The complexity of game isomorphism

We address the question of whether two multiplayer strategic games are equivalent and the computationalcomplexity of deciding such a property. We introduce two notions of isomorphisms, strong and weak. Each oneof those isomorphisms preserves a different structure of the game. Strong isomorphism are defined to preservethe utility functions and Nash equilibria. Weak isomorphism preserve only the player’s preference relationsand thus pure Nash equilibria. We show that the computational complexity of the game isomorphismproblem depends on the level of succinctness of the description of the input games but it is independent onwhich of the two types of isomorphisms is considered. Utilities in games can be given succinctly by Turingmachines, boolean circuits or boolean formulas, or explicitly by tables. Actions can be given also explicitlyor succinctly. When the games are given in general form, we asume a explicit description of actions anda succinct description of utilities. We show that the game isomorphism problem for general form gamesis equivalent to the circuit isomorphism when utilities are described by TMs and to the boolean formulaisomorphism problem when utilities are described by formulas. When the game is given in explicit form, weshow that the game isomorphism problem is equivalent to the graph isomorphism problem.