Finite approximation of zero-sum games played in staircase-function continuous spaces

Abstract

Background. There is a known method of approximating continuous zero-sum games, wherein an approximate solution
is considered acceptable if it changes minimally by changing the sampling step minimally. However, the method cannot
be applied straightforwardly to a zero-sum game played with staircase-function strategies. Besides, the independence of
the player’s sampling step selection should be taken into account.
Objective. The objective is to develop a method of finite approximation of zero-sum games played in staircase-function
continuous spaces by taking into account that the players are likely to independently sample their pure strategy sets.
Methods. To achieve the said objective, a zero-sum game, in which the players’ strategies are staircase functions of time,
is formalized. In such a game, the set of the player’s pure strategies is a continuum of staircase functions of time, and
the time is thought of as it is discrete. The conditions of sampling the set of possible values of the player’s pure strategy
are stated so that the game becomes defined on a product of staircase-function finite spaces. In general, the sampling
step is different at each player and the distribution of the sampled points (function-strategy values) is non-uniform.
Results. A method of finite approximation of zero-sum games played in staircase-function continuous spaces is pre-
sented. The method consists in irregularly sampling the player’s pure strategy value set, solving smaller-sized matrix
games, each defined on a subinterval where the pure strategy value is constant, and stacking their solutions if they are
consistent. The stack of the smaller-sized matrix game solutions is an approximate solution to the initial staircase game.
The (weak) consistency of the approximate solution is studied by how much the payoff and optimal situation change as
the sampling density minimally increases by the three ways of the sampling increment: only the first player’s increment,
only the second player’s increment, both the players’ increment. The consistency is decomposed into the payoff, opti-
mal strategy support cardinality, optimal strategy sampling density, and support probability consistency. It is practically
reasonable to consider a relaxed payoff consistency.
Conclusions. The suggested method of finite approximation of staircase zero-sum games consists in the independent
samplings, solving smaller-sized matrix games in a reasonable time span, and stacking their solutions if they are con-
sistent. The finite approximation is regarded appropriate if at least the respective approximate (stacked) solution is
e-payoff consistent.