PUBLIC PROTESTS IN GDR, SEPTEMBER 1989– FEBRUARY 1990
Date Number of Events Turnout Average Turnout
September 1989 7 16,500 3,300
October 1989 32 1,431,500 49,347
November 1989 28 3,268,900 136,204
December 1989 21 903,082 5,443
January 1990 26 1,782,567 81,026
February 1990 20 464,346 35,719
5.4 Selecting among Nash Equilibria MANY OF THE GAMES examined in this chapter have multiple Nash equilibria. Multiplicity is problematic because the more equilibria there are, the less pre- cise are our statements about behavior. This is a matter that has drawn much effort within the game theory community and will continually draw our
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138 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
*The section is designed to be read without having read what precedes it in Chapter 5. References to games in Chapter 5 have been placed in clearly demarcated digressions.
**The economist Joel Sobel suggested that, in practice, it should be the person who makes the initial call who calls back, because that person is assured of having the other person’s phone number.
attention in this book. In this section, we raise some general issues regarding how to handle multiple equilibria.*
It is important to distinguish various types of multiplicity, since some are innocuous and some are not. Consider first the multiplic- ity of Nash equilibria in the telephone game (Section 4.2), which is reproduced in FIGURE 5.9. This game has two Nash equilibria, one in which Colleen calls (and Winnie waits) and the other with Winnie calling (and Colleen waiting). The outcome of the game is the same for the two equilibria in the sense that one person calls and the other person waits, although the roles played by the players varies. Such a
multiplicity is not of great concern to us (with one caveat, discussed shortly), because the general description of the outcome is the same across equilibria. Other games with a similar type of multiplicity include Chicken (Section 4.1) and driving conventions (Section 4.2).
This same type of multiplicity arose in The Sneetches and the Internship game. In the latter, any strategy profile with 4 students applying to JP Morgan and 6 to Legg Mason is a Nash equilibrium. There are, in fact, 210 Nash equilibria, since there are 210 ways to allocate 10 students so that 4 apply to JP Morgan. Both of these games have a congestion effect, and indeed, multiple asymmetric Nash equilibria are typical in symmetric games with congestion effects. Because the payoff for using a strategy decreases with the number of players that use it, it is often the case that there is an asymmetric equilibrium—and in a symmetric game, if there is one asymmetric equilibrium, then there are more, all with the same mix of strategies (so many players use strategy 1, so many use strategy 2, etc.) and all differing only in terms of which player uses which strategy.
Multiplicity of Nash equilibria in which the distribution of strategies is the same—just the identity of the player using the strategy varies—does raise one source of concern, however: How do players coordinate on a particular Nash equilibrium when there are more than one? How do Colleen and Winnie co- ordinate as to who calls?** How do the drivers in Chicken coordinate on who swerves? If players were to play this game again and again, some common set of beliefs might develop and would result in play converging on a particular
Nash equilibrium. But that approach doesn’t help us if our task is to predict behavior the first time that they play. There is no resolution of this issue at this time.
Now consider the modified driving conventions game shown in FIGURE 5.10 between two drivers who are accus- tomed to driving on the right side of the road. There are two Nash equilibria—both drivers drive on the right and both drivers drive on the left—but the outcomes are different in terms of the payoffs they deliver. Both drivers would prefer to drive on the right, though driving on the left is also an equi- librium. Multiplicity also arose in the American Idol fandom
DIGRESSION
FIGURE 5.9 Telephone
Colleen
Winnie
Call Wait
Call
Wait
0,0
3,2
2,3
1,1
FIGURE 5.10 Driving Conventions with Two American Drivers
Driver 1
Driver 2
Right Left
5,5
1,1
1,1
2,2
Right
Left
5.4 Selecting among Nash Equilibria 139
and voting games (Section 4.4), and, similarly, it made a big difference in the outcome. In these cases, different equilibria have quite different predictions.
This type of multiplicity tends to occur in games with tipping. It can be an equilibrium for all players to choose one particular strategy and an equilibrium for them all to choose some other strategy, because much of what influences the attractiveness of a strategy is its popularity. Such is the case in the operating systems game: Everyone choosing Windows is an equilibrium, as is everyone choosing a Mac; consumers, however, prefer the latter equilibrium.
When equilibria have different implications, game theory offers a few tricks that can, in some such games, result in a selection from the set of Nash equilibria. Consider the game illustrated in FIGURE 5.11. This game has three Nash equilibria: one with both Scooter and Brownie choosing low, a second with Scooter choosing high and Brownie choosing medium, and a third with Scooter choosing medium and Brownie choosing high. Before you are tempted to select (low, low) because it is symmetric—and the game is symmetric—take note that the strategy low is weakly dominated by the strat- egy high. Thus, a cautious player ought to avoid low.
One selection device is to give preference to undominated Nash equilib- ria: Nash equilibria in which players are not using weakly dominated strate- gies. This approach results in the elimination of (low, low), although there are still two Nash equilibria that remain. Returning to the voting game (Section 4.4), we see that it had five Nash equilibria, but all but one of them had one or more voters using a weakly dominated strategy. In that game, there is a unique undominated Nash equilibrium, and it has shareholders 1 and 2 vot- ing for option B and shareholder 3 voting for C. As practice, I’d recommend going back to the voting game and verifying that claim.
Another selection criterion commonly used is based upon payoff dominance and what economists call the Pareto criterion. A strategy profile satisfies pay- off dominance if there is no other strategy profile for which each player has a strictly higher payoff. Although Nash equilibrium is based upon individual rationality—in that each player is doing what is best for him, given what oth- ers are doing—payoff dominance is based upon collective rationality—in that it involves choosing a strategy profile that is best for everyone. The selection criterion is that one use payoff dominance in conjunction with Nash equilib- rium. To only use the former could mean choosing a strategy profile that vio- lates individual rationality. The proposed criterion is called equilibrium pay- off dominance, which means focusing on Nash equilibria for which there is no other Nash equilibrium in which each player has a strictly higher payoff. An equilibrium satisfying this property is called a payoff-dominant Nash equilibrium.
Equilibrium payoff dominance tells us to focus on the equilibrium (right, right) for the game in Figure 5.10, because it results in a higher payoff for both drivers relative to the equilibrium (left, left). By this criterion, the unique so- lution in the American Idol fandom game is that all three girls wear the let- tered T-shirts.
DIGRESSION
FIGURE 5.11
2,2 1,1
1,1
2,0 4,1
3,3Scooter
Brownie
Low
Medium
High
Low Medium High
0,2
0,0
1,4
140 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
The equilibrium payoff dominance criterion has a lot of power in the airline security game. That game has seven Nash equilibria, which can be ranked in terms of payoffs. The equilibrium with all airlines choosing the most intense security measures has the highest payoff for each player among all of the Nash equilibria. This criterion can also be used to yield a unique solution in the operating systems game.
One rationale for the use of equilibrium payoff dominance is that if players were to meet beforehand and communicate, it would be in their mutual in- terest to coordinate on an equilibrium with that property. They know that agreeing on anything but Nash equilibrium is not credible; that is, one or more players would diverge from the agreement. Players should then be real- istic and consider only those agreements that are self-enforcing, which means that it is in each player’s interest to comply and thus doesn’t require institu- tions such as the courts and the police to enforce it. Among all the self- enforcing agreements (i.e., the Nash equilibria), all players can agree that they should focus on the payoff-dominant ones.
The game in FIGURE 5.12 has two Nash equilibria: (top, left) and (bottom, right). (top, left) is the payoff- dominant Nash equilibrium. However, bottom weakly dominates top for player 1, and right weakly dominates left for player 2, so (bottom, right) is the undominated Nash equilibrium. Which equilibrium do you think is more compelling?
In his 1961 landmark book, Strategy of Conflict, Nobel Laureate Thomas Schelling introduced the concept of a focal point. Like pornography and jazz, a focal point is difficult to formally de- fine. As Justice Potter Stewart said in a judicial decision, “I can’t
define it for you, but I know pornography when I see it,” and as the legendary trumpeter Louis Armstrong proclaimed, “If you have to ask what jazz is, you’ll never know.” In spite of a focal point not being easily definable, it is a power- fully compelling concept.
To understand what is meant by a focal point and how it can help with the selection of a Nash equilibrium, one must first recognize that games do not occur in a vacuum; they occur with rich societal context. The players are peo- ple with a history, and some of this history is common knowledge to them. Although strategies may be assigned labels such as “top” and “bottom,” in ac- tual contexts they have concrete descriptions, like “Meet at the train station” and “Meet at the ballpark.” In terms of who the players are and what strate- gies they might adopt, there may be one prominent Nash equilibrium, which thereby becomes an obvious selection, or focal point. But for players to set- tle upon it, it’s not sufficient that each player perceive it as conspicuous: the conspicuousness itself must be common knowledge.
To see how this idea might work, let’s return to our driving conventions game, but now suppose the issue is on which side of the sidewalk to walk. In London, since the law is to drive on the left, the focal point would be to walk on the left. Indeed, that is what seems to be the case, and I have had many awkward moments on London pavements to prove it!
DIGRESSION
FIGURE 5.12 Payoff Dominance vs. Weak Dominance
Player 1
Player 2
Left Right
Bottom 3,1 2,2
1,33,3Top
Find the payoff-dominant Nash equilibria in the game of Civil Unrest.
5.4 CHECK YOUR UNDERSTANDING
Exercises 141
Summary This chapter has tackled games in which all players face the same choice, whether it be a consumer choosing the type of computer system to buy, a citi- zen deciding whether to protest, a sneetch contemplating having a star tattoo, or a company deciding whether to enter a market. Much of the analysis was done without specifying exactly how many players there are. Some strategy pro- files (such as buying a Mac) are equilibria regardless of how many players there are, while others (such as buying Windows) depend on the number of players.
If, in addition to all players having the same choice, it is assumed that they all have the same payoffs, then the game is symmetric. A symmetric game need not imply that equilibrium play has everyone doing the same thing. Also, the presence of a symmetric equilibrium can depend on whether payoff func- tions have the property of tipping or congestion. In tipping, the relative at- tractiveness of using a strategy is greater when more players use it, while in congestion, a strategy is less attractive when more players use it. Games with tipping tend to result in equilibria in which everyone (or almost everyone) does the same thing. For example, if enough people buy a Mac, then it’ll be more attractive to also be a Mac user, there will be lots of software written for the Mac, and you can easily exchange files with other users. An equilibrium then has everyone being Mac people. Similarly, if enough people buy Windows, then that’ll be the preferred operating system. By contrast, when congestion is present, identical people can make different choices. For in- stance, students applying for an internship don’t all want to apply to the same investment banking firm, because the competition will then be fierce. Equilibrium entails students applying to different companies. Though faced with the same situation, people may act differently.
Multiplicity of Nash equilibria is a common property of the games examined in this chapter. Indeed, each had more than one Nash equilibrium. More gen- erally, multiplicity is routinely encountered in game theory. We explored a few criteria for selecting among multiple equilibria with the goal of identifying a unique solution. Among the criteria that have been proposed are to give pref- erence to undominated Nash equilibria—thereby ruling out equilibria in which players use weakly dominated strategies—and to give preference to payoff-dominant Nash equilibria—thereby dropping equilibria for which there is another equilibrium that delivers a higher payoff to every player. In some games a unique solution emerges, but in many games that is not the case.
Identifying a criterion that always delivers a unique solution is the “holy grail” of game theory. Some game theorists find this task unrealistic and sug- gest that pursuing it turns one into a Don Quixote, while others see it as a chal- lenge of the first order and worthy of King Arthur. Regardless of your view, all game theorists would prize such a criterion because yielding precise predic- tions and explanations is ultimately what game theory is all about.
1. The magazine Science 84 planned to announce a contest in which any- one could submit a request for either $20 or $100. If fewer than 20% of the submissions requested $100, then everybody would receive what
EXERCISES
142 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
they requested. If 20% or more asked for $100, then everybody would get nothing. Although the magazine wound up not running the contest, be- cause Lloyds of London was unwilling to insure against losses, we can still analyze what equilibrium would predict. Suppose 100,000 people might participate in the contest, and assume that payoffs are measured in money. a. Assume that every possible participant submits a request for either
$20 or $100. Find a Nash equilibrium. b. Now suppose that a request form comes only with the purchase of
Science 84 and that the magazine costs $21.95. Then each person’s strategy set has three elements: do not buy the magazine, buy the mag- azine and submit a request for $20, and buy the magazine and submit a request for $100. Suppose zero value is attached to the magazine. Find a Nash equilibrium.
c. Consider again the situation described in part (b), but now suppose the magazine costs $19.95. Find a Nash equilibrium.
2. Someone at a party pulls out a $100 bill and announces that he is going to auction it off. There are other people at the party who are po- tential bidders. The owner of the $100 bill puts forth the following pro- cedure: All bidders simultaneously submit a written bid. Everyone (not just the highest bidder) pays his bid, and the bidder with the highest bid gets the $100 bill (assuming that the highest bid is positive). If m people submit the highest bid, then each receives a share of the $100. Each person’s strategy set is so bidding can go as high as $1,000. If denotes the bid of player j and is the max- imum (or highest) of the n bids, then the payoff to player i is
where m is the number of bidders whose bid equals Find all Nash equilibria.
3. It is the morning commute in Congestington, DC. There are 100 drivers, and each driver is deciding whether to take the toll road or take the back roads. The toll for the toll road is $10, while the back roads are free. In deciding on a route, each driver cares only about income, denoted y, and his travel time, denoted t. If a driver’s final income is y and his travel time is t, then his payoff is assumed to be (where we have made the dollar value of one unit of travel time equal to 1). A driver’s income at the start of the day is $1,000. If m drivers are on the toll road, the travel time for a driver on the toll road is assumed to be m (in dollars). In con- trast, if m drivers take the back roads, the travel time for those on the back roads is 2m (again, in dollars). Drivers make simultaneous deci- sions as to whether to take the toll road or the back roads. a. Derive each player’s payoff function (i.e., the expression that gives us
a player’s payoff as a function of her strategy profile.) b. Find a Nash equilibrium.
4. Return to The Sneetches in Section 5.2. Of the n sneetches, let k denote the number of sneetches born with stars and assume Now suppose Sylvester McMonkey McBean charges a price of p to either get a star removed (if the sneetch was born
k 6 n2.
y � t
max5b1, . . . , bn6. �bi if bi 6 max5b1, . . . , bn6 100 m � bi if bi � max5b1, . . . , bn6
max5b1, . . . , bn6bj 50, 1, 2, . . . , 1,0006, 1 m
n � 2
Exercises 143
with one) or added (if the sneetch was born without one). Assume that A sneetch’s payoff is as described in Table 5.1, except that
you have to subtract p if he used either the Star-On or Star-Off machine. Find all Nash equilibria.
5. Suppose several friends go out to dinner with the understanding that the bill will be divided equally. The problem is that someone might order something expensive, knowing that part of the cost will be paid by oth- ers. To analyze such a situation, suppose there are n diners and, for sim- plicity, they have the same food preferences. The accompanying table states the price of each of three dishes on the menu and how much each person values it. Value is measured by the maximum amount the person would be willing to pay for the meal.
0 6 p 6 1.
DINING DILEMMA
Dish Value Price Surplus
Pasta Primavera $21.00 $14.00 $7.00
Salmon $26.00 $21.00 $5.00
Filet Mignon $29.00 $30.00 �$1.00
Surplus is just the value assigned to the meal, less the meal’s price. The pasta dish costs $14 and each diner assigns it a value of $21. Thus, if a diner had to pay for the entire meal, then each diner would buy the pasta dish, since the surplus of $7 exceeds the surplus from either salmon or steak. In fact, a diner would prefer to skip dinner then to pay the $30 for the steak, as reflected by a negative surplus. A player’s payoff equals the value of the meal she eats, less the amount she has to pay. The latter is assumed to equal the total bill divided by the number of diners. For ex- ample, if there are three diners and each orders a different meal, then the payoff to the one ordering the pasta dish is
the payoff for the person ordering the salmon is
and the payoff to whoever is ordering the steak is
Not surprisingly, the people who order the more expensive meal do bet- ter, since all pay the same amount. a. Suppose there are two diners What will they order (at a Nash
equilibrium)? b. Suppose there are four diners What will they order (at a
Nash equilibrium)? (n � 4).
(n � 2).
29 � a14 � 21 � 30 3
b � 7.33.
26 � a14 � 21 � 30 3
b � 4.33,
21 � a14 � 21 � 30 3
b � 21 � 21.67 � �0.67,
144 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
6. Consider again the entry game from Section 5.3, but now suppose the five potential entrants are identical in that each faces the same entry cost of $300. Given the total number of companies in the market, the accom- panying table reports a company’s net profit (or payoff) if it enters. As be- fore, the payoff from staying out of the market is zero, and each company can choose either enter or do not enter. Find all Nash equilibria.