# Non-atomic game

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 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 [1]). Non-atomic games are amenable to the general classification customary in game theory (cf. Games, theory of), and the basic game-theoretic principles of optimality (see Core in the theory of games; 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 [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 [1]). There are two directions of research — the theory of cooperative non-atomic games (see [1], [3], [4]) and the theory of non-cooperative non-atomic games (see [2]).

A cooperative non-atomic game, in analogy with an ordinary 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 [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) and the related results carry over to cooperative non-atomic games (see [3], [4]).

The definition of a non-cooperative non-atomic game is analogous to that of a classical non-cooperative game. There are also analogues of the 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 [2]).

#### References

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