SITUATION: APPLYING FOR AN INTERNSHIP
It’s the middle of the spring semester, and you’re thinking about applying for a summer internship. You’ve narrowed your prospects down to two investment banking firms: JP Morgan (JPM) in New York and Legg Mason (LM) in Baltimore. You, as well as everyone else, prefer the JPM internship, but you are hesitant to apply for two reasons. First, there is only one summer intern posi- tion at JPM for someone from your school, while there are three at LM. Second, you know that everyone else finds the JPM internship more attractive, and this is likely to make it harder to land a position. Suppose, due to time constraints, you can apply to only one of the internships. Which one should you choose?
Suppose 10 students from your school are interested in an investment banking internship at either JPM or LM. Everyone has the same preferences, and each assigns a value of 200 to a JPM internship and a value of 100 to an LM internship. A student’s payoff from applying for a JPM internship is 200 only if she is assured of getting it, which is the case only if she is the lone per- son to apply. The payoff is lower than 200 when more than one apply and gen- erally decreases the more that apply. Analogously, a student’s payoff from ap- plying for an LM internship is 100 only if she is assured of getting it, which is the case only if no more than three people apply. (Recall that there are three openings at LM.) When there are more than three applicants, the payoff from applying decreases with an increasing number of applicants.
The payoffs for the internship game are listed in TABLE 5.3 and plotted in FIGURE 5.6. The more students who apply to JPM, the lower is the payoff to each
TABLE 5.3 PAYOFFS FOR THE INTERNSHIP GAME
Number of Applicants Payoff to a JPM Payoff to an LM to JPM Applicant Applicant
0 — 30
1 200 35
2 100 40
3 65 45
4 50 50
5 40 60
6 35 75
7 30 100
8 25 100
9 20 100
10 15 —
5.2 Symmetric Games 129
of those applicants. Since more applicants to JPM means fewer applicants to LM, the payoff for applying to LM increases with the number of students com- peting for a position at JPM. Contrast this game with the Operating Systems game. As shown in Figure 5.5, the more consumers who choose Windows, the higher is the payoff to each of them. Thus, although tipping is at work in the Operating Systems game, congestion characterizes the Internship game.
To derive a Nash equilibrium, let’s first suppose that no one applies to JPM. Then the payoff to each of those 10 LM applicants is 30, which is considerably less than the payoff for being the lone JPM applicant, which is 200. Hence, all students applying to LM is not an equilibrium. Next, consider a strategy pro- file in which 1 student applies to JPM and the other 9 apply to LM. Again, any of those LM applicants would do better by applying to JPM: Applying to JPM raises the payoff from 35 (the payoff to an LM applicant when there is only one JPM application) to 100 (the payoff to a JPM applicant when there are two JPM applicants).
More generally, in considering a strategy profile in which m students apply to JPM, a student who is intending to apply to LM is comparing the payoff from being one of applicants to LM and one of applicants to JPM. As depicted in Figure 5.6, when the payoff for applying to JPM is higher, in which case it is not optimal for this applicant to apply to LM. As long as the number of applicants to JPM is less than 4, we do not have an equilibrium.
Now let’s start with the other extreme: suppose all 10 students apply to JPM. Then each has a payoff of 15, which falls well short of the payoff of 100 from applying to LM. Indeed, as long as more than 4 students apply to JPM, an applicant to JPM would do better by applying to LM. If the strategy profile has m students applying to JPM, then, when the payoff for being one of m applicants to JPM is less than the payoff for being one of ap- plicants to LM, in which case a JPM applicant ought to switch his application.
We’ve shown that any strategy profile in which fewer than 4 or more than 4 students apply to JPM is not a Nash equilibrium. This leaves one remaining
10 � m � 1 m 7 4,
m 6 4, m � 110 � m
FIGURE 5.6 Payoffs to Applying to JP Morgan and to Legg Mason
Pa yo
ff
200
180
160
140
120
100
80
60
40
20
0 1 2 3 4 5 6 7 8 9 10
Payoff to JPM applicant
Payoff to LM applicant
Number of applicants to JPM
130 CHAPTER 5: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE n-PLAYER GAMES
possibility: exactly 4 students apply to JPM and 6 apply to LM. In that case, the payoff to both a JPM applicant and an LM applicant is 50. If one of the students who is intending to apply to JPM switches to LM, then her payoff de- clines to 45 (see Figure 5.6), and if one of those students who is intending to apply to LM switches to JPM, then his payoff declines to 40. Thus, because any student is made worse off by changing her strategy, exactly 4 students apply- ing to JPM is a Nash equilibrium.
Hence, in this scenario, although all students have the same options and the same preferences, they make different choices in the equilibrium situation. Four students apply to JP Morgan—and compete for the one available posi- tion—and 6 apply to Legg Mason—and compete for the three available slots there. Asymmetric behavior emerges from a symmetric game because of con- gestion effects. The more students who apply for a position, the tougher it is to land a position (as reflected in a lower payoff), and thus the less attractive it becomes to apply.
5.3 Asymmetric Games A GAME CAN BE ASYMMETRIC because players have distinct roles and thus differ- ent strategy sets. In a kidnapping scenario (such as that explored in Chapter 2), the kidnapper’s choices are whether to release or kill the victim, while the vic- tim’s kin has to decide whether or not to pay ransom. Even when all players face the same set of alternatives, a game can be asymmetric because the play- ers have different payoffs. For example, in choosing between operating sys- tems, all consumers face the same choices, but they may differ in how they evaluate them. Some may attach a lot of value to a Mac, while others put a lot of weight on having the most popular system, whatever that might be. That is the source of asymmetry we explore in this section: players have different pay- offs, while facing the same choices and the same information.