Vector Representation of Zero-Determinant Strategies and Social Welfare Control

Vector representation of zero-determinant (ZD) strategies and social welfare control for repeated finite games are investigated in this paper. First, strategy extraction vector is introduced, which can be used to extract a given player's strategy from Markov profile transition matrix. By virtue of the strategy extraction vector, the ZD strategy can be regard as an additivity of three vectors: the first one is a vector with the same elements, the second one is strategy extraction vector, and the third one is the STP between payoff vector of the n -player game and n -dimensional adjustable elements vector. Secondly, existence conditions of effective ZD strategies is provided. This paper reveals that only when the maximum difference between the elements is no more than 1, the designed ZD strategies are effective. Finally, weighted social welfare control via ZD strategies are investigated. For social welfare with given weights, the weighted social welfare only can be controlled to an interval with its interval length less than 1 by the master player (player using ZD strategies). If there are multiple master players, the controlled weighted social welfare should be unique. For social welfare with adjustable weights, the optimization of weighted social welfare is proved to be equivalent to solve a linear programming problem. For repeated common interest game, the ZD strategies are determined by two adjustable parameters.