Will there be a kidnapping?
If so, will ransom be paid? Will Orlando sur- vive? Solving this game means answering these questions by se
lecting among the eight possible pairs of strategies. We need to weed out unreasonable and implausible strategy profiles and, ideally, identify a unique compelling one. The fewer solutions there are, the more precise is our prediction about be- havior. To derive a solution, we’ll need to assume something about how a
player selects among his or her various strategies. Of course, what makes this task challenging (and results in a long textbook!) is the fact that how a player selects a strategy may well depend on how she thinks other players are select- ing. This is a complex undertaking not quickly dispensed with, and it is best that we start at the beginning and start simple.
The plan is to progressively make more assumptions about players and ex- plore what we can say about how they’ll behave. We begin with assuming that players are rational (Section 3.2), then further assume that each player be- lieves that all players are rational (Section 3.3), and then assume on top of that that each player believes that all players believe that all players are rational (Section 3.3). We conclude the chapter by generalizing this sequence of solu- tion techniques in Section 3.4.
3.2 Solving a Game when Players Are Rational IN MODELING A PLAYER’S SELECTION of a strategy, we’ll begin by assuming that play- ers are rational. A player is rational when she acts in her own best interests. More specifically, given a player’s beliefs as to how other players will behave, the player selects a strategy in order to maximize her payoff. Note that ration- ality has nothing to say about what are reasonable beliefs to hold regarding what others will do; rationality just says that a player chooses the strategy that maximizes her payoff, given her beliefs as to the strategies of other players.