Fragility of the Commons under Prospect-Theoretic Risk Attitudes

We study a common-pool resource game where the resource experiences failure with a probability that grows with the aggregate investment in the resource. To capture decision making under such uncertainty, we model each player's risk preference according to the value function from prospect theory. We show the existence and uniqueness of a pure strategy Nash equilibrium when the players have arbitrary (potentially heterogeneous) risk preferences and under natural assumptions on the rate of return and failure probability of the resource. Greater competition, vis-a-vis the number of players, increases the failure probability at the Nash equilibrium, and we quantify this effect by obtaining (tight) upper bounds on the failure probability at the equilibrium for a large number of players with respect to the failure probability under investment by a single player. We further examine the effects of heterogeneity in risk preferences of the players with respect to two characteristics of the prospect-theoretic value function: loss aversion and diminishing sensitivity. Heterogeneity in attitudes towards loss aversion always leads to higher failure probability of the resource at the equilibrium when compared to the case where players have identical risk preferences, whereas there is no clear trend under heterogeneity in the diminishing sensitivity parameter.