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Some topics in two-person games. (English) Zbl 0126.16204
The paper contains the following independent sections (most of the results are invariant under order-preserving transformations of payoffs):
A characterization of a matrix of zero-sum two-person game which is symmetric in the players (such a matrix can be decomposed into an array of square blocks each of which has constant diagonals and size which is a power of 2 etc.), a remark about solutions and corresponding results for non-zero-sum case are given.
An \(m\)-by-\(n\) matrix \(A\) has a saddlepoint if (a) there exist \(r,s,2\leqq r\leqq m, 2\leqq s\leqq n\) such that each \(r\)-by-\(s\) submatrix has a saddlepoint and (b) no two points in \(A\) in the same row or column are equal. For \(r=s=2\) (b) can be thrown away.
An order matrix \(A\) is a class of numerical matrices which are of the same size and the elements of the corresponding rows and columns are ordered alike. A saddle \(S(A)\) is a submatrix of \(A\) such that each row disjoint with \(S(A)\) is strictly majorized by one incident with \(S(A)\) for all the points lying in columns incident with \(S(A)\) (the same holds for columns with minorizing) and no other submatrix of \(S(A)\) has these properties. A center \(C(A)\) is the union of all submatrices each of them is a submatrix of pure active strategies for at least one numerical matrix of \(A\). \(R(A)\) is the union of 1-by-1 submatrices each of which is a result of a sequence of deleting dominated row or column or row with minimal or column with maximal payoff. Every \(A\) has a unique \(S(A)\). It holds \(C(A)\subseteq S(A)\) and \(R(A)\subseteq S(A)\) but \(R(A)\subseteq C(A)\) is not always true (for certain \(A\) by means of modifying the order matrix concept the validity of the last inclusion is proved). A method of determining \(S(A)\), a result about \(S(-A)\) and a lot of interesting examples are given.
A certain noisy duel (in which each player shoots once only but can choose from among several types of guns) is studied by reduction to a finite matrix game whose payoffs are results either of pairs of simultaneous actions at the beginning of the play or (in row and column with the label “Wait”) subgames after the beginning. The conditions for the existence of a value of the duel (with examples without a value) and the definition of a formal value in the indeterminate case are given.
A class of non-zero-sum two-person 3-by-3 matrix games (with a numerical example) is presented for which the sequence of mixed-strategy pairs generated by the iterative learning process (in which at each stage the player chooses such a pure strategy that yields the best result if employed against all past choices of its opponent) does not converge to the unique equilibrium point.
Reviewer: V. Polák

91A05 2-person games
game theory