Nash Equilibria Mappings

This project was originally designed to study the relations between differential geometry and basic game theory, that is, how games are related to space in which they are played. We discovered that by examining particular surjective strategy maps and special payoff maps between games that Nash Equilibria are invariant under a "redistribution" of the payoff. More generally, we derived and proved a simple theorem which claimed that this map preserves all Nash Equilibria as a subset of the new game. We did this by drawing the digraph interpretation of the mapping matrix. We also established a loose correspondence between the rank payoff mapping matrix and the number of "sets" of exchanges. We also found that our stipulations on the mapping allowed the number of sharing sets to diminish and did not allow them to increase. Further research would study the implications of the theorem for larger games and potentially create a more generalized version while considering the forms that these "sets" take.