Convex game

A non-cooperative game for $n$ persons with a non-empty set of players $A$ such that for each player $i\in A$ the set of his pure strategies $X_i$ is convex, while the pay-off function (cf. Gain function) $K_i(x_1,\dots,x_n)$ is concave with respect to $x_i\in X_i$ for all values $x_k$, $k\neq i$. 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 $A$ utilize pure strategies. A convex game is called finite if each $X_i$ is compact and is contained in some Euclidean space $E^{n_i}$ while the functions $K_i$ are multi-linear. In particular, a finite zero-sum convex game is specified by a triplet $\langle R,S,K\rangle$, where $R\subset E^m$, $S\subset E^n$ and the function $K$ has the form

$$K(r,s)=\sum a_{ij}r_is_j,\quad r_i\in R,\quad s_j\in S.$$

If $\mu$ and $\nu$ are the dimensions of the sets of optimal strategies of the players I and II, respectively, while $\rho$ is the rank of the matrix $\|a_{ij}\|$, then $\mu+\nu\leq m+n-\rho$. Accordingly, if the matrix $\|a_{ij}\|$ is non-degenerate, $\mu+\nu\leq\max(m,n)$. Finite convex games are closely connected with degenerate (separable) games (cf. Degenerate game).

Let $\Gamma=\langle X,Y,K\rangle$ be a zero-sum game on the unit square whose pay-off function is concave with respect to $x\in X$ for each $y\in Y$ and is continuous on the square $X\times Y$. Player I will then have an optimal pure strategy $x_0\in X$, 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 $A$. A natural generalization of a convex game on the unit square are generalized convex games, which are defined by the fact that, for some $n$, the inequality $\partial^nK(x,y)/\partial x^n\leq0$ is met for $x\in X$, $y\in Y$. 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 $n/2$ points, while player II will have an optimal measure the support of which consists of at most $n$ points.

References

[1]	H. Nikaido, K. Isoda, "Note on non-cooperative convex games" Pacific J. Math. , 5 (1955) pp. 807–815
[2]	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
[3]	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