A Fragile multi-CPR Game

TL;DR

This paper extends the analysis of fragile common-pool resource games to multiple resources, demonstrating the existence of generalized Nash equilibria and characterizing their measure when players outnumber resources.

Contribution

It introduces a multi-resource extension of the Fragile CPR Game and proves the existence of generalized Nash equilibria under mild assumptions.

Findings

Existence of generalized Nash equilibrium in multi-CPR games.

Set of equilibria has Lebesgue measure zero when players exceed resources.

Extension of previous single-resource results to multiple resources.

Abstract

A Fragile CPR Game is an instance of a resource sharing game where a common-pool resource, which is prone to failure due to overuse, is shared among several players. Each player has a fixed initial endowment and is faced with the task of investing in the common-pool resource without forcing it to fail. The return from the common-pool resource is subject to uncertainty and is perceived by the players in a prospect-theoretic manner. It is shown in [A.~R.~Hota, S.~Garg, S.~Sundaram, \textit{Fragility of the commons under prospect-theoretic risk attitudes}, Games and Economic Behavior \textbf{98} (2016) 135--164.] that, under some mild assumptions, a Fragile CPR Game admits a unique Nash equilibrium. In this article we investigate an extended version of a Fragile CPR Game, in which players are allowed to share multiple common-pool resources that are also prone to failure due to overuse. We refer to this game as a Fragile multi-CPR Game. Our main result states that, under some mild assumptions, a Fragile multi-CPR Game admits a Generalized Nash equilibrium. Moreover, we show that, when there are more players than common-pool resources, the set consisting of all Generalized Nash equilibria of a Fragile multi-CPR Game is of Lebesgue measure zero.