View source for Convex game

A [[Non-cooperative game|non-cooperative game]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262601.png" /> persons with a non-empty set of players <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262602.png" /> such that for each player <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262603.png" /> the set of his pure strategies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262604.png" /> is convex, while the pay-off function (cf. [[Gain function|Gain function]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262605.png" /> is concave with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262606.png" /> for all values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262607.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262608.png" />. If the pay-off functions of all players in the convex game are continuous while the sets of pure strategies are compact, there exists an equilibrium point in which the players of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c0262609.png" /> utilize pure strategies. A convex game is called finite if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626010.png" /> is compact and is contained in some Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626011.png" /> while the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626012.png" /> are multi-linear. In particular, a finite zero-sum convex game is specified by a triplet <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626015.png" /> and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626016.png" /> has the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626017.png" /></td> </tr></table>

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626019.png" /> are the dimensions of the sets of optimal strategies of the players I and II, respectively, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626020.png" /> is the rank of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626022.png" />. Accordingly, if the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626023.png" /> is non-degenerate, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626024.png" />. Finite convex games are closely connected with degenerate (separable) games (cf. [[Degenerate game|Degenerate game]]).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626025.png" /> be a zero-sum [[Game on the unit square|game on the unit square]] whose pay-off function is concave with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626026.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626027.png" /> and is continuous on the square <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626028.png" />. Player I will then have an optimal pure strategy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626029.png" />, while player II will have an optimal measure (a mixed strategy) the support of which consists of at most two points. It is thus possible in a convex game to obtain certain information about the properties of the strategies of the players not belonging to the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626030.png" />. A natural generalization of a convex game on the unit square are generalized convex games, which are defined by the fact that, for some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626031.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626032.png" /> is met for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626034.png" />. In such a case, if it is agreed to assign a weight 1/2 to the terminal point of the segment, player I has an optimal measure the support of which consists of at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626035.png" /> points, while player II will have an optimal measure the support of which consists of at most <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c026/c026260/c02626036.png" /> points.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Nikaido,   K. Isoda,   "Note on non-cooperative convex games"  ''Pacific J. Math.'' , '''5'''  (1955)  pp. 807–815</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Dresher,   S. Karlin,   "Solution of convex games as fixed points"  H.W. Kuhn (ed.)  A.W. Tucker (ed.) , ''Contributions to the theory of games'' , '''2''' , Princeton Univ. Press  (1953)  pp. 75–83</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H.F. Bohnenblust,   S. Karlin,   L.S. Shapley,   "Games with continuous, convex pay-off"  H.W. Kuhn (ed.)  A.W. Tucker (ed.) , ''Contributions to the theory of games'' , '''1''' , Princeton Univ. Press  (1950)  pp. 181–192</TD></TR></table>