UDC 518.9

MATHEMATICS

G. N. LYUBIN

ON THE SET OF GAMES ON THE UNIT SQUARE WITH A UNIQUE SOLUTION

(Presented by Academician Yu. V. Linnik on 15 V 1968)

Let \(C\) be the standard metrized space of continuous functions on the square \(X \times Y\), where \(X, Y = [0,1]\). Each function \(K \in C\) determines a two-person zero-sum game \(\Gamma_K\) on the unit square, in which \(K(x,y)\) is the payoff of the first player if he has chosen \(x\), and the second player has chosen \(y\). As Ville showed \((^{4})\) (see also \((^{1})\)), every such game has an equilibrium situation in pure strategies. We shall say that a game has a unique solution if each player has only one optimal strategy. A game with a unique solution will be called a game with finite spectrum if the spectrum of the players’ optimal strategies is finite. All necessary definitions may be found in \((^{1})\). As was shown in \((^{1,2})\), the set of matrices of a given size that determine a game with a unique solution is an open and dense subset of the set of all matrices of the given size.

Theorem 1. The set of functions \(K \in C\) that determine a game with finite spectrum is a dense subset of \(C\).

Theorem 2. The set of functions \(K \in C\) for which the game \(\Gamma_K\) has the following properties:

1) \(\Gamma_K\) has a unique solution;
2) the optimal distribution functions are continuous;
3) the spectrum of the optimal strategy of each player is a nowhere dense perfect closed set of Lebesgue measure zero, containing an everywhere dense subset of type \(G_\delta\).

Here we shall give a proof of a weaker theorem.

Theorem \(2'\). The set of functions \(K \in C\) that determine a game with a unique solution is everywhere dense of type \(G_\delta\).

It is easy to construct an example showing that the set of functions \(K \in C\) determining a game with a unique solution is not open.

Proof of Theorem 1. Denote by \(\{K\}_n\) the class of all continuous functions that are linear on the triangles with vertices \((i/n, j/n)\), \(((i+1)/n, j/n)\), \(((i+1)/n, (j+1)/n)\), and on the triangles with vertices \((i/n, j/n)\), \((i/n, (j+1)/n)\), \(((i+1)/n, (j+1)/n)\), \(i,j = 0,1,\ldots,n-1\). Next put

\[ A_n=\{x\in X \mid x=i/n,\ i=0,1,\ldots,n\}, \]

\[ B_n=\{y\in Y \mid y=j/n,\ j=0,1,\ldots,n\}. \]

Then, for any \(\varepsilon>0\), one can find a number \(n\) and a function \(K_n \in \{K\}_n\) such that the following conditions are satisfied:

1) the game \(\Gamma_n\), specified on \(A_n \times B_n\) by the payoff function \(K_n(x,y)\), has a unique solution.

2) \(\rho(K_n,K)<\varepsilon\).

The existence of such a function follows from the uniform continuity of the function \(K\), and also from the fact that the set of matrix games with a unique solution is everywhere dense (see, for example, \((^{1})\)). Let \(F_n\) and \(G_n\) be optimal—

strategies of the players in the game \(\Gamma_n\). Since \(A_n \subset X\), and \(B_n \subset Y\), the strategies \(F_n\) and \(G_n\) determine in a natural way measures on \(X\) and \(Y\). We shall denote them also by \(F_n\) and \(G_n\). These strategies are optimal in the game \(\Gamma_{K_n}\). Let us verify this. Indeed, the minimum of the function
\[ \varphi(y)=\int K_n(x,y)\,dF_n(x) \]
is attained on \(B_n\), since on each interval complementary to \(B_n\) the function \(\varphi(y)\) is linear, and therefore the value of the minimum is not less than the value of the game \(\Gamma_n\), because \(F_n\) is an optimal strategy in the game \(\Gamma_n\). Thus, by using the strategy \(F_n\), the first player can guarantee himself a payoff not less than \(V_{\Gamma_n}\), where \(V_{\Gamma_n}\) is the value of the game \(\Gamma_n\). It is shown analogously that, by using the strategy \(G_n\), the second player cannot lose more than \(V_{\Gamma_n}\). This means that the strategies \(F_n\) and \(G_n\) are optimal strategies in the game \(\Gamma_{K_n}\).

Let \(\Phi_n(x,y)\) be a continuous function on \(X \times Y\), equal to zero on \(A_n \times B_n\). Suppose, moreover, that \(\Phi_n(x,y)>0\) on \(A_n \times Y \setminus A_n \times B_n\) and \(\Phi_n(x,y)<0\) on \(X \times B_n \setminus A_n \times B_n\). Obviously, such a function exists. The value of the game \(\Gamma_{\Phi_n}\) is equal to zero, and the optimal strategies of the players are derivative measures respectively on \(A_n\) and \(B_n\), including \(F_n\) and \(G_n\). Consequently, \(F_n\) and \(G_n\) are optimal strategies in the game \(\Gamma_P\), where \(P_n=K_n+\alpha_n\Phi_n\), \(\alpha_n\) is an arbitrary positive number, and the value of the game is equal to \(V_{\Gamma_n}\). These strategies are the unique optimal strategies in the game \(\Gamma_{P_n}\). Indeed,
\[ \int P_n(x,y)\,dF_n(x)>V_{\Gamma_n} \]
on \(Y\setminus B_n\). Therefore the support of any optimal strategy of the second player is contained in \(B_n\). It is shown analogously that the support of any optimal strategy of the first player is contained in \(A_n\). Thus, all optimal strategies of the players in the game \(\Gamma_{P_n}\) are optimal strategies in the game \(\Gamma_n\). The game \(\Gamma_n\) has a unique solution; hence the game \(\Gamma_{P_n}\) has a unique solution. Since \(\varepsilon\) and \(\alpha_n\) are arbitrary positive numbers, the theorem is proved.

Proof of Theorem \(2'\). Let \(\hat{\rho}(F_1,F_2)\) be some metric on the set of distribution functions, convergence in which is equivalent to convergence in the main. For example, as \(\hat{\rho}\) one may take the Lévy metric (see \((^3)\)). It can also be defined as
\[ \max_x \left|\int_0^x F_1(t)\,dt-\int_0^x F_2(t)\,dt\right|. \]
We shall say that the game \(\Gamma_K\) has an \(\varepsilon\)-unique solution if the set of optimal strategies of the first player can be enclosed in an open ball of radius \(\varepsilon\), and the set of optimal strategies of the second player can be enclosed in an open ball of radius \(\varepsilon\). Let \(M_\varepsilon\) be the set of continuous functions \(K\) for which the game \(\Gamma_K\) has an \(\varepsilon\)-unique solution. We shall prove that \(M_\varepsilon\) is an open subset of \(C\). Indeed, let \(K \in M_\varepsilon\), \(K_m \in C\setminus M_\varepsilon\), and suppose that the sequence \(K_m\) converges to \(K\). Moreover, let \(A_j,\ j=1,2\), be an open ball of radius \(\varepsilon\) which contains all optimal strategies of player \(j\) in the game \(\Gamma_K\). We may assume that there exists a player \(j\) such that, for every \(m\), in the game \(\Gamma_{K_m}\) he has an optimal strategy \(F_m\) not contained in \(A_j\); otherwise one could choose a subsequence of the sequence \(K_m\) possessing this property. From the sequence \(F_m\) one can choose a subsequence converging in the main, and hence also in the metric \(\hat{\rho}\), to some optimal strategy of player \(j\) in the game \(\Gamma_K\), contained in \(A_j\), which is impossible, since \(F_m\notin A_j\), and \(A_j\) is an open set. The contradiction obtained proves the openness of \(M_\varepsilon\). The set of all functions \(K\in C\) for which the game \(\Gamma_K\) has a unique solution is equal to \(\bigcap M_{1/n}\). Each set \(M_{1/n}\) is open and contains all \(K\) that define a game with a unique solution,

and therefore, by Theorem 1, is dense in \(C\). Thus, the set of all \(K\) for which the game \(\Gamma_K\) has a unique solution is the intersection of a countable number of open sets that are everywhere dense. Theorem \(2'\) is proved.

In conclusion, the author expresses his gratitude to N. N. Vorob’ev and M. L. Gromov for valuable advice and comments.

Leningrad Branch
of the Central Economics and Mathematics Institute
of the Academy of Sciences of the USSR

Received
12 V 1968

REFERENCES

1. S. Karlin, Mathematical Methods in the Theory of Games, Programming, and Economics, Moscow, 1964.
2. H. Bohnenblust, S. Karlin, L. Shapley, in: Matrix Games, Moscow, 1961, p. 17.
3. V. V. Gnedenko, A. N. Kolmogorov, Limit Theorems for Sums of Independent Random Variables, Moscow, 1949.
4. J. Ville, Traité du Calcul des Probabilités et de ses Applications, par E. Borel et collaborateurs, Paris, 2, fasc. 5, 1938, p. 105.