View source for Non-atomic game

{{TEX|done}}
A game with the following property: If $I$ denotes the set of all players, there is a given $\sigma$-algebra of subsets $\mathcal C$ on $I$ and a [[Non-atomic measure|non-atomic measure]] on $\mathcal C$ such that sets of players $C\in\mathcal C$ of zero measure have no effect on the outcome of the game. Non-atomic games serve as models for situations in which there is a large quantity of very "small" individuals, like customers in an economic system, so the development of non-atomic games is closely connected with the study of economic models with large numbers of participants (see [[#References|[1]]]). Non-atomic games are amenable to the general classification customary in game theory (cf. [[Games, theory of|Games, theory of]]), and the basic game-theoretic principles of optimality (see [[Core in the theory of games|Core in the theory of games]]; [[Shapley value|Shapley value]]) carry over to them in a natural way. In relation to non-atomic games, however, the non-cooperative principles of optimality are usually implemented without the usual convexity assumptions (see [[#References|[2]]]), and the different optimality principles turn out to be more strongly interconnected. For example, for a broad range of non-atomic models of market type, the set of competitive equilibria coincides with the core, which consists of a single element — the value of the game (the analogue of the Shapley value for non-atomic games; see [[#References|[1]]]). There are two directions of research — the theory of cooperative non-atomic games (see [[#References|[1]]], [[#References|[3]]], [[#References|[4]]]) and the theory of non-cooperative non-atomic games (see [[#References|[2]]]).

A cooperative non-atomic game, in analogy with an ordinary [[Cooperative game|cooperative game]], is a triple $\langle J,v,H\rangle$, where $J=(I,\mathcal C)$ is a measurable space of players; the elements $C\in\mathcal C$ are called coalitions; $v$ is a real-valued function on $\mathcal C$, called the characteristic function; and $H$ is a certain subset of the set $FA$ of all finitely-additive measures of bounded variation on $\mathcal C$ (it is usually assumed that $\mu(I)=v(I)$ for all $\mu\in H$). In the simplest case, $v$ is a non-atomic measure function on $\mathcal C$. It is assumed that the space $J$ is standard (i.e. $(I,C)$ is isomorphic to the unit interval, with the Borel subsets), and that the set function $v$ is of bounded variation (i.e. is expressible as the difference of two monotone functions). Descriptions have been given [[#References|[1]]] of various subspaces of the space $BV$ of all functions of bounded variation on $\mathcal C$ for which an analogue of the Shapley value can be constructed (as a positive linear operator with values in $FA$).

The concepts of a balanced game, a market game, a game without side payments (see [[Cooperative game|Cooperative game]]) and the related results carry over to cooperative non-atomic games (see [[#References|[3]]], [[#References|[4]]]).

The definition of a non-cooperative non-atomic game is analogous to that of a classical [[Non-cooperative game|non-cooperative game]]. There are also analogues of the [[Nash theorem (in game theory)|Nash theorem (in game theory)]], as well as general results concerning the existence of equilibrium situations without convexity assumptions, in contrast with the case of games with finitely many players (see [[#References|[2]]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  R.J. Aumann,   L.S. Shapley,   "Values of non-atomic games" , Princeton Univ. Press  (1974)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A.Ya. Kiruta,   "Equilibrium points in non-atomic non-cooperative games"  ''Mat. Met. Sots. Nauk.'' , '''6'''  (1975)  pp. 18–71  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J. Rosenmüller,   "Kooperative Spiele und Märkte" , Springer  (1971)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J. Rosenmüller,   "Large games without side-payments"  ''Operat. Res. Verfahren'' , '''20'''  (1975)  pp. 107–128</TD></TR></table>



====Comments====


====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Hildenbrand,   "Core and equilibria of a large economy" , Princeton Univ. Press  (1974)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Owen,   "Game theory" , Acad. Press  (1982)</TD></TR>
</table>

[[Category:Game theory, economics, social and behavioral sciences]]