If Bonds did take steroids, was it a reaction to remaining competitive with the other top home-run hitters in baseball?
Suppose each of the athletes is rational and each believes that the others believe that he is rational. However, each athlete is unsure whether the other athletes are rational. For example, Carl is rational and believes that Maurice and Ben believe he is rational, but Carl isn’t sure whether Maurice and Ben are rational. Do these beliefs make sense? What can you say about what each athlete will do?
FIGURE 3.19 Doping Game when Ben Will Use Steroids. Maurice Can Now Deduce That Carl’s Dominant Strategy Is to Use Steroids
Maurice
Carl
Steroids
No steroids
No steroidsSteroids
2, 3 ,3
1, 4 ,5
3, 1 ,5
5, 2 ,6
FIGURE 3.20 Doping Game when Both Ben and Carl Choose to Use Steroids
2,3,3
1,4,5 Maurice
Steroids
No steroids
Steroids
3.4.2 Iterative Deletion of Strictly Dominated Strategies Up to now, we have progressively used additional levels of knowledge about ra- tionality in order to solve games. The Tosca game was solved with only the as- sumption that players are rational. The Existence-of-God game required not only that the players were rational, but also that each player believed that all players were rational. (Specifically, we needed God and Man to be rational and Man to believe that God is rational). And with the Doping game, the athletes’ decisions regarding steroid use could be derived only when all the players were assumed to be rational, each player believed that all players were rational, and each player believed that all players believed that all players were rational. These are all examples of a more general procedure for solving a game—a pro- cedure known as the iterative deletion of strictly dominated strategies (IDSDS).