A Note on Payoff Equivalence of the Volunteer's Dilemma and the Stag Hunt Game and Inferiority of Intermediate Thresholds

Abstract

The launch of a public project requires "enough" support from a group of 'n' players, i.e., a certain threshold has to be passed. The players may be differently important for passing the threshold; they may have different costs of support and different benefits if the project is launched. If players have only binary decision sets (participate or not, vote approvingly or not) this game is called a Binary Threshold Public Goods game (BTPG). We compare the expected equilibrium payoffs in BTPGs with the same costs and benefits but different thresholds. Applying two principles of equilibrium selection, the least and the most demanding threshold, namely "one supporting player is sufficient" (Volunteer's Dilemma) and "support by all players is necessary" (Stag Hunt game) are payoff equivalent for all players. Compared with the Stag Hunt game, all intermediate thresholds are connected with Pareto-inferior payoffs.