The problem is that these are the same game, only with the labels for one players actions switched
For two objects, in this case games, to be considered equivalent you usually want not merely a bijection between them but an isomorphism. A game G will be a set of labels L(G) with extra structure: Morphisms G->H will be particular functions L(G)->L(H). The structure your morphisms preserve will be what one can reason. The unusual fact you describe is that your function L(G)->M corresponds to no morphism G->H.
What is and isn’t an isomorphism depends on what you want to be preserved under isomorphism. If you want everything thats game-theoretically relevant to be preserved, then of course those games won’t turn out equivalent. But that doesn’t explain anything. If my argument had been that the correct action in the prisoners dilemma depends on sunspot activity, you could have written your comment just as well.
It’s easy to get confused between similar equivalence relations, so it’s useful to formally distinguish them. See the other thread’s arguing about sameness. Category theory language is relevant here because it gives a short description of your anomaly, so it may give you the tools to address it. And it is in fact unusual: For the cases of the underlying sets of a graph, group, ring, field, etc., one can find a morphism for every function.
We can construct a similar anomaly for the case of rings by saying that every ring’s underlying set contains 0 and 1, and that these are its respective neutral elements. Then a function that swaps 0 and 1 would have no corresponding ring morphism. The corresponding solution for your case would be to encode the structure not in the names of the elements of the underlying set, but in something that falls away when you go to the set. This structure would encode such knowledge as which decision is called heads and which tails. Then for any game and any function from its underlying set you could push the structure forward.
For two objects, in this case games, to be considered equivalent you usually want not merely a bijection between them but an isomorphism. A game G will be a set of labels L(G) with extra structure: Morphisms G->H will be particular functions L(G)->L(H). The structure your morphisms preserve will be what one can reason. The unusual fact you describe is that your function L(G)->M corresponds to no morphism G->H.
What is and isn’t an isomorphism depends on what you want to be preserved under isomorphism. If you want everything thats game-theoretically relevant to be preserved, then of course those games won’t turn out equivalent. But that doesn’t explain anything. If my argument had been that the correct action in the prisoners dilemma depends on sunspot activity, you could have written your comment just as well.
It’s easy to get confused between similar equivalence relations, so it’s useful to formally distinguish them. See the other thread’s arguing about sameness. Category theory language is relevant here because it gives a short description of your anomaly, so it may give you the tools to address it. And it is in fact unusual: For the cases of the underlying sets of a graph, group, ring, field, etc., one can find a morphism for every function.
We can construct a similar anomaly for the case of rings by saying that every ring’s underlying set contains 0 and 1, and that these are its respective neutral elements. Then a function that swaps 0 and 1 would have no corresponding ring morphism. The corresponding solution for your case would be to encode the structure not in the names of the elements of the underlying set, but in something that falls away when you go to the set. This structure would encode such knowledge as which decision is called heads and which tails. Then for any game and any function from its underlying set you could push the structure forward.