How often have I said to you that when you have eliminated the impossible, whatever remains, however improbable, must be the truth?
—SHERLOCK HOLMES IN A STUDY IN SCARLET, BY ARTHUR CONAN DOYLE
3.1 Introduction IN CHAPTER 2, WE LEARNED how to construct a game, in both the extensive and strategic forms. But having built a game, what do we do with it? I say, Let’s play amateur sleuth and investigate people’s behavior in strategic scenarios. To do so, we need to know how to solve a game, which is what this chapter is all about.
FIGURE 3.1 presents the strategic form of the kidnapping situation whose ex- tensive form was illustrated in Figure 2.1. Recall that Guy, the kidnapper, has four strategies. He can either not kidnap Orlando, but in the event that he does, then kill him (Do not kidnap/Kill); not kidnap Orlando, but in the event that he does, then release him (Do not kidnap/Release); kidnap Orlando and kill him (Kidnap/Kill); or kidnap Orlando and release him (Kidnap/Release). Vivica, who is Orlando’s kin, can either pay the ransom or not. The first num- ber in a cell is the row player’s payoff (Guy’s payoff), and the second number is the column player’s payoff (Vivica’s payoff). Recall that payoff numbers are rank ordered from the least to the most preferred outcome.
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Eliminating the Impossible: Solving a Game when Rationality Is Common Knowledge
FIGURE 3.1 Strategic Form of the Kidnapping Game
3,5 3,5
3,5 3,5Guy (kidnapper)
Vivica (kin of victim)
Pay ransom Do not pay ransom
4,1 2,2
5,3 1,4
Do not kidnap/Kill
Do not kidnap/Release
Kidnap/Kill
Kidnap/Release