Abstract
Abstract Approximation in solving the infinite two-person non-cooperative games is studied in the paper. An approximation approach with conversion of infinite game into finite one is suggested. The conversion is fulfilled in three stages. Primarily the players’ payoff functions are sampled variously according to the stated requirements to the sampling. These functions are defined on unit hypercube of the appropriate Euclidean finite-dimensional space. The sampling step along each of hypercube dimensions is constant. At the second stage, the players’ payoff multidimensional matrices are reshaped into ordinary two-dimensional matrices, using the reversible index-to-index reshaping. Thus, a bimatrix game as an initial infinite game approximation is obtained. At the third stage of the conversion, the player’s finite equilibrium strategy support is checked out for its weak consistency, defined by five types of inequalities within minimal neighbourhood of every specified sampling step. If necessary, the weakly consistent solution of the bimatrix game is checked out for its consistency, strengthened in that the cardinality of every player’s equilibrium strategy support and their densities shall be non-decreasing within minimal neighbourhood of the sampling steps. Eventually, the consistent solution certifies the game approximation acceptability, letting solve even games without any equilibrium situations, including isomorphic ones to the unit hypercube game. A case of the consistency light check is stated for the completely mixed Nash equilibrium situation.
Highlights
Approximation in solving the infinite two-person non-cooperative games is studied in the paper
The player, personifying a side of the conflict, often possesses an infinite set of pure strategies, usually being multiparametric. These sets, if constituting the compacts, predetermine non-cooperative compact games [2], [16], [17], which are isomorphic to the two-person non-cooperative game on a hyperparallelepiped of the appropriate Euclidean finitedimensional space (EFDS)
Let zur S 0 be the r -th player’s pure strategy, corresponding to its pure strategy in initial infinite game (3) with components in (15), whose indices are included into set (16)
Summary
Exact methods of solving finite two-person non-cooperative games are based on linear programming or linear inequality manipulations, implying algorithms of Lemke – Howson [19], [20] or of Vorobyov [21] and Kuhn [22]. Such games can be solved approximately [23]. There is an infinite two-person non-cooperative game
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