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 where

1. players,

2. is a compact set, in a metric space th player's set of pure strategies,

to be the set of Borel probability measures, giving us the mixed strategy space of player i.

. As with discrete games, we can define a best response correspondence's strategies, such that each element of

. Define

1. .

is a Nash equilibrium 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.'s which are not compact, or if we allow non-continuous utility functions.

Separable games

A separable game can be expressed in the sum-of-products form:

1. are continuous.

A polynomial game 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:

1. For any separable game there exists at least one Nash equilibrium where player i pure strategies.

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 where

1. .

The pure strategy best response relations are:

as a linear combination of the first and second moments of the probability distributions of X and Y:

and similarly for Y).

(with similar constraints for y,) are given by Hausdorff as:

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

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 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:

.

Requiring a Cantor distribution

Consider a zero-sum 2-player game between players X and Y where

1. .

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.

Further reading

• H. W. Kuhn and A. W. Tucker, eds. (1950). Contributions to the Theory of Games: Vol. II. Annals of Mathematics Studies 28. Princeton University Press. .

See also

• Nonatomic game - a game with continuously many players, but a finite set of actions.

• Graph continuous

References