Continuous game

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A continuous game is a mathematical generalization, used in game theory. 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[edit]

Define the n-player continuous game where

is the set of players,
where each is a compact metric space corresponding to the th player's set of pure strategies,
where is the utility function of player
We define to be the set of Borel probability measures on , giving us the mixed strategy space of player i.
Define the strategy profile where

Let be a strategy profile of all players except for player . As with discrete games, we can define a best response correspondence 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 been proven using Irving Glicksberg's generalization of the Kakutani fixed point theorem.[1] 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[edit]

A separable game is a continuous game where, for any i, the utility function can be expressed in the sum-of-products form:

, where , , , and the functions are continuous.

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:

For any separable game there exists at least one Nash equilibrium where player i mixes at most pure strategies.[2]

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[edit]

Separable games[edit]

A polynomial game[edit]

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:

(where and similarly for Y).

The constraints on and (with similar constraints for y,) 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[edit]

A rational pay-off function[edit]

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[3] 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[edit]

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.[4]

Further reading[edit]

  • 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. ISBN 0-691-07935-8.

See also[edit]

References[edit]

  1. ^ I.L. Glicksberg. A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proceedings of the American Mathematical Society, 3(1):170–174, February 1952.
  2. ^ N. Stein, A. Ozdaglar and P.A. Parrilo. "Separable and Low-Rank Continuous Games". International Journal of Game Theory, 37(4):475–504, December 2008. http://arxiv.org/abs/0707.3462
  3. ^ Glicksberg, I. & Gross, O. (1950). "Notes on Games over the Square." Kuhn, H.W. & Tucker, A.W. eds. Contributions to the Theory of Games: Volume II. Annals of Mathematics Studies 28, p.173–183. Princeton University Press.
  4. ^ Gross, O. (1952). "A rational payoff characterization of the Cantor distribution." Technical Report D-1349, The RAND Corporation.