A continuous game is a mathematical concept, used in game theory, that generalizes the idea of an ordinary game like tic-tac-toe or checkers. 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 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. As with discrete games, we can define a best response for player,. is a relation from the set of all probability distributions over opponent player profiles to a set of player 's strategies, such that each element of is a best response to. 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, '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 is a compact interval on 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 and as and respectively. Define the utility functions where The pure strategy best response relations are: and 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: . The constraints on and are given by Hausdorff as: Each pair of constraints defines a compact convex subset in the plane. Since 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 will never give both 0 and 1. However 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 pay-off function
Consider a zero-sum 2-player game between players X and Y, with. Denote elements of and as and 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 probability density functions: The value of the game is.
Requiring a Cantor distribution
Consider a zero-sum 2-player game between players X and Y, with. Denote elements of and as and 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.
