# Difference between revisions of "Infinite game"

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

− | be an infinite game with | + | 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} | ||

+ | 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|Game on the unit square]]). | 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|Game on the unit square]]). |

## Revision as of 23:12, 19 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 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