SITUATION: TEAM-PROJECT GAME
Stanford is sort of a big, incredibly smart high school, the high school that we never had. We’ve got the jocks, the nerds, the sorority girls, the frat boys, the indie kids, the preps, the `whatever’ college kids. . . . —TAM VO IN THE STANFORD DAILY
Consider a college class with a diverse array of students, and let’s indulge ourselves with a few stereotypes. Some of the students are underachieving jocks who, as long as it means minimal studying, are content to get a grade of C (fondly known as the “hook”—for looking like one—at my alma mater, the University of Virginia). Then there are the frat boys and sorority girls who are satisfied with a B, but are willing to work hard to avoid a lower grade and the dissatisfaction of their parents. And let us not forget the overachieving nerds who work hard to get an A and find that the best place for their nose is buried in a book. (Is there anyone I have not offended?)
Determining how much effort a student will exert is fairly straightforward when it comes to an individual assignment such as an exam. The nerd will study hard, the frat boy will study moderately, and the jock will study just enough to pass. But what happens when they are thrown together in a team project? The
quality of the project (and thereby the grade) depends on what all of the team members do. How much effort a student should exert may well depend on how hard other team members are expected to work.
To keep things simple, let’s consider two-person team projects and initially examine a team made up of a nerd and a jock. The associated payoff matrix is shown in FIGURE 3.10. Each student has three levels of effort: low, moderate, and high. The grade on the proj- ect is presumed to increase as a function of the effort of both students. Hence, a student’s payoff is always
68 CHAPTER 3: ELIMINATING THE IMPOSSIBLE: SOLVING A GAME WHEN RATIONALITY IS COMMON KNOWLEDGE
FIGURE 3.10 Team-Project Game with a Nerd and a Jock
3,1 4,2
2,2
1,3 2,4
3,3Jock
Nerd
Low
Moderate
High
ModerateLow High
5,3
3,5
4,4
3.3 Solving a Game when Players Are Rational and Players Know That Players Are Rational 69
increasing with the effort of the other student, as an increasing effort by the other student means a better grade without having to work harder.
Jocks strongly dislike academic work, so their payoffs are ordered to reflect a distaste for effort. Regardless of the effort exerted by her nerdy partner (yes, there are female jocks!), the jock’s payoff is lower when she works harder. For example, if the nerd exerts a moderate effort, then the jock’s payoff falls from 4 to 3 to 2 as her effort goes from low to moderate to high. You can confirm that low is the jock’s dominant strategy, since exerting a low effort yields a higher payoff than any other strategy, regardless of the effort chosen by her partner.
What about the nerd? The nerd’s payoff increases with effort. Regardless of the effort of his partner, a nerd prefers to work harder in order to improve the project’s grade. Thus, a high effort is the dominant strategy for the nerd. The outcome of the game in Figure 3.10 is then clear: If students are rational (and sober), then the jock will exert a low effort and the nerd will exert a high ef- fort. The jock gets a payoff of 5—she does great because she’s matched up with someone who is willing to work hard—and the nerd gets a payoff of 3 (while muttering “stupid lazy jock” under his breath).
Next, consider a frat boy and a nerd being matched up. The payoff matrix is presented in FIGURE 3.11. As before, the nerd’s pay- offs increase with effort. The frat boy is a bit more complicated than the nerd and the jock. He wants a reasonably good grade and is willing to work hard to get it if that is what is required, but he isn’t willing to work hard just to go from a B to an A. The frat boy then lacks a dominant strategy. If his partner is lazy, then the frat boy is willing to work hard in order to get that B. If his partner “busts his buns,” then the frat boy is content to do squat, as he’ll still get the B. And if the partner exerts a moderate effort then the frat boy wants to do the same.
Simply knowing that the frat boy is rational doesn’t tell us how he’ll behave. Can we solve this game if we assume more than just that players are rational? Remember that the game is characterized by common knowledge: the frat boy knows that he’s matched up with a nerd. (It’s pretty apparent from the tape around the bridge of his glasses.) Suppose the frat boy not only is rational, but knows that his partner is rational. Since a rational player uses a dominant strat- egy when he has one, the frat boy can infer from his partner’s being rational (and a nerd) that he will exert a high effort. Then, given that his partner exerts a high effort, the frat boy should exert a low effort. Thus, when a nerd and a frat boy are matched, the nerd will hunker down and the frat boy will lounge about. In order to derive this conclusion, we needed to assume that the nerd and the frat boy are rational and that the frat boy knows that the nerd is rational.
Finally, suppose the frat boy is matched up with his female counterpart, the sorority girl. The payoff matrix is given in FIGURE 3.12. Assuming that the players are rational and that each player knows that the other is rational is not enough to solve this game. The trick that solves the game