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 $u_{i}:\mathbf {C} \to \mathbb {R}$ 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 $u_{i}:\mathbf {C} \to \mathbb {R}$ 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 $\mathbf {s} \in \mathbf {C}$, $s_{i}\in C_{i}$, $a_{i\,,\,k_{1}\ldots k_{n}}\in \mathbb {R}$, and the functions $f_{i\,,\,k}:C_{i}\to \mathbb {R}$ are continuous.

A polynomial game is a separable game where each $C_{i}\,$ is a compact interval on $\mathbb {R} \,$ 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 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.

Examples

Separable games

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:

${\begin{aligned}\mu _{X1}\geq \mu _{X2}\\\mu _{X1}^{2}\leq \mu _{X2}\end{aligned}}\qquad {\begin{aligned}\mu _{Y1}\geq \mu _{Y2}\\\mu _{Y1}^{2}\leq \mu _{Y2}\end{aligned}}$

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.

Non-Separable Games

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:

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

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.

Further reading

• 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

• Graph continuous