 Mathematical foundations Incorporating players raises new questions about game equivalence. When agency matters, adequate notions of equivalence cannot stop at preserving properties of the underlying game form. Rather, player-dependent equivalences will also require preservation of players’ beliefs, preferences or reasoning types. Incorporating such additional parameters makes it harder for two games to be equivalent, as new space for variation comes into play. On the other hand, agent limitations might also create new simpler game equivalences that can be studied by the tools of this entry.