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"Zero Sum" is a misnomer.

In order for the standard rationality assumptions used in game theory to apply, the payouts of a game must be utilities, not resources such as money, power, or personal property. Zero-sum transfer of resources is often far from zero-sum in utility.

Hm, I feel like when I talk about game theory I don't usually use those assumptions? Admittedly I've never studied game theory in depth. But in particular, the concept of a Nash equilibrium only seems to rely on "each player has a preference order for payouts".

Actually, I'm not really sure what assumptions you mean. I assume "the players are indifferent between a certain payout of x and a 50% chance of 2x" is one, but I don't know if there's anything missing. More questions about these assumptions:

IIUC, if utility is logarithmic in a resource, then it's roughly linear in small changes of that resource. If I have £100 then I value a 50% chance of an extra £100 noticeably differently from a certain chance of an extra £50, but if I have £10000 it's about the same. Is it mostly reasonable to act as though the axioms work for resources, provided the amounts at stake are "small" for all players? (And when people talk about game theory over resources, does that tend to be the case, implicitly or explicitly?)

What do you lose if the assumptions are violated? Broadly speaking I assume many theorems about mixed and iterated games no longer apply.