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A '''continuous game''' is a mathematical generalization, used in [[game theory]]. 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 [[uncountable|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 space|compact]] and the utility functions [[continuous function|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==
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Define the ''n''-player continuous game <math> G = (P, \mathbf{C}, \mathbf{U}) </math> where
:: <math>P = {1, 2, 3,\ldots, n}</math> is the set of <math>n\, </math> players,
:: <math>\mathbf{C}=  (C_1, C_2, \ldots, C_n) </math> where each <math>C_i\, </math> is a [[compact space|compact]] [[metric space]] corresponding to the <math>i\, </math> ''th'' player's set of pure strategies,
:: <math>\mathbf{U}=  (u_1, u_2, \ldots, u_n) </math> where <math>u_i:\mathbf{C}\to \R</math> is the utility function of player <math>i\, </math>
: We define <math>\Delta_i\, </math> to be the set of Borel [[probability measure]]s on <math>C_i\, </math>, giving us the mixed strategy space of player ''i''.
: Define the strategy profile <math>\boldsymbol{\sigma} = (\sigma_1, \sigma_2, \ldots, \sigma_n)</math> where <math>\sigma_i \in \Delta_i\, </math>
Let <math>\boldsymbol{\sigma}_{-i}</math> be a strategy profile of all players except for player <math>i</math>. As with discrete games, we can define a [[best response]] [[correspondence (mathematics)|correspondence]] for player <math>i\, </math>, <math>b_i\ </math>. <math>b_i\, </math> is a relation from the set of all probability distributions over opponent player profiles to a set of player <math>i</math>'s strategies, such that each element of
 
:<math>b_i(\sigma_{-i})\, </math>
is a best response to <math>\sigma_{-i}</math>. Define
 
:<math>\mathbf{b}(\boldsymbol{\sigma}) = b_1(\sigma_{-1}) \times b_2(\sigma_{-2}) \times \cdots \times b_n(\sigma_{-n})</math>.
A strategy profile <math>\boldsymbol{\sigma}*</math> is a [[Nash equilibrium]] if and only if
<math>\boldsymbol{\sigma}* \in \mathbf{b}(\boldsymbol{\sigma}*)</math>
The existence of a Nash equilibrium for any continuous game with continuous utility functions can been proven using  [[Irving Glicksberg]]'s generalization of the [[Kakutani fixed point theorem]].<ref>I.L. Glicksberg. A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points.  Proceedings of the American Mathematical Society, 3(1):170–174, February 1952.</ref> In general, there may not be a solution if we allow allow strategy spaces, <math>C_i\, </math>'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 <math>u_i:\mathbf{C}\to \R</math> can be expressed in the sum-of-products form:
: <math>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)</math>, where <math>\mathbf{s} \in \mathbf{C}</math>, <math>s_i \in C_i</math>, <math>a_{i\, ,\, k_1\ldots k_n} \in \R</math>, and the functions <math>f_{i\, ,\, k}:C_i \to \R</math> are continuous.
A '''polynomial game''' is a separable game where each <math>C_i\, </math> is a compact interval on <math>\R\, </math> 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 <math>m_i+1\, </math> pure strategies.<ref>N. Stein, A. Ozdaglar and P.A. Parrilo. "Separable and Low-Rank Continuous Games". ''International Journal of Game Theory'', 37(4):475–504, December 2008. http://arxiv.org/abs/0707.3462</ref>
Whereas an equilibrium strategy for a non-separable game may require an [[uncountable set|uncountably infinite]] [[Support (mathematics)|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 <math>C_X = C_Y = \left [0,1 \right ] </math>. Denote elements of <math>C_X\, </math> and <math>C_Y\, </math> as <math>x\, </math> and <math>y\, </math> respectively.  Define the utility functions <math>H(x,y) = u_x(x,y) = -u_y(x,y)\, </math> where
 
:<math>H(x,y)=(x-y)^2\, </math>.
 
The pure strategy best response relations are:
 
:<math>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}</math>
 
:<math>b_Y(x) = x\, </math>
 
<math>b_X(y)\, </math>  and <math>b_Y(x)\, </math>  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, <math> v = \mathbb{E} [H(x,y)]</math> as a [[linear]] combination of the first and second [[moment (mathematics)|moments]] of the probability distributions of '''X''' and '''Y''':
 
: <math> v = \mu_{X2} - 2\mu_{X1} \mu_{Y1} + \mu_{Y2}\, </math>
 
(where <math>\mu_{XN} = \mathbb{E} [x^N]</math> and similarly for ''Y'').
 
The constraints on <math>\mu_{X1}\, </math> and <math>\mu_{X2}</math> (with similar constraints for ''y'',) are given by [[Hausdorff moment problem|Hausdorff]] as:
 
: <math>
\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}
</math>
 
Each pair of constraints defines a compact convex subset in the plane.  Since <math>v\, </math> 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
 
: <math>\mu_{i1} = \mu_{i2} \text{ or } \mu_{i1}^2 = \mu_{i2} </math>
 
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 <math>\mu_{i1} = \mu_{i2}\, </math>, it will lie on the whole line, so that both 0 and 1 are a best response. <math>b_Y(\mu_{X1},\mu_{X2})\, </math> simply gives the pure strategy <math>y = \mu_{X1}\, </math>, so <math>b_Y\, </math> will never give both 0 and 1.
However <math>b_x\, </math> gives both 0 and 1 when y = 1/2.
A Nash equilibrium exists when:
 
: <math> (\mu_{X1}*, \mu_{X2}*, \mu_{Y1}*, \mu_{Y2}*) = (1/2, 1/2, 1/2, 1/4)\, </math>
 
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 <math>C_X = C_Y = \left [0,1 \right ] </math>.  Denote elements of <math>C_X\, </math> and <math>C_Y\, </math> as <math>x\, </math> and <math>y\, </math> respectively. Define the utility functions <math>H(x,y) = u_x(x,y) = -u_y(x,y)\, </math> where
 
:<math>H(x,y)=\frac{(1+x)(1+y)(1-xy)}{(1+xy)^2}. </math>
 
This game has no pure strategy Nash equilibrium.  It can be shown<ref>Glicksberg, I. & Gross, O. (1950). "Notes on Games over the Square." Kuhn, H.W. & Tucker, A.W. eds. ''Contributions to the Theory of Games: Volume II.'' Annals of Mathematics Studies '''28''', p.173–183. Princeton University Press.</ref> that a unique mixed strategy Nash equilibrium exists with the following pair of [[probability density function]]s:
 
: <math>f^*(x) = \frac{2}{\pi \sqrt{x} (1+x)} \qquad g^*(y) = \frac{2}{\pi \sqrt{y} (1+y)}. </math>
 
The value of the game is <math>4/\pi</math>.
 
====Requiring a Cantor distribution====
Consider a zero-sum 2-player game between players '''X''' and '''Y''', with <math>C_X = C_Y = \left [0,1 \right ] </math>. Denote elements of <math>C_X\, </math> and <math>C_Y\, </math> as <math>x\, </math> and <math>y\, </math> respectively. Define the utility functions <math>H(x,y) = u_x(x,y) = -u_y(x,y)\, </math> where
:<math>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 )</math>.
This game has a unique mixed strategy equilibrium where each player plays a mixed strategy with the [[cantor function|cantor singular function]] as the [[cumulative distribution function]].<ref>Gross, O. (1952). "A rational payoff characterization of the Cantor distribution." Technical
Report D-1349, The RAND Corporation.</ref>
 
==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==
<references />
 
[[Category:Game theory]]

Latest revision as of 05:23, 9 February 2014


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