An $n$-player prisoner's dilemma where a coalition of 2 players is better off defecting
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 $\frac{1}{n} &lt; c &lt; 1$. In these games it is individually rational to defect (set $x_i = 0$) but the Pareto optimal outcome is to cooperate (meaning $\forall i: x_i = \bar{x}$).

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.