Assortative Matching with Inequality in Voluntary Contribution Games

Table 1 In this table we show which equilibria exist for homogeneous and heterogeneous players depending on the value of \(\gamma \)

For homogeneous players, for any payoff such that \(\gamma \) is bigger than a threshold value \( \bar{\gamma }\), there exist one Nash Equilibrium in which all players contribute nothing (indicated as PSL) and almost Pareto optimal Nash Equilibria where almost all players contribute everything and few (less than the group size) contribute the within group NE \({\bar{\alpha }}\). (PSH). For \(\gamma < \bar{\gamma }\), the only Nash Equilibirum is for everybody to contribute \({\bar{\alpha }}\). In these cases, there exist no mixed strategy (MS) Nash Equilibrium. For heterogeneous players the situation is different for different values of \(\gamma \). For sublinear payoffs (\(\gamma < 1\)) there exist no pure strategy equilibria and hence the only Nash Equilibrium is in mixed strategies. For superlinear payoffs (\(\gamma \ge 1\)), the only pure strategy Nash Equilibrium is non-contribution by all players