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Difference between revisions of "Infinite game"

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A [[Non-cooperative game|non-cooperative game]], in particular a [[Two-person zero-sum game|two-person zero-sum game]], with infinite sets of player strategies. Let
 
A [[Non-cooperative game|non-cooperative game]], in particular a [[Two-person zero-sum game|two-person zero-sum game]], with infinite sets of player strategies. Let
  
 
$$\Gamma = (X_1,X_2,...,X_n,H_1,H_2,...,H_n)$$
 
$$\Gamma = (X_1,X_2,...,X_n,H_1,H_2,...,H_n)$$
  
be an infinite game with $n$ participants. It was shown by C. Berge [[#References|[1]]] that if $X_1,X_2,...X_n$ are locally convex compact linear topological Hausdorff spaces, if the pay-off functions $H_i$ are continuous on $\Pi_{i=1]^n X_i$ and are quasi-concave for $x_i \in X_i$, $i=1,2,...,n$, then the game $\Gamma$ has equilibrium points (solutions). It was also shown [[#References|[2]]] that if the $X_i$ are compact Hausdorff spaces and the $H_i$ are continuous on $\Pi_{i=1]^n X_i$, $i=1,2,...,n$, then $\Gamma$ has equilibrium points in mixed strategies. However, not all infinite games have equilibrium points, even in mixed strategies. For example, for the two-person zero-sum game in which the sets of player strategies are sets of integers, while the pay-off function has the form
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be an infinite game with $n$ participants. It was shown by C. Berge [[#References|[1]]] that if $X_1,X_2,...X_n$ are locally convex compact linear topological Hausdorff spaces, if the pay-off functions $H_i$ are continuous on $\Pi_{i=1}^n X_i$ and are quasi-concave for $x_i \in X_i$, $i=1,2,...,n$, then the game $\Gamma$ has equilibrium points (solutions). It was also shown [[#References|[2]]] that if the $X_i$ are compact Hausdorff spaces and the $H_i$ are continuous on $\Pi_{i=1}^n X_i$, $i=1,2,...,n$, then $\Gamma$ has equilibrium points in mixed strategies. However, not all infinite games have equilibrium points, even in mixed strategies. For example, for the two-person zero-sum game in which the sets of player strategies are sets of integers, while the pay-off function has the form
  
 
$$H(m,n)= \begin{cases}
 
$$H(m,n)= \begin{cases}
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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Berge,  "Théorie génerale des jeux à <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086015.png" /> personnes" , Gauthier-Villars  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.L. Gliksberg,  "A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points"  ''Proc. Amer. Math. Soc.'' , '''3''' :  1  (1952)  pp. 170–174</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Berge,  "Théorie génerale des jeux à $n$ personnes" , Gauthier-Villars  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.L. Gliksberg,  "A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points"  ''Proc. Amer. Math. Soc.'' , '''3''' :  1  (1952)  pp. 170–174</TD></TR></table>

Latest revision as of 17:56, 25 November 2012

A non-cooperative game, in particular a two-person zero-sum game, with infinite sets of player strategies. Let

$$\Gamma = (X_1,X_2,...,X_n,H_1,H_2,...,H_n)$$

be an infinite game with $n$ participants. It was shown by C. Berge [1] that if $X_1,X_2,...X_n$ are locally convex compact linear topological Hausdorff spaces, if the pay-off functions $H_i$ are continuous on $\Pi_{i=1}^n X_i$ and are quasi-concave for $x_i \in X_i$, $i=1,2,...,n$, then the game $\Gamma$ has equilibrium points (solutions). It was also shown [2] that if the $X_i$ are compact Hausdorff spaces and the $H_i$ are continuous on $\Pi_{i=1}^n X_i$, $i=1,2,...,n$, then $\Gamma$ has equilibrium points in mixed strategies. However, not all infinite games have equilibrium points, even in mixed strategies. For example, for the two-person zero-sum game in which the sets of player strategies are sets of integers, while the pay-off function has the form

$$H(m,n)= \begin{cases} 1, & m>n\\ 0, & m=n\\ -1, & m<n\\ \end{cases}$$


no value exists. The best studied classes of infinite games in normal form are infinite two-person zero-sum games and, in particular, games on the unit square (cf. Game on the unit square).

References

[1] C. Berge, "Théorie génerale des jeux à $n$ personnes" , Gauthier-Villars (1957)
[2] I.L. Gliksberg, "A further generalization of the Kakutani fixed point theorem with application to Nash equilibrium points" Proc. Amer. Math. Soc. , 3 : 1 (1952) pp. 170–174
How to Cite This Entry:
Infinite game. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Infinite_game&oldid=28797
This article was adapted from an original article by E.B. Yanovskaya (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article