SITUATION: AMERICAN IDOL FANDOM
Alicia, Kaitlyn, and Lauren are ecstatic. They’ve just landed tickets to attend this week’s segment of American Idol. The three teens have the same favorite among the nine contestants that remain: Ace Young. They’re determined to take this opportunity to make a statement. While IMing, they come up with a plan to wear T-shirts that spell out “ACE” in large letters. Lauren is to wear a T-shirt with an big “A,” Kaitlyn with a “C,” and Alicia with an “E.” If they pull this stunt off, who knows—they might end up on national television! OMG!
While they all like this idea, each is tempted to wear instead an attractive new top just purchased from their latest shopping expedition at Bebe. It’s now an hour before they have to leave to meet at the studio, and each is at home trying to decide between the Bebe top and the lettered T-shirt. What should each wear?
In specifying the strategic form of this game, we assign a payoff of 2 if they all wear their lettered T-shirts (and presumably remember to sit in the right sequence). This payoff is higher than the one they get from wearing the Bebe top, which is 1. Finally, wearing a lettered T-shirt when one or both of the other girls do not yields a payoff of 0, as the wearer realizes the worst of all worlds: she doesn’t look alluring and they don’t spell out ACE.
The strategic form is shown in FIGURE 4.14.* Lauren’s choice is represented as selecting a row—either wearing the T-shirt with the letter “A” or her Bebe top—while Kaitlyn chooses a column and Alicia chooses a matrix.
FIGURE 4.14 American Idol Fandom
Alicia chooses E Alicia chooses Bebe
2,2,2 0,1,0
1,0,0 1,1,0 Lauren
Kaitlyn
A
Bebe
BebeC
Lauren
Kaitlyn
0,0,1 0,1,1
1,0,1 1,1,1
A
Bebe
BebeC
Using the best-reply method to solve this game, consider the situation faced by Lauren. If Alicia wears her T-shirt with E and Kaitlyn wears hers with C, then Lauren’s best reply is to do her part and wear the T-shirt with A. So, we circle Lauren’s payoff of 2 in the cell associated with strategy profile (A, C, E), as shown in FIGURE 4.15. If, instead, Kaitlyn chooses Bebe and Alicia wears E, then Lauren’s best reply is to wear her Bebe top and receive a payoff of 1, so we circle that payoff for Lauren. If Alicia wears her Bebe top and Kaitlyn wears C, then Lauren’s best reply is again to wear her Bebe top, so we circle Lauren’s payoff of 1. Finally, if both of the other two girls choose their Bebe
*This game is a 21st-century teen girl version of the Stag Hunt game due to Jean-Jacques Rousseau in On the Origins and Foundations of Inequality among Men (1755). In that setting, hunters can work together to catch a stag (rather than spell out ACE) or hunt individually for hare (rather than wear a Bebe top).
102 CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO OR THREE PLAYERS
tops, then Lauren optimally does so as well, which means that we now circle Lauren’s payoff of 1 in that instance. Performing this same exercise for Kaitlyn and Alicia, we end up with Figure 4.15.
Examining the figure, we find that there are two Nash equilibria—that is, two strategy profiles in which all three payoffs are circled—signifying that all three teens are choosing best replies. One equilibrium is when Lauren wears A, Kaitlyn wears C, and Alicia wears E, which, to the delight of Ace Young, spells out his name on American Idol. The other equilibrium occurs when each tosses her lettered T-shirt aside and instead wears that eye-catching top from Bebe.
Note that the players rank the Nash equilibria the same: all prefer wearing the T-shirts spelling out ACE than wearing Bebe shirts. The situation differs from that pertaining to driving conventions, in which players are indifferent among equilibria, and also from that in the Chicken and telephone games, in which players ranked equilibria differently.