Environment and Statecraft: The Strategy of Environmental Treaty-Making

By Scott Barrett | Go to book overview

4 Games with Multiple Equilibria

Trust is not a commodity in abundant supply among diplomats. Henry Kissinger, Diplomacy (1994)


4.1 INTRODUCTION

The students who play the card game discussed in Chapter 1 have a strong incentive to hold on to their red cards. This incentive is derived from the payoffs of the prisoners' dilemma (PD).

Imagine now a different card game, called majority 1 (or M1 for short). The mechanics of this game are the same as for the game described in Chapter 1 : each student is given a red and a black card and is required to hand back one of these without anyone else knowing which card was handed in. What distinguishes this game from the PD are its payoffs. In this new game, each student gets $20 if she hands in her red card. If she keeps her red card (and so hands in her black card) she gets $0 provided a minority of students hand in a red card and she gets $30 if a majority of students hand in a red card (to avoid a tie, assume that there are an odd number of students). Suppose there are 35 students in the class. Then the equilibrium of this game will require that 18 students hand in their red cards, getting $20 each, and that 17 hand in their black cards, getting $30 each. The equilibrium would probably not be reached the first time the game is played, but in contrast to the game described in Chapter 1 , it is likely that behavior will converge towards the equilibrium pretty quickly. This is because in this game there does not exist a tension between a player's individual interests and the group's collective interests; the equilibrium is efficient.

Like the PD, it appears that M1 has a unique equilibrium. However, the equilibrium of this game is unique only in terms of the number of students who keep or hand in their red cards. The equilibrium is not unique as regards the identities of the students who hand in or keep their red cards. If identities matter, there is a much large number of equilibria.

Moreover, though the equilibria of M1 are efficient, there may still exist a tension of sorts between the students who hand in their red cards and get $20 and the students who keep their red cards and get $30. The equilibrium attained in any play of the game may not seem fair. Why should some students get more money than others?

Now consider a different majority game, called M2. Again, the mechanics are the same as before. But now every student gets $10 for keeping the same card that the majority of students keep (and assume again that there are an odd number of

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