A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe (noughts and crosses) or checkers (draughts). In other words, it extends the notion of a discrete game, where the players choose from a finite set of pure strategies. The continuous game concepts allows games to include more general sets of pure strategies, which may be uncountably infinite. In general, a game with uncountably infinite strategy sets will not necessarily have a Nash equilibrium solution. If, however, the strategy sets are required to be compact and the utility functions continuous, then a Nash equilibrium will be guaranteed; this is by Glicksberg's generalization of the Kakutani fixed point theorem. The class of continuous games is for this reason usually defined and studied as a subset of the larger class of infinite games (i.e. games with infinite strategy sets) in which the strategy sets are compact and the utility functions continuous.

Formal definition

Define the n-player continuous game where Let be a strategy profile of all players except for player i. As with discrete games, we can define a best response correspondence for player i,, b_i. b_i, is a relation from the set of all probability distributions over opponent player profiles to a set of player i's strategies, such that each element of is a best response to \sigma_{-i}. Define A strategy profile is a Nash equilibrium if and only if The existence of a Nash equilibrium for any continuous game with continuous utility functions can be proven using Irving Glicksberg's generalization of the Kakutani fixed point theorem. In general, there may not be a solution if we allow strategy spaces, C_i,'s which are not compact, or if we allow non-continuous utility functions.

Separable games

A separable game is a continuous game where, for any i, the utility function can be expressed in the sum-of-products form: A polynomial game is a separable game where each C_i, is a compact interval on \R, and each utility function can be written as a multivariate polynomial. In general, mixed Nash equilibria of separable games are easier to compute than non-separable games as implied by the following theorem: Whereas an equilibrium strategy for a non-separable game may require an uncountably infinite support, a separable game is guaranteed to have at least one Nash equilibrium with finitely supported mixed strategies.

Examples

Separable games

A polynomial game

Consider a zero-sum 2-player game between players X and Y, with. Denote elements of C_X, and C_Y, as x, and y, respectively. Define the utility functions where The pure strategy best response relations are: b_X(y), and b_Y(x), do not intersect, so there is no pure strategy Nash equilibrium. However, there should be a mixed strategy equilibrium. To find it, express the expected value, as a linear combination of the first and second moments of the probability distributions of X and Y: (where and similarly for Y). The constraints on \mu_{X1}, and \mu_{X2} (with similar constraints for y,) are given by Hausdorff as: Each pair of constraints defines a compact convex subset in the plane. Since v, is linear, any extrema with respect to a player's first two moments will lie on the boundary of this subset. Player i's equilibrium strategy will lie on Note that the first equation only permits mixtures of 0 and 1 whereas the second equation only permits pure strategies. Moreover, if the best response at a certain point to player i lies on, it will lie on the whole line, so that both 0 and 1 are a best response. simply gives the pure strategy, so b_Y, will never give both 0 and 1. However b_x, gives both 0 and 1 when y = 1/2. A Nash equilibrium exists when: This determines one unique equilibrium where Player X plays a random mixture of 0 for 1/2 of the time and 1 the other 1/2 of the time. Player Y plays the pure strategy of 1/2. The value of the game is 1/4.

Non-Separable Games

A rational payoff function

Consider a zero-sum 2-player game between players X and Y, with. Denote elements of C_X, and C_Y, as x, and y, respectively. Define the utility functions where This game has no pure strategy Nash equilibrium. It can be shown that a unique mixed strategy Nash equilibrium exists with the following pair of cumulative distribution functions: Or, equivalently, the following pair of probability density functions: The value of the game is 4/\pi.

Requiring a Cantor distribution

Consider a zero-sum 2-player game between players X and Y, with. Denote elements of C_X, and C_Y, as x, and y, respectively. Define the utility functions where This game has a unique mixed strategy equilibrium where each player plays a mixed strategy with the Cantor singular function as the cumulative distribution function.