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.
If we pretend “resource games” always have utility logarithmic in resources, then we can save almost everything, since that’s just a transform. We might lose some results about iterated games, since normally iterated games are assumed to be worth a discounted sum (just like rewards in reinforcement learning).
One major case where game theory is applied to a resource is evolutionary game theory, where the payoffs are assumed to be reproductive success. I don’t think they do anything logarithmic, and I’m confused about whether they should or not. (I think they should? Reproductive strategies should be evaluated like Kelly betting? But I could be missing something.)
Another case is mechanism design, where payoffs are often thought of as monetary. I think there, too, they often assume (erroneously) than utility is linear in money, when a logarithmic assumption is more appropriate. However, I’m not sure where this makes a real difference vs a cosmetic difference.
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.
It’s a good question.
If we pretend “resource games” always have utility logarithmic in resources, then we can save almost everything, since that’s just a transform. We might lose some results about iterated games, since normally iterated games are assumed to be worth a discounted sum (just like rewards in reinforcement learning).
One major case where game theory is applied to a resource is evolutionary game theory, where the payoffs are assumed to be reproductive success. I don’t think they do anything logarithmic, and I’m confused about whether they should or not. (I think they should? Reproductive strategies should be evaluated like Kelly betting? But I could be missing something.)
Another case is mechanism design, where payoffs are often thought of as monetary. I think there, too, they often assume (erroneously) than utility is linear in money, when a logarithmic assumption is more appropriate. However, I’m not sure where this makes a real difference vs a cosmetic difference.