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 $G = (P, \mathbf, \mathbf)$ where

::$P =$ is the set of $n\,$ players,
:: $\mathbf= (C_1, C_2, \ldots, C_n)$ where each $C_i\,$ is a compact set, in a metric space, corresponding to the $i\,$ ''th'' player's set of pure strategies,
:: $\mathbf= (u_1, u_2, \ldots, u_n)$ where $u_i:\mathbf\to \R$ 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 $\boldsymbol = (\sigma_1, \sigma_2, \ldots, \sigma_n)$ where $\sigma_i \in \Delta_i\,$
Let $\boldsymbol_$ 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

:$b_i(\sigma_)\,$
is a best response to $\sigma_$. Define

:$\mathbf(\boldsymbol) = b_1(\sigma_) \times b_2(\sigma_) \times \cdots \times b_n(\sigma_)$.
A strategy profile $\boldsymbol*$ is a Nash equilibrium if and only if
$\boldsymbol* \in \mathbf(\boldsymbol*)$
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 $u_i:\mathbf\to \R$ can be expressed in the sum-of-products form:
: $u_i(\mathbf) = \sum_^ \ldots \sum_^ a_ f_1(s_1)\ldots f_n(s_n)$, where $\mathbf \in \mathbf$, $s_i \in C_i$, $a_ \in \R$, and the functions $f_:C_i \to \R$ 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.

Examples

Separable games

A polynomial game

Consider a zero-sum 2-player game between players X and Y, with C_X = C_Y = \left ,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
1, & \mboxy \in \left ,1/2 \right ) \\
0\text1, & \mboxy = 1/2 \\
0, & \mbox y \in \left (1/2,1 \right
\end

:$b_Y(x) = x\,$

$b_X(y)\,$ 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 (x,y)/math> as a linear combination of the first and second moments of the probability distributions of X and Y:

: $v = \mu_ - 2\mu_ \mu_ + \mu_\,$

(where \mu_ = \mathbb ^N/math> and similarly for ''Y'').

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

: $\begin \mu_ \ge \mu_ \\ \mu_^2 \le \mu_ \end \qquad \begin \mu_ \ge \mu_ \\ \mu_^2 \le \mu_ \end$

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

: $\mu_ = \mu_ \text \mu_^2 = \mu_$

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 $\mu_ = \mu_\,$, it will lie on the whole line, so that both 0 and 1 are a best response. $b_Y(\mu_,\mu_)\,$ simply gives the pure strategy $y = \mu_\,$, 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:

: $(\mu_*, \mu_*, \mu_*, \mu_*) = (1/2, 1/2, 1/2, 1/4)\,$

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 pay-off function

Consider a zero-sum 2-player game between players X and Y, with C_X = C_Y = \left ,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

:$H(x,y)=\frac.$

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:

: $f^*(x) = \frac \qquad g^*(y) = \frac.$

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 ,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
:$H(x,y)=\sum_^\infty \frac\left(2x^n-\left (\left(1-\frac \right )^n-\left (\frac\right)^n \right ) \right ) \left(2y^n - \left (\left(1-\frac \right )^n-\left (\frac\right)^n \right ) \right )$.
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.Gross, O. (1952). "A rational payoff characterization of the Cantor distribution." Technical
Report D-1349, The RAND Corporation.

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

References

Game theory game classes