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Ganguly, Chirantan; Ray, Indrajit
doi: 10.1007/s00182-023-00857-0pmid: N/A
We consider a version of the Battle of the Sexes with private information and allow cheap talk regarding the players’ types before the game. We show that a desirable type-coordination property is achieved at the unique fully revealing symmetric equilibrium (when it exists). Type-coordination is also obtained in a partially revealing equilibrium that exists when the fully revealing equilibrium does not. We further prove that truthfully revealed messages, followed by actions that depend meaningfully on these messages, are not equilibrium profiles with one-sided cheap talk. Finally, fully revealing equilibria do not exist under sequential communication either.
doi: 10.1007/s00182-023-00849-0pmid: N/A
Often preferences in a group of agents are such that any sensible goal must admit a tie between all alternatives. The standard formulation in mechanism design demands that in this case all alternatives must be equilibrium outcomes of the decision making mechanism. However, as far as the idea of an equilibrium is to predict the outcome, we could equally well require that there are no equilibria at all. Although this may seem innocent, it allows the mechanism designer to implement goals that are impossible to enforce with any other implementation concept, like mixed Nash implementation, subgame perfect implementation, or Nash implementation using undominated strategies.
Kerber, Manfred; Rowat, Colin; Yoshihara, Naoki
doi: 10.1007/s00182-023-00859-ypmid: N/A
We study pillage games (Jordan in J Econ Theory 131(1):26–44, 2006), which model unstructured power contests. To enable empirical tests of pillage game theory, we relax a symmetry assumption that agents’ intrinsic contributions to a coalition’s power is identical. We characterise the core for all n. In the three-agent game: (i) only eight configurations are possible for the core, which contains at most six allocations; (ii) for each core configuration, the stable set is either unique or fails to exist; (iii) the linear power function creates a tension between a stable set’s existence and the interiority of its allocations, so that only special cases contain strictly interior allocations. Our analysis suggests that non-linear power functions may offer better empirical tests of pillage game theory.
doi: 10.1007/s00182-023-00858-zpmid: N/A
This paper proposes a parsimonious model of network formation with introductions in the presence of intermediation rents. Introductions allow two nodes to form a new connection on favorable terms with the help of a common neighbor. The decision to form links via introductions is subject to a trade-off between the gains from having a direct connection at lower cost and the potential losses for the introducer from lower intermediation rents. When nodes take advantage of introductions, stable networks tend to exhibit a minimum amount of clustering. At the same time, intermediary nodes have incentives to protect their position, and stable networks can exhibit nodes exploiting structural holes, that is, bridges across otherwise unconnected parts of the network earning intermediation rents.
doi: 10.1007/s00182-023-00846-3pmid: N/A
This paper studies an infinite horizon oligopoly model in markets with network effects and segmented demands. In each period, three firms make compatibility decisions before competing in prices for a newly arrived consumer. The firm that made a sale in the last period provides a better product quality in terms of an installed base consumer, which can be shared with its rivals through compatibility. We show that compatibility can be used as an exclusionary device even though it intensifies short-run price competition when firms are sufficiently patient. Under certain conditions, this is the only stable prediction with respect to a dynamic analog of strong stability in network formation games (Dutta and Mutuswami in J Econ Theory 76:322–344, 1997).
Liang, Dong; Wang, Yunlong; Cao, Zhigang; Yang, Xiaoguang
doi: 10.1007/s00182-023-00853-4pmid: N/A
The Colonel Blotto game is one of the most classical zero-sum games, with diverse applications in auctions, political elections, etc. We consider the discrete two-battlefield Colonel Blotto Game, a basic case that has not yet been completely characterized. We study three scenarios where at least one player’s resources are indivisible (discrete), and compare them with a benchmark scenario where the resources of both players are arbitrarily divisible (continuous). We present the equilibrium values for all three scenarios, and provide a complete equilibrium characterization for the scenario where both players’ resources are indivisible. Our main finding is that, somewhat surprisingly, the distinction between continuous and discrete strategy spaces generally has no effect on players’ equilibrium values. In some special cases, however, the larger continuous strategy space when resources are divisible does bring the corresponding player a higher equilibrium value than when resources are indivisible, and this effect is more significant for the stronger player who possesses more resources than for the weaker player.
doi: 10.1007/s00182-023-00863-2pmid: N/A
We study the distribution of the number of mixed strategy Nash equilibria in two-player games where each player’s payoffs are independently drawn from an identical distribution. When the payoff distributions are sufficiently right fat-tailed, we characterize the Nash equilibria by best reply cycles of pure strategies, and we show that the expected number of Nash equilibria is approximately πmn/m+n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\sqrt{\pi mn/\left( m+n\right) }$$\end{document} in a random m×n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$m\times n$$\end{document} asymmetric game and approximately n/2 in a random n×n\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\times n$$\end{document} symmetric game. We also provide new lower bounds for the expected number of Nash equilibria in a random game with any type of payoff distribution.
doi: 10.1007/s00182-023-00860-5pmid: N/A
We study optimal strategies in two-player stochastic games that are played on a finite graph, equipped with a general payoff function. The existence of optimal strategies that do not make use of memory and randomisation is a desirable property that vastly simplifies the algorithmic analysis of such games. Our main theorem gives a sufficient condition for the maximizer to possess such a simple optimal strategy. The condition is imposed on the payoff function, saying the payoff does not depend on any finite prefix (shift-invariant) and combining two trajectories does not give higher payoff than the payoff of the parts (submixing). The core technical property that enables the proof of the main theorem is that of the existence of ϵ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\epsilon$$\end{document}-subgame-perfect strategies when the payoff function is shift-invariant. Furthermore, the same techniques can be used to prove a finite-memory transfer-type theorem: namely that for shift-invariant and submixing payoff functions, the existence of optimal finite-memory strategies in one-player games for the minimizer implies the existence of the same in two-player games. We show that numerous classical payoff functions are submixing and shift-invariant.
doi: 10.1007/s00182-023-00867-ypmid: N/A
We present a general framework of evolutionary dynamics under persistent heterogeneity in payoff functions and revision protocols, allowing continuously many types in a game with finitely many strategies. Unlike existing literature, we do not assume anonymity of the game nor aggregability of the dynamic. The dynamic is formulated as a differential equation of a joint probability measure of types and strategies. To establish a foundation for this framework, we clarify regularity assumptions on the revision protocol, the game, and the type distribution to guarantee the existence of a unique solution trajectory as well as those to guarantee the existence of an equilibrium in a heterogeneous population game. We further verify equilibrium stationarity in general and stability in potential games under admissible dynamics. Our framework encompasses a wide range of possible applications, including incomplete information games and spatial evolution.
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