Continuous game
Define the "n"-player continuous game formula_1 where
formula_2 is the set of formula_3 players,
 formula_4 where each formula_5 is a compact set, in a metric space, corresponding to the formula_6 "th" player's set of pure strategies,
 formula_7 where formula_8 is the utility function of player formula_6
 We define formula_10 to be the set of Borel probability measures on formula_5, giving us the mixed strategy space of player "i". Define the strategy profile formula_12 where formula_13
Let formula_14 be a strategy profile of all players except for player formula_15. As with discrete games, we can define a best response correspondence for player formula_6, formula_17. formula_18 is a relation from the set of all probability distributions over opponent player profiles to a set of player formula_15's strategies, such that each element of
formula_20
is a best response to formula_21. Define
formula_22.
A strategy profile formula_23 is a Nash equilibrium if and only if
formula_24
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, formula_5's which are not compact, or if we allow non-continuous utility functions. A separable game is a continuous game where, for any i, the utility function formula_8 can be expressed in the sum-of-products form:
 formula_27, where formula_28, formula_29, formula_30, and the functions formula_31 are continuous. A polynomial game is a separable game where each formula_5 is a compact interval on formula_33 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 formula_34 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. Consider a zero-sum 2-player game between players X and Y, with formula_35. Denote elements of formula_36 and formula_37 as formula_38 and formula_39 respectively. Define the utility functions formula_40 where
formula_41. The pure strategy best response relations are:
formula_42
formula_43
formula_44 and formula_45 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, formula_46 as a linear combination of the first and second moments of the probability distributions of X and Y:
 formula_47
(where formula_48 and similarly for "Y"). The constraints on formula_49 and formula_50 (with similar constraints for "y",) are given by Hausdorff as:
 formula_51
Each pair of constraints defines a compact convex subset in the plane. Since formula_52 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
 formula_53
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 formula_54, it will lie on the whole line, so that both 0 and 1 are a best response. formula_55 simply gives the pure strategy formula_56, so formula_57 will never give both 0 and 1.
However formula_58 gives both 0 and 1 when y = 1/2. A Nash equilibrium exists when:
 formula_59
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. Consider a zero-sum 2-player game between players X and Y, with formula_35. Denote elements of formula_36 and formula_37 as formula_38 and formula_39 respectively. Define the utility functions formula_40 where
formula_66
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:
 formula_67
The value of the game is formula_68. Consider a zero-sum 2-player game between players X and Y, with formula_35. Denote elements of formula_36 and formula_37 as formula_38 and formula_39 respectively. Define the utility functions formula_40 where
formula_75.
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.
Formal definition
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.