# 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 | ||

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

be an infinite game with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508602.png" /> participants. It was shown by C. Berge [[#References|[1]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508603.png" /> are locally convex compact linear topological Hausdorff spaces, if the pay-off functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508604.png" /> are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508605.png" /> and are quasi-concave for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508607.png" />, then the game <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508608.png" /> has equilibrium points (solutions). It was also shown [[#References|[2]]] that if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508609.png" /> are compact Hausdorff spaces and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086010.png" /> are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086013.png" /> 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 | be an infinite game with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508602.png" /> participants. It was shown by C. Berge [[#References|[1]]] that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508603.png" /> are locally convex compact linear topological Hausdorff spaces, if the pay-off functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508604.png" /> are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508605.png" /> and are quasi-concave for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508606.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508607.png" />, then the game <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508608.png" /> has equilibrium points (solutions). It was also shown [[#References|[2]]] that if the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i0508609.png" /> are compact Hausdorff spaces and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086010.png" /> are continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086011.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086012.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/i/i050/i050860/i05086013.png" /> 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 |

## Revision as of 23:06, 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 participants. It was shown by C. Berge [1] that if are locally convex compact linear topological Hausdorff spaces, if the pay-off functions are continuous on and are quasi-concave for , , then the game has equilibrium points (solutions). It was also shown [2] that if the are compact Hausdorff spaces and the are continuous on , , then 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

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=28796