Pair-valued bi-matrix games and lexicographic equilibrium strategy profiles

Abstract

For pair-valued bi-matrix games we show that a necessary and sufficient condition for a strategy profile to be a lexicographic equilibrium strategy profile is that the strategy profile must be an equilibrium strategy profile for the bi-matrix game corresponding to the first coordinates and if the expected value of an action for a player in this bi-matrix game is equal to the expected value of this bi-matrix game for the player, then the expected value of the same action for the same player in the bi-matrix game corresponding to the second coordinates is less than or equal to its expected value in this second bi-matrix game. This result is used in an example of a pair-valued bi-matrix game to show that it has no lexicographic equilibrium strategy profile. Subsequently we prove a necessary and sufficient condition in terms of a bi-linear programming problem with a real-valued objective function and constraints which are also real-valued. We show that a possible resolution of the coordination problem in 2X2 coordination games could be achieved by introducing the concept of a dominant player, thereby embedding the coordination game in a pair-valued bi-matrix game and then choosing the lexicographic equilibrium strategy profile of the pair-valued bi-matrix game.