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I'm wondering if there's a name for a $n$-player symmetric game, such that the payoff for player $i$ playing strategy $j$ only depends on the number of other players which played $j$.

Such family of games would capture many real-life games, such as career choosing (where the profit of becoming, say, a programmer, depends on the number of people studying programming, and it is not effected by the ratio of, e.g., agronomists to athletes).

There seems to be quite a lot shared properties for such games, especially if the profit $U_i(j)$ is decreasing as the number of players that chose $j$ increases, so I was wondering if such family was researched before and if it has a name.

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Are you familiar with super/submodular games? I can't tell if what you want is more complicated than these, but you might start here Or clarify why that isn't sufficient. – Pburg Dec 8 '14 at 16:21
@Pburg - I've looked at both papers referenced from that wiki page, and it doesn't seem quite what I'm after. They assume that each player player some real number $x\in [0,E]$ (could be, for example, the number of units he produce), and that the payoff for the player is monotonically increasing in $x$ and decreasing in the strategy of the other players. I'm looking at a simpler setting - every player picks a strategy from a discrete strategy set, and his payoff depends on the number of other players playing the exact same strategy. Unfortunately, I see no connection. – R B Dec 8 '14 at 17:16
@Pburg - think of the career selection example I gave, do you see a way to model it as a submodular game? – R B Dec 8 '14 at 17:18
@Pburg - I agree discreteness is not the problem (it's just something that might make it even easier), but in my game class there is no notion of "bigger" strategy. The strategies of the player depends only on how many other players played the same strategy, regardless of which strategy they picked, should they play a different strategy. That's a key different.. – R B Dec 8 '14 at 17:30
Your description reminds me of congestion games: – Erel Segal-Halevi Dec 8 '14 at 19:07

OK, Erel Segal Halevi's comments have placed me on the right track for finding what I was looking for;

This family of games is called "Resource Selection Games", and it has quite a few papers written on it (such as this one).

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