What will these athletes do?
What will these athletes do? Rationality doesn’t shed any light on what Carl and Maurice will do, as their usage depends on what the other athletes are ex- pected to do. However, Ben has a dominant strategy of taking steroids, as shown in FIGURE 3.18. If neither Carl nor Maurice uses steroids, then Ben’s payoff from steroid use is 6, which exceeds his payoff of 4 from abstaining. If one of the other athletes uses steroids (either Carl or Maurice), then steroid use for Ben means a payoff of 5, versus a payoff of 2 from staying off of them. Finally, if both Carl and Maurice take steroids, then Ben’s payoff from using steroids is 3, ver- sus 1 from abstention. Thus, rationality implies that Ben will use steroids.
3.4 Solving a Game when Rationality Is Common Knowledge 75
Let us assume not only that these athletes are ra- tional, but also that each believes that the other two athletes are rational. This assumption implies that both Carl and Maurice believe that Ben will use steroids, since the rationality of Ben implies steroid use. From the perspective of Carl and Maurice, the game then looks like that shown in FIGURE 3.19, where we’ve eliminated the no steroids strategy for Ben. Carl now has a dominant strategy of taking steroids. Given that he knows that Ben is going to use them (because Carl knows that Ben is rational and that steroids is the dominant strategy for Ben), it follows that Carl should do so as well, because it is the best strategy for him, regardless of whether Maurice uses steroids. Maurice still lacks a dominant strategy.
Thus far, we know that Ben will use steroids—because he is rational—and that Carl will use steroids—because Carl is ra- tional and Carl knows that Ben is rational. So, what will Maurice do? Let us make the assumption that each athlete knows that athletes know that athletes are rational. What this assumption buys us is that Maurice knows that Carl knows that Ben is rational. Hence, Maurice knows that Carl knows that Ben will use steroids, and, therefore, Maurice knows that Carl will use steroids. Thus, Maurice eliminates no steroids for Carl so the situation Maurice faces is as shown in FIGURE 3.20. Given that Maurice then expects both Ben and Carl to resort to taking steroids, Maurice finds it optimal to use steroids as well, since it gives a payoff of 2 as opposed to 1.
We conclude that if (1) all athletes are rational, (2) each athlete believes that the other athletes are rational, and (3) each athlete believes that the other ath- letes believe that the other athletes are rational, then all three of the athletes use steroids. What is depressing about this conclusion is that two of the three athletes don’t even want to take steroids and do so only because others are tak- ing them. Ben’s strong temptation to enhance his performance through chem- icals results in the other two athletes succumbing as well. This is the challenge that sports faces today.
This solution has a ring of truth to it. In Game of Shadows, the authors contend that Bonds turned to taking steroids only after the 1998 season, when Mark McGwire and Sammy Sosa were center stage, battling to break Roger Maris’s single-season home-run record of 61. Both McGwire and Sosa did in fact surpass 61 home runs; McGwire, who has since been accused of being “juiced” with steroids, set the new record of 70 home runs. Three years later, Bonds broke that record with 73 dingers.