathsf{out}(P) \vdash \gamma _i$| and not |$\mathsf{out}(Q) \vdash \gamma _i$|⁠. Proposition 4.1 Let |$G = (N, \gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game, two profiles |$P \in \mathsf{ch}(G)$| and |$Q \in \mathsf{ch}(G)$| and a player |$i \in N$|⁠. The problem of deciding whether |$Q \prec _i P$| is in |$\mathsf{PTIME}$| when provability in LOG is in |$\mathsf{PTIME}$|⁠. It is |$\mathsf{NP \land coNP} = \mathsf{B\mathsf{H_2}}$|-complete when provability in LOG is |$\mathsf{NP}$|-complete. It is |$\mathsf{PSPACE}$|-complete when provability in LOG is |$\mathsf{PSPACE}$|-complete. Proof. The language corresponding to the problem is |$L = \{ (P,Q) \mid Q \prec _i P \} = L_1 \cap L_2$| with |$L_1 = \{ (P,Q) \mid \mathsf{out}(P) \vdash \gamma _i \}$|⁠, and |$L_2 = \{ (P,Q) \mid \ \textrm{not}\ \mathsf{out}(Q) \vdash \gamma _i\}$|⁠. In particular, when the problem of provability in LOG is in |$\mathsf{NP}$|⁠, we clearly have that |$L_1$| is a |$\mathsf{NP}$| language and |$L_2$| is a |$\mathsf{coNP}$| language. For hardness, we consider a newly fabricated decision problem that we call PROV-NONPROV. The problem PROV-NONPROV takes in input two sequents of LOG|$\varGamma _1 \vdash \varDelta _1$| and |$\varGamma _2 \vdash \varDelta _2$|⁠, and outputs true iff |$\varGamma _1 \vdash \varDelta _1$| is provable and |$\varGamma _2 \vdash \varDelta _2$| is not provable. It is easy to see that if LOG is |$\mathsf{NP}$|-complete, then PROV-NONPROV is |$\mathsf{B\mathsf{H_2}}$|-complete, and if LOG is |$\mathsf{PSPACE}$|-complete, then PROV-NONPROV is |$\mathsf{PSPACE}$|-complete. We propose a reduction of PROV-NONPROV into the problem of deciding whether in an individual resource game, a profile is strongly preferred to another profile by a player. Let |$\varGamma _1 \vdash \varDelta _1$| and |$\varGamma _2 \vdash \varDelta _2$| be two sequents of LOG. We can prove using |$\bot L$|⁠, |$\bot R$|⁠, (cut) and (E) that |$\varGamma \vdash \varDelta $| iff |$\varGamma \vdash \varDelta , \bot $|⁠. Thus, we have |$\varGamma _1 \vdash \varDelta _1$| iff |$\varGamma _1, \mathop \sim \varDelta _1 \vdash \bot $|⁠, and we have |$\varGamma _2 \vdash \varDelta _2$| iff |$\varGamma _2, \mathop \sim \varDelta _2 \vdash \bot $|⁠.5 Now we construct the game |$G = (\{1\}, \gamma _1 = \bot , \epsilon _1 = \varGamma _1 \uplus \mathop \sim \varDelta _1 \uplus \varGamma _2 \uplus \mathop \sim \varDelta _2)$|⁠. It is now easy to see that PROV-NONPROV instantiated with |$\varGamma _1 \vdash \varDelta _1$| and |$\varGamma _2 \vdash \varDelta _2$| returns true iff Player |$1$| strongly prefers |$(\varGamma _1 \uplus \mathop \sim \varDelta _1)$| over |$(\varGamma _2 \uplus \mathop \sim \varDelta _2)$| in |$G$|⁠. 4.1 Finding Nash equilibria We study the complexity of the problem NASH EQUILIBRIUM with dichotomous preferences. 4.1.1 Hardness We are about to prove the lower bound of the complexity NE with dichotomous preferences. Before we do so, observe that by applying the rules |$L\mathop \sim $| and |$R\mathop \sim $|⁠, $$\begin{equation*}A_1, \ldots, A_n \vdash B_1, \ldots, B_m \text{ iff } A_1, \ldots, A_n, \mathop \sim B_2, \ldots, \mathop \sim B_m \vdash B_1\end{equation*}$$ is immediate. Hence, we can, without loss of generality, consider only the intuitionistic sequents of LOG in the many-to-one reductions of this paper. Proposition 4.2 NE is as hard as the problem of checking sequent provability in LOG, even when there is only one player. Proof. We reduce the problem of sequent provability for the logic LOG. W.l.o.g., we only consider intuitionistic sequents. Let |$\varGamma \vdash \delta $| be the intuitionistic sequent where |$\varGamma $| is an arbitrary multiset of formulas of LOG and |$\delta $| is an arbitrary formula. We can construct the individual resource game |$G$| such that |$G = (\{1\},\delta , \varGamma \cup \{\delta \})$|⁠. |$G$| is thus the one-player individual resource game where Player |$1$|’s goal is to achieve |$\delta $|⁠, and Player |$1$| is endowed with |$\varGamma \cup \{\delta \}$| (this is a set union but we could have chosen the endowment |$\varGamma \uplus \{\delta \}$| as well). A profile in |$G$| is a choice of Player |$1$|⁠, i.e. a subset |$C_1$| of |$\varGamma \cup \{\delta \}$|⁠. In this case for any profile |$P$| in |$G$|⁠, |$\mathsf{out}(P) = P$|⁠. We show that |$\varGamma \vdash \delta $| iff |$\varGamma \in NE(G)$|⁠. From left to right, suppose that |$\varGamma \vdash \delta $|⁠. We need to show that |$\varGamma \in NE(G)$|⁠. That is, for all |$C_1 \subseteq \varGamma \cup \{\delta \}$|⁠, if |$C_1 \vdash \delta $| then |$\varGamma \vdash \delta $|⁠. Since we supposed |$\varGamma \vdash \delta $|⁠, this is trivially true. From right to left, suppose that |$\varGamma \in NE(G)$|⁠. This means that for all |$C_1 \subseteq \varGamma \cup \{\delta \}$|⁠, if |$C_1 \vdash \delta $| then |$\varGamma \vdash \delta $|⁠. Let in particular |$C_1 = \{\delta \}$|⁠. Indeed, |$C_1 \subseteq \varGamma \cup \{\delta \}$|⁠. Moreover, by (ax) we have |$\delta \vdash \delta $|⁠. Hence, |$\varGamma \vdash \delta $| follows. 4.1.2 Algorithms To establish an upper bound on the complexity of NE let us first outline an algorithm for solving its complement. That is, checking whether a profile is not a Nash equilibrium. Let |$P \in \mathsf{ch}(G)$| be a profile. To determine whether |$P \not \in NE(G)$|⁠, we can employ a simple non-deterministic algorithm, showed as Algorithm 1. Algorithm 1 General algorithm for co-NE 1: non-deterministically guess |$(i,C^{\prime}_i) \in N \times \mathsf{ch}_i(G)$|⁠. 2: return |$P \prec _i (P_{-i},C^{\prime}_i)$|⁠. Proposition 4.3 If the problem of provability in LOG is in |$\mathsf{PTIME}$| then NE is in |$\mathsf{coNP}$|⁠. If the problem of provability in LOG is in |$\mathsf{NP}$| then NE is in |$\mathsf{coNP^{B\mathsf{H_2}}}$| and indeed in |$\mathsf{\varPi _2^p}$|⁠. If the problem of provability in LOG is in |$\mathsf{PSPACE}$| then NE is in |$\mathsf{PSPACE}$|⁠. Proof. Consider Algorithm 1. If sequent provability in LOG is in |$\mathsf{NP}$|⁠, we can check |$P \prec _i (P_{-i},C^{\prime}_i)$| in |$\mathsf{B\mathsf{H_2}}$| (Proposition 4.1). Thus, we can check whether |$P \not \in NE(G)$| in |$\mathsf{NP^{B\mathsf{H_2}}}$|⁠. Finally, we can solve NE in |$\mathsf{coNP^{B\mathsf{H_2}}}$|⁠. It is the case that |$\mathsf{B\mathsf{H_2}} \subseteq \mathsf{\varDelta _2^p}$|⁠, and also that |$\mathsf{NP^{\varDelta _2^p}} = \mathsf{\varSigma _2^p}$| so we can solve NE in |$\mathsf{\varPi _2^p}$|⁠. The proofs for the cases of sequent provability in |$\mathsf{PTIME}$| and |$\mathsf{PSPACE}$| proceed with similar considerations about Algorithm 1. Affine logic admits the rule of weakening |$(W)$|⁠, which allows one to discard resources. In this setting, if a player can achieve her goal with the resources |$\varGamma $|⁠, she can as well achieve her goal with the resources |$\varGamma \cup \{A\}$|⁠. A consequence is the following lemma, which will have a significant impact on the computational complexity of NE. Lemma 4.4 Let |$G = (N,\gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game. When LOG is affine, |$P \not \in NE(G)$| iff |$\exists i \in N: P \prec _i (P_{-i}, \epsilon _i)$|⁠. Proof. Suppose |$P \not \in NE(G)$|⁠. There is |$i \in N$| and |$C_i \in \mathsf{ch}_i(G)$| s.t. |$P \prec _i (P_{-i}, C_i)$|⁠. By definition, |$\mathsf{out}((P_{-i}, C_i)) \vdash \gamma _i$| and |$\mathsf{out}(P) \not \vdash \gamma _i$|⁠. We have |$C_i \subseteq \epsilon _i$|⁠, so by applying weakening |$(W)$| with every instance of formulas in |$\epsilon _i \setminus C_i$|⁠, we can prove that |$\mathsf{out}((P_{-i}, \epsilon _i)) \vdash \gamma _i$|⁠. We thus have that there is |$i \in N$| s.t. |$P \prec _i (P_{-i}, \epsilon _i)$|⁠. The other way around is immediate from the definition of Nash equilibria. It means that, in a profile, if no player has an incentive to deviate by making available their whole endowment, then the profile is a Nash equilibrium. The very profile where all the players make available their whole endowment is trivially such a profile. The next proposition follows immediately: Proposition 4.5 Let |$G = (N, \gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game. When LOG is affine: |$NE(G) \not = \emptyset $| and |$(\epsilon _1, \ldots , \epsilon _n) \in NE(G)$|⁠. Lemma 4.4 also helps us to establish the following result. Proposition 4.6 When LOG is affine, if the problem of sequent provability in LOG is in |$\mathsf{PTIME}$| then NE is in |$\mathsf{PTIME}$|⁠. If the problem of sequent provability in LOG is in |$\mathsf{NP}$| then NE is in |$\mathsf{P^{NP||}}$|⁠. If the problem of sequent provability in LOG is in |$\mathsf{PSPACE}$| then NE is in |$\mathsf{PSPACE}$|⁠. Proof. Let |$G = (N, \gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game and let |$P \in \mathsf{ch}(G)$| be a profile. One can check whether |$P \in NE(G)$| with Algorithm 2. Algorithm 2 Algorithm for NE with dichotomous preferences and affine LOG 1: for each |$i \in N$| do: 2: if (⁠|$\mathsf{out}(P) \vdash \gamma _i$|⁠): 3: continue; 4: else if (⁠|$\mathsf{out}((P_{-i}, \epsilon _i)) \vdash \gamma _i$|⁠): 5: return false. 6: return true. For correctness, note that the instructions of the lines |$2-4$| are equivalent to a test of whether |$\mathsf{out}(P) \not \vdash \gamma _i$| and |$\mathsf{out}((P_{-i}, \epsilon _i)) \vdash \gamma _i$|⁠, i.e. |$P \prec _i (P_{-i}, \epsilon _i)$|⁠. Lemma 4.4 ensures that exactly when there is an |$i \in N$| such that |$P \prec _i (P_{-i}, \epsilon _i)$| we can conclude that |$P$| is not a Nash equilibrium. Suppose sequent provability in LOG is in |$\mathsf{NP}$|⁠. The algorithm can be simulated by a deterministic oracle Turing machine in polynomial time with |$2n$| non-adaptive queries to an |$\mathsf{NP}$| oracle. Indeed, |$P \in NE(G)$| is thus a |$\mathsf{P^{NP||[\text{$2n$}]}}$| predicate. The problem is in |$\mathsf{P^{NP||}}$|⁠. When sequent provability in LOG is in |$\mathsf{PTIME}$| (resp., |$\mathsf{PSPACE}$|⁠), the algorithm runs in polynomial time (resp., polynomial space). 4.2 Elimination A very simple illustration of RATIONAL ELIMINATION is given by the individual resource game |$G^\epsilon = (\{1,2\}, \gamma _1 = B, \gamma _2 = A, \{A\}, \{B\})$|⁠. There are two players. Player |$1$| wants |$B$| but is endowed with |$\{A\}$|⁠, while Player |$2$| wants |$A$| but is endowed with |$\{B\}$|⁠. The game |$G^\epsilon $| can be represented as on Figure 1. (We indicate the realized objectives assuming that LOG is affine.) Figure 1 Open in new tabDownload slide The game |$G^\epsilon $|⁠. |$\gamma _1$| and |$\gamma _2$| indicate that Player |$1$| and Player |$2$| have their goals satisfied, assuming LOG is affine. The symbol |${}{\square }$| denotes a Nash equilibrium. Figure 1 Open in new tabDownload slide The game |$G^\epsilon $|⁠. |$\gamma _1$| and |$\gamma _2$| indicate that Player |$1$| and Player |$2$| have their goals satisfied, assuming LOG is affine. The symbol |${}{\square }$| denotes a Nash equilibrium. One can readily check that all profiles are Nash equilibria. However, the profile |$(\{A\}, \{B\})$| is more ‘socially desirable’ than the others since it satisfies both players’ goal. A centralized authority could effectively eliminate the others by redistributing the resources present in |$G^{\epsilon }$| so as to obtain |$G^{\epsilon ^{\prime}} = (\{1,2\}, \gamma _1 = B, \gamma _2 = A, \{B\}, \{A\})$|⁠. The game |$G^{\epsilon ^{\prime}}$| can be represented as on Figure 2. Figure 2 Open in new tabDownload slide The game |$G^{\epsilon ^{\prime}}$|⁠. Figure 2 Open in new tabDownload slide The game |$G^{\epsilon ^{\prime}}$|⁠. The only Nash equilibrium is now the one with outcome |$\{B, A\}$|⁠. 4.2.1 Algorithms As a consequence of Proposition 4.5, we already know that Proposition 4.7 Let |$G = (N, \gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game. When LOG is affine, the profile |$P$| such that |$\mathsf{out}(P) = \biguplus _j \epsilon _j$| is not rationally eliminable. This is very specific to the affine case (and dichotomous preferences), and even then, it is of course not true of all Nash equilibria. To decide whether some outcome is rationally eliminable, one naïve approach consists in trying all possible redistributions and check whether the outcome is a Nash equilibrium in the resulting individual resource game. Instead, we are going to exploit a pleasant property, analogous to [18, Corollary |$4$|]. Let |$G^\epsilon = (N, \gamma _1, \ldots , \gamma _n, \epsilon (1), \ldots , \epsilon (n))$| be an individual resource game. For each player |$i \in N$|⁠, we define |$G^{[\epsilon \vartriangleright i]}$| where |$[\epsilon \vartriangleright i]$| is the redistribution of |$\epsilon $| where all resources are assigned to |$i$|⁠, i.e. $$\begin{equation*} [\epsilon\vartriangleright i](j) = \begin{cases} \biguplus_{k \in N} \epsilon(k) & \textrm{when}\ j = i\\ \emptyset & \text{otherwise.} \end{cases}\end{equation*}$$ Because there is only one active player in |$G^{[\epsilon \vartriangleright i]}$|⁠, we will sometimes write a profile of |$G^{[\epsilon \vartriangleright i]}$| as |$(C_i)$| with |$C_i \in \mathsf{ch}_i(G^{[\epsilon \vartriangleright i]})$| instead of |$(\emptyset , \ldots , \emptyset , C_i, \emptyset , \ldots , \emptyset )$|⁠, by abuse of notation. Lemma 4.8 Let |$G^\epsilon $| be an individual resource game and |$P \in \mathsf{ch}(G^\epsilon )$|⁠. |$P$| is rationally eliminable iff there is a player |$i \in N$| and a profile |$Q \in \mathsf{ch}(G^{[\epsilon \vartriangleright i]})$|⁠, such that |$\mathsf{out}(Q) = \mathsf{out}(P)$| and |$Q \not \in NE(G^{[\epsilon \vartriangleright i]})$|⁠. Proof. From right to left. Suppose |$Q \not \in NE(G^{[\epsilon \vartriangleright i]})$| for some |$i \in N$|⁠. Let also |$P \in \mathsf{ch}(G^\epsilon )$| be a profile and assume |$\mathsf{out}(P) = \mathsf{out}(Q)$|⁠. When there is at most one player with a non-empty endowment, as in |$[\epsilon \vartriangleright i]$|⁠, there is a one-to-one correspondence between the set of profiles and the set of outcomes. Thus, there is one and only one profile in |$G^{[\epsilon \vartriangleright i]}$| with outcome |$\mathsf{out}(P)$| and it is |$Q$|⁠. So there is a redistribution of |$\epsilon $|⁠, namely |$[\epsilon \vartriangleright i]$|⁠, such that for all profiles |$Q \in \mathsf{ch}(G^{[\epsilon \vartriangleright i]})$| with outcome |$\mathsf{out}(P)$|⁠, we have |$Q \not \in NE(G^{[\epsilon \vartriangleright i]})$|⁠. So |$P$| is rationally eliminable. From left to right. Suppose that |$P$| is rationally eliminable. Thus, there is a redistribution |$\epsilon ^{\prime}$| of |$\epsilon $| such that for all |$P^{\prime} \in \mathsf{ch}(G^{\epsilon ^{\prime}})$|⁠, if |$\mathsf{out}(P^{\prime}) = \mathsf{out}(P)$| then |$P^{\prime} \not \in NE(G^{\epsilon ^{\prime}})$|⁠. So let |$R \in \mathsf{ch}(G^{\epsilon ^{\prime}})$| be an arbitrary profile with |$\mathsf{out}(R) = \mathsf{out}(P)$|⁠. By assumption, we have that |$R \not \in NE(G^{\epsilon ^{\prime}})$|⁠. By definition of Nash equilibria, this means that there is |$i \in N$| and |$C_i^{\prime} \in \mathsf{ch}_i(G^{\epsilon ^{\prime}})$| such that |$R \prec _i (R_{-i},C_i^{\prime})$|⁠. Now consider the game |$G^{[\epsilon \vartriangleright i]}$|⁠. We have |$\mathsf{out}(R) \in \mathsf{ch}_i(G^{[\epsilon \vartriangleright i]})$| and |$\mathsf{out}((R_{-i},C_i^{\prime})) \in \mathsf{ch}_i(G^{[\epsilon \vartriangleright i]})$|⁠. Let the profile |$R^1 \in \mathsf{ch}(G^{[\epsilon \vartriangleright i]})$| with |$R^1_i = \mathsf{out}(R)$| and |$R^1_j = \emptyset $| when |$j \not = i$|⁠. Let |$R^2 \in \mathsf{ch}(G^{[\epsilon \vartriangleright i]})$| be the profile with |$R^2_i = \mathsf{out}((R_{-i},C_i^{\prime}))$| and |$R^2_j = \emptyset $| when |$j \not = i$|⁠. Since, |$R \prec _i (R_{-i},C_i^{\prime})$|⁠, we also have |$R^1 \prec _i R^2$|⁠. So |$R^1 \not \in NE(G^{[\epsilon \vartriangleright i]})$|⁠. The profile |$R^1$| is the only profile of |$G^{[\epsilon \vartriangleright i]}$| with outcome |$\mathsf{out}(P)$|⁠. So we can conclude. We establish an upper bound on the complexity of RE when LOG does not admit the weakening rule. Proposition 4.9 When LOG is linear, RE is in |$\mathsf{NP}$| when provability in LOG is in |$\mathsf{PTIME}$|⁠, in |$\mathsf{NP^{B\mathsf{H_2}}}$| and indeed in |$\mathsf{\varSigma _2^p}$| when LOG is in |$\mathsf{NP}$|⁠, and in |$\mathsf{PSPACE}$| when LOG is in |$\mathsf{PSPACE}$|⁠. Proof. Let |$P \in \mathsf{ch}(G^\epsilon )$| be a profile. To determine whether |$P$| is rationally eliminable, we can use Algorithm 3. Algorithm 3 General algorithm for RE 1: non-deterministically guess |$(i,C^{\prime}_i) \in N \times \mathsf{ch}_i(G^{[\epsilon \vartriangleright i]})$|⁠. 2: return |$P \prec _i (P_{-i},C^{\prime}_i)$|⁠. Straightforwardly, it guesses a player |$i$| and a deviation in the game |$G^{[\epsilon \vartriangleright i]}$| for Player |$i$| from the profile |$(\mathsf{out}(P)) \in \mathsf{ch}(G^{[\epsilon \vartriangleright i]})$| and checks whether Player |$i$| has an incentive to do this deviation. By Lemma 4.8, if such a player and deviation exist and only if they exist, the profile |$P$| is rationally eliminable in |$G^\epsilon $|⁠. So the algorithm is correct. It can of course be simulated by a non-deterministic oracle Turing machine with one call to an oracle for |$P \prec _i (P_{-i},C^{\prime}_i)$|⁠. Proposition 4.1 informs us of a containing class of this oracle. When LOG admits the weakening rule, we can propose a surprisingly simple algorithm, which takes advantage of both Lemmas 4.4 and 4.8. Proposition 4.10 When LOG is affine, RE is in |$\mathsf{PTIME}$| when provability in LOG is in |$\mathsf{PTIME}$|⁠, in |$\mathsf{P^{NP||}}$| when LOG is in |$\mathsf{NP}$|⁠, and in |$\mathsf{PSPACE}$| when LOG is in |$\mathsf{PSPACE}$|⁠. Proof. Let |$G = (N,\gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game and let |$P \in \mathsf{ch}(G)$| be a profile. Consider Algorithm 4. Algorithm 4 Algorithm for RE with dichotomous preferences and affine LOG 1: for each |$i \in N$| do: 2: if (⁠|$P \prec _i ([\epsilon \vartriangleright i](i))$|⁠): 3: return true. 4: return false. The algorithm is correct. Indeed, by Lemma 4.8, |$P$| is eliminable in |$G$| iff there is |$i \in N$| where |$(\mathsf{out}(P)) \not \in NE(G^{[\epsilon \vartriangleright i]})$|⁠. By Lemma 4.4, we know that |$(\mathsf{out}(P)) \not \in NE(G^{[\epsilon \vartriangleright i]})$| iff |$P \prec _i ([\epsilon \vartriangleright i](i))$|⁠. Notice that the test of line 2 is equivalent to |$P \not \vdash \gamma _i$| and |$[\epsilon \vartriangleright i](i) \vdash \gamma _i$|⁠. Thus, it can be simulated by a deterministic oracle Turing machine in polynomial time with at most |$2n$| non-adaptive queries to an oracle for the problem of sequent provability. When the problem of sequent provability in LOG is in |$\mathsf{NP}$| it yields a complexity of |$\mathsf{P^{NP||}}$|⁠. When it is in |$\mathsf{PTIME}$| (resp., |$\mathsf{PSPACE}$|⁠), it yields a complexity of |$\mathsf{PTIME}$| (resp., |$\mathsf{PSPACE}$|⁠). 4.2.2 Hardness The linear and affine cases both use the same proof strategy which we present at once. Proposition 4.11 RE is as hard as the problem of checking sequent non-provability in LOG. Proof. Let |$\varGamma \vdash \delta $| be an arbitrary intuitionistic sequent. Let |$\varphi = \varGamma ^* \multimap \delta $|⁠. (Remember that |$\varGamma ^* = \bigotimes _{A \in \varGamma } A$|⁠.) Let |$G^\epsilon = (\{1,2\}, \varphi , \textbf{1}, \emptyset , \{\varphi \})$| be an individual resource game. So, we have |$\epsilon _1 = \emptyset $| and |$\epsilon _2 = \{\varphi \}$|⁠. There is only one other distinct redistribution |$\epsilon ^{\prime}$| of |$\epsilon $| where |$\epsilon ^{\prime}_1 = \{\varphi \}$| and |$\epsilon ^{\prime}_2 = \emptyset $|⁠. It is the case that |$\mathsf{redis}(\epsilon ) = \{\epsilon , \epsilon ^{\prime}\}$|⁠. Let |$G^{\epsilon ^{\prime}} = (\{1,2\}, \varphi , \textbf{1}, \{\varphi \}, \emptyset )$| be the individual resource game resulting from the redistribution |$\epsilon ^{\prime}$|⁠. Both games are represented on Figure 3. We show that both in the case of linear and of affine logics, we have |$\varGamma \not \vdash \delta $| iff |$(\emptyset , \emptyset )$| is rationally eliminable in |$G^\epsilon $|⁠. We first show that $$\begin{equation} \varGamma \vdash \delta \ \textrm{iff}\ \emptyset \vdash \varphi \enspace. \end{equation}$$(1) From left to right, suppose |$\varGamma \vdash \delta $|⁠. By applying (⁠|$\otimes $|L) enough times, we obtain |$\varGamma ^* \vdash \delta $|⁠. Then we obtain |$\emptyset \vdash \varGamma ^* \multimap \delta $| using (⁠|$\multimap $|R). From right to left, suppose |$\emptyset \vdash \varGamma ^* \multimap \delta $|⁠. With (ax) and |$\otimes $|R we can show |$\varGamma \vdash \varGamma ^*$|⁠. Using |$\otimes $|R on the sequents |$\varGamma \vdash \varGamma ^*$| and |$\emptyset \vdash \varGamma ^* \multimap \delta $|⁠, we obtain $$\begin{equation} \varGamma \vdash \varGamma^* \otimes \varGamma^* \multimap \delta \enspace. \end{equation}$$(2) Without assumption, we can also show $$\begin{equation} \varGamma^* \otimes \varGamma^* \multimap \delta \vdash \delta \enspace, \end{equation}$$(3) using the rules (ax), (⁠|$\multimap $|L), and (⁠|$\otimes $|L). We conclude that |$\varGamma \vdash \delta $| using (cut) on the sequents 2 and 3. We can proceed. Suppose |$\varGamma \not \vdash \delta $|⁠. We show that |$(\emptyset , \emptyset )$| is not a Nash equilibrium in |$G^{\epsilon ^{\prime}}$|⁠. Since |$\varGamma \not \vdash \delta $|⁠, we also have |$\emptyset \not \vdash \varphi $| (by Equation 1). On the other hand, using (ax), we have |$\{\varphi \} \vdash \varphi $|⁠. So in the profile |$(\emptyset ,\emptyset )$|⁠, Player |$1$| has an incentive to deviate to the profile |$(\{\varphi \},\emptyset )$|⁠. So |$(\emptyset ,\emptyset )$| is not a Nash equilibrium in |$G^{\epsilon ^{\prime}}$|⁠. Suppose |$\varGamma \vdash \delta $|⁠. We show that |$(\emptyset , \emptyset )$| is a Nash equilibrium both in |$G^{\epsilon }$| and in |$G^{\epsilon ^{\prime}}$|⁠. In |$G^{\epsilon }$|⁠, we have |$\emptyset \vdash \textbf{1}$| from |$\textbf{1}$|R, so Player |$2$| has no incentive to deviate from the profile |$(\emptyset ,\emptyset )$| in |$G^{\epsilon }$|⁠. Moreover, Player |$1$| is dummy in |$G^{\epsilon }$|⁠. So |$(\emptyset ,\emptyset )$| is a Nash equilibrium in |$G^{\epsilon }$|⁠. In |$G^{\epsilon ^{\prime}}$|⁠, since |$\varGamma \vdash \delta $|⁠, we also have |$\emptyset \vdash \varphi $| (by Equation 1), so Player |$1$| has no incentive to deviate from the profile |$(\emptyset ,\emptyset )$| in |$G^{\epsilon ^{\prime}}$|⁠. Moreover, Player |$2$| is dummy in |$G^{\epsilon ^{\prime}}$|⁠. So |$(\emptyset ,\emptyset )$| is a Nash equilibrium in |$G^{\epsilon ^{\prime}}$|⁠. Figure 3 Open in new tabDownload slide Games |$G^{\epsilon }$| and |$G^{\epsilon ^{\prime}}$|⁠. The profile |$(\emptyset ,\emptyset )$| is a Nash equilibrium in |$G^{\epsilon }$|⁠. The profile |$(\emptyset ,\emptyset )$| is a Nash equilibrium in |$G^{\epsilon ^{\prime}}$| iff |$\varGamma \vdash \delta $|⁠. (The profile |$(\{\varphi \}, \emptyset )$| is a Nash equilibrium in |$G^{\epsilon ^{\prime}}$|⁠. Depending on whether |$\varGamma \vdash \delta $| and whether LOG is linear or affine, |$(\emptyset , \{\varphi \})$| may or may not be Nash equilibria in |$G^{\epsilon }$|⁠. This is inconsequential for the reduction in the proof of Proposition 4.11.) Figure 3 Open in new tabDownload slide Games |$G^{\epsilon }$| and |$G^{\epsilon ^{\prime}}$|⁠. The profile |$(\emptyset ,\emptyset )$| is a Nash equilibrium in |$G^{\epsilon }$|⁠. The profile |$(\emptyset ,\emptyset )$| is a Nash equilibrium in |$G^{\epsilon ^{\prime}}$| iff |$\varGamma \vdash \delta $|⁠. (The profile |$(\{\varphi \}, \emptyset )$| is a Nash equilibrium in |$G^{\epsilon ^{\prime}}$|⁠. Depending on whether |$\varGamma \vdash \delta $| and whether LOG is linear or affine, |$(\emptyset , \{\varphi \})$| may or may not be Nash equilibria in |$G^{\epsilon }$|⁠. This is inconsequential for the reduction in the proof of Proposition 4.11.) 4.3 Construction For elimination, Lemma 4.8 provided a remarkable necessary and sufficient condition for the rational eliminability of a profile. For the rational constructibility of a profile, we can only indicatively provide sufficient conditions. Let |$G = (N,\gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an IRG, and let |$P \in \mathsf{ch}(G)$| be a profile in |$G$|⁠. If there is a player |$i \in N$| such that |$\mathsf{out}(P) \vdash \gamma _i$|⁠, then |$P$| can be rationally constructed by redistributing all the resources to Player |$i$|⁠. Also, if there is a player |$i \in N$| such that |$\biguplus _{k \in N} \epsilon _k \not \vdash \gamma _i \otimes \top $|⁠, then |$P$| can be rationally constructed by redistributing all the resources to Player |$i$|⁠. We tackle the complexity of RATIONAL CONSTRUCTION with dichotomous preferences. 4.3.1 Hardness We prove a lower bound of the problem RC in presence of dichotomous preferences. Proposition 4.12 RC is as hard as the problem of checking sequent provability in LOG. Proof. Let |$\varphi = \varGamma ^* \multimap \delta $| and |$G = (\{1\}, \varphi , \epsilon _1 = \{\varphi \})$|⁠. We can see that |$(\emptyset ) \in NE(G)$| iff |$\emptyset \vdash \varphi $|⁠, i.e. |$\varGamma \vdash \delta $|⁠. As |$\mathsf{redis}(\epsilon ) = \{\epsilon \}$|⁠, we conclude that for every sequent |$\varGamma \vdash \delta $|⁠, |$(\emptyset )$| is rationally constructible in |$G$| iff |$\varGamma \vdash \delta $| is provable. 4.3.2 Algorithms Let |$G^\epsilon $| be an individual resource game, and let |$P \in \mathsf{ch}(G^\epsilon )$|⁠. To decide whether the profile |$P$| can be rationally constructed we can use Algorithm 5. This algorithm will serve for all cases of rational construction in this paper. Algorithm 5 General algorithm for RC 1: non-deterministically guess |$(\epsilon ^{\prime}, P^{\prime}) \in \mathsf{redis}(\epsilon ) \times \mathsf{ch}(G^{\epsilon ^{\prime}})$|⁠. 2: return |$\mathsf{out}(P^{\prime}) = \mathsf{out}(P)$| and |$P^{\prime} \in NE(G^{\epsilon ^{\prime}})$|⁠. The algorithmic analysis is rather simple: we use the problem NE as a blackbox, for which complexity upper bounds have been established in Propositions 4.3 and 4.6. Proposition 4.13 When LOG is in |$\mathsf{PTIME}$|⁠, RC is in |$\mathsf{\varSigma _2^p}$|⁠. When LOG is in |$\mathsf{NP}$|⁠, RC is in |$\mathsf{\varSigma _3^p}$|⁠. When LOG is in |$\mathsf{PSPACE}$|⁠, RC is in |$\mathsf{PSPACE}$|⁠. Proof. When LOG is in |$\mathsf{PTIME}$|⁠, from Proposition 4.3, we know that the test of line |$2$| is in |$\mathsf{coNP}$|⁠. So RC is in |$\mathsf{NP^{coNP}} = \mathsf{\varSigma _2^p}$|⁠. Similarly, when LOG is in |$\mathsf{NP}$|⁠, from Proposition 4.3, we know that the test of line |$2$| is in |$\mathsf{\varPi _2^p}$|⁠. So RC is in |$\mathsf{NP^{\varPi _2^p}} = \mathsf{\varSigma _3^p}$|⁠. The case for LOG in |$\mathsf{PSPACE}$| is analogous. Again, an affine LOG seems to bring some relative algorithmic ease. Proposition 4.14 If LOG is affine, when provability in LOG is in |$\mathsf{PTIME}$|⁠, then RC is in |$\mathsf{NP}$|⁠. When LOG is in |$\mathsf{NP}$|⁠, RC is in |$\mathsf{\varSigma _2^p}$|⁠. When LOG is in |$\mathsf{PSPACE}$|⁠, RC is in |$\mathsf{PSPACE}$|⁠. Proof. The proof is similar to the one of Proposition 4.13, using the result of Proposition 4.6 and, for the case of |$\mathsf{NP}$| the fact that |$\mathsf{NP^{P^{NP||}}} \subseteq \mathsf{NP^{\varDelta _2^p}} = \mathsf{\varSigma _2^p}$|⁠. 5 Parsimonious preferences Weakening |$(W)$| is sometimes a desirable property of LOG and of our preferences of resources. However, it has the untoward consequence of incentivizing players to spend all their resources in individual resource games with dichotomous preferences. This is well exemplified for instance by Proposition 4.5. We can teach our players parsimony by attaching to them finer preferences that take into account the realization of their objective, but also the optimality of their contribution. In an individual resource game |$G = (N,\gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$|⁠, we now say that player |$i \in N$| (parsimoniously) strongly prefers |$P \in \mathsf{ch}(G)$| over |$Q \in \mathsf{ch}(G)$| (noted |$Q \prec _i P$|⁠) iff one of the following conditions is satisfied: 1. not |$\mathsf{out}(P) \vdash \gamma _i$| and not |$\mathsf{out}(Q) \vdash \gamma _i$| and |$P_i \subset Q_i$|⁠; 2. |$\mathsf{out}(P) \vdash \gamma _i$| and not |$\mathsf{out}(Q) \vdash \gamma _i$|⁠; 3. |$\mathsf{out}(P) \vdash \gamma _i$| and |$\mathsf{out}(Q) \vdash \gamma _i$| and |$P_i \subset Q_i$|⁠. Similar preferences have been called pseudo-dichotomous in the literature. We recognize that the second condition corresponds to profile |$P$| being dichotomously strongly preferred by Player |$i$| to profile |$Q$|⁠. The following proposition is a simple consequence. Lemma 5.1 If Player |$i$| dichotomously strongly prefers |$P$| over |$Q$| then Player |$i$| parsimoniously strongly prefers |$P$| over |$Q$|⁠. This has another immediate consequence on Nash equilibria. Lemma 5.2 If a profile |$P$| is a Nash equilibrium in presence of parsimonious preferences, then |$P$| is a Nash equilibrium in presence of dichotomous preferences. Proof. Let |$\prec _i^d$| (resp., |$\prec _i^p$|⁠) denote Player |$i$|’s parsimonious (resp., dichotomous) preferences; let |$NE_d(G)$| (resp., |$NE_p(G)$|⁠) denote the set of Nash equilibria in |$G$| when considering dichotomous (resp., parsimonious) preferences. Now suppose that |$P \in NE_p(G)$|⁠. That is, for every |$i \in N$| and for every |$C_i \in \mathsf{ch}_i(G)$| we have not |$P \prec _i^p (C_i, P_{-i})$|⁠, and by Lemma 5.1, we have not |$P \prec _i^d (C_i, P_{-i})$|⁠. So |$P \in NE_d(G)$|⁠. Lemma 5.2 indicates that every Nash equilibrium in presence of parsimonious preferences is also a Nash equilibrium in presence of dichotomous preferences. The next proposition, which will help us later to prove some hardness result, says that the other way around holds when the profile is the one where every player plays the empty set of resources. Lemma 5.3 The profile |$(\emptyset ,\ldots ,\emptyset )$| is a Nash equilibrium in presence of parsimonious preferences iff it is a Nash equilibrium in presence of dichotomous preferences. Proof. Left to right is a consequence of Lemma 5.2. For right to left, assume |$(\emptyset ,\ldots ,\emptyset )$| is in |$NE_d(G)$|⁠. With parsimonious preferences, the only incentive to deviate from a Nash equilibrium in presence of dichotomous preferences would be to play a smaller multiset of resources. This is impossible in |$(\emptyset ,\ldots ,\emptyset )$|⁠. We now address the complexity of the decision problem of deciding whether a player parsimoniously strongly prefers a profile over another profile. Proposition 5.4 Let |$G = (N, \gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game. Let also |$P \in \mathsf{ch}(G)$| and |$Q \in \mathsf{ch}(G)$| be two profiles, and |$i \in N$| be a player. The problem of deciding whether |$Q \prec _i P$| is in |$\mathsf{PTIME}$| when provability in LOG is in |$\mathsf{PTIME}$|⁠. It is in |$\mathsf{P^{NP||[2]}}$| when provability in LOG is |$\mathsf{NP}$|-complete. It is in |$\mathsf{PSPACE}$| when provability in LOG is |$\mathsf{PSPACE}$|-complete. Proof. First, we can evaluate |$P_i \subseteq Q_i$| efficiently. We store the result in the Boolean variable |$v_{\subseteq }$|⁠. We can then perform two non-adaptive queries to an oracle to solve sequent validity in LOG on |$\mathsf{out}(P) \vdash \gamma _i$| and on |$\mathsf{out}(Q) \vdash \gamma _i$| and store the results in the Boolean variables |$v_P$| and |$v_Q$| respectively. The formula |$((\lnot v_p \land \lnot v_q \land v_{\subseteq }) \lor (v_p \land \lnot v_q) \lor (v_p \land v_q \land v_{\subseteq }))$| is true iff |$Q \prec _i P$|⁠. This yields a correct algorithm for deciding |$Q \prec _i P$| in |$\mathsf{PTIME}$| when LOG is in |$\mathsf{PTIME}$|⁠, in |$\mathsf{P^{NP||[2]}}$| when LOG is in |$\mathsf{NP}$|⁠, and in |$\mathsf{PSPACE}$| when LOG is in |$\mathsf{PSPACE}$|⁠. To compare the complexity of dichotomous and parsimonious preferences, remember from Proposition 4.1 that when LOG is in |$\mathsf{NP}$|⁠, the same problem for dichotomous preferences is in |$\mathsf{B\mathsf{H_2}}$|⁠. From [21] we know that |$\mathsf{P^{NP||[1]}} \subseteq \mathsf{B\mathsf{H_2}} \subseteq \mathsf{P^{NP||[2]}}$|⁠. It is not known whether these inclusions are strict. 5.1 Illustration of redistribution and parsimony Consider again the individual resource game of Section 4.2. (Unless stated otherwise, suppose we are in the affine case.) With parsimonious preferences, we have |$NE(G) = \{ (\emptyset , \emptyset )\}$|⁠. The profile |$(\{A\}, \{B\})$| is not a Nash equilibrium as it was with dichotomous preferences. It would be more desirable from a social welfare point of view than any other outcome (it satisfies both players), but the players would nonetheless not be individually rational by choosing it. They have indeed no bearing upon the outcome that satisfies them and thus are rational in withholding their resources. Nonetheless, like in the case of dichotomous preference, we can effectively eliminate the current Nash equilibrium in |$G^{\epsilon }$|and construct the Nash equilibrium yielding |$\{A, B\}$| by redistributing the resources present in |$G^{\epsilon }$| so as to obtain |$G^{\epsilon ^{\prime}} = (\{1,2\}, \gamma _1 = B, \gamma _2 = A, \{B\}, \{A\})$|⁠. The only Nash equilibrium is now |$(\{B\}, \{A\})$|⁠. Unlike dichotomous preferences, parsimonious preferences do not ensure the existence of a Nash equilibrium in the affine case. Consider the individual resource game |$H^\epsilon = (\{1,2\}, \gamma _1 = A, \gamma _2 = A \otimes A, \{A\}, \{A\})$|⁠. There are two players. The game |$H^\epsilon $| can be represented as on Figure 4. Figure 4 Open in new tabDownload slide The game |$H^\epsilon $|⁠. There is no Nash equilibrium under parsimonious preferences. Figure 4 Open in new tabDownload slide The game |$H^\epsilon $|⁠. There is no Nash equilibrium under parsimonious preferences. The game |$H^\epsilon $| has no Nash equilibrium: at |$(\emptyset ,\emptyset )$|⁠, Player |$1$| does not realize her objective, but she can deviate and play |$\{A\}$| to satisfy it. At |$(\{A\},\emptyset )$|⁠, Player |$2$| has an incentive to deviate and play |$\{A\}$| to realize her objective. At |$(\{A\},\{A\})$|⁠, Player |$1$| has an incentive to deviate and play |$\emptyset $|⁠. (In the affine case, this is because she can still satisfy her objective by contributing less. In the linear case, this is because she can satisfy her objective while she does not before deviating.) At |$(\emptyset ,\{A\})$|⁠, Player |$2$| does not satisfy her objective and thus has an incentive to deviate to play |$\emptyset $|⁠. However, we can construct the Nash equilibrium yielding |$\{A, A\}$|⁠. Let |$\epsilon ^{\prime}$| be the redistribution of |$\epsilon $| such that |$\epsilon ^{\prime}(2) = \{A,A\}$| and |$\epsilon ^{\prime}(1) = \emptyset $|⁠. We obtain the game depicted on Figure 5. Figure 5 Open in new tabDownload slide The game |$H^{\epsilon ^{\prime}}$|⁠. The symbol |${}{\blacksquare }$| denotes a Nash equilibrium. Figure 5 Open in new tabDownload slide The game |$H^{\epsilon ^{\prime}}$|⁠. The symbol |${}{\blacksquare }$| denotes a Nash equilibrium. In |$H^{\epsilon ^{\prime}}$|⁠, by assigning all the resources to Player |$2$|⁠, the profile |$(\emptyset , \{A,A\})$| is a Nash equilibrium and the only one. In affine logics, both players satisfy their objectives, but only Player |$2$| does when the logic is linear. 5.2 Finding Nash equilibria We study the complexity of NASH EQUILIBRIUM with parsimonious preferences. 5.2.1 Hardness We are now getting used to many-to-one reductions from sequent (non-)provability. It was a fruitful problem in presence of dichotomous preference, and it will remain one in presence of parsimonious preferences. We prove a complexity lower bound for the problem of NE in presence of parsimonious preferences. Proposition 5.5 The problem NE is as hard as the problem of checking sequent non-provability in LOG, even when there is only one player. Proof. As before, we consider w.l.o.g. only the intuitionistic sequents of LOG in the following reduction. Let |$\varGamma \vdash \delta $| be an intuitionistic sequent of LOG. We define |$\varphi = \varGamma ^* \multimap \delta $|⁠. We can construct the individual resource game |$G$| such that |$G = (\{1\},\varphi , \{ \varphi \})$|⁠. In |$G$|⁠, Player 1 has exactly two choices: |$\mathsf{ch}_i(G) = \{\emptyset , \{\varphi \}\}$|⁠. We show that |$\varGamma \vdash \delta $| iff |$\varphi \not \in NE(G)$|⁠. Suppose |$(\{\varphi \}) \not \in NE(G)$|⁠. So |$(\{\varphi \}) \prec _1 (\emptyset )$|⁠. Since by (ax) |$\varphi \vdash \varphi $| (the profile |$(\{\varphi \})$| satisfies Player |$1$|’s objectives) and |$\emptyset \subset \{\varphi \}$| (Player |$1$|’s contribution is strictly less in the profile |$(\emptyset )$| than it is in |$(\{\varphi \})$|⁠), it must be that |$\emptyset \vdash \varphi $|⁠. We infer |$\varGamma \vdash \delta $|⁠, as we did in part of the proof of Proposition 4.11. Suppose |$\varGamma \vdash \delta $|⁠. We obtain |$\varGamma ^* \vdash \delta $| by using (⁠|$\otimes $|L) enough times, and we deduce |$\vdash \varphi $| with (⁠|$\multimap $|R). We thus have |$\emptyset \vdash \varphi $| and |$\emptyset \subset \{\varphi \}$|⁠. So |$(\{\varphi \}) \prec _1 (\emptyset )$| and |$(\{\varphi \}) \not \in NE(G)$|⁠. 5.2.2 Algorithms In the individual resource game |$G = (N, \gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$|⁠, we can use Algorithm 1 to check whether a profile |$P \not \in NE(G)$|⁠, even for parsimonious preferences. We have a result analogous to Proposition 4.3 for parsimonious preferences. Proposition 5.6 If the problem of sequent provability in LOG is in |$\mathsf{PTIME}$| then NE is in |$\mathsf{coNP}$|⁠. If the problem of sequent provability in LOG is in |$\mathsf{NP}$| then NE is in |$\mathsf{\varPi _2^p}$|⁠. If the problem of sequent provability in LOG is in |$\mathsf{PSPACE}$| then NE is in |$\mathsf{PSPACE}$|⁠. Proof. We use Proposition 5.4 and, for the case of |$\mathsf{NP}$|⁠, the fact that |$\mathsf{coNP^{{P}^{NP||[2]}[1]}} \subseteq \mathsf{coNP^{{P}^{NP}}} = \mathsf{coNP^{\varDelta _2^p}} = \mathsf{co\varSigma _2^p} = \mathsf{\varPi _2^p}$|⁠. When LOG is affine, we can do better than using Algorithm 1. We first state a technical lemma which is analogous to Lemma 4.4. Lemma 5.7 Let |$G = (N, \gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game. When LOG is affine, |$P \not \in NE(G)$| iff |$\exists i \in N:$| s.t. either: 1. |$\mathsf{out}(P) \not \vdash \gamma _i$| and |$P_i \not = \emptyset $|⁠; 2. |$\mathsf{out}(P) \not \vdash \gamma _i$| and |$\mathsf{out}((P_{-i},\epsilon _i)) \vdash \gamma _i$|⁠; 3. |$\mathsf{out}(P) \vdash \gamma _i$| and |$\exists A \in P_i$|⁠: |$\mathsf{out}((P_{-i}, P_i \setminus \{A\})) \vdash \gamma _i$|⁠. Proof. Right to left is immediate. From left to right, suppose |$P \not \in NE(G)$|⁠. So there exists |$i \in N$| and |$C_i \in \mathsf{ch}_i(G)$| such that |$P\prec _i (P_{-i}, C_i)$|⁠. There are three cases to consider: 1. not |$\mathsf{out}((P_{-i}, C_i)) \vdash \gamma _i$| and not |$\mathsf{out}(P) \vdash \gamma _i$| and |$C_i \subset P_i$|⁠; 2. |$\mathsf{out}((P_{-i}, C_i)) \vdash \gamma _i$| and not |$\mathsf{out}(P) \vdash \gamma _i$|⁠; 3. |$\mathsf{out}((P_{-i}, C_i)) \vdash \gamma _i$| and |$\mathsf{out}(P) \vdash \gamma _i$| and |$C_i \subset P_i$|⁠. Suppose (1) is the case. It implies that there is |$C_i \subset P_i$| and thus that |$P_i \not = \emptyset $|⁠. Suppose (2) is the case. We essentially use the same argument as the one used in the proof of Lemma 4.4. We have |$\mathsf{out}((P_{-i}, C_i)) \vdash \gamma _i$|⁠. By applying weakening |$(|\epsilon _i| - |C_i|)$| times, we easily obtain that |$\mathsf{out}((P_{-i}, \epsilon _i)) \vdash \gamma _i$|⁠. Suppose (3) is the case. We thus have |$\mathsf{out}((P_{-i}, C_i)) \vdash \gamma _i$| with |$C_i \subset P_i$|⁠. Take a formula |$A \in P_i \setminus C_i$|⁠. Then, by applying weakening |$(|P_i| - |C_i| - 1)$| times, we easily obtain that |$\mathsf{out}((P_{-i}, P_i \setminus \{A\})) \vdash \gamma _i$|⁠. Algorithm 6 can then be used to check whether |$P \in NE(G)$|⁠.6 Algorithm 6 Algorithm for NE with parsimonious preferences and affine LOG 1: for each |$i \in N$| do: 2: if (⁠|$\mathsf{out}(P) \vdash \gamma _i$|⁠): { 3: for each |$A \in P_i$| do: 4: if (⁠|$\mathsf{out}((P_{-i},P_i\setminus \{A\})) \vdash \gamma _i$|⁠): 5: return false. 6: } else { 7: if (⁠|$\mathsf{out}((P_{-i},\epsilon _i)) \vdash \gamma _i$|⁠): 8: return false. 9: if (⁠|$P_i \not = \emptyset $|⁠): 10: return false. 11: } 12: return true. Proposition 5.8 When LOG is affine, if the problem of sequent provability in LOG is in |$\mathsf{PTIME}$|⁠, then NE is in |$\mathsf{PTIME}$|⁠. If the problem of sequent provability in LOG is in |$\mathsf{NP}$|⁠, then NE is in |$\mathsf{P^{NP||}}$|⁠. If the problem of sequent provability in LOG is in |$\mathsf{PSPACE}$|⁠, then NE is in |$\mathsf{PSPACE}$|⁠. Proof. Lemma 5.7 justifies the correctness of Algorithm 6. The algorithm can be simulated by a deterministic oracle Turing machine in polynomial time with less than |$\varSigma _{i\in N} (1 + |P_i|)$| non-adaptive queries to an oracle for sequent provability in LOG. When the complexity of sequent provability in LOG is in |$\mathsf{NP}$| it yields a complexity of |$\mathsf{P^{NP||}}$|⁠. 5.3 Elimination We study the complexity of RATIONAL ELIMINATION with parsimonious preferences. 5.3.1 Algorithms Lemma 4.8 also holds for parsimonious preferences. It is easy to see that the proof carries over. Algorithm 3 can still be used in the case of parsimonious preferences because Lemma 4.8 is still granted. We thus have the analog to Proposition 4.9 for parsimonious preferences. Proposition 5.9 When LOG is linear, RE is in |$\mathsf{NP}$| when sequent provability in LOG is in |$\mathsf{PTIME}$|⁠, in |$\mathsf{\varSigma _2^p}$| when LOG is in |$\mathsf{NP}$|⁠, and in |$\mathsf{PSPACE}$| when LOG is in |$\mathsf{PSPACE}$|⁠. Proof. We use Proposition 5.4 and, in the case of |$\mathsf{NP}$|⁠, the fact that |$\mathsf{NP^{{P}^{NP||[2]}[1]}} \subseteq \mathsf{NP^{{P}^{NP}}} = \mathsf{NP^{\varDelta _2^p}} = \mathsf{\varSigma _2^p}$|⁠. Let |$G = (N,\gamma _1, \ldots , \gamma _n, \epsilon _1, \ldots , \epsilon _n)$| be an individual resource game and let |$P \in \mathsf{ch}(G)$| be a profile. We can use Algorithm 7 to check whether a profile |$P \in \mathsf{ch}(G)$| is rationally eliminable. Algorithm 7 Algorithm for RE with parsimonious preferences and affine LOG 1: for each |$i \in N$| do: 2: if (⁠|$(\mathsf{out}(P)) \prec _i ([\epsilon \vartriangleright i](i))$|⁠): 3: return true. 4: for each |$A \in \mathsf{out}(P)$|⁠: 5: if (⁠|$(\mathsf{out}(P)) \prec _i (\mathsf{out}(P)\setminus \{A\})$|⁠): 6: return true. 7: return false. Proposition 5.10 When LOG is affine, RE is in |$\mathsf{PTIME}$| when provability in LOG is in |$\mathsf{PTIME}$|⁠. It is in |$\mathsf{P^{NP||}}$| when LOG is in |$\mathsf{NP}$|⁠. It is in |$\mathsf{PSPACE}$| when LOG is in |$\mathsf{PSPACE}$|⁠. Proof. Lemma 4.8 which still holds with parsimonious preferences ensures that it is enough to consider the redistributions |$[\epsilon \vartriangleright i]$| for some player |$i$|⁠. Algorithm 7, then checks for each of these redistributions whether Player |$i$| has an incentive to deviate in the game |$G^{[\epsilon \vartriangleright i]}$| from the profile |$(\mathsf{out}(P)) \in \mathsf{ch}(G^{[\epsilon \vartriangleright i]})$| to any one of |$([\epsilon \vartriangleright i](i)) \in \mathsf{ch}(G^{[\epsilon \vartriangleright i]})$| and |$(\mathsf{out}(P) \setminus \{A\}) \in \mathsf{ch}(G^{[\epsilon \vartriangleright i]})$| for some |$A \in \mathsf{out}(P)$|⁠. It is weakening |$(W)$| that justifies that it is enough to consider these profiles, because |$X \not \vdash \gamma _i$| implies |$Y \not \vdash \gamma _i$| for any couple of multisets |$Y \subseteq X$|⁠. The correctness of Algorithm 7 follows. The tests of line 2 and line 5 only involve the following instances of the sequent provability decision problem: |$(\mathsf{out}(P)) \vdash \gamma _i$| and |$([\epsilon \vartriangleright i](i)) \vdash \gamma _i$| for very Player |$i \in N$|⁠, and |$(\mathsf{out}(P)\setminus \{A\}) \vdash \gamma _i$|⁠, for every Player |$i \in N$| and every formula |$A \in \mathsf{out}(P)$|⁠. The algorithm can thus be simulated by a deterministic oracle Turing machine in polynomial time with at most |$|N|(|\mathsf{out}(P)|+2)$| non-adaptive calls to an oracle for sequent provability. 5.3.2 Hardness After Lemma 5.3 and the proof of Proposition 4.11, the following proposition does not come as a surprise. Proposition 5.11 RE is as hard as the problem of checking sequent non-provability in LOG. Proof. Let |$\varGamma \vdash \delta $| be an arbitrary intuitionistic sequent. We construct the same game as in the proof of Proposition 4.11. Let |$\varphi = \varGamma ^* \multimap \delta $|⁠. Let |$G^\epsilon = (\{1,2\}, \varphi , \textbf{1}, \emptyset , \{\varphi \})$|⁠. In the proof of Proposition 4.11, we showed that, in presence of dichotomous preferences, both in the case of linear and of affine logics, we have |$\varGamma \not \vdash \delta $| iff |$(\emptyset , \emptyset )$| is rationally eliminable in |$G^\epsilon $|⁠. Now with Lemma 5.3, we know that |$(\emptyset , \emptyset )$| is a Nash equilibrium in presence of dichotomous preferences iff it is a Nash equilibrium in presence of parsimonious preferences (both in |$G^{\epsilon }$| and |$G^{\epsilon ^{\prime}}$|⁠, and no matter if LOG is linear or affine, or if |$\varGamma \vdash \delta $| or |$\varGamma \not \vdash \delta $|⁠). Hence, we have |$\varGamma \not \vdash \delta $| iff |$(\emptyset , \emptyset )$| is rationally eliminable in |$G^\epsilon $|⁠, also in presence of parsimonious preferences. 5.4 Construction Finally, we tackle the complexity of RATIONAL CONSTRUCTION with parsimonious preferences. 5.4.1 Hardness We establish a complexity lower bound for the problem of RC in presence of parsimonious preferences. Proposition 5.12 RC is as hard as the problem of checking sequent non-provability in LOG. Proof. Consider the games in the proof of Proposition 5.11. We can see that both for linear and affine logics we have that |$\varGamma \not \vdash \delta $| iff |$(\{\varphi \}, \emptyset )$| can be rationally constructed in |$G^{\epsilon ^{\prime}}$|⁠. 5.4.2 Algorithms Our algorithmic analysis is very similar to the analysis we made when the preferences are dichotomous in Section 4.3.2. Let |$G^\epsilon $| be an individual resource game and |$P \in \mathsf{ch}(G^\epsilon )$|⁠. To decide whether |$P$| can be rationally constructed, we can reuse Algorithm 5. Again, we use the problem NE as a blackbox, for which complexity upper bounds have been established in Propositions 5.6 and 5.8. Proposition 5.13 When sequent provability in LOG is in |$\mathsf{PTIME}$|⁠, RC is in |$\mathsf{\varSigma _2^p}$|⁠. When LOG is in |$\mathsf{NP}$|⁠, RC is in |$\mathsf{\varSigma _3^p}$|⁠. When LOG is in |$\mathsf{PSPACE}$|⁠, RC is in |$\mathsf{PSPACE}$|⁠. Proof. The proof is similar to the one of Proposition 4.13, using the result of Proposition 5.6. The next proposition also comes without surprise. Proposition 5.14 If LOG is affine, when LOG is in |$\mathsf{PTIME}$|⁠, RC is in |$\mathsf{NP}$|⁠. When LOG is in |$\mathsf{NP}$|⁠, RC is in |$\mathsf{\varSigma _2^p}$|⁠. When LOG is in |$\mathsf{PSPACE}$|⁠, RC is in |$\mathsf{PSPACE}$|⁠. Proof. The proof is similar to the one of Proposition 4.14, using the result of Proposition 5.8. 6 Examples We present more thorough examples. They involve several resources and objectives that are modelled with a variety of logical operands. We take the opportunity to present fully the important formal proofs of the realized objectives. We start with a toy example, simple but rich enough, upon which we can demonstrate all the frameworks and problems addressed in the paper. Then, we formally study the divorce arbitration scenario of Example 1.2, as well as a three-player variant of the scenario of interconnected economies from Example 1.1. 6.1 Alan and the fish We first introduce the resources involved and how they are built in the logical language. • Basic resources: – one mole of dioxygen: |$\mathsf{O_2}$| – one mole of dihydrogen: |$\mathsf{H_2}$| – one mole of water: |$\mathsf{H_2O}$| – one ‘token’ of thirst: |$\mathsf{thirst}$| • Anti-resources can be captured via the linear negation: – one thirst quencher: |$\mathop \sim \mathsf{thirst}$| • Resource transformation processes: – one process of electrolysis: |$\mathsf{elec} = \mathsf{H_2O} \otimes \mathsf{H_2O} \multimap \mathsf{H_2} \otimes \mathsf{H_2} \otimes \mathsf{O_2}$| – one process of drinking water: |$\mathsf{drink} = \mathsf{H_2O} \multimap \mathop \sim \mathsf{thirst}$| Game definition. Let |$G^\epsilon _{\textit{af}} = (\{a,f\},\gamma _a, \gamma _f, \epsilon _a, \epsilon _f)$| be the individual resource game with two players, Alan |$a$| and the Fish |$f$|⁠. The fish wants one mole of dioxygen: |$\gamma _f = \mathsf{O_2}$|⁠. Alan wants one mole of dioxygen for his fish and wants to quench his thirst: |$\gamma _a = \mathsf{O_2} \otimes \mathop \sim \mathsf{thirst}$|⁠. In the game |$G^\epsilon _{\textit{af}}$|⁠, Alan is endowed with |$\epsilon _a = \{\mathsf{drink}, \mathsf{elec}\}$|⁠. He can drink once and can electrolyse water once. The fish is endowed with three tokens of water |$\epsilon _f = \{\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}\}$|⁠. We suppose that LOG is affine. For this example, we will consider both cases of dichotomous and parsimonious preferences. As we did before, we will represent a Nash equilibrium under dichotomous preferences with the symbol |${}{\square }$| and under parsimonious preferences with the symbol |${}{\blacksquare }$|⁠. By Lemma 5.2, the latter implies the former. Then, when a profile is a Nash equilibrium under both dichotomous and parsimonious preferences we will use the symbol |${}{\blacksquare }$|⁠. The game |$G^\epsilon _{\textit{af}}$| and the realized objectives can be depicted as on Figure 6. Figure 6 Open in new tabDownload slide The game |$G^{\epsilon }_{\textit{af}}$|⁠. Alan plays rows, and the fish plays columns. LOG is affine. The symbol |${}{\square }$| marks the Nash equilibria under dichotomous preferences. The symbol |${}{\blacksquare }$| marks the profiles that are also Nash equilibria under both dichotomous and parsimonious preferences. Figure 6 Open in new tabDownload slide The game |$G^{\epsilon }_{\textit{af}}$|⁠. Alan plays rows, and the fish plays columns. LOG is affine. The symbol |${}{\square }$| marks the Nash equilibria under dichotomous preferences. The symbol |${}{\blacksquare }$| marks the profiles that are also Nash equilibria under both dichotomous and parsimonious preferences. Appendix C provides the detailed proofs of the realized objectives. Dichotomous preferences: eliminations of bad equilibria. If the preferences are dichotomous, there are plenty Nash equilibria in |$G^\epsilon _{\textit{af}}$|⁠. They are |$(\emptyset , \emptyset )$|⁠, |$(\emptyset , \{\mathsf{H_2O}\})$|⁠, |$(\emptyset , \{\mathsf{H_2O}, \mathsf{H_2O}\})$|⁠, |$(\{\mathsf{drink}\}, \emptyset )$|⁠, |$(\{\mathsf{drink}\}, \{\mathsf{H_2O}\})$|⁠, |$(\{\mathsf{drink}\}, \{\mathsf{H_2O}, \mathsf{H_2O}\})$|⁠, |$(\{\mathsf{elec}\}, \{\mathsf{H_2O},\mathsf{H_2O}\})$|⁠, |$(\{\mathsf{drink},\mathsf{elec}\}, \{\mathsf{H_2O},\mathsf{H_2O}\})$|⁠, and |$(\{\mathsf{drink},\mathsf{elec}\}, \{\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}\})$|⁠. However, only the profile |$(\{\mathsf{drink},\mathsf{elec}\}, \{\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}\})$|⁠, whose outcome is |$\{\mathsf{drink},\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}, \mathsf{elec}\}$|⁠, satisfies the objectives of both players. It would thus be desirable to eliminate the other profiles. To do so, let |$\epsilon ^{\prime}$| be the endowment such that |$\epsilon ^{\prime}_a = \{\mathsf{drink},\mathsf{elec},\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}\}$| and |$\epsilon ^{\prime}_f = \emptyset $|⁠. The game |$G^{\epsilon ^{\prime}}_{\textit{af}}$| and the realized objectives can be (partially) depicted as on Figure 7. Figure 7 Open in new tabDownload slide The game |$G^{\epsilon ^{\prime}}_{\textit{af}}$|⁠. Figure 7 Open in new tabDownload slide The game |$G^{\epsilon ^{\prime}}_{\textit{af}}$|⁠. It is readily seen that in |$G^{\epsilon ^{\prime}}_{\textit{af}}$|⁠, when preferences are dichotomous, only the profile |$(\{\mathsf{drink},\mathsf{elec},\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}\}, \emptyset )$| whose outcome is |$\{\mathsf{drink},\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O},\mathsf{elec}\}$|⁠, is a Nash equilibrium. Parsimonious preferences: construction of a good equilibrium. If the preferences are parsimonious, the profile |$(\emptyset , \emptyset )$| is a Nash equilibrium in the game |$G^{\epsilon }_{\textit{af}}$|⁠, and is the only one. One can nonetheless redistribute the resources so as to construct an equilibrium where Alan and the fish both realize their objectives. That is, one can construct the profile |$(\{\mathsf{drink}, \mathsf{elec}\}, \{\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}\})$|⁠. To do so, let |$\epsilon ^{\prime\prime}$| be the endowment such that |$\epsilon ^{\prime\prime}_a = \{\mathsf{drink},\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}\}$| and |$\epsilon ^{\prime\prime}_f = \{\mathsf{elec}\}$|⁠. The game |$G^{\epsilon ^{\prime\prime}}_{\textit{af}}$| and the realized objectives can be depicted as on Figure 8. Figure 8 Open in new tabDownload slide The game |$G^{\epsilon ^{\prime\prime}}_{\textit{af}}$|⁠. Figure 8 Open in new tabDownload slide The game |$G^{\epsilon ^{\prime\prime}}_{\textit{af}}$|⁠. When preferences are parsimonious, the profiles |$(\emptyset , \emptyset )$| and |$(\{\mathsf{drink},\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O}\}, \{\mathsf{elec}\})$| are Nash equilibria in |$G^{\epsilon ^{\prime\prime}}_{\textit{af}}$| and are the only ones. Notice that, the redistribution |$\epsilon ^{\prime}$| would also effectively construct the profile, although at the price of a more draconian redistribution. It would also eliminate |$(\emptyset , \emptyset )$|⁠. 6.2 Ann and Bernard get a divorce We formalize Example 1.2. We will only consider parsimonious preferences. We also assume that LOG is affine MLL. We introduce the resources involved in the example. • the lease agreement: |$\mathsf{shop}$| • the resource of flour for a year: |$\mathsf{flour}$| • the resource of one year worth of bread: |$\mathsf{bread}$| • the bread making equipment is the resource transformation process: |$\mathsf{flour} \multimap \mathsf{bread}$| Using these as basic resources, we formalize Example 1.2 as the game |$G^{\epsilon }_{\textit{ab}}$|⁠. Game definition. Let |$G^\epsilon _{\textit{ab}} = (\{a,b\},\gamma _a, \gamma _b, \epsilon _a, \epsilon _b)$| be the individual resource game with two players, Ann |$a$| and Bernard |$b$|⁠. Ann wants enough bread for a year: |$\gamma _a = \mathsf{bread}$|⁠. Bernard wants the lease agreement: |$\gamma _b = \mathsf{shop}$|⁠. In the game |$G^\epsilon _{\textit{ab}}$|⁠, Ann is endowed with the lease agreement: |$\epsilon _a = \{\mathsf{shop}\}$|⁠. Bernard is endowed with enough flour to make bread for two years and with the bread making equipment: |$\epsilon _b = \{\mathsf{flour}, \mathsf{flour}, \mathsf{\mathsf{flour} \multimap \mathsf{bread}}\}$|⁠. The game |$G^{\epsilon }_{\textit{ab}}$| and the realized objectives can be depicted as on Figure 9. Figure 9 Open in new tabDownload slide The game |$G^{\epsilon }_{\textit{ab}}$|⁠. Bernard plays rows, and Ann plays columns. The profile |$(\emptyset ,\emptyset ) \in \mathsf{ch}_b \times \mathsf{ch}_a$| is the only Nash equilibrium in presence of parsimonious preferences. Figure 9 Open in new tabDownload slide The game |$G^{\epsilon }_{\textit{ab}}$|⁠. Bernard plays rows, and Ann plays columns. The profile |$(\emptyset ,\emptyset ) \in \mathsf{ch}_b \times \mathsf{ch}_a$| is the only Nash equilibrium in presence of parsimonious preferences. All the formal proofs of the realized objectives are trivial. An undesirable equilibrium. One can see on Figure 9 that the profiles |$(\{\mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}, \{\mathsf{shop}\})$| and |$(\{\mathsf{flour}, \mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}, \{\mathsf{shop}\})$| in |$\mathsf{ch}_b \times \mathsf{ch}_a$| would satisfy both Ann and Bernard. However, in both of them, Bernard has an incentive to provide less resources from his endowment and to deviate to |$\emptyset \in \mathsf{ch}_b$|⁠. In turn, in |$(\emptyset , \{\mathsf{shop}\}) \in \mathsf{ch}_b \times \mathsf{ch}_a$|⁠, Ann is not satisfied, and so has an incentive to retain her resources as well, deviating to her choice |$\emptyset \in \mathsf{ch}_a$|⁠. The profiles |$(\{\mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}, \emptyset )$| and |$(\{\mathsf{flour}, \mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}, \emptyset \})$| in |$\mathsf{ch}_b \times \mathsf{ch}_a$| satisfy Ann’s objective but do not satisfy Bernard’s. Hence, Bernard has an incentive to deviate to |$\emptyset \in \mathsf{ch}_b$|⁠. The profile |$(\emptyset , \emptyset )$| is the only Nash equilibrium of |$G^{\epsilon }_{\textit{ab}}$|⁠, but it satisfies neither Ann’s objective, nor Bernard’s. On the other hand, the outcome of the profile |$(\{\mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}, \{\mathsf{shop}\}) \in \mathsf{ch}_b \times \mathsf{ch}_a$| would satisfy them both. A desirable redistribution. So the arbitrator redistributes the resources that are available. He assigns the bread making equipment and half the flour to Ann. He assigns the lease agreement and half the flour to Bernard. That is, |$\epsilon ^{\prime}_a = \{\mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}$| and |$\epsilon ^{\prime}_b = \{\mathsf{flour}, \mathsf{shop}\}$|⁠. This redistribution yields the game |$G^{\epsilon ^{\prime}}_{\textit{ab}}$|⁠. It can be depicted as on Figure 10. Figure 10 Open in new tabDownload slide The game |$G^{\epsilon ^{\prime}}_{\textit{ab}}$|⁠. The profile |$(\{\mathsf{shop}\}, \{\mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}) \in \mathsf{ch}_b \times \mathsf{ch}_a$| is the only Nash equilibrium in presence of parsimonious preferences. Figure 10 Open in new tabDownload slide The game |$G^{\epsilon ^{\prime}}_{\textit{ab}}$|⁠. The profile |$(\{\mathsf{shop}\}, \{\mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}) \in \mathsf{ch}_b \times \mathsf{ch}_a$| is the only Nash equilibrium in presence of parsimonious preferences. In |$G^{\epsilon ^{\prime}}_{\textit{ab}}$|⁠, the profile |$(\emptyset , \emptyset )$| is not a Nash equilibrium and so has been eliminated from |$G^{\epsilon }_{\textit{ab}}$|⁠. Indeed, it does not satisfy Bernard, and he has an incentive to deviate to the profile |$(\{\mathsf{shop}\}, \emptyset ) \in \mathsf{ch}_b \times \mathsf{ch}_a$| in which his objective is satisfied. But |$(\{\mathsf{shop}\}, \emptyset )$| is not a Nash equilibrium either. Indeed, it does not satisfy Ann, and she has an incentive to deviate to the profile |$(\{\mathsf{shop}\}, \{\mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}) \in \mathsf{ch}_b \times \mathsf{ch}_a$|⁠. From here, nobody has an incentive to deviate, and it is a Nash equilibrium. It is in fact the only Nash equilibrium in |$G^{\epsilon ^{\prime}}_{\textit{ab}}$|⁠. One can readily see that the profile |$(\{\mathsf{flour}, \mathsf{shop}\}, \{\mathsf{flour}, \mathsf{flour} \multimap \mathsf{bread}\}) \in \mathsf{ch}_b \times \mathsf{ch}_a$|⁠, even though it satisfies both Ann and Bernard, is not a Nash equilibrium. Bernard has an incentive to provide less resources. The same can be said about the profile |$(\{\mathsf{flour}, \mathsf{shop}\}, \{\mathsf{flour} \multimap \mathsf{bread}\}) \in \mathsf{ch}_b \times \mathsf{ch}_a$|⁠. 6.3 An interconnected economy We present a three-player variant of Example 1.1. The setting, which we remind briefly, is analogous. In a local telecom industry, three companies must by regulation accept traffic from each other’s customers. Moreover, activating a network at some capacity has a cost, and companies can privately activate and deactivate networks on the fly. Company |$A$| manages a 3G network of comprised capacity |$3$| (bundled as capacities |$1$| and |$2$|⁠). Company |$B$| manages a 4G network of capacity |$3$| (bundled as capacities |$1$| and |$2$|⁠). Company |$A$| needs to offer their customers 3G at capacity |$2$| and 4G at capacity |$1$|⁠. Company |$B$| needs to offer their customers 3G at capacity |$2$| and 4G at capacity |$2$|⁠. A new company, Company |$C$| is entering in this interconnected economy. It has some capital, say, two token of an arbitrary unit; one token being fair price for a mobile network antenna. However, Company |$C$| does not manage any network. Company |$C$| needs to offer their customers 3G at capacity |$1$| and 4G at capacity |$1$|⁠. Again, we will only consider parsimonious preferences and assume that LOG is MULT. Since we are using this modest fragment, we trust that formal proofs would be more than superfluous and will be omitted. We introduce the resources involved in the scenario: • the resource of one capacity of 3G network: |$\mathsf{3G}$| • the resource of one capacity of 4G network: |$\mathsf{4G}$| • the resource of one token of capital: |$\mathsf{cap}$| Game definition. Let |$G^\epsilon _{\textit{ie}} = (\{a,b,c\},\gamma _a, \gamma _b, \gamma _c, \epsilon _a, \epsilon _b, \epsilon _c)$| be the individual resource game with three players, Company |$A$|⁠, |$B$| and |$C$| being represented by |$a$|⁠, |$b$| and |$c$|⁠, respectively. In the game |$G^\epsilon _{\textit{ie}}$|⁠, we have |$\epsilon _a = \{ \mathsf{3G}, \mathsf{3G} \otimes \mathsf{3G}\}$|⁠. |$\epsilon _b = \{ \mathsf{4G}, \mathsf{4G} \otimes \mathsf{4G}\}$|⁠, and |$\epsilon _c = \{\mathsf{cap}, \mathsf{cap}\}$| for endowments. The objectives are as follows: |$\gamma _a = \mathsf{3G} \otimes \mathsf{3G} \otimes \mathsf{4G}$|⁠, |$\gamma _b = \mathsf{3G} \otimes \mathsf{3G} \otimes \mathsf{4G} \otimes \mathsf{4G}$|⁠, and |$\gamma _c = \mathsf{3G} \otimes \mathsf{4G}$|⁠. Two equilibria. The game |$G^{\epsilon }_{\textit{ie}}$| and the realized objectives can be depicted as on Figure 11. Company |$B$|⁠, Player |$b$|⁠, plays rows; Company |$A$|⁠, Player |$a$|⁠, plays column. For simplicity, we do not represent all Company |$C$|’s choices because they do not bear on the players’ objectives. We only represent Player |$c$|’s choice |$\emptyset $|⁠. With other choices different from |$\emptyset $|⁠, the realized objectives are exactly the same. Assuming parsimonious preferences, no profile where Company |$C$|’s action is different from |$\emptyset $| is a Nash equilibrium. Figure 11 Open in new tabDownload slide Partial representation of the game |$G^{\epsilon }_{\textit{ie}}$|⁠. Figure 11 Open in new tabDownload slide Partial representation of the game |$G^{\epsilon }_{\textit{ie}}$|⁠. There are two Nash equilibria in the IRG |$G^{\epsilon }_{\textit{ie}}$|⁠, namely, |$(\emptyset , \emptyset , \emptyset )$| and |$(\{\mathsf{3G} \otimes \mathsf{3G}\}, \{\mathsf{4G} \otimes \mathsf{4G}\}, \emptyset )$|⁠. In the latter, all agents realize their objective. In the former, none of them do. Eliminating the bad equilibrium. In the IRG |$G^{\epsilon }_{\textit{ie}}$|⁠, the profile |$(\emptyset , \emptyset , \emptyset )$| is an arguably undesirable equilibrium. An arbitrator could however advise the three companies to redistribute their endowments to eliminate |$(\emptyset , \emptyset , \emptyset )$|⁠. The arbitrator could propose the redistribution |$\epsilon ^{\prime}$| of |$\epsilon $|⁠, where |$\epsilon ^{\prime}_a = \{ \mathsf{3G} \otimes \mathsf{3G}, \mathsf{cap}\}$|⁠, |$\epsilon ^{\prime}_b = \{ \mathsf{4G} \otimes \mathsf{4G}, \mathsf{cap}\}$| and |$\epsilon ^{\prime}_c = \{\mathsf{3G}, \mathsf{4G}\}$|⁠. The game |$G^{\epsilon ^{\prime}}_{\textit{ie}}$| and the realized objectives can be depicted as on Figure 12, when Player |$c$|’s choice is |$\emptyset $|⁠. The choices containing the resource |$\mathsf{cap}$| are not represented. The resource |$\mathsf{cap}$| has no bearing on the player’s objectives, and no profile containing it is a Nash equilibrium. Figure 12 Open in new tabDownload slide Partial representation of the game |$G^{\epsilon ^{\prime}}_{\textit{ie}}$|⁠. Figure 12 Open in new tabDownload slide Partial representation of the game |$G^{\epsilon ^{\prime}}_{\textit{ie}}$|⁠. After the redistribution, Company |$C$| manages a 3G and a 4G network, both at capacity |$1$|⁠. Activating both of them would be enough to satisfy Company |$C$|’s objective. In |$G^{\epsilon ^{\prime}}_{\textit{ie}}$|⁠, Player |$c$| thus has an incentive to deviate from |$(\emptyset , \emptyset , \emptyset )$|⁠. Hence, the arbitrator’s advice permits the elimination of the bad equilibrium: |$(\emptyset , \emptyset , \emptyset )$| is not a Nash equilibrium in |$G^{\epsilon ^{\prime}}_{\textit{ie}}$|⁠. In the profile |$(\emptyset , \emptyset , \{\mathsf{3G}, \mathsf{4G}\})$|⁠, Player |$1$| has an incentive to deviate and play |$\{\mathsf{3G} \otimes \mathsf{3G}\}$|⁠, in order to realize its objective. In turn, in the profile |$(\{\mathsf{3G} \otimes \mathsf{3G}\}, \emptyset , \{\mathsf{3G}, \mathsf{4G}\})$|⁠, Player |$2$| has an incentive to deviate and play |$\{\mathsf{4G} \otimes \mathsf{4G}\}$| to realize its objective. (Player |$3$|⁠, by parsimony, has also an incentive to withdraw the resource |$\mathsf{3G}$|⁠.) In the profile |$(\{\mathsf{3G} \otimes \mathsf{3G}\}, \{\mathsf{4G} \otimes \mathsf{4G}\}, \{\mathsf{3G}, \mathsf{4G}\})$|⁠, by parsimony, Player |$3$| has an incentive to deviate to the choice |$\emptyset $|⁠. Every player is satisfied in |$(\{\mathsf{3G} \otimes \mathsf{3G}\}, \{\mathsf{4G} \otimes \mathsf{4G}\}, \emptyset )$|⁠, and none of them have an incentive to withdraw any resources. Hence, the good equilibrium of |$G^{\epsilon }_{\textit{ie}}$|⁠, |$(\{\mathsf{3G} \otimes \mathsf{3G}\}, \{\mathsf{4G} \otimes \mathsf{4G}\}, \emptyset )$|⁠, is still a Nash equilibrium in |$G^{\epsilon ^{\prime}}_{\textit{ie}}$|⁠. In addition, this is the unique Nash equilibrium in |$G^{\epsilon ^{\prime}}_{\textit{ie}}$|⁠. 7 Conclusions We presented a class of games of resources that exploits the formalisms and reasoning methods for resource-sensitive logics. The language of linear logic allows us to represent in a harmonious way simultaneous resources, deterministic and non-deterministic choice and resource-transforming capacities. In individual resource games, each player of a game is endowed with a multiset of resources and has an objective represented by a resource. In this context, we studied three decision problems, the first of which is to decide whether a profile is a Nash equilibrium. Some profiles that are not equilibria can have desirable outcomes from the point of view of an external authority. Some equilibria can have outcomes that are undesirable. We thus studied redistribution schemes which can be used by a central authority to enforce some behavior, either by disincentivizing a behavior or incentivizing a behavior. This yielded two related decision problems: rational elimination and rational construction of profiles. We illustrated the models and the decision problems with two examples. We considered dichotomous or parsimonious preferences and showed striking algorithmic differences when the logic employed admits or not the weakening rule. Summary of the complexity results. For all decision problems, for both types of preferences, we have studied six cases where LOG can have the following properties along two dimensions: (i) affine vs. linear, and (ii) in |$\mathsf{PTIME}$| vs. in |$\mathsf{NP}$| vs. in |$\mathsf{PSPACE}$|⁠. When LOG is |$\mathsf{NP}$|-complete, we sum up precisely the results in Table 4. Table 4 Complexity results when the problem of provability in LOG is in |$\mathsf{NP}$|⁠. . . Linear . Affine . |$\mathsf{NP}$|-hard (Prop. 4.2) |$\mathsf{NP}$|-hard (Prop. 4.2) NE in |$\mathsf{\varPi _2^p}$| (Prop. 4.3) in |$\mathsf{P^{NP||}}$| (Prop. 4.6) |$\mathsf{coNP}$|-hard (Prop. 4.11) |$\mathsf{coNP}$|-hard (Prop. 4.11) Dichotomous RE in |$\mathsf{\varSigma _2^p}$| (Prop. 4.9) in |$\mathsf{P^{NP||}}$| (Prop. 4.10) |$\mathsf{NP}$|-hard (Prop. 4.12) |$\mathsf{NP}$|-hard (Prop. 4.12) RC in |$\mathsf{\varSigma _3^p}$| (Prop. 4.13) in |$\mathsf{\varSigma _2^p}$| (Prop. 4.14) |$\mathsf{coNP}$|-hard (Prop. 5.5) |$\mathsf{coNP}$|-hard (Prop. 5.5) NE in |$\mathsf{\varPi _2^p}$| (Prop. 5.6) in |$\mathsf{P^{NP||}}$| (Prop. 5.8) |$\mathsf{coNP}$|-hard (Prop. 5.11) |$\mathsf{coNP}$|-hard (Prop. 5.11) Parsimonious RE in |$\mathsf{\varSigma _2^p}$| (Prop. 5.9) in |$\mathsf{P^{NP||}}$| (Prop 5.10) |$\mathsf{coNP}$|-hard (Prop. 5.12) |$\mathsf{coNP}$|-hard (Prop. 5.12) RC in |$\mathsf{\varSigma _3^p}$| (Prop. 5.13) in |$\mathsf{\varSigma _2^p}$| (Prop. 5.14) . . Linear . Affine . |$\mathsf{NP}$|-hard (Prop. 4.2) |$\mathsf{NP}$|-hard (Prop. 4.2) NE in |$\mathsf{\varPi _2^p}$| (Prop. 4.3) in |$\mathsf{P^{NP||}}$| (Prop. 4.6) |$\mathsf{coNP}$|-hard (Prop. 4.11) |$\mathsf{coNP}$|-hard (Prop. 4.11) Dichotomous RE in |$\mathsf{\varSigma _2^p}$| (Prop. 4.9) in |$\mathsf{P^{NP||}}$| (Prop. 4.10) |$\mathsf{NP}$|-hard (Prop. 4.12) |$\mathsf{NP}$|-hard (Prop. 4.12) RC in |$\mathsf{\varSigma _3^p}$| (Prop. 4.13) in |$\mathsf{\varSigma _2^p}$| (Prop. 4.14) |$\mathsf{coNP}$|-hard (Prop. 5.5) |$\mathsf{coNP}$|-hard (Prop. 5.5) NE in |$\mathsf{\varPi _2^p}$| (Prop. 5.6) in |$\mathsf{P^{NP||}}$| (Prop. 5.8) |$\mathsf{coNP}$|-hard (Prop. 5.11) |$\mathsf{coNP}$|-hard (Prop. 5.11) Parsimonious RE in |$\mathsf{\varSigma _2^p}$| (Prop. 5.9) in |$\mathsf{P^{NP||}}$| (Prop 5.10) |$\mathsf{coNP}$|-hard (Prop. 5.12) |$\mathsf{coNP}$|-hard (Prop. 5.12) RC in |$\mathsf{\varSigma _3^p}$| (Prop. 5.13) in |$\mathsf{\varSigma _2^p}$| (Prop. 5.14) Open in new tab Table 4 Complexity results when the problem of provability in LOG is in |$\mathsf{NP}$|⁠. . . Linear . Affine . |$\mathsf{NP}$|-hard (Prop. 4.2) |$\mathsf{NP}$|-hard (Prop. 4.2) NE in |$\mathsf{\varPi _2^p}$| (Prop. 4.3) in |$\mathsf{P^{NP||}}$| (Prop. 4.6) |$\mathsf{coNP}$|-hard (Prop. 4.11) |$\mathsf{coNP}$|-hard (Prop. 4.11) Dichotomous RE in |$\mathsf{\varSigma _2^p}$| (Prop. 4.9) in |$\mathsf{P^{NP||}}$| (Prop. 4.10) |$\mathsf{NP}$|-hard (Prop. 4.12) |$\mathsf{NP}$|-hard (Prop. 4.12) RC in |$\mathsf{\varSigma _3^p}$| (Prop. 4.13) in |$\mathsf{\varSigma _2^p}$| (Prop. 4.14) |$\mathsf{coNP}$|-hard (Prop. 5.5) |$\mathsf{coNP}$|-hard (Prop. 5.5) NE in |$\mathsf{\varPi _2^p}$| (Prop. 5.6) in |$\mathsf{P^{NP||}}$| (Prop. 5.8) |$\mathsf{coNP}$|-hard (Prop. 5.11) |$\mathsf{coNP}$|-hard (Prop. 5.11) Parsimonious RE in |$\mathsf{\varSigma _2^p}$| (Prop. 5.9) in |$\mathsf{P^{NP||}}$| (Prop 5.10) |$\mathsf{coNP}$|-hard (Prop. 5.12) |$\mathsf{coNP}$|-hard (Prop. 5.12) RC in |$\mathsf{\varSigma _3^p}$| (Prop. 5.13) in |$\mathsf{\varSigma _2^p}$| (Prop. 5.14) . . Linear . Affine . |$\mathsf{NP}$|-hard (Prop. 4.2) |$\mathsf{NP}$|-hard (Prop. 4.2) NE in |$\mathsf{\varPi _2^p}$| (Prop. 4.3) in |$\mathsf{P^{NP||}}$| (Prop. 4.6) |$\mathsf{coNP}$|-hard (Prop. 4.11) |$\mathsf{coNP}$|-hard (Prop. 4.11) Dichotomous RE in |$\mathsf{\varSigma _2^p}$| (Prop. 4.9) in |$\mathsf{P^{NP||}}$| (Prop. 4.10) |$\mathsf{NP}$|-hard (Prop. 4.12) |$\mathsf{NP}$|-hard (Prop. 4.12) RC in |$\mathsf{\varSigma _3^p}$| (Prop. 4.13) in |$\mathsf{\varSigma _2^p}$| (Prop. 4.14) |$\mathsf{coNP}$|-hard (Prop. 5.5) |$\mathsf{coNP}$|-hard (Prop. 5.5) NE in |$\mathsf{\varPi _2^p}$| (Prop. 5.6) in |$\mathsf{P^{NP||}}$| (Prop. 5.8) |$\mathsf{coNP}$|-hard (Prop. 5.11) |$\mathsf{coNP}$|-hard (Prop. 5.11) Parsimonious RE in |$\mathsf{\varSigma _2^p}$| (Prop. 5.9) in |$\mathsf{P^{NP||}}$| (Prop 5.10) |$\mathsf{coNP}$|-hard (Prop. 5.12) |$\mathsf{coNP}$|-hard (Prop. 5.12) RC in |$\mathsf{\varSigma _3^p}$| (Prop. 5.13) in |$\mathsf{\varSigma _2^p}$| (Prop. 5.14) Open in new tab For instance, one can quickly gather that when LOG is affine MLL (whose sequent provability is |$\mathsf{NP}$|-complete) and we consider parsimonious preferences, RATIONAL ELIMINATION is in |$\mathsf{P^{NP||}}$|⁠. We proved the same problem to be in |$\mathsf{\varSigma _2^p}$| when LOG is linear MLL. When LOG is in |$\mathsf{PTIME}$|⁠, we sum up precisely the results in Table 5. Table 5 Complexity results when the problem of provability in LOG is in |$\mathsf{PTIME}$|⁠. . . Linear . Affine . NE in |$\mathsf{coNP}$| (Prop. 4.3) in |$\mathsf{PTIME}$| (Prop. 4.6) Dichotomous RE in |$\mathsf{NP}$| (Prop. 4.9) in |$\mathsf{PTIME}$| (Prop. 4.10) RC in |$\mathsf{\varSigma _2^p}$| (Prop. 4.13) in |$\mathsf{NP}$| (Prop. 4.14) NE in |$\mathsf{coNP}$| (Prop. 5.6) in |$\mathsf{PTIME}$| (Prop. 5.8) Parsimonious RE in |$\mathsf{NP}$| (Prop. 5.9) in |$\mathsf{PTIME}$| (Prop 5.10) RC in |$\mathsf{\varSigma _2^p}$| (Prop. 5.13) in |$\mathsf{NP}$| (Prop. 5.14) . . Linear . Affine . NE in |$\mathsf{coNP}$| (Prop. 4.3) in |$\mathsf{PTIME}$| (Prop. 4.6) Dichotomous RE in |$\mathsf{NP}$| (Prop. 4.9) in |$\mathsf{PTIME}$| (Prop. 4.10) RC in |$\mathsf{\varSigma _2^p}$| (Prop. 4.13) in |$\mathsf{NP}$| (Prop. 4.14) NE in |$\mathsf{coNP}$| (Prop. 5.6) in |$\mathsf{PTIME}$| (Prop. 5.8) Parsimonious RE in |$\mathsf{NP}$| (Prop. 5.9) in |$\mathsf{PTIME}$| (Prop 5.10) RC in |$\mathsf{\varSigma _2^p}$| (Prop. 5.13) in |$\mathsf{NP}$| (Prop. 5.14) Open in new tab Table 5 Complexity results when the problem of provability in LOG is in |$\mathsf{PTIME}$|⁠. . . Linear . Affine . NE in |$\mathsf{coNP}$| (Prop. 4.3) in |$\mathsf{PTIME}$| (Prop. 4.6) Dichotomous RE in |$\mathsf{NP}$| (Prop. 4.9) in |$\mathsf{PTIME}$| (Prop. 4.10) RC in |$\mathsf{\varSigma _2^p}$| (Prop. 4.13) in |$\mathsf{NP}$| (Prop. 4.14) NE in |$\mathsf{coNP}$| (Prop. 5.6) in |$\mathsf{PTIME}$| (Prop. 5.8) Parsimonious RE in |$\mathsf{NP}$| (Prop. 5.9) in |$\mathsf{PTIME}$| (Prop 5.10) RC in |$\mathsf{\varSigma _2^p}$| (Prop. 5.13) in |$\mathsf{NP}$| (Prop. 5.14) . . Linear . Affine . NE in |$\mathsf{coNP}$| (Prop. 4.3) in |$\mathsf{PTIME}$| (Prop. 4.6) Dichotomous RE in |$\mathsf{NP}$| (Prop. 4.9) in |$\mathsf{PTIME}$| (Prop. 4.10) RC in |$\mathsf{\varSigma _2^p}$| (Prop. 4.13) in |$\mathsf{NP}$| (Prop. 4.14) NE in |$\mathsf{coNP}$| (Prop. 5.6) in |$\mathsf{PTIME}$| (Prop. 5.8) Parsimonious RE in |$\mathsf{NP}$| (Prop. 5.9) in |$\mathsf{PTIME}$| (Prop 5.10) RC in |$\mathsf{\varSigma _2^p}$| (Prop. 5.13) in |$\mathsf{NP}$| (Prop. 5.14) Open in new tab We thus obtained some positive results when the resources are expressed in the fragment MULT, which is suitable to represent and reason about multisets of resources. Theorem 7.1 When LOG is affine MULT, with dichotomous or parsimonious preferences, the problems NASH EQUILIBRIUM and RATIONAL ELIMINATION can be solved in polynomial time. It is interesting to note that, although weakening usually does not change the complexity of the problem of sequent provability of the logics we considered,7 we have always been able to capitalize on its presence to simplify our solutions to the problems we studied here. Putting the results of this paper together, it is also easy to see that we have this theorem. Theorem 7.2 When LOG is MALL, linear or affine, with dichotomous or with parsimonious preferences, all three decision problems are |$\mathsf{PSPACE}$|-complete. First-order MLL is one of these logics whose complexity of sequent provability is in |$\mathsf{NP}$|⁠. On the other hand, sequent provability for first-order MALL is |$\mathsf{NEXPTIME}$|-complete. It is routine to adapt our proofs to show this theorem. Theorem 7.3 When LOG is first-order MALL, linear or affine, with dichotomous or with parsimonious preferences, all three decision problems are |$\mathsf{NEXPTIME}$|-complete. Comparison with the related literature. The research in artificial intelligence, multiagent systems and computer science has shown some interest in the formal and computational aspects of resource-conscious agents (e.g. [1, 10, 17, 18, 33, 35, 43, 44, 47]). Boolean games [4, 19] are games based on classical logic. Each player controls a set of Boolean variables and produces truth values which can be used without restriction towards the Boolean goals, expressed as classical propositional formula. Somehow, also in Boolean games do the players produce and consume ‘resources’. But there are no immediate natural correspondences between IRGs and Boolean games. As in Boolean games, we could force the endowments to be non-overlapping (for exclusive control over a resource). Moreover, we could allow the players in our games to have preferences about the absence of a resource. Under these conditions, and using classical propositional logic as LOG, a connection would then exist. Electric Boolean games [18] are an extension of Boolean games where playing a certain action has a numeric cost, and agents are endowed with a certain amount of ‘energy’. Deciding whether a profile is a Nash equilibrium in a Boolean game is |$\mathsf{coNP}$|-complete [4]. In electric Boolean games, deciding whether a profile is rationally eliminable is |$\mathsf{NP}$|-complete, while deciding whether a profile is rationally constructible is |$\mathsf{coNP}$|-hard and in |$\mathsf{\varDelta _2^p}$|⁠. In Boolean games, goals of players are expressed as classical propositional formulas. Moreover, game outcomes or profiles are in fact models of classical propositional logic, i.e. valuations. Checking whether the goal of a player is satisfied in a game profile is thus an easy problem in Boolean games. This is also true in electric Boolean games. In contrast in resource games, checking whether the goal of a player is satisfied in a game profile is as hard as provability in LOG. Unsurprisingly, when working with the fragments MLL or MALL, the trend is that the complexity of decision problems in individual resource games is higher than for their counterparts in electric Boolean games. An obvious exception is the problem to decide whether an individual resource game admits a Nash equilibrium when LOG is affine and we consider dichotomous preferences. The problem is trivial by Proposition 4.5 (there is always a Nash equilibrium), while it is |$\mathsf{\varSigma _2^p}$|-complete in Boolean games [4]. Moreover, in individual resource games, there is no one-to-one correspondence between profiles and outcomes. This is another difference with electric Boolean game. As a consequence, the notions of elimination and construction in individual resource games add a bit of complexity by having to consider a set of profiles with the same outcomes. On the other hand, the fragment MULT is one instance of LOG in which it is easy to check whether a goal of a player is satisfied in a game profile (Proposition 2.5). In this context, and as shown on Table 5 and compared to the realm of Boolean games, reasoning about IRGs remains a relatively easy task. It can even be tractable if one considers affine MULT. Affine logic should be used when we can assume that a player satisfied with an outcome would be satisfied with a more sizeable outcome, which is often a very acceptable assumption. Congestion games (CGs) [39] (see also potential games [27]; exact potential games correspond to CGs up to an isomorphism) are a celebrated class of games where the players interact in resource-sensitive environments. Despite some apparent similarities between IRGs and CGs, they are rather superficial. Players in CGs do not have endowments per se. Players’ actions in CGs consist in choosing a subset of an already available common pool of resources to use. In CGs, the players are only consumers. In IRGs, players are consumers but also producers of resources; their actions consist in making resources available in the common pool. In CGs, these resources are exclusively atomic resources while in IRGs they can be any logical formula in LOG. With the decision problems of rational elimination and rational construction, there is a dimension of social choice theory and mechanism design. Formal frameworks concerned with redistribution schemes and economic policies can be found for instance in [18] again, or [11, 25, 28]. Our games bear some resemblance with combinatorial exchanges [23] and with mixed multi-unit combinatorial auctions (MMUCAs) [8, 15], where the agents can be both sellers and buyers. Interestingly, in MMUCAs, sets of goods can be transformed into different sets of goods. Resource-transforming capacities are central, as the agents are allowed to bid on transformation services. Determining the sequences of bids to be accepted by an auctioneer is generally intractable in MMUCAs; [13] identifies tractable classes for the winner determination problem. Finally, we focused on individual games and looked at Nash equilibria. Nonetheless, the setting allows one to easily build classes of coalition games, reminiscent of coalitional resource games [10, 47] and of coalition skill games [3]. In [43], we have started the study of what we called rich coalitional resource games (RCRGs). Individual resource games are essentially one-goal RCRGs. Perspectives. We have obtained tight complexity results when LOG is |$\mathsf{PSPACE}$|-complete. However, this is lacking when LOG is in |$\mathsf{NP}$| and in |$\mathsf{PTIME}$|⁠. We suspect that the complexity of the diverse decision problems generally lie above the lower bounds we have obtained. It is more likely that some proposed upper bounds are tight. One perspective will thus be to investigate whether some decision problems could be proven hard for some complexity class in the polynomial or Boolean hierarchy, for instance using the techniques from [45] of raising |$\mathsf{NP}$| lower bounds to lower bounds for classes above |$\mathsf{NP}$|⁠. Resource games based on resource-sensitive logics become all the more significant when the resources are subject to transforming activities. We can exploit the existing research on these resource-sensitive logics about their proof theory. In particular, through the Curry-Howard correspondence between proofs and programs (see, e.g. [14]), an exciting perspective is the possibility to interpret the logical proofs as rigorous programs to be executed by the players. We can expect to obtain some results for the automated generation of plans, where the resources can be subjected to a series of transforming activities by the agents. Similar ideas have already been defended in multiagent systems (see, e.g. [24]). Our models are agnostic about how the contributed resources are distributed. Instead of having preferences about a raw profile |$P$|⁠, the player’s preferences could be raised over the result of the (fair, envy-free, efficient, etc.) allocation of the resources [5] contributed in |$\mathsf{out}(P)$|⁠. These are interesting extensions that are just a step away to get the models more fit for application, although at the expense of mathematical simplicity. We are interested in using resource games in problems of gamification. Gamification refers to the broad application of game-design techniques in contexts that do not otherwise present game-like features [9, 36]. Gamification aims at incentivizing an intended behavior by introducing rewards for specific tasks. Rewards often present themselves as virtual resources such as achievement badges. Formally, they might be nothing more than distinguished tokens of resources. In Example 1.1, we saw that the profile where all companies refrain from providing any resources, |$(\emptyset , \emptyset )$|⁠, is a Nash equilibrium. This is an undesirable behaviour that policy makers might be able to anticipate by using the analytical tools defined in this paper, and to avoid by using advanced gamification methods. Acknowledgements I thank an anonymous reviewer for making a number of suggestions that greatly helped to improve the presentation. I am grateful to Jamie Gabbay for his enthusiasm and his encouragements. A Sequent rules of affine MALL We present the sequent rules for Affine MALL. In what follows, |$A$|⁠, |$B$|⁠, |$A_0$| and |$A_1$| are formulas. |$\varGamma $|⁠, |$\varGamma ^{\prime}$|⁠, |$\varDelta $| and |$\varDelta ^{\prime}$| are sequences of zero or more formulas. A sequent rule has an upper and a lower part. The upper part is composed of zero, one or two sequents. The lower part is composed of one sequent. If there is a proof of all the sequents of the upper part, then the rule can be used to obtain a proof of the sequent of the lower part. Identities |$ $| Structural rules |$ $| Negation |$ $| Multiplicatives |$ $| Additives (In |$\oplus $|R, and &L, |$i$| stands for either |$0$| or |$1$|⁠.) |$ $| B Elements of computational complexity We need to assume some familiarity with computational complexity. This appendix only introduces some elements of terminology and some definitions about complexity theory. The reader familiar with these notions can use this section for quick reference. Another reader can use it as a starting point and move to a more complete introduction. A classic introduction to computational complexity is [31]. All elementary complexity classes used in this paper are presented in [40]. A decision problem (or problem for short) is a problem that is posed as ‘yes’/‘no’ question of the values of the input. The class |$\mathsf{PTIME}$|⁠, also noted |$\mathsf{P}$|⁠, is the class of decision problems that can be solved in deterministic polynomial time (wrt. the size of the input). The class |$\mathsf{NP}$| is the class of problems that can be solved in non-deterministic polynomial time. The class |$\mathsf{PSPACE}$| is the class of problems that can be solved using a polynomial amount of space. The complement of a decision problem is the decision problem resulting from reversing the ‘yes’ and ‘no’ answers. For every class of complexity |$\mathsf{C}$|⁠, we denote |$\mathsf{coC}$| the class populated with the complements of the problems in |$\mathsf{C}$|⁠. Given two classes of complexity |$\mathsf{C_1}$| and |$\mathsf{C_2}$|⁠, the class |$\mathsf{C_1^{C_2}}$| is the class of problems that are in |$\mathsf{C_1}$| if we assume the availability of an oracle to solve the problems in |$\mathsf{C_2}$|⁠. An oracle for |$\mathsf{C_2}$| is a black box capable to solve every problem in |$\mathsf{C_2}$| in a single operation. Queries to an oracle can be adaptive (also called serial), or non-adaptive (also called parallel). A query is adaptive when it depends on the answer of a previous query. Non-adaptive queries on the other hand can be chosen in advance and computed from the start and are asked in parallel. For every class of complexity |$\mathsf{C}$|⁠, we denote |$\mathsf{P^C}$| (resp. |$\mathsf{NP^C}$|⁠) the class of problems solvable on a deterministic (resp. non-deterministic) polynomial-time bounded oracle Turing machine using an oracle set |$\mathsf{C}$|⁠. We denote |$\mathsf{P^{C[k]}}$| and |$\mathsf{NP^{C[k]}}$| when at most |$k$| adaptive queries to |$\mathsf{C}$| can be used. We denote |$\mathsf{P^{C||[k]}}$| and |$\mathsf{NP^{C||[k]}}$| when at most |$k$| non-adaptive queries to |$\mathsf{C}$| can be used. We denote |$\mathsf{P^{C||}}$| (resp. |$\mathsf{NP^{C||}}$|⁠) the class of problems solvable on a deterministic (resp. non-deterministic) polynomial-time bounded oracle Turing machine with non-adaptive queries to |$\mathsf{C}$|⁠. The class |$\mathsf{P^{NP||}}$| is also referred to as |$\mathsf{\varTheta _2^p}$|⁠. The polynomial hierarchy. The polynomial hierarchy contains a family of complexity classes that are smaller than |$\mathsf{PSPACE}$|⁠. The class |$\mathsf{P}$| lies at the bottom of the polynomial hierarchy. Then, for every positive integer |$i$|⁠, we can define |$\mathsf{\varDelta _{i}^p}$|⁠, |$\mathsf{\varSigma _{i}^p}$|⁠, and |$\mathsf{\varPi _{i}^p}$| recursively as follows: • |$\mathsf{\varDelta _{0}^p} = \mathsf{\varSigma _{0}^p} = \mathsf{\varPi _{0}^p} = \mathsf{P}$|⁠; • |$\mathsf{\varDelta _{i+1}^p} = \mathsf{P^{\varSigma _{i}^p}}$|⁠; • |$\mathsf{\varSigma _{i+1}^p} = \mathsf{NP^{\varSigma _{i}^p}}$|⁠; • |$\mathsf{\varPi _{i}^p} = \mathsf{co\varSigma _{i}^p}$|⁠. The Boolean hierarchy over |$\mathsf{NP}$|⁠. The Boolean hierarchy has been studied in [21, 45, 46]. The Boolean hierarchy over |$\mathsf{NP}$| contains a family of complexity classes that are smaller than |$\mathsf{\varDelta _{2}^p}$|⁠. The class |$\mathsf{NP}$| lies at the bottom of the Boolean hierarchy over |$\mathsf{NP}$|⁠. Here, we are better off looking at complexity classes not as classes of decision problems, but as classes of languages. A language is the formal realization of a decision problem. Let |$p$| be a decision problem with |$k$| inputs. A language of |$p$| is the language |$L_p = \{ (a_1, \ldots , a_k) \mid p \text{ answers `yes$^{\prime}$ of the input} (a_1, \ldots , a_k) \}$|⁠. Given a class of complexity |$\mathsf{C}$|⁠, we say that |$L_p \in \mathsf{C}$| iff |$p \in \mathsf{C}$|⁠. Then, given two classes of complexity |$\mathsf{C_1}$| and |$\mathsf{C_2}$|⁠, each representing a set of languages and the decision problems they formalize, we define |$\mathsf{C_1} \land \mathsf{C_2} = \{ L_1 \cap L_2 \mid L_1 \in \mathsf{C_1} \ \textrm{and}\ L_2 \in \mathsf{C_2} \}$| and |$\mathsf{C_1} \lor \mathsf{C_2} = \{ L_1 \cup L_2 \mid L_1 \in \mathsf{C_1} \ \textrm{and}\ L_2 \in \mathsf{C_2} \}$|⁠. In this context, the class |$\mathsf{NP}$| is the class of languages that can be recognised in non-deterministic polynomial time. Then, for every positive integer |$i$|⁠, we can define |$\mathsf{BH_i}$| recursively as follows: • |$\mathsf{BH_0} = \mathsf{NP}$|⁠; • |$\mathsf{BH_{2k}} = \mathsf{coNP} \land \mathsf{BH_{2k-1}}$|⁠; • |$\mathsf{BH_{2k+1}} = \mathsf{NP} \lor \mathsf{BH_{2k}}$|⁠. The class |$\mathsf{B\mathsf{H_2}} = \mathsf{NP} \land \mathsf{coNP}$| is the ‘difference class’ |$\mathsf{D^P}$| presented in [32]. Useful properties. Besides the definitions, the following properties are useful: • |$\mathsf{C_1^{coC_2}} = \mathsf{C_1^{C_2}}$| (for all two classes |$\mathsf{C_1}$| and |$\mathsf{C_2}$|⁠); • |$\mathsf{NP^{\varSigma _{i}^p}} = \mathsf{\varSigma _{i+1}^p}$|⁠; • |$\mathsf{co{\varSigma _{i}^p}} = \mathsf{\varPi _{i}^p}$|⁠; • |$\mathsf{NP^{\varDelta _{i}^p}} = \mathsf{\varSigma _{i}^p}$|⁠; • |$\mathsf{P^{\varDelta _{i}^p}} = \mathsf{\varDelta _{i}^p}$|⁠; • |$\mathsf{\varSigma _{i}^p} \subseteq \mathsf{PSPACE}$|⁠; • |$\mathsf{PSPACE} = \mathsf{coPSPACE} = \mathsf{P^{PSPACE}} = \mathsf{NP^{PSPACE}}$|⁠; • |$\mathsf{BH_i} \subseteq \mathsf{\varDelta _{2}^p}$|⁠; • |$\mathsf{P^{NP||[k]}} \subseteq \mathsf{BH_{k+1}} \subseteq \mathsf{P^{NP||[k+1]}}$|⁠. C Proofs of the realized objectives in the Example of Section 6.1 The proof of |$\mathsf{H_2O},\mathsf{H_2O},\mathsf{elec} \vdash \gamma _f$| will be instrumental for the subsequent proofs. We label it Proof |$\star $| for reuse. |$ $| The other realized objectives of the fish are immediate using Proof |$\star $| and the weakening rule. We prove that |$\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O},\mathsf{elec} \vdash \gamma _f$|⁠, |$\mathsf{drink},\mathsf{H_2O},\mathsf{H_2O},\mathsf{elec} \vdash \gamma _f$| and |$\mathsf{drink},\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O},\mathsf{elec} \vdash \gamma _f$|⁠. |$ $| Finally, we prove |$\mathsf{drink},\mathsf{elec},\mathsf{H_2O},\mathsf{H_2O},\mathsf{H_2O} \vdash \gamma _a$|⁠. The proof also uses Proof |$\star $|⁠. |$ $| Footnotes 1 Indeed, |$X \vdash \gamma _i$| indicates that the resources |$X$| are sufficient to produce |$\gamma _i$|⁠, and |$X \vdash \gamma _j$| indicates that the resources |$X$| are sufficient to produce |$\gamma _j$|⁠. It may be however that the resources |$X$| are not sufficient to produce |$\gamma _i$| and |$\gamma _j$| simultaneously. 2 We use |$\biguplus $| for the multiset union and |$\bigcup $| for the set union. 3 See Appendix B for some elements of complexity that will be useful in the proofs in this paper. 4 Individual resource games were called ideal resource games in [42]. 5 For |$\varGamma = \{A_1, \ldots , A_k\}$|⁠, we note |$\mathop \sim \varGamma $| the set |$\{\mathop \sim A_1, \ldots , \mathop \sim A_k\}$|⁠. 6 Algorithm 6 corrects an omission in [42,Algorithm |$5$|] by adding ‘if (⁠|$P_i \not = \emptyset $|⁠): return false’ lines |$9$| and |$10$|⁠. 7 We did not consider full propositional linear logic, which also contains so-called ‘exponentials’. Weakening does make a difference: sequent provability in full propositional linear logic is undecidable [26], while sequent provability in full propositional affine logic is decidable [22]. References [1] N. Alechina , N. Bulling, B. 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TI - Individual resource games and resource redistributions
JF - Journal of Logic and Computation
DO - 10.1093/logcom/exaa031
DA - 2020-07-23
UR - https://www.deepdyve.com/lp/oxford-university-press/individual-resource-games-and-resource-redistributions-0X8jYKbpP0
SP - 1
VL - Advance Article
IS - 
DP - DeepDyve
ER -