Whoever has the highest performance is promoted
Whoever has the highest performance is promoted. Each contestant has one unit of effort that she can allocate in three possible ways: she can use it to enhance her own performance (which we’ll refer to as a “positive” effort) or to denigrate one of the two competing players (which we’ll refer to as a “negative” effort).
Before the competition begins, player i’s performance equals vi. If a player ex- erts a positive effort, then she adds 1 to her performance. If exactly one player ex- erts a negative effort against player i, then player i’s performance is reduced by 1. If both players go negative against her, then player i’s performance is reduced by 4. Hence, the marginal impact of a second person’s being negative is more detrimental than the impact of one person’s being negative. This idea seems plau- sible, since one person making negative remarks may be dismissed as a fabrica- tion, but two people saying the same thing could be perceived as credible.
How effort affects performance is summarized in TABLE 4.4. For example, if player i exerts a positive effort and the other two players exert a negative effort
4.4 Three-Player Games 107
against her, then her final performance is vi � 3, going up 1 through her posi- tive effort, but down 4 by the two units of negative effort directed against her.
Suppose players care intrinsically, not about performance, but rather about promotion. More specifically, a player’s payoff is specified to be the probability that she is promoted. If a player ends up with a performance higher than those of the other two players, then she is promoted with probability 1, so her payoff is 1. If her performance is highest, but she is tied with one other player, then each has probability of being promoted, and thus each has a payoff of . If all three players end up with the same performance, then each receives a payoff of . Finally, if a player’s performance is below that of another player, then her payoff is 0, since her probability of gaining the promotion is 0.
Assume that v1 � 2 and v2 � 0 � v3 so that player 1 is the front-runner. To start the analysis, let’s be a bit idealistic and consider the “no dirty tricks” strategy profile, in which each player exerts a positive effort, so that player i’s final performance is vi � 1. This scenario translates into a final performance of 3 for player 1 (since she began with 2) and 1 for both players 2 and 3 (since each of them began with 0). Hence, player 1 is promoted. We see, then, that if all exert a positive effort in order to boost their own performances, then the player who was initially ahead will end up ahead and thus be promoted. Let’s now assess whether this is a Nash equilibrium:
■ Player 1: First note that player 1’s strategy is clearly optimal, since her payoff is 1 (recall that it is the probability of being promoted) and that is the highest feasible payoff. Thus, there can’t be a strategy for player 1 that delivers a higher payoff.
■ Player 2: Player 2’s payoff from a positive effort is 0, since he is definitely not promoted, as his performance of 1 falls short of player 1’s perform- ance of 3. Alternatively, he could exert a negative effort against player 3, but that isn’t going to help, since the real competition for player 2 is player 1 and going negative against 3 doesn’t affect 1’s performance. The final alternative is for player 2 to exert a negative effort against player 1, in which case player 1’s performance is 2 instead of 3, while player 2’s performance is 0 instead of 1 (since he is no longer exerting a positive ef- fort on his own behalf). In that case, player 2 is still not promoted. We then find that player 2 is indifferent among all three of his strategies, since all deliver a zero payoff. Thus, because there is no strategy that
1 3
1 2
1 2
TABLE 4.4 PERFORMANCE OF PLAYER i
Amount of Positive Amount of Negative Effort by i Effort Against i Performance
0 0 vi
0 1 vi � 1
0 2 vi � 4
1 0 vi � 1
1 1 vi
1 2 vi � 3
yields a strictly higher payoff, player 2 is satisfied with exerting a posi- tive effort.
■ Player 3: The situation of player 3 is identical to that of player 2. They face the same payoffs and are choosing the same strategy. Thus, if going positive is optimal for player 2, then it is optimal for player 3.
■ In sum, all three players choosing a positive effort is a Nash equilibrium and results in the front-runner gaining the promotion.
In now considering a strategy profile in which some negative effort is ex- erted, let’s think about the incentives of players and what might be a natural strategy profile. It probably doesn’t make much sense for player 2 to think about denigrating player 3, because the “person to beat” is player 1, as she is in the lead at the start of the competition. An analogous argument suggests that player 3 should do the same. Player 1 ought to focus on improving her own per- formance, since she is in the lead and the key to winning is maintaining that lead.
Accordingly, let us consider the strategy profile in which player 1 promotes herself, while players 2 and 3 denigrate player 1. The resulting performance is �1 for player 1 (because her performance, which started at 2, is increased by 1 due to her positive effort and lowered by 4 due to the negative effort of the other two players) and 0 for players 2 and 3 (since no effort—positive or negative—is directed at them, so that their performance remains at its initial level). Because players 2 and 3 are tied for the highest performance, the pay- offs are 0 for player 1 and each for players 2 and 3. Now let’s see whether we have a Nash equilibrium:
■ Player 1: Unfortunately for player 1, there’s not much she can do about her situation. If she exerts a negative effort against player 2, then she lowers 2’s performance to �1 and her own to �2. Player 3’s performance of 0 results in her own promotion, so player 1 still loses out. An analo- gous argument shows that player 1 loses if she engages instead in a neg- ative effort targeted at player 3: now player 2 is the one who wins the promotion. Thus, there is no better strategy for player 1 than to exert a positive effort.
■ Player 2: If, instead of denigrating player 1, player 2 goes negative against player 3, then player 1’s performance is raised from �1 to 2, player 2’s per- formance remains at 0, and player 3’s performance is lowered from 0 to �1. Since player 1 now wins, player 2’s payoff is lowered from to 0, so player 2’s being negative about player 1 is preferred to player 2’s being neg- ative about player 3. What about player 2’s being positive? This does raise his performance to 1, so that he now outperforms player 3 (who still has a performance of 0), but it has also raised player 1’s performance to 2, since only one person is being negative against her. Since player 1 has the high- est performance, player 2’s payoff is again 0. Thus, player 2’s strategy of being negative against player 1 is strictly preferred to either player 2’s being negative against player 3 or player 2’s being positive.
■ Player 3: By an argument analogous to that used for player 2, player 3’s strategy of being negative against player 1 is optimal.
■ In sum, player 1’s going positive and players 2 and 3 denigrating player 1 is a Nash equilibrium. Doing so sufficiently lowers the performance of
1 2
1 2
108 CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO OR THREE PLAYERS
4.5 Foundations of Nash Equilibrium 109
player 1 (Zhao Ziyang?) such that the promotion goes to either player 2 (Jiang Zemin?) or player 3 (Li Peng?). The front-runner loses. As Wayne Campbell, of Wayne’s World, would say, “Promotion . . . denied!”
The promotion game, then, has multiple Nash equilibria (in fact, there are many more than we’ve described), which can have very different implications. One equilibrium has all players working hard to enhance their performance, and the adage “Let the best person win” prevails. But there is a darker solu- tion in which the weaker players gang up against the favorite and succeed in knocking her out of the competition. The promotion then goes to one of those weaker players. Perhaps the more appropriate adage in that case is the one at- tributed to baseball player and manager Leo Durocher: “Nice guys finish last”.
For the game in FIGURE 4.19, find all Nash equilibria.
FIGURE 4.19
4.2 CHECK YOUR UNDERSTANDING
Player 3: I
Player 1
Player 2
a
b
c
yx z
2,1,2
0,3,1
1,1,1
2,2,4
0,0,2
3,2,1
3,1,0
1,2,3
2,2,2
Player 3: II
Player 1
Player 2
a
b
c
yx z
2,1,3
1,2,1
1,2,1
3,3,3
1,0,3
1,0,0
1,1,1
1,0,4
2,1,2