<p>A public goods game is a generalization of the prisoners dilemma that could cover this.</p>

<p>In a public goods game all players set their individual contribution level $x_i \in [0,\bar{x}]$ where $\bar{x}$ is a given positive parameter. Sometimes instead of an interval there are several discrete options to choose from. Player $i$'s payoff is defined as
$$
- x_i + c \cdot \sum_{j = 1}^{n} x_j
$$
where $c$ is a parameter such that $\frac{1}{n} &lt; c &lt; 1$. In these games it is individually rational to defect (set $x_i = 0$) but the Pareto optimal outcome is to cooperate (meaning $\forall i: x_i = \bar{x}$).</p>

<p>Seems to me that you could adapt this to your problem by narrowing the range of $c$ to $\frac{1}{n} &lt; c &lt; \frac{1}{2}$. In this case a player would prefer even dual defection to himself having to cooperate. Of course these games do not cover the whole game class you described, by they are part of it, and there is a lot of literature on public goods games, though mostly in behavioral economics.</p>

CC BY-SA 3.0
2016-12-08T19:54:00.017Z
I like this answer, but I suspect modelling it in this way (i.e. non-cooperatively) would not capture some of the cooperative aspects the OP is interested in. I think the OP needs some [cooperative game theory](https://en.wikipedia.org/wiki/Cooperative_game_theory), which often isn't a part of most economists' toolkits.
Theoretical Economist
CC BY-SA 3.0
2016-12-08T20:58:45.053Z
@TheoreticalEconomist I don't think so, because OP specifically names two actions: "co-operate" and "defect", which implies that there are actions. In co-op game theory there are no actions, only coalitions. Maybe you have a co-op game theory based answer in mind that goes around this problem. Do post it, a competition of ideas is always welcome.
Giskard
CC BY-SA 3.0
2016-12-08T21:04:39.157Z
I unfortunately don't know enough cooperative game theory to give a reasonably well-developed answer. You're right that classic cooperative game theory doesn't quite work here -- I guess what I meant more is that the OP needs game theory with some cooperative elements built in. I guess I was thinking of something along the lines of the equilibrium concept developed [here](http://levine.sscnet.ucla.edu/archive/refs4786969000000001288.pdf).
Theoretical Economist
CC BY-SA 3.0
2016-12-07T21:01:42.863Z
2016-12-08T08:15:03.253Z
2016-12-08T08:15:03.253Z
Giskard
<p>I'm interested in a game kind of like the prisoner's dilemma. The important thing about the 2x2 prisoner's dilemma is that it is efficient if all cooperate, but this is not an equilibrium, because both players have a dominant strategy to defect. </p>

<p>I have in mind a variant. Suppose $n$ players can choose to cooperate or defect. It is group efficient if all cooperate, but any player would like to defect from that. Further, any coalition of 2 (more generally $k$) players would prefer to defect together. So we could imagine payoffs $u(\text{all defect})=0$, and switching to cooperate increases all individual payoffs by 1, but at a private cost of $2+\epsilon$ (more generally $k+\epsilon&lt;n$). So, it is always better for the group if someone cooperates, but only for a large enough group. </p>

<p>Is there a name for this kind of thing? Are there any papers or other resources that explore how this differs than the classic prisoner's dilemma?</p>

CC BY-SA 3.0
2016-12-08T09:49:49.737Z
Prisoner's dilemma are subjective to terms/resources set and rationality. Under sub-optimum conditions, everyone are subject to defect.
mootmoot
<reference-request><game-theory>
An $n$-player prisoner's dilemma where a coalition of 2 players is better off defecting
Pburg
# An $n$-player prisoner's dilemma where a coalition of 2 players is better off defecting

(Asked by: username_0 on 2016-12-07)

I'm interested in a game kind of like the prisoner's dilemma. The important thing about the 2x2 prisoner's dilemma is that it is efficient if all cooperate, but this is not an equilibrium, because both players have a dominant strategy to defect.

I have in mind a variant. Suppose $n$ players can choose to cooperate or defect. It is group efficient if all cooperate, but any player would like to defect from that. Further, any coalition of 2 (more generally $k$) players would prefer to defect together. So we could imagine payoffs $u(\text{all defect})=0$, and switching to cooperate increases all individual payoffs by 1, but at a private cost of $2+\epsilon$ (more generally $k+\epsilon<n$). So, it is always better for the group if someone cooperates, but only for a large enough group.

Is there a name for this kind of thing? Are there any papers or other resources that explore how this differs than the classic prisoner's dilemma?

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

> **Comments:**
> **username_1**: Prisoner's dilemma are subjective to terms/resources set and rationality. Under sub-optimum conditions, everyone are subject to defect.  

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## 1 answers

### Answer by username_2 on 2016-12-07. Score: 1

A public goods game is a generalization of the prisoners dilemma that could cover this.

In a public goods game all players set their individual contribution level $x\_i \in [0,\bar{x}]$ where $\bar{x}$ is a given positive parameter. Sometimes instead of an interval there are several discrete options to choose from. Player $i$'s payoff is defined as
$$
- x\_i + c \cdot \sum\_{j = 1}^{n} x\_j
$$
where $c$ is a parameter such that $\frac{1}{n} < c < 1$. In these games it is individually rational to defect (set $x\_i = 0$) but the Pareto optimal outcome is to cooperate (meaning $\forall i: x\_i = \bar{x}$).

Seems to me that you could adapt this to your problem by narrowing the range of $c$ to $\frac{1}{n} < c < \frac{1}{2}$. In this case a player would prefer even dual defection to himself having to cooperate. Of course these games do not cover the whole game class you described, by they are part of it, and there is a lot of literature on public goods games, though mostly in behavioral economics.

> **Comments:**
> **username_3**: I like this answer, but I suspect modelling it in this way (i.e. non-cooperatively) would not capture some of the cooperative aspects the OP is interested in. I think the OP needs some [cooperative game theory](https://en.wikipedia.org/wiki/Cooperative_game_theory), which often isn't a part of most economists' toolkits.  
> **username_2**: @username_3 I don't think so, because OP specifically names two actions: "co-operate" and "defect", which implies that there are actions. In co-op game theory there are no actions, only coalitions. Maybe you have a co-op game theory based answer in mind that goes around this problem. Do post it, a competition of ideas is always welcome.  
> **username_3**: I unfortunately don't know enough cooperative game theory to give a reasonably well-developed answer. You're right that classic cooperative game theory doesn't quite work here -- I guess what I meant more is that the OP needs game theory with some cooperative elements built in. I guess I was thinking of something along the lines of the equilibrium concept developed [here](http://levine.sscnet.ucla.edu/archive/refs4786969000000001288.pdf).  

---

