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.
Contents
- Formal definition
- Separable games
- A polynomial game
- A rational pay off function
- Requiring a Cantor distribution
- References
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
Let
is a best response to
A strategy profile
Separable games
A separable game is a continuous game where, for any i, the utility function
A polynomial game is a separable game where each
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 mostWhereas 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.
A polynomial game
Consider a zero-sum 2-player game between players X and Y, with
The pure strategy best response relations are:
no pure strategy Nash equilibrium. However, there should be a mixed strategy equilibrium. To find it, express the expected value,
(where
The constraints on
Each pair of constraints defines a compact convex subset in the plane. Since
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
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.
A rational pay-off function
Consider a zero-sum 2-player game between players X and Y, with
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
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.