Games with Random Payoff

Title: Games with Random Payoff

Research Question: The study aims to compute the fraction of all possible 2x2 symmetric non-zero-sum games with random payoff matrices that fall into certain classes defined by Nash equilibrium conditions and Pareto optimality conditions.

Methodology: The researchers used strategic form to represent the games, where /Gmir= ({A, B},{0,1}⊗{0,1},$A:{0,1}⊗{0,1} →R,$B:{0,1}⊗{0,1} →R) represents the game, {A, B} is the set of players, and {0,1} is the space of strategies of any of the players. The payoff functions, $A and $B, are the payoff matrices with labels of the strategies as indexes. The study focused on symmetric games, where there is a symmetry condition on the payoffs, ensuring equality of conditions for the two players.

Results: The study found that the essential and unnecessary features of the payoff bimatrices can be represented in a compact set. The researchers introduced a geometric representation of the games, where every possible game in the considered type, defined by a payoff matrix, has a "normalized" representative in the surface of the unit sphere (except for the trivial one with the same payoff for any situation). The fraction of the surface of the unit sphere enclosed by the planes corresponding to the conditions was found to be the fraction of the possible games set that fulfills those conditions.

Implications: The study provides a cartography of the space of possible payoff matrices, guiding the selection of models based on dynamic models with 2x2 symmetric non-zero-sum games. The geometric representation of the games allows for a better understanding of the relationships between the different kinds of 2x2 non-zero symmetric games. The findings also contribute to the field of game theory, providing a new approach to studying and analyzing strategic form games.

Link to Article: https://arxiv.org/abs/0312005v1 Authors: arXiv ID: 0312005v1