SITUATION: CIVIL UNREST
How is it that a small group of people can oppress a large population? Having the support of the military is certainly important, as is having con- trol of the media and economic resources. But what is arguably most im- portant is coordination failure. In a nondemocratic society, the controlling faction is always a small fraction of a country’s population. If the populace were to rise up and oppose the ruling body, few dictatorships could survive. The challenge is having people coordinate so that mass demonstrations happen.
Suppose a country is composed of 500 citizens, each of whom is deciding whether to protest. The benefit to protesting is where m is the num- ber of citizens who participate in the protest. This specification captures the reasonable notion that a bigger protest will be more influential. The benefit of protesting is shared by all those who protest and is the feeling of empower- ment that it yields.
The cost of protesting is a personal one—for example, the risk of being im- prisoned. Here, we’ll suppose that people are different as to how they weigh this cost. To keep things simple, suppose there are three types of individuals in this society (see TABLE 5.7.) There are 100 radicals in the country, and they have the lowest cost of protesting, namely, 6,000. The progressives, who also number 100, each incur a cost of 8,000 by protesting. Finally, there are 300 bourgeois who, while still desiring revolt, are not too bad off and have the
50 � m,
Eliminate company 1 from the Entry game so that only companies 2, 3, 4, and 5 simultaneously decide whether to enter. The payoffs are still as stated in Table 5.6. Find all Nash equilibria.
5.3 CHECK YOUR UNDERSTANDING
5.3 Asymmetric Games 135
most to lose from a failed revolution. To them, the cost of participating in demonstrations is 20,000.
The payoff to a citizen equals zero if he doesn’t protest and is the benefit, less his cost if he does protest. This means that a radical will protest
if and only if
Solving this expression for m, we find that Thus, if a radical believes that another 120 people will protest, then she’ll protest as well and receive a payoff of 50 (If 119 other people are expected to protest, then her payoff from protesting is zero and she’s indifferent between protest- ing or not.) Analogously, a progressive will protest when she expects m to be large enough to satisfy the inequality
or equivalently,
Hence, due to the higher cost incurred, a progressive requires a bigger antici- pated demonstration to draw him out than does a radical. Finally, it takes an anticipated protest size of 400 to induce the bourgeois to protest:
or equivalently,
The minimum protest size necessary to induce a person to attend demonstra- tions will be referred to as her critical mass. The relevant information is shown in Table 5.7.
In solving for Nash equilibria, a useful property to note is that if it is op- timal for a progressive to protest, then it is also optimal for a radical to do so. In other words, you won’t see progressives at a demonstration without also seeing radicals. To establish this claim, recall that a progressive will find it optimal to protest when he expects 160 people to protest, while it takes only 120 people to induce a radical to protest. Thus, if there are enough people to bring out the progressives, there also are enough to draw out the radicals. Intuitively, a progressive and a radical realize the same benefit, but since the latter has a lower cost, a radical will always protest when a progressive does. Using the same type of argument, we can see that if a bourgeois finds it optimal to protest, then so do radicals and progressives.
From the preceding scenario, the candidates for Nash equilibrium are that (1) no one protests, (2) only radicals protest, (3) only radicals and progressives
m � 400.50 � m � 20,000 � 0
m � 160.50 � m � 8,000 � 0,
(� 50 � 121 � 6,000).
m � 120.
50 � m � 6,000 � 0.
50 � m,
TABLE 5.7 CIVIL UNREST
Type of Citizen Number of Citizens Personal Cost Critical Mass
Radicals 100 6,000 120
Progressives 100 8,000 160
Bourgeois 300 20,000 400
136 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
protest, and (4) everyone protests. The case of no one protesting is clearly an equilibrium: if a radical expects no one to participate, then his payoff from protesting is which is worse than the zero payoff from staying home. Protesting is even less attractive to a progressive and a bourgeois.