What is a strategy for the kidnapper?
To be more concrete as to the nature of a strategy in game theory, let us re- turn to the Kidnapping game in Figure 2.1. What is a strategy for the kidnap- per? As we’ve just said, a strategy is a complete decision rule—one that pre- scribes an action for every situation that a player can find himself in. Guy (the kidnapper) can find himself in three situations: (1) contemplating whether to kidnap Orlando (i.e., the initial node); (2) having kidnapped Orlando, with ran- som having been paid by Vivica, and deciding whether to kill or release Orlando; and (3) having kidnapped Orlando, with ransom not having been paid by Vivica, and deciding whether to kill or release Orlando. It is not coinciden- tal that Guy can find himself in three scenarios and he has three information sets: A “situation” for a player is defined as finding himself at an information set; hence, a strategy assigns one action to each of a player’s information sets.
A template for Guy’s strategy is, then,
At the initial node, _____ [ fill in kidnap or do not kidnap].
If a kidnapping occurred and ransom was paid, then _____ [ fill in kill or release].
If a kidnapping occurred and ransom was not paid, then _____ [ fill in kill or release].
There are as many strategies as ways in which to fill in those three blanks. Exhausting the possibilities, we have eight feasible strategies:
2.4 What Is a Strategy? 35
1. At the initial node, kidnap. If a kidnapping occurred and ransom was paid, then release. If a kidnapping occurred and ransom was not paid, then kill.
2. At the initial node, kidnap. If a kidnapping occurred and ransom was paid, then release. If a kidnapping occurred and ransom was not paid, then release.
3. At the initial node, kidnap. If a kidnapping occurred and ransom was paid, then kill. If a kidnapping occurred and ransom was not paid, then release.
4. At the initial node, kidnap. If a kidnapping occurred and ransom was paid, then kill. If a kidnapping occurred and ransom was not paid, then kill.
5. At the initial node, do not kidnap. If a kidnapping occurred and ransom was paid, then release. If a kidnapping occurred and ransom was not paid, then kill.
6. At the initial node, do not kidnap. If a kidnapping occurred and ransom was paid, then release. If a kidnapping occurred and ransom was not paid, then release.
7. At the initial node, do not kidnap. If a kidnapping occurred and ransom was paid, then kill. If a kidnapping occurred and ransom was not paid, then release.
8. At the initial node, do not kidnap. If a kidnapping occurred and ransom was paid, then kill. If a kidnapping occurred and ransom was not paid, then kill.
Analogously, we can define a strategy template for Vivica:
If a kidnapping occurred, then _____ [fill in pay ransom or do not pay ransom].
Since Vivica has just one information set, her strategy is just a single action. With only two feasible actions and one information set, she then has two fea- sible strategies:
1. If a kidnapping occurred, then pay ransom.
2. If a kidnapping occurred, then do not pay ransom.
The strategy set for a player is defined to be the collection of all feasible strategies for that player. In this example, the strategy set for Guy comprises the eight strategies just listed for him, and the strategy set for Vivica is made up of two strategies. There are then 16 possible strategy pairs for this game.
As previously mentioned, all of the hard thinking goes into choosing a strat- egy, and once one is chosen, play arises from the implementation of that strat- egy. To see this point more clearly, suppose Guy chooses the following strategy:
At the initial node, kidnap.
If a kidnapping occurred and ransom was paid, then release.
If a kidnapping occurred and ransom was not paid, then kill.
Suppose also that Vivica chooses the following strategy:
If a kidnapping occurred, then pay ransom.
So what will happen? According to Guy’s strategy, he kidnaps Orlando. Vivica then pays the ransom (as instructed by her strategy), and in response to the
36 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
ransom being paid, Guy releases Orlando (reading from his strategy). Similarly, you can consider any of the 16 possible strategy pairs and figure out what the ensuing sequence of actions is. It’s just a matter of following instructions.
Before moving on, notice a peculiar feature about some of Guy’s strategies, namely, that strategies 5 through 8 prescribe do not kidnap and then tell Guy what to do if he chose kidnap. In other words, it tells him to do one thing, but also what to do if he doesn’t do what he should have done. In spite of how
strange that might sound, we’ll allow for this possibility in a player’s strategy set, for three reasons. First, it’s simpler to define a strategy as any way in which to assign feasible actions to information sets than to try to come up with a more complicated definition that rules out these “silly” strategies. Second, inclusion of the silly strategies is, at
worst, some harmless detritus that won’t affect the conclusions that we draw. And the third reason, which is the most important, I can’t tell you now. It’s not that I don’t want to, but you’ll need to know a bit about solving games before you can understand what I want to say. I’ll clue you in come Chapter 4.
2.5 Strategic Form Games THE EXTENSIVE FORM IS one type of scaffolding around which a game can be con- structed. Its appeal is that it describes (1) a concrete sequence with which players act, (2) what actions they have available and what they know, and (3) how they evaluate the outcomes, where an outcome is a path through the de- cision tree. In this section, we introduce an alternative scaffolding that, though more abstract, is easier to work with than the extensive form. In the next section, we’ll show how you can move back and forth between these two game forms so that you may work with either one.
A strategic form game (which, in the olden days of game theory, was referred to as the normal form) is defined by three elements that address the following questions: (1) Who is making decisions? (2) Over what are they making deci- sions? and (3) How do they evaluate different decisions? The answer to the first question is the set of players, the answer to the second question is the players’ strategy sets, and the answer to the third question is players’ payoff functions.
The set of players refers to the collection of individuals who have decisions to make. The decision is with regard to a strategy, which is defined exactly as in the previous section. A player’s strategy set is the collection of strategies from which he can choose. Finally, a player’s payoff function tells us how the player evaluates a strategy profile, which is a collection of strategies, one for each of the players. A higher payoff means that a player is better off, and when we get to solving a game, the presumption will be that each player tries to maximize his or her payoff.
Although a player does not intrinsically value strategies—for they are just decision rules, and you can’t eat, wear, caress, or live in a decision rule—a strategy profile determines the outcome of the game (e.g., whether there is a kidnapping), and a player does care about the outcome. One final piece of jar- gon before we move on: The term n-tuple refers to n of something—for ex- ample, an n-tuple of strategies in a game with n players. Two of something is a pair, three of something is a triple, and n of something is an n-tuple. With all of this jargon, you can now talk like a game theorist!
For the revised Kidnapping game in Figure 2.6, write down the strategy sets for Guy and Vivica.
2.3 CHECK YOUR UNDERSTANDING
2.5 Strategic Form Games 37
� SITUATION: TOSCA
The force of my desire has two aims, and the rebel’s head is not the more pre- cious. Ah, to see the flames of those victorious eyes smoulder, aching with love! Caught in my arms, smouldering with love. One to the gallows, the other in my arms! —BARON SCARPIA FROM THE OPERA TOSCA
Giacomo Puccini was arguably the last great operatic composer. He died in 1924 after a career that produced such spectacular successes as La Bohème (the plot of which was recycled for the Broadway musical Rent), Madame Butterfly, and Turandot. Puccini’s music is the type that leads you to hum or whistle it after you leave the theater. It clearly runs counter to the popular def- inition of opera as two heavy-set people 6 inches apart screaming at the top of their lungs.
One of his most popular operas is Tosca, which is a story of love, devotion, corruption, lechery, and murder—in other words, perfect fodder for learning game theory!4 The main characters are Baron Vitellio Scarpia, the local chief of police; an attractive woman named Floria Tosca; and Mario Cavaradossi, her lover. Scarpia has lustful designs on Tosca and has devised a diabolical plot to act on them. He first has Cavaradossi arrested. He then tells Tosca that Cavaradossi is to go before the firing squad in the morning and he (Scarpia) can order the squad to use real bullets—and Cavaradossi will surely die—or blanks—in which case Cavaradossi will survive. After then hearing Scarpia’s sexual demands, Tosca must decide whether or not to concede to them.
Scarpia and Tosca meet that evening after Scarpia has already given his or- ders to the firing squad. Tosca faces Scarpia and—knowing that Scarpia has decided, but not knowing what he has decided—chooses between consenting to his lustful desires or thrusting the knife she has hidden in her garments into the heart of this heartless man.
In writing down the strategic form game, we have our two players, Scarpia and Tosca. The strategy set for Scarpia has two strategies—use real bullets or use blanks—while Tosca can either consent or stab Scarpia. As de- picted in FIGURE 2.11, the two strategies for Tosca correspond to the two rows, while the two strategies for Scarpia correspond to the two columns. Thus, Tosca’s choosing a strategy is equivalent to her choosing a row.
The final element to the strategic form game are the payoffs. The first number in a cell is Tosca’s payoff and the second num- ber is Scarpia’s payoff. (We will use the convention that the row player’s payoff is the first number in a cell.) For example, if Tosca chooses stab and Scarpia chooses blanks, then Tosca’s payoff is 4 and Scarpia’s payoff is 1. We have chosen the payoffs so that Tosca ranks the four possible strategy pairs as follows (going from best to worst): stab and blanks, consent and blanks, stab and real, and consent and real. Due to her love for Cavaradossi, the most important thing to her is that Scarpia use blanks, but it is also the case that she’d rather kill him than con- sent to his lascivious libido. From the information in the opening quote, Scarpia’s payoffs are such that his most preferred strategy pair is consent and real, as he then gets what he wants from Tosca and eliminates Cavaradossi as a future rival for Tosca. His least preferred outcome is, not surprisingly, stab and blanks.
FIGURE 2.11 Tosca
2,2 4,1
1,4 3,3 Tosca
Scarpia
Stab
Consent
Real Blanks
38 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
Figure 2.11 is known as a payoff matrix and succinctly contains all of the elements of the strategic form game. Tosca is a reinterpretation of the Prisoners’ Dilemma, which is the most famous game in the entire kingdom of game theory. I’ll provide the original description of the Prisoners’ Dilemma in Chapter 4.