Mathematics

E. B. Yanovskaya

QUASI-INVARIANT KERNELS IN ANTAGONISTIC GAMES

(Presented by Academician V. I. Smirnov on 16 III 1963)

The most general results concerning the question of the existence of a solution of antagonistic games on the unit square with a bounded payoff function (for unexplained concepts and notation, see, for example, (¹)) were obtained by S. Karlin (²). In the present paper a description is given of a class on the unit square with an unbounded payoff function \(K(x,y)\), for which the equality

\[ \sup_F \inf_G \int_0^1 \int_0^1 K(x,y)\, dF(x)\, dG(y) = \inf_G \sup_F \int_0^1 \int_0^1 K(x,y)\, dF(x)\, dG(y), \tag{*} \]

holds, where \(F\) and \(G\) denote distributions on the interval \([0,1]\).

Let \(\varphi(x)\) be a one-to-one mapping of the interval \([0,1]\) onto itself. For any distribution function \(F\), by \(\varphi(F)\) we denote the function defined by the equality

\[ \varphi(F(x)) = \int_{\varphi^{-1}\{[0,x]\}} dF(x). \]

This function is also a distribution function.

Let \(\varphi_\delta(x)\), \(\psi_\delta(y)\) \((0 \leq \delta \leq \Delta)\) be two families of one-to-one transformations of the interval \([0,1]\) onto itself. A kernel \(K(x,y)\) is called quasi-invariant with respect to the families of transformations \(\varphi_\delta(x)\), \(\psi_\delta(y)\), if for every \(\varepsilon > 0\) there exists a \(\delta' > 0\) such that for all \(\delta \in [0,\delta']\), \(x\), and \(y\),

\[ \left| K(\varphi_\delta(x),y) - K(x,\psi_\delta(y)) \right| < \varepsilon . \]

Consider a game \(\Gamma\) on the unit square, the payoff function \(K(x,y)\) of which has the following properties:

1. The function \(K(x,y)\) is measurable in each variable for every fixed value of the other.

2. The integrals \(\displaystyle \int_0^1 K(x,y)\,dx\) and \(\displaystyle \int_0^1 K(x,y)\,dy\) converge uniformly for all \(y\) and \(x\), respectively.

3. For all \(n > 1\) the games \(\Gamma_n\) with kernels \(K_n(x,y)\) have solutions, where the functions \(K_n(x,y)\) are defined as follows:

\[ K_n(x,y)= \begin{cases} K(x,y), & |K(x,y)| < n,\\ n, & K(x,y) \geq n,\\ -n, & K(x,y) \leq -n. \end{cases} \]

1. There exist such families of transformations of the interval \([0,1]\) onto itself, \(\varphi_\delta\) and \(\psi_\delta\), with respect to which the function \(K(x,y)\) is quasi-invariant and for which, for all \(\delta_1,\delta_2 \in [0,\Delta]\), \(x \in [0,1]\), the inequalities

\[ |\varphi_{\delta_1}(x)-\varphi_{\delta_2}(x)| > C|\delta_1-\delta_2|, \]

\[ |\psi_{\delta_1}(x)-\psi_{\delta_2}(x)| > C|\delta_1-\delta_2| \]

hold.

Theorem 1. If the function \(K(x,y)\) satisfies conditions 1–4, then equality \((*)\) holds.

Moreover, one can show that for every \(\varepsilon>0\) there exist \(N\) and \(\delta'>0\) such that, for all \(n>N\), \(\delta\in[0,\delta']\), the distribution functions

\[ \overline{\varphi}_{\delta}\bigl(\widetilde{F}_{n}(x)\bigr) = \frac{1}{\delta}\int_{0}^{\delta}\varphi_{\delta}\bigl(\widetilde{F}_{n}(x)\bigr)\,d\delta, \]

\[ \overline{\psi}_{\delta}\bigl(\widetilde{G}_{n}(y)\bigr) = \frac{1}{\delta}\int_{0}^{\delta}\psi_{\delta}\bigl(\widetilde{G}_{n}(y)\bigr)\,d\delta \]

\(\bigl(\widetilde{F}_{n}(x),\ \widetilde{G}_{n}(y)\)—optimal strategies of players I and II respectively in the games \(\Gamma_n\bigr)\) are \(\varepsilon\)-optimal strategies of the players in the game \(\Gamma\).

From this theorem there immediately follow the facts established in \((^{3})\). A less trivial consequence is the following result.

Theorem 2. Let the function \(K(x,y)\) satisfy conditions 1–3 of Theorem 1 and, in addition, on the set of points of discontinuity be monotonically increasing in \(x\) and monotonically decreasing in \(y\). Suppose, moreover, that the function \(K(x,y)\) is quasi-invariant with respect to the following two families of transformations of subsets of the unit interval into the unit interval:

\[ \varphi_{\delta}^{(1)}(x)=\frac{x}{1-\delta},\qquad x\in[0,1-\delta]; \qquad \psi_{\delta}^{(1)}(y)=(1-\delta)y,\qquad y\in[0,1]; \]

\[ \varphi_{\delta}^{(2)}(x)=x-\delta,\qquad x\in[0,1]; \qquad \psi_{\delta}^{(2)}(y)=y+\delta,\qquad y\in[0,1-\delta]. \]

Then equality \((*)\) holds.

This theorem asserts the existence, for every \(\varepsilon>0\), of \(\varepsilon\)-equilibrium situations for games with kernels of the type

\[ K(x,y)= \begin{cases} h_1(x,y)\ln(x-y), & x>y,\\ f(x), & x=y,\\ h_2(x,y)\ln(y-x), & x<y, \end{cases} \]

where \(h_1,h_2\) are nonnegative functions continuous in both variables, defined respectively in the closed triangles \(0\le y\le x\le 1\) and \(0\le x\le y\le 1\); the functions \(h_1\) and \(h_2\) are monotonically increasing in \(x\) and monotonically decreasing in \(y\), and the function \(f(x)\) is bounded.

Leningrad Branch
of the V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
5 III 1963

CITED LITERATURE

\(^{1}\) N. N. Vorob'ev, Infinite Antagonistic Games, 1963, p. 7.
\(^{2}\) S. Karlin, ibid., p. 47.
\(^{3}\) E. B. Yanovskaya, ibid., p. 77.