The Complexity of game isomorphism

Abstract

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