Continuous game

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


 * $$P = {1, 2, 3,\ldots, n}$$ is the set of $$n\, $$ players,
 * $$\mathbf{C}= (C_1, C_2, \ldots, C_n) $$ where each $$C_i\, $$ is a compact set, in a metric space, corresponding to the $$i\, $$ th player's set of pure strategies,
 * $$\mathbf{U}= (u_1, u_2, \ldots, u_n) $$ where $$u_i:\mathbf{C}\to \R$$ is the utility function of player $$i\, $$
 * We define $$\Delta_i\, $$ to be the set of Borel probability measures on $$C_i\, $$, giving us the mixed strategy space of player i.
 * Define the strategy profile $$\boldsymbol{\sigma} = (\sigma_1, \sigma_2, \ldots, \sigma_n)$$ where $$\sigma_i \in \Delta_i\, $$

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


 * $$b_i(\sigma_{-i})\, $$

is a best response to $$\sigma_{-i}$$. Define


 * $$\mathbf{b}(\boldsymbol{\sigma}) = b_1(\sigma_{-1}) \times b_2(\sigma_{-2}) \times \cdots \times b_n(\sigma_{-n})$$.

A strategy profile $$\boldsymbol{\sigma}*$$ is a Nash equilibrium if and only if $$\boldsymbol{\sigma}* \in \mathbf{b}(\boldsymbol{\sigma}*)$$ 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, $$C_i\, $$'s which are not compact, or if we allow non-continuous utility functions.

Separable games
A separable game is a continuous game where, for any i, the utility function $$u_i:\mathbf{C}\to \R$$ can be expressed in the sum-of-products form:
 * $$u_i(\mathbf{s}) = \sum_{k_1=1}^{m_1} \ldots \sum_{k_n=1}^{m_n} a_{i\, ,\, k_1\ldots k_n} f_1(s_1)\ldots f_n(s_n)$$, where $$\mathbf{s} \in \mathbf{C}$$, $$s_i \in C_i$$, $$a_{i\, ,\, k_1\ldots k_n} \in \R$$, and the functions $$f_{i\, ,\, k}:C_i \to \R$$ are continuous.

A polynomial game is a separable game where each $$C_i\, $$ is a compact interval on $$\R\, $$ 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 player i mixes at most $$m_i+1\, $$ 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.

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


 * $$H(x,y)=(x-y)^2\, $$.

The pure strategy best response relations are:


 * $$b_X(y) =

\begin{cases} 1, & \mbox{if  }y \in \left [0,1/2 \right ) \\  0\text{ or }1, & \mbox{if }y = 1/2 \\  0, & \mbox{if  } y \in \left (1/2,1 \right ]

\end{cases}$$


 * $$b_Y(x) = x\, $$

$$b_X(y)\, $$ and  $$b_Y(x)\, $$  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, $$ v = \mathbb{E} [H(x,y)]$$ as a linear combination of the first and second moments of the probability distributions of X and Y:


 * $$ v = \mu_{X2} - 2\mu_{X1} \mu_{Y1} + \mu_{Y2}\, $$

(where $$\mu_{XN} = \mathbb{E} [x^N]$$ and similarly for Y).

The constraints on $$\mu_{X1}\, $$ and $$\mu_{X2}$$ (with similar constraints for y,) are given by Hausdorff as:



\begin{align} \mu_{X1} \ge \mu_{X2} \\ \mu_{X1}^2 \le \mu_{X2} \end{align} \qquad \begin{align} \mu_{Y1} \ge \mu_{Y2} \\ \mu_{Y1}^2 \le \mu_{Y2} \end{align} $$

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


 * $$\mu_{i1} = \mu_{i2} \text{ or } \mu_{i1}^2 = \mu_{i2} $$

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


 * $$ (\mu_{X1}*, \mu_{X2}*, \mu_{Y1}*, \mu_{Y2}*) = (1/2, 1/2, 1/2, 1/4)\, $$

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.

A rational payoff function
Consider a zero-sum 2-player game between players X and Y, with $$C_X = C_Y = \left [0,1 \right ] $$. Denote elements of $$C_X\, $$ and $$C_Y\, $$ as $$x\, $$ and $$y\, $$ respectively. Define the utility functions $$H(x,y) = u_x(x,y) = -u_y(x,y)\, $$ where


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

This game has no pure strategy Nash equilibrium. It can be shown that a unique mixed strategy Nash equilibrium exists with the following pair of cumulative distribution functions:


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

Or, equivalently, the following pair of probability density functions:


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

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

Requiring a Cantor distribution
Consider a zero-sum 2-player game between players X and Y, with $$C_X = C_Y = \left [0,1 \right ] $$. Denote elements of $$C_X\, $$ and $$C_Y\, $$ as $$x\, $$ and $$y\, $$ respectively. Define the utility functions $$H(x,y) = u_x(x,y) = -u_y(x,y)\, $$ where
 * $$H(x,y)=\sum_{n=0}^\infty \frac{1}{2^n}\left(2x^n-\left (\left(1-\frac{x}{3} \right )^n-\left (\frac{x}{3}\right)^n \right ) \right ) \left(2y^n - \left (\left(1-\frac{y}{3} \right )^n-\left (\frac{y}{3}\right)^n \right ) \right )$$.

This game has a unique mixed strategy equilibrium where each player plays a mixed strategy with the Cantor singular function as the cumulative distribution function.