Reports of the Academy of Sciences of the USSR

1. Volume 144, No. 1

MATHEMATICS

I. V. ROMANOVSKII

ON REDUCING A GAME WITH PERFECT RECALL TO A MATRIX GAME

(Presented by Academician V. I. Smirnov on 25 XII 1961)

1. In this paper we consider the problem of solving finite zero-sum positional games of two persons with perfect recall (for notions and notation not explained, see (¹)). It is known that any finite positional game can be normalized, and our task is to make use of the advantages of games with perfect recall that follow from Kuhn’s theorem on behavior strategies (²).

We shall reduce the problem of finding optimal behavior strategies to the solution of a matrix game with constraints which, as shown in (⁴), is effectively reducible to a problem of linear programming. Apparently, analogous reduction methods can also be used in finding signaling and decomposed strategies (³, ⁵).

1. To define our matrix game, we shall construct auxiliary graphs—“information trees” of the first and second player. In doing so, we shall use the partial ordering, existing under perfect recall, of the set consisting of the player’s information sets and of the alternatives in these information sets.

We shall call an information tree \(T_1\) of the first player the tree formed according to the following rules:

1) To each information set \(U_k^{(1)}\) of player I in the original game there corresponds one (its own) vertex \(u_k\) of the information tree. We shall call such vertices basic.

2) To each alternative of each information set \(U_k^{(1)} : v\) of player I in the original game there corresponds one (its own) vertex \(v_{k,v}\) of the information tree. We shall call such vertices auxiliary.

3) If the initial position \(0\) of the original game tree does not belong to \(I_1\), then one more auxiliary vertex \(v_0\), corresponding to it and preceding all the other vertices, is introduced.

4) Each basic vertex \(u_k\) is followed immediately by auxiliary vertices \(v_{k,v}\), whose number \(t_k\) is equal to the number of alternatives of the information set \(U_k^{(1)}\).

5) The number of vertices immediately following an auxiliary vertex \(v_{k,v}\) is determined as follows: a) if there are no information sets \(U_l^{(1)}\) following \(U_k^{(1)} : v\), then the vertex \(v_{k,v}\) is terminal; b) the vertex \(v_{k,v}\) is immediately followed by as many basic vertices as there are pairwise incomparable information sets of player I following \(U_k^{(1)} : v\) and preceding all the other information sets of player I that follow \(U_k^{(1)} : v\). In addition, if there exists a terminal position \(W\) following \(U_k^{(1)} : v\) and such that the path from \(U_k^{(1)} : v\) to \(W\) does not intersect player I’s information sets, then the vertex \(v_{k,v}\) is immediately followed by one more auxiliary vertex, which is terminal.

6) The player’s information tree preserves the partial ordering of this player’s information sets and of the alternatives in them.

Similarly, the information tree of player II, \(T_2\), is defined.

Each terminal position \(\mathfrak w_i\) of the tree \(T_i\) (the set of which we shall denote by \(\mathfrak w_i\)) corresponds to a certain set of terminal positions \(W\) of the original tree for which \(\mathfrak P_i(W)\) coincide. We shall denote this set by \(W(\mathfrak w_i)\). Note that, whatever \(\mathfrak w_1 \in \mathfrak w_1\) and \(\mathfrak w_2 \in \mathfrak w_2\) may be, the set \(W(\mathfrak w_1)\cap W(\mathfrak w_2)\) contains no more than one element.

1. Let a nonnegative number \(p_x \ge 0\) be assigned to each vertex \(x\) of the tree \(T_1\), and suppose that the following conditions are satisfied: a)

\[ p_{u_k}=\sum_{\nu=1}^{t_k} p_{v_k,\nu}, \]

for all basic vertices \(u_k\); b) for any auxiliary vertex \(v\), \(p_v=p_x\) for all \(x\in f_{T_1}^{-1}(v)\); c) for the initial position \(x_0\), \(p_{x_0}=1\).

It is easy to see that every such set can be reconstructed from the numbers \(p_{\mathfrak w_1}\) corresponding to the terminal vertices \(\mathfrak w_1\). A set \(\pi_1=\{p_{\mathfrak w_1}\}\), \(\mathfrak w_1\in\mathfrak w_1\), from which the numbers \(p_x\) can be reconstructed for all vertices \(x\) of the tree \(T_1\), will be called a quasistrategy of player I. The set of quasistrategies \(\Pi_1\) of player I will be a bounded convex polyhedron. The quasistrategy \(\pi_2=\{q_{\mathfrak w_2}\}\), \(\mathfrak w_2\in\mathfrak w_2\), of player II is defined analogously.

We shall call a quasigame \(\Gamma\) the problem of finding the minimax of the bilinear form

\[ \pi_1 K\pi_2=\sum_{\mathfrak w_1\in\mathfrak w_1}\sum_{\mathfrak w_2\in\mathfrak w_2} K(\mathfrak w_1,\mathfrak w_2)p_{\mathfrak w_1}q_{\mathfrak w_2} \]

(where \(K(\mathfrak w_1,\mathfrak w_2)=h_1(W)\) if \(W\in W(\mathfrak w_1)\cap W(\mathfrak w_2)\), and \(K(\mathfrak w_1,\mathfrak w_2)=0\) if \(W(\mathfrak w_1)\cap W(\mathfrak w_2)=\Lambda\)) for \(\pi_1\in\Pi_1\) and \(\pi_2\in\Pi_2\).

It is known \((^{4,5})\) that

\[ \min_{\Pi_2}\max_{\Pi_1}\pi_1K\pi_2 = \max_{\Pi_1}\min_{\Pi_2}\pi_1K\pi_2 \]

and that the problem of finding optimal quasistrategies \(\pi_1\) and \(\pi_2\) is equivalent to a linear-programming problem.

1. Finding optimal quasistrategies in the quasigame turns out to be equivalent to finding optimal behavior strategies in the original game, as follows from the following theorems.

Theorem 1. To each quasistrategy \(\pi_i\) of player \(i\) there corresponds, in a one-to-one way, his behavior strategy in the original positional game \(\mu_i(\pi_i)\).

Theorem 2. The value of the bilinear form \(\pi_1K\pi_2\) is equal to the payoff in the original positional game when the behavior strategies \(\mu_1(\pi_1)\) and \(\mu_2(\pi_2)\) are used.

1. In conclusion let us consider an example. The tree of the positional game is shown in Fig. 1. Player I has four information sets, player II three. The 17 terminal positions have been renumbered. The payoff to player I in position \(k\) is equal to \(h_k\).

Fig. 1. Game tree

The information trees of the players are shown in Figs. 2 and 3. The basic positions are circled. The terminal positions of \(T_1\) are denoted by Latin letters, and those of \(T_2\) by Greek letters.

Here the quasistrategy \(\pi_1=(p_a,p_b,\ldots,p_g)\) of player I satisfies the conditions

\[ p_a+p_b=p_c,\qquad p_d+p_e=p_f+p_g,\qquad p_c+p_d+p_e=1, \]

quasi-strategy \(\pi_2=(q_\alpha,q_\beta,\ldots,q_\eta)\) of player II—the conditions

\[ q_\alpha+q_\beta=q_\gamma+q_\delta,\qquad q_\alpha+q_\beta+q_\varepsilon+q_\zeta+q_\eta=1. \]

The matrix \(K\) and the sets \(W(w_1)\) and \(W(w_2)\) are as follows:

6. Remark 1. The same reduction method is also applicable to games with an intermediary, as well as to games in which only one player has memory.

Fig. 2. Information tree of player I

Fig. 3. Information tree of player II

Remark 2. The use of the proposed method does not preclude other techniques for reducing the dimensions of the game. Thus, in the example considered one can separately determine the optimal behavior of the players on the sets \(U^{(1)}_2\) and \(U^{(2)}_2\), by solving a \(2\times2\) matrix game.

Remark 3. In the case where no more than one information set immediately follows each alternative of each information set of player \(i\), one can construct such a quasi-game in which \(\Pi_i\) coincides with the simplex of probability vectors. Such games reduce especially simply to a linear programming problem.

Leningrad State University
named after A. A. Zhdanov

Received
8 XII 1961

References

1. N. N. Vorob’ev, Uspekhi Mat. Nauk, 14, No. 4 (1959).
2. H. W. Kuhn, Contributions to the Theory of Games, 2, Princeton, 1953.
3. G. L. Thompson, ibid.
4. A. Charnes, Proc. Nat. Acad. Sci. USA, 39, No. 7 (1953).
5. N. N. Vorob’ev, Theory of Probability and Its Applications, 5, No. 4 (1960).