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

<p>This family of games is called "Resource Selection Games", and it has quite a few papers written on it (<a href="http://web.mit.edu/iashlagi/www/papers/rsg_F3.pdf" rel="noreferrer">such as this one</a>).</p>

CC BY-SA 3.0
2014-12-09T13:21:32.107Z
2014-12-09T13:21:32.107Z
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<p>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 <strong>number</strong> of other players which played $j$.</p>

<p>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).</p>

<p>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.</p>

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2014-12-08T16:21:38.137Z
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 http://en.wikipedia.org/wiki/Strategic_complements. Or clarify why that isn't sufficient.
Pburg
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2014-12-08T17:16:23.783Z
@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.
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CC BY-SA 3.0
2014-12-08T17:18:10.017Z
@Pburg - think of the career selection example I gave, do you see a way to model it as a submodular game?
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2014-12-08T17:22:22.127Z
Not yet sure if it will work in a general problem with many dimensions to the strategy. However, I don't think discreteness is a problem, as modularity properties work on a lattice. Better notes might be here: http://ocw.mit.edu/courses/economics/14-126-game-theory-spring-2010/lecture-notes/MIT14_126S10_lec10b.pdf
Pburg
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2014-12-08T17:30:26.357Z
@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..
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CC BY-SA 3.0
2014-12-08T17:34:36.167Z
Yes, but you can still have partial orders within lattices so this doesn't require an exact and complete ordering like you suggest. I'm just not immediately surely how your scenario might work--hence the comment and not an answer.
Pburg
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2014-12-08T19:07:38.320Z
Your description reminds me of congestion games: https://en.wikipedia.org/wiki/Congestion_game
Erel Segal-Halevi
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2014-12-09T07:45:40.743Z
@ErelSegalHalevi - I was also thinking about congestion games, but it doesn't quite seems to fit. In CG, the goal of every player is to minimize his delay, and the delay on each element is monotonically increasing in the number of players. Do you see a way to use any of the CG results for these games?
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CC BY-SA 3.0
2014-12-09T09:34:56.460Z
As far as I understand, congestion games can be used both when the payoff is negative (decreasing in the number of players) or positive (increasing in the number of players). In both cases this is a potential game so the same results apply.
Erel Segal-Halevi
<game-theory><reference-request><definition>
Is there a name for this family of games (Binomial games?)?
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# Is there a name for this family of games (Binomial games?)?

(Asked by: username_0 on 2014-12-08)

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.

**Tags:** game-theory, reference-request, definition

> **Comments:**
> **username_1**: 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 http://en.wikipedia.org/wiki/Strategic_complements. Or clarify why that isn't sufficient.  
> **username_0**: @username_1 - 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.  
> **username_0**: @username_1 - think of the career selection example I gave, do you see a way to model it as a submodular game?  
> **username_1**: Not yet sure if it will work in a general problem with many dimensions to the strategy. However, I don't think discreteness is a problem, as modularity properties work on a lattice. Better notes might be here: http://ocw.mit.edu/courses/economics/14-126-game-theory-spring-2010/lecture-notes/MIT14_126S10_lec10b.pdf  
> **username_0**: @username_1 - 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..  
> **username_1**: Yes, but you can still have partial orders within lattices so this doesn't require an exact and complete ordering like you suggest. I'm just not immediately surely how your scenario might work--hence the comment and not an answer.  
> **username_2**: Your description reminds me of congestion games: https://en.wikipedia.org/wiki/Congestion_game  
> **username_0**: @ErelSegalHalevi - I was also thinking about congestion games, but it doesn't quite seems to fit. In CG, the goal of every player is to minimize his delay, and the delay on each element is monotonically increasing in the number of players. Do you see a way to use any of the CG results for these games?  
> **username_2**: As far as I understand, congestion games can be used both when the payoff is negative (decreasing in the number of players) or positive (increasing in the number of players). In both cases this is a potential game so the same results apply.  

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## 1 answers

### Answer by username_0 on 2014-12-09. Score: 5

OK, username_2'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](http://web.mit.edu/iashlagi/www/papers/rsg_F3.pdf)).


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