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Acontinuous game is a mathematical concept, used ingame 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 beuncountably infinite.

In general, a game with uncountably infinite strategy sets will not necessarily have aNash equilibrium solution. If, however, the strategy sets are required to becompact and the utility functionscontinuous, then a Nash equilibrium will be guaranteed; this is by Glicksberg's generalization of theKakutani 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.

Define then-player continuous game${\displaystyle G=(P,\mathbf{C},\mathbf{U})}$ where

${\displaystyle P=1,2,3,\dots ,n}$ is the set of${\displaystyle n}$ players,

${\displaystyle \mathbf{C}=(C_1,C_2,\dots ,C_n)}$ where each${\displaystyle C_i}$ is acompact set, in ametric space, corresponding to the${\displaystyle i}$th player's set of pure strategies,

${\displaystyle \mathbf{U}=(u_1,u_2,\dots ,u_n)}$ where${\displaystyle u_i:\mathbf{C}\to \mathbb{R}}$ is the utility function of player${\displaystyle i}$

We define${\displaystyle {\mathrm{\Delta }}_i}$ to be the set of Borelprobability measures on${\displaystyle C_i}$, giving us the mixed strategy space of playeri.

Define the strategy profile${\displaystyle \mathit{\sigma }=({\sigma }_1,{\sigma }_2,\dots ,{\sigma }_n)}$ where${\displaystyle {\sigma }_i\in {\mathrm{\Delta }}_i}$

Let${\displaystyle {\mathit{\sigma }}_{-i}}$ be a strategy profile of all players except for player${\displaystyle i}$. As with discrete games, we can define abest responsecorrespondence for player${\displaystyle i}$,${\displaystyle b_i}$.${\displaystyle b_i}$ is a relation from the set of all probability distributions over opponent player profiles to a set of player${\displaystyle i}$'s strategies, such that each element of

${\displaystyle b_i({\sigma }_{-i})}$

is a best response to${\displaystyle {\sigma }_{-i}}$. Define

${\displaystyle \mathbf{b}(\mathit{\sigma })=b_1({\sigma }_{-1})\times b_2({\sigma }_{-2})\times \cdots \times b_n({\sigma }_{-n})}$.

A strategy profile${\displaystyle \mathit{\sigma }\ast }$ is aNash equilibrium if and only if${\displaystyle \mathit{\sigma }\ast \in \mathbf{b}(\mathit{\sigma }\ast )}$The existence of a Nash equilibrium for any continuous game with continuous utility functions can be proven usingIrving Glicksberg's generalization of theKakutani fixed point theorem.^[1] In general, there may not be a solution if we allow strategy spaces,${\displaystyle C_i}$'s which are not compact, or if we allow non-continuous utility functions.

Aseparable game is a continuous game where, for any i, the utility function${\displaystyle u_i:\mathbf{C}\to \mathbb{R}}$ can be expressed in the sum-of-products form:

${\displaystyle u_i(\mathbf{s})=\sum _{k_1=1}^{m_1}\dots \sum _{k_n=1}^{m_n}{a}_{i,k_1\dots k_n}{f}_1(s_1)\dots f_n(s_n)}$, where${\displaystyle \mathbf{s}\in \mathbf{C}}$,${\displaystyle s_i\in C_i}$,${\displaystyle a_{i,k_1\dots k_n}\in \mathbb{R}}$, and the functions${\displaystyle f_{i,k}:C_i\to \mathbb{R}}$ are continuous.

Apolynomial game is a separable game where each${\displaystyle C_i}$ is a compact interval on${\displaystyle \mathbb{R}}$ 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 playeri mixes at most${\displaystyle m_i+1}$ pure strategies.^[2]

Whereas an equilibrium strategy for a non-separable game may require anuncountably infinitesupport, 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 playersX andY, with${\displaystyle C_X=C_Y=\left[0,1\right]}$. Denote elements of${\displaystyle C_X}$ and${\displaystyle C_Y}$ as${\displaystyle x}$ and${\displaystyle y}$ respectively. Define the utility functions${\displaystyle H(x,y)=u_x(x,y)=-u_y(x,y)}$ where

${\displaystyle H(x,y)=(x-y{)}^2}$.

The pure strategy best response relations are:

${\displaystyle b_Y(x)=x}$

${\displaystyle b_X(y)}$ and${\displaystyle b_Y(x)}$ 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,${\displaystyle v=\mathbb{E}[H(x,y)]}$ as alinear combination of the first and secondmoments of the probability distributions ofX andY:

${\displaystyle v={\mu }_{X2}-2{\mu }_{X1}{\mu }_{Y1}+{\mu }_{Y2}}$

(where${\displaystyle {\mu }_{XN}=\mathbb{E}[x^N]}$ and similarly forY).

The constraints on${\displaystyle {\mu }_{X1}}$ and${\displaystyle {\mu }_{X2}}$ (with similar constraints fory,) are given byHausdorff as:

${\displaystyle \begin{array}{r}{\mu }_{X1}\ge {\mu }_{X2}\\ {\mu }_{X1}^2\le {\mu }_{X2}\end{array}\begin{array}{r}{\mu }_{Y1}\ge {\mu }_{Y2}\\ {\mu }_{Y1}^2\le {\mu }_{Y2}\end{array}}$

Each pair of constraints defines a compact convex subset in the plane. Since${\displaystyle v}$ 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

${\displaystyle {\mu }_{i1}={\mu }_{i2}\text{ or }{\mu }_{i1}^2={\mu }_{i2}}$

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${\displaystyle {\mu }_{i1}={\mu }_{i2}}$, it will lie on the whole line, so that both 0 and 1 are a best response.${\displaystyle b_Y({\mu }_{X1},{\mu }_{X2})}$ simply gives the pure strategy${\displaystyle y={\mu }_{X1}}$, so${\displaystyle b_Y}$ will never give both 0 and 1.However${\displaystyle b_x}$ gives both 0 and 1 when y = 1/2.A Nash equilibrium exists when:

${\displaystyle ({\mu }_{X1}\ast ,{\mu }_{X2}\ast ,{\mu }_{Y1}\ast ,{\mu }_{Y2}\ast )=(1/2,1/2,1/2,1/4)}$

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 playersX andY, with${\displaystyle C_X=C_Y=\left[0,1\right]}$. Denote elements of${\displaystyle C_X}$ and${\displaystyle C_Y}$ as${\displaystyle x}$ and${\displaystyle y}$ respectively. Define the utility functions${\displaystyle H(x,y)=u_x(x,y)=-u_y(x,y)}$ where

${\displaystyle H(x,y)=\frac{(1+x)(1+y)(1-xy)}{(1+xy{)}^2}.}$

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 ofcumulative distribution functions:

${\displaystyle F^{\ast }(x)=\frac{4}{\pi }\arctan \sqrt{x}{G}^{\ast }(y)=\frac{4}{\pi }\arctan \sqrt{y}.}$

Or, equivalently, the following pair ofprobability density functions:

${\displaystyle f^{\ast }(x)=\frac{2}{\pi \sqrt{x}(1+x)}{g}^{\ast }(y)=\frac{2}{\pi \sqrt{y}(1+y)}.}$

The value of the game is${\displaystyle 4/\pi }$.

Consider a zero-sum 2-player game between playersX andY, with${\displaystyle C_X=C_Y=\left[0,1\right]}$. Denote elements of${\displaystyle C_X}$ and${\displaystyle C_Y}$ as${\displaystyle x}$ and${\displaystyle y}$ respectively. Define the utility functions${\displaystyle H(x,y)=u_x(x,y)=-u_y(x,y)}$ where

${\displaystyle H(x,y)=\sum _{n=0}^{\mathrm{\infty }}\frac{1}{2^n}\left(2x^n-\left({\left(1-\frac{x}{3}\right)}^n-{\left(\frac{x}{3}\right)}^n\right)\right)\left(2y^n-\left({\left(1-\frac{y}{3}\right)}^n-{\left(\frac{y}{3}\right)}^n\right)\right)}$.

This game has a unique mixed strategy equilibrium where each player plays a mixed strategy with theCantor singular function as thecumulative distribution function.^[4]

• H. W. Kuhn and A. W. Tucker, eds. (1950).Contributions to the Theory of Games: Vol. II. Annals of Mathematics Studies28. Princeton University Press.ISBN 0-691-07935-8.

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