Positivity 6: 433–453, 2002.
© 2002 Kluwer Academic Publishers. Printed in the Netherlands.
Static and Dynamic Equilibria in Games With
Continuum of Players
Institute of Applied Mathematics and Mechanics, Warsaw University and Institute of Computer
Science, Polish Academy of Sciences, Warsaw, Poland E-mail email@example.com
Received 13 August 2000; accepted 20 May 2001
Abstract. This paper is a study of a general class of deterministic dynamic games with an atomless
measure space of players and an arbitrary time space. The payoffs of the players depend on their own
strategy, a trajectory of the system and a function with values being ﬁnite dimensional statistics of
static proﬁles. The players’ available decisions depend on trajectories of the system.
The paper deals with relations between static and dynamic open-loop equilibria as well as their
existence. An equivalence theorem is proven and theorems on the existence of a dynamic equilibrium
are shown as consequences.
90D13, 90D44, 90D2, 90D25, 90A14, 90A58:
Key words: best response correspondence, differential games, dynamic games, equilibrium, games
with continuum of players, multistage games
This paper is an attempt to join two streams in game theory: atomless games or
games with a continuum of players (ﬁrst deﬁned by Schmeidler , studied,
e.g., by Mas-Colell , Balder , Wieczorek ) and dynamic games with
payoffs depending on ﬁnitely dimensional statistics of proﬁles and trajectories of
There have been a few papers in the literature dealing with dynamic games with
a continuum of players, mainly concerning cases of particular interest, among them
Kwiatkowski , Karatzas et al. , Wieczorek and Wiszniewska , but it seems
that the general theory of such games has to be still developed. Some attempts of
the author in this direction are included in [15–17], in which a differential game
of extraction of common, renewable ecosystem by players constituting an atom-
less measure space was examined. A survey on games modelling exploitation of
common ecosystem can be found in e.g. . Further research can be found in
The paper is organized as follows: in Section 2 the deﬁnition of game is given, in
Subsection 2.1 the notion of static and dynamic equilibrium is introduced, Section
3 deals with the existence of a static equilibrium (Theorem 3.1), in Section 5 the