# 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 ). 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 ), 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 ). There are two directions of research — the theory of cooperative non-atomic games (see , , ) and the theory of non-cooperative non-atomic games (see ).
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  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 $FA$).