KIDNAPPING GAME AND PAYOFFS
Outcome Guy (Violent) Guy Vivica
No kidnapping 3 3 5
Kidnapping, ransom is paid, Orlando is killed 4 5 1
Kidnapping, ransom is paid, Orlando is released 5 4 3
Kidnapping, ransom is not paid, Orlando is killed 2 2 2
Kidnapping, ransom is not paid, Orlando is released 1 1 4
The objective of our model is to make some predictions about how Guy and Vivica will behave. Although solving a game won’t be tackled until the next chapter, in fact we don’t have enough information to solve it even if we knew how. To describe how someone will behave, it’s not enough to know what they can do (e.g., kill or release) and what they know (e.g., whether ran- som has been paid); we also need to know what these people care about. What floats their boat? What rings their bell? What tickles their fancy? You get the idea.
A description of what a player cares about takes the form of a ranking of the five outcomes of the game. Suppose Guy is someone who really just wants the money and kills only out of revenge for the ransom not being paid. Then Guy’s best outcome is to perform the kidnapping, Vivica pays the ransom, and he releases Orlando. Because we assume that he is willing to kill in exchange for money, his second-best outcome is to perform the kidnapping, have the ransom paid, and kill Orlando. The third-best outcome is not to kidnap Orlando, since Guy prefers not to run the risk of kidnapping when ransom is not to be paid. Of the two remaining outcomes, suppose that if he kidnaps Orlando and ransom is not paid, then he prefers to kill Orlando (presumably out of spite for not receiving the ransom). The least preferred outcome is then that there is a kidnapping, ransom is not paid, and Orlando is released.
To concisely include Guy’s preferences in our description of the game, we’ll assign a number to each outcome, with a higher number indicating a more preferred outcome for a player. This ranking is done in Table 2.1 under the col- umn labeled “Guy.” These numbers are referred to as payoffs and are in- tended to measure the well-being (or utility, or welfare, or happiness index) of a player. For example, the highest payoff, 5, is assigned to the best outcome: the kidnapping takes place, ransom is paid, and Orlando is released. The worst
20 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
outcome—the kidnapping takes place, ransom is not paid, and Orlando is re- leased—receives the lowest payoff, 1.
Suppose, contrary to what was just assumed, that Guy felt that his chances of getting caught would be less if Orlando were dead, so that he now always prefers killing Orlando to releasing him. Then Guy’s payoffs would be as shown in the column “(Violent) Guy.” The highest payoff is now assigned to the outcome in which Guy kidnaps and kills Orlando and the ransom is paid.
What about Vivica? If she cares about Orlando more than she cares about money, then her most preferred outcome is no kidnapping, and we’ll assign that the highest payoff of 5. Her least preferred outcome is that Orlando is kid- napped and killed and ransom is paid, so it receives the lowest payoff of 1. The payoffs for the other outcomes are shown in the table.
To ensure that the depiction in Figure 2.1 contains all of the relevant infor- mation, the payoffs have been included. Each terminal node corresponds to a particular outcome of the game, and listed below a terminal node are the pay- offs that Guy and Vivica assign to that outcome; the top number is Guy’s pay- off and the bottom number is Vivica’s payoff. While we could also list Orlando’s payoffs—he is surely not indifferent about what happens—that would be extraneous information. Because our objective is to say something about behavior, and this model of kidnapping allows only the kidnapper and the victim’s kin to act, only their payoffs matter.
This step of assigning a payoff to an outcome is analogous to what was done in Chapter 1. There we began with a person’s preferences for certain items (in our example, it was cell phone providers), and we summarized those preferences by assigning a number—known as utility—to each item. A per- son’s preferences were summarized by the resulting utility function, and her behavior was described as making the choice that yielded the highest utility. We’re performing the same step here, although game theory calls the number a payoff; still, it should be thought of as the same as utility.
The scenario depicted in Figure 2.1 is an example of an extensive form game. An extensive form game is depicted as a decision tree with decision nodes, branches, and terminal nodes. A decision node is a location in the tree at which one of the players has to act. Let us think about all of the informa- tion embodied in Figure 2.1. It tells us which players are making decisions (Guy and Vivica), the sequence in which they act (first Guy then, possibly, Vivica, and then Guy again), what choices are available to each player, and how they evaluate the various outcomes of the game. This extensive form game has four decision nodes: the initial node at which Guy decides whether to kidnap Orlando, the decision node at which Vivica decides whether to pay ransom, and Guy’s two decision nodes concerning whether to kill or release Orlando (one decision node for when Vivica pays ransom and one for when she does not). Extending out of each decision node are branches, where a branch represents an action available to the player who is to act at that deci- sion node. More branches mean more choices.
We refer to the decision node at the top of the tree as the initial node (that is where the game starts) and to a node corresponding to an end to the game as a terminal node (which we have not bothered to represent as a dot in the figure). There are five terminal nodes in this game, since there are five possi- ble outcomes. Terminal nodes are distinct from decision nodes, as no player acts at a terminal node. It is at a terminal node that we list players’ payoffs,
2.2 Extensive Form Games: Perfect Information 21
Orioles’ manager
Yankees’ manager
MR
1
3
3
1
2 Orioles’ manager
2 Yankees’ manager
JG
RJ
JL
where a payoff describes how a player evaluates an outcome of the game, with a higher number indicating that the player is better off.
� SITUATION: BASEBALL, I
Good pitching will always stop good hitting and vice-versa. —CASEY STENGEL
One of the well-known facts in baseball is that right-handed batters generally perform better against left-handed pitch- ers and left-handed batters generally perform better against right-handed pitchers. TABLE 2.2 documents this claim.1 If you’re not familiar with baseball, batting average is the percentage of official at bats for which a batter gets a hit (in other words, a batter’s success rate). Right-handed batters got a hit in 25.5% of their attempts against a right- handed pitcher, or, as it is normally stated in baseball, their batting average was .255. However, against left-handed pitchers, their batting average was significantly higher, namely, .274. There is an analogous pattern for left-handed batters, who hit .266 against left-handed pitchers but an impressive .291 against right-handed pitching. Let’s explore the role that this simple fact plays in a commonly occurring strategic situation in baseball.
It is the bottom of the ninth inning and the game is tied between the Orioles and the Yankees. The pitcher on the mound for the Yankees is Mariano Rivera, who is a right-hander, and the batter due up for the Orioles is Javy Lopez, who is also a right-hander. The Orioles’ manager is thinking about whether to sub- stitute Jay Gibbons, who is a left-handed batter, for Lopez. He would prefer to have Gibbons face Rivera in order to have a lefty–righty matchup and thus a better chance of getting a hit. However, the Yankees’ manager could respond to Gibbons pinch-hitting by substituting the left-handed pitcher Randy Johnson for Rivera. The Orioles’ manager would rather have Lopez face Rivera than have Gibbons face Johnson. Of course, the Yankees’ manager has the exact opposite preferences.
The extensive form of this situation is shown in FIGURE 2.2. The Orioles’ manager moves first by deciding whether to substitute Gibbons for Lopez. If he does make the substi- tution, then the Yankees’ manager decides whether to substitute Johnson for Rivera. Encompassing these preferences, the Orioles’ manager assigns the highest payoff (which is 3) to when Gibbons bats against Rivera and the lowest payoff (1) to when Gibbons bats against Johnson. Because each manager is presumed to care only about winning, what makes the Orioles better off must make the Yankees worse off. Thus, the best outcome for the Yankees’ manager is when Gibbons bats against Johnson, and the worst is when Gibbons bats against Rivera.
TABLE 2.2
Batter Pitcher Batting Average
Right Right .255
Right Left .274
Left Right .291
Left Left .266
FIGURE 2.2 Baseball
22 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION