ENTRY GAME WITH IDENTICAL COMPANIES
Number of Firms Gross Profit per Firm Net Profit per Firm
1 1,000 700
2 400 100
3 250 �50
4 150 �150
5 100 �200
7. There is a rough neighborhood with residents. Each resident has to decide whether to engage in the crime of theft. If an individual chooses to be a thief and is not caught by the police, he receives a pay- off of W. If he is caught by the police, his payoff is Z. If he chooses not to commit theft, he receives a zero payoff. Assume that All n residents simultaneously decide whether or not to commit theft. The probability of a thief being caught equals , where m is the number of residents who choose to engage in theft. Thus, the probability of being caught is lower when more crimes are committed and the police have more crimes to investigate. The payoff from being a thief, given that
other people have also chosen to be thieves, is then
Find all Nash equilibria.
8. Consider the game in FIGURE PR5.8. a. Find all Nash equilibria. b. Provide an argument for selecting among those equilibria.
9. Consider the game in FIGURE PR5.9. a. Find all Nash equilibria. b. Provide an argument for selecting among those equilibria.
am � 1 m
b W � a 1 m b Z.
m � 1
1 m
W 7 0 7 Z.
n � 2
2,2 1,2
0,0
2,1
3,1
2,2Player 1
Player 2
0,0
0,0
1,3
x y z
a
b
c
FIGURE PR5.8 FIGURE PR5.9
1,0 1,2
1,2
0,0
1,1
2,1Player 1
Player 2
0,1
0,1
3,3
x y z
a
b
c
References 145
10. Consider a country with n citizens, and let be the value that citizen i at- taches to protesting. Enumerate the citizens so that citizen 1 attaches more value to protesting than citizen 2, who attaches more value than cit- izen 3, and so forth: where citizen n attaches no value to protesting. Assume that the cost of protesting is the same for all citizens and is where and m is the number of protestors. Then the payoff to citizen i from protesting is while the payoff from not protesting is zero. Assume that Find all Nash equilibria.
11. n pre-med students are planning to take the MCAT. Each student must decide whether to take a preparatory course prior to taking the test. Let
denote the choice of student i, where indicates that she will not take the course and indicates that she will take the course. A stu- dent cares about her ranking in terms of her MCAT score and whether or not she took the prep course. Let denote student i‘s MCAT score and
denote the ranking of student i among the n students who took the test. Specifically, equals 1 plus the number of students who scored strictly higher than student i. To clarify this specification, here are three examples: If for all then (In other words, if nobody’s score is higher than that of student i, then her rank is 1.) If for all
then (In other words, if student i has the lowest score, then her rank is n.) Finally, if then
Now, assume that student i’s payoff equals where Note that taking the prep course entails a cost to
a student equal to c. Note also that a student adds to her payoff by an amount b if her rank increases by 1. Student i’s score is assumed to be determined from the formula where and is related to the innate ability of the student and is what she would score if she did not take the prep course. If she takes the prep course, she adds to her score by an amount z. Assume that
This means that student 1 is, in a sense, smarter than student 2, student 2 is smarter than student 3, . . . , student is smarter than student and students and n are equally smart. The final assumption is
In this game, there are n students simultaneously deciding whether or not to take the MCAT preparatory course. Derive a Nash equilibrium.
ai�1 � z 7 ai for all i � 1, 2, . . . , n � 1.
n � 1 n � 1,n � 2
a1 7 a2 7 p 7 an�1 � an.
aiz 7 0.ai 7 0si � ai � xiz,
b 7 c 7 0.xic, b(n � ri) �r3 � r4 � 3, r5 � 5.
r1 � 1, r2 � 2,s1 7 s2 7 s3 � s4 7 s5, ri � n.j i,
si 6 sj ri � 1.j i,si � sj
ri ri
si
xi � 1 xi � 0xi
v1 � c 6 0. vi � (
c m),
c 7 0cm
v1 7 v2 7 p 7 vn(� 0),
vi
REFERENCES 1. The phenomenon of tipping is investigated more broadly in Malcolm
Gladwell, The Tipping Point: How Little Things Can Make a Big Difference (Boston: Little, Brown, 2000).
2. This quote and the ensuing quotes in this example are from Dr. Seuss, “The Sneetches,” in The Sneetches and Other Stories (New York: Random House, 1961).
3. NewsHour with Jim Lehrer, December 24, 2001. <www.pbs.org/newshour>
4. Howard Kunreuther and Geoffrey Heal, “Interdependent Security,” Journal of Risk and Uncertainty, 26 (2003), 231–49.
5. The results that follow are from John B. Van Huyck, Raymond C. Battalio, and Richard O. Beil, “Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure,” American Economic Review, 80 (1990), 234–48.
6. Remarks at Microsoft MIX06 Conference, Las Vegas, Nevada, Mar. 20, 2006. <http://www.microsoft.com/billgates/speeches/2006/03-20MIX.asp>
7. This insightful observation was made in Michael Suk-Young Chwe, Rational Ritual: Culture, Coordination, and Common Knowledge (Princeton, NJ: Princeton University Press, 2001).
8. Figure 5.7 is from G. Day, A. Fein, and G. Ruppersberger, “Shakeouts in Digital Markets: Lessons from B2B Exchanges,” California Management Review, 45 (2003), 131–50.
9. Figure 5.8 is from Homer B. Vanderblue, “Pricing Policies in the Automobile Industry,” Harvard Business Review, 39 (1939), 385–401.
10. This discussion is based on Susanne Lohmann, “The Dynamics of Informational Cascades: The Monday Demonstrations in Leipzig, East Germany, 1989–91,” World Politics, 47 (1994), 42–101.
11. Lohmann (1994), pp. 70–71.
146 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
6.1 Introduction IN THE GAMES EXAMINED thus far in this book, the number of strategies has been severely limited. In some contexts, this is quite natural: A driver decides whether to drive on the right or left (the driving conventions discussed in Chapter 2) or whether to swerve or hang tough (the game of Chicken in Chapter 2); a citizen either joins a protest or not (the game of Civil Unrest in Chapter 5); and a com- pany either enters a market or not (Chapter 5). In other contexts, a large array of options is more natural. For example, when we explored bidding at an auc- tion (Chapter 2), a bidder was restricted to choosing from among five possible bids, but in reality, there are many feasible bids. Similarly, in the Team Project game (Chapter 3), there were three levels of effort that a student could exert— low, medium, and high—but if we measure effort by the amount of time spent on a project, there are, in fact, many possible levels. In particular, when a per- son is choosing an amount of money or an amount of time, she’ll typically have many options available.
In this chapter, we explore games that allow for many strategies. What do we mean by “many”? The numerical system of the Bushmen of the Botswana contains the numbers one through six, with numbers in excess of six lumped into the category of “many.”1 That notion of “many” will hardly suffice for our purposes. Our interpretation of “many” will far exceed seven; in fact, it will be infinity! Of course, an infinite number of strategies can be rather complicated to keep track of, so we’ll assume that strategies have the property that they can be ordered from lowest to highest; more specifically, a strategy set is a set of numbers. A number may represent a price, how many hours to work, or even something less obviously quantifiable, such as the quality of a company’s product. As long as we can rank the various options—for example, from low to high quality—we can then assign numbers to them, where, say, a higher number means higher quality.
One example of an infinite set of numbers is the set of natural numbers: 1, 2, 3, . . . . However, the notion of “many” that we want to assume has the ad- ditional property that between any two strategies is a third strategy. This prop- erty is not a property of the natural numbers; for example, there is no natural number between 73 and 74. A set of numbers that does have this property is the real numbers—for instance, all of the numbers (not just the integers) be- tween 2 and 10. Denoted as [2,10], this interval includes such numbers as 5.374, 8/3 and (i.e., 3.141592653589793238462643383279502884197 . . .).
The real numbers comprise both the rational and the irrational numbers. A rational number is any number that can be represented by a fraction of two
p
6
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