Peter Vanderschraaf

I resolve a previously unnoticed anomaly in the analysis of collective action problems. Some political theorists apply game theory to analyze the paradox of anarchy: War is apparently inevitable in anarchy even though all warring parties prefer peace over war. Others apply tipping threshold analysis to resolve the paradox of revolution: Joining a revolution is apparently always irrational even when an overwhelming majority of the population wish to replace their regime. The usual game theoretic analysis of anarchy yields the conclusion that the suboptimal equilibrium of war is inevitable. The usual tipping threshold analysis of revolution yields the conclusion that the optimal equilibrium of successful revolution is possible. Yet structurally the collective action problems of anarchy and potential revolution are much the same. This suggests that tipping threshold analysis and game theory are incompatible methodologies, despite their widespread use in the social sciences. I argue that there is no real tension between game theory and tipping threshold analysis, even though these methodologies have developed largely independently of each other. I propose a Variable Belief Threshold model of collective action that combines elements of game theory and tipping threshold analysis. I show by example that one can use this kind of hybrid model to give compatible explanations of conflict in anarchy and successful revolution.1. Introduction2. Two Classic Problems, and Two Popular Analyses2.1. The paradox of anarchy2.2. The paradox of revolution3. Restating the Puzzle4. Evaluating the Prisoners’ Dilemma and S-Curve Models5. The Variable Belief Threshold ModelExample 5.1. A population of moderates with independent deviationsExample 5.2 A heterogeneous population with independent deviationsExample 5.3 A heterogeneous population with coordinated deviations6. ConclusionAppendix: Computer Simulations.

We introduce a dynamic model for evolutionary games played on a network where strategy changes are correlated according to degree of influence between players. Unlike the notion of stochastic stability, which assumes mutations are stochastically independent and identically distributed, our framework allows for the possibility that agents correlate their strategies with the strategies of those they trust, or those who have influence over them. We show that the dynamical properties of evolutionary games, where such influence neighborhoods appear, differ dramatically from those where all mutations are stochastically independent, and establish some elementary convergence results relevant for the evolution of social institutions.