CONFLICT AND MUTUAL INTEREST IN GAMES
Rock–Paper–Scissors is a game of pure conflict. What do I mean by that? Well, take note of an interesting property of the payoff matrix in Figure 4.8: Players’ payoffs always sum to the same number (which happens to be zero). For example, if both Bart and Lisa choose rock, then each gets zero, so the sum of their payoffs is zero. If Bart chooses paper and Lisa chooses rock, then Bart gets 1 and Lisa gets �1, which once again sums to zero. For every strategy pair, the sum of their payoffs is zero. This type of game is known as a constant-sum game, because the payoffs always sum to the same number. When that number happens to be zero, the game is called a zero-sum game.
So think about what this implies. Since payoffs must sum to the same num- ber, if some strategy pair results in a higher payoff for Bart, then it must re- sult in a lower payoff for Lisa. Thus, what makes Bart better off has to make Lisa worse off, and analogously, what makes Lisa better off has to make Bart worse off. It is in that sense that Rock–Paper–Scissors is a game of pure con- flict. In fact, all constant-sum games have this property.
Contrast this game with driving conventions. Here we have the opposite of Rock–Paper–Scissors, in the sense that there is no conflict at all. A strategy pair that makes driver 1 better off—such as (left, left) compared with (left, right)—also makes driver 2 better off; they both get a payoff of 1 rather than 0. This is a game of mutual interest, as the rankings of strategy pairs by their payoffs coincides for the players.
Chicken and the telephone game lie between these two extremes. Those strategic settings do provide grounds for mutual interest. In Chicken, both players want to avoid (hang tough, hang tough); they both prefer (swerve, hang tough) and (hang tough, swerve). But there is also room for conflict, as they disagree as to how they rank (swerve, hang tough) and (hang tough, swerve); driver 1 prefers the latter and driver 2 prefers the for- mer. Similarly, with the telephone game, both Colleen and Winnie agree that one of them calling is preferable to either both of them waiting or both calling, but they disagree as to who should call. Colleen prefers that it be Winnie, while Winnie prefers it to be Colleen. They share a common interest in coordinating on exactly one person calling, but their interests depart—they are in conflict—when it comes to who that person should be.
Knowing whether players’ interests are entirely in conflict, partially in conflict, or entirely in common can provide some insight into which strategy profiles are Nash equilibria. So, when you come to a game, think about the interests of the players before launching into a robotic search for solutions. Your ruminations may offer some valuable shortcuts.
4.3 The Best-Reply Method AS THE CELEBRATED TV chef Emeril Lagasse would say, “Let’s kick it up a notch!” by adding a third player to the mix. But before doing so, I’ll share a useful shortcut with you for deriving Nash equilibria.
For the game in FIGURE 4.9, find all Nash equilibria.
FIGURE 4.9
4.1 CHECK YOUR UNDERSTANDING
0,1 0,1
1,2
3,0
2,1
1,0
0,1
2,2 Player 1
Player 2
a
b
c
d
1,0
1,1
1,2
4,0
3,2
3,1
1,0
0,2
w x y z
100 CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO OR THREE PLAYERS
Recall that a player’s best reply is a strategy that maximizes his pay- off, given the strategies used by the other players. We can then think of a Nash equilibrium as a strategy profile in which each player’s strategy is a best reply to the strategies the other players are using. Stemming from this perspective, the best-reply method offers a way of finding all of the Nash equilibria. Rather than describe it in the abstract, let’s walk through the method for the two-player game in FIGURE 4.10.
For each strategy of Diane, we want to find Jack’s best replies. If Diane uses x, then Jack has two best replies—b and c—each of which gives a payoff of 2 that exceeds the payoff of 1 from the other possible strategy, a. If Diane uses y, then Jack has a unique best reply of a. And if Diane uses
z, then c is Jack’s only best reply. To keep track of these best replies, circle those of Jack’s payoffs associated with his best replies, as shown in FIGURE 4.11.
Next, perform the same exercise on Diane by finding her best replies in re- sponse to each of Jack’s strategies. If Jack uses a, then both x and y are Diane’s best replies. If Jack uses b, then Diane’s best reply is x. Finally, if Jack uses c, then y is Diane’s best reply. Circling the payoffs for Diane’s best replies, we now have FIGURE 4.12.
FIGURE 4.10 Jack and Diane’s Strategies
1,1
2,1
2,3
2,1
1,2
0,2Jack
Diane
a
b
c
2,0
3,0
2,1
x y z
FIGURE 4.12 Diane’s and Jack’s Best Replies (Circled). Two Strategy Pairs Are Nash Equilbria: (b,x) and (a,y)
FIGURE 4.11 Jack’s Best Replies (Circled)
1,1 2 ,1
2 ,1
2 ,3
1,2
0,2Jack
Diane
a
b
c
2,0
3 ,0
2,1
x y z
Jack
Diane
a
b
c
x y z
2,0
3 ,0
2,1
1, 1
2 ,1
2 , 3
2 , 1
1, 2
0,2
Since a Nash equilibrium is a strategy pair in which each player’s strategy is a best reply, we can identify Nash equilibria in Figure 4.12 as those strategy pairs in which both pay- offs in a cell are circled. Thus, (b,x) and (a,y) are Nash equilibria. We have just used the best-reply method to derive all Nash equilibria.
Before we explore how the best-reply method is used in three-player games, let’s deploy it in Rock– Paper–Scissors. Marking each of Lisa’s and Bart’s best replies, we have FIGURE 4.13. For example, if Lisa chooses rock, then Bart’s best reply is paper, so we cir- cle Bart’s payoff of 1 earned from the strategy pair (paper, rock). Note that no cell has two circled payoffs, indicating that there is no Nash equilibrium; this is the same result we derived earlier.
FIGURE 4.13 Best Replies (Circled) for Bart and Lisa’s Game of Rock– Paper–Scissors. There Is No Nash Equilibrium
Bart
Lisa
Rock
Paper
Scissors
PaperRock Scissors
0,0
1 ,�1
�1, 1
0,0
1 ,�1
�1, 1
0,0
1 ,�1
�1, 1
4.4 Three-Player Games 101