SITUATION: VOTING, SINCERE OR DEVIOUS?
A company has three shareholders. Shareholder 1 controls 25% of the shares, shareholder 2 controls 35%, and shareholder 3 controls 40%. The company has offers from two other companies, denoted A and B, to purchase it. There is also a third option, which is to decline both offers. Shareholder 1 ranks the three choices, from the most to least preferred, as follows: accept A’s offer, ac- cept B’s offer, and accept neither offer (which we’ll denote option C). Shareholder 2’s ranking is B, then C, then A; and shareholder 3’s ranking is C, then B, then A. The rankings are summarized in TABLE 4.2.
Assume that a shareholder gets a payoff of 2 if his most preferred choice is implemented, a payoff of 1 for his second choice, and a payoff of 0 for his third
TABLE 4.2 SHAREHOLDERS’ PREFERENCES
Shareholder 1st Choice 2nd Choice 3rd Choice
1 A B C
2 B C A
3 C B A
FIGURE 4.15 The Best-Reply Method Applied to American Idol Fandom. There Are Two Nash Equilibria (Two Strategy Profiles in Which all Three Payoffs Are Circled)
Lauren
Kaitlyn
A
Bebe
Bebe
0,1,0
1 , 1 ,0
C
2 , 2 , 2
1,0,0 1 , 1 , 1
Alicia chooses E
Lauren
Kaitlyn
A
Bebe
C
0,0,1
1 ,0, 1
Bebe
0, 1 , 1
Alicia chooses Bebe
4.4 Three-Player Games 103
choice. The three shareholders cast their votes simultaneously. There are 100 votes, allocated according to share ownership, so shareholder 1 has 25 votes, shareholder 2 has 35 votes, and shareholder 3 has 40 votes. Shareholders are required to allocate their votes as a bloc. For example, shareholder 1 has to cast all of her 25 votes for A, B, or C; she cannot divvy them up among the projects. The strategy set for a player is then composed of A, B, and C. Plurality voting is used, which means that the alternative with the most votes is implemented.
To derive the payoff matrix, let us first determine how votes translate into a plurality winner. For example, if shareholders 1 and 2 vote for alternative B, then B is the winner, with either 60 votes (if shareholder 3 votes instead for A or C) or 100 votes (if 3 votes for B as well). FIGURE 4.16 shows the plurality win- ner for each of the 27 different ways in which the three players can vote.
The next step is to substitute the associated payoff vector for each alterna- tive in a cell in Figure 4.16. For example, if B is the winner, then shareholder 1’s payoff is 1 (since B is his second choice), shareholder 2’s payoff is 2 (since B is his first choice), and shareholder 3’s payoff is 1 (since B is his second choice). Substitution, then, gives us FIGURE 4.17.
In making statements about how these shareholders might vote, a natural possibility to consider is what political scientists call sincere voting. The term
FIGURE 4.17 Strategic Form of the Voting Game
FIGURE 4.16 Plurality Winners
3 votes for A
1
2
A
B
C
A
A
A
A
B
A
A
A
C
BA C
3 votes for B
1
2
A
B
C
A
B
B
B
B
B
B
B
C
BA C
3 votes for C
1
2
A
B
C
A
C
C
C
B
C
C
C
C
BA C
3 votes for A
1
2
A
B
C
2,0,0
BA C
2,0,0
2,0,0
1,2,1
2,0,0
2,0,0
2,0,0
2,0,0
0,1,2
3 votes for B
1
2
A
B
C
2,0,0
BA C
1,2,1
1,2,1
1,2,1
1,2,1
1,2,1
1,2,1
1,2,1
0,1,2
3 votes for C
1
2
A
B
C
2,0,0
BA C
0,1,2
0,1,2
1,2,1
0,1,2
0,1,2
0,1,2
0,1,2
0,1,2
104 CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO OR THREE PLAYERS
is used when a voter casts his vote for his first choice. In this case, it would mean that shareholder 1 casts her 25 votes for A, shareholder 2 casts his 35 votes for B, and shareholder 3 casts her 40 votes for C. As a result, choice C would be approved, since it received the most votes. But is sincere voting a Nash equilibrium? Is it optimal for shareholders to vote sincerely? Actually, no. Note that shareholder 1 prefers choice B over C. Given that shareholders 2 and 3 are voting sincerely, shareholder 1 can instead engage in (shall we call it) devious voting and vote for choice B rather than A. Doing so means that B ends up with 60 votes—being supported by both shareholders 1 and 2—and thus is approved. Shifting her votes from her most preferred alternative, A, to her next most preferred alternative, B, raises shareholder 1’s payoff from 0 to 1. Hence, sincere voting is not a Nash equilibrium for this game.
Although it can be shown that it is always optimal to vote sincerely when there are only two alternatives on the ballot, it can be preferable to vote for something other than the most preferred option when there are three or more options, as we just observed. The intuition behind this assertion is that the most preferred option may not be viable—that is, it won’t win, regardless of how you vote. In the event that your first choice can’t win, its reasonable to start think-
ing about which remaining choices could prevail, depend- ing on various voting scenarios, and, among those choices, vote for the one that is most preferred. In the case we have just examined, with shareholders 2 and 3 voting for B and C, respectively, shareholder 1 can cause B to win (by cast- ing her votes for B) or cause C to win (by casting her votes for either A or C). The issue, then, is whether she prefers B or C. Since she prefers B, she ought to use her votes strate- gically to make that option the winner.
Having ascertained that sincere voting does not pro- duce a Nash equilibrium, let’s see if the best-reply method can derive a strategy profile that is a Nash equilibrium. Start with shareholder 1. If shareholders 2 and 3 vote for A, then shareholder 1’s payoff is 2, whether she votes for A, B, or C. (This statement makes sense, since alternative A receives the most votes, regardless of how shareholder 1 votes.) Thus, all three strategies for shareholder 1 are best replies, and in FIGURE 4.18 we’ve circled her payoff of 2 in the column associated with shareholder 2’s choosing A and the matrix associated with shareholder 3’s choosing A. If shareholder 2 votes for B and shareholder 3 votes for A, then shareholder 1’s best reply is to vote for A or C (thereby ensuring that A wins); the associated payoff of 2 is then circled. If shareholder 2 votes for C and share- holder 3 votes for A, then, again, shareholder 1’s best replies are A and B. Continuing in this manner for share- holder 1 and then doing the same for shareholders 2 and 3, we get Figure 4.18.
Now look for all strategy profiles in which all three pay- offs are circled. Such a strategy profile is one in which each player’s strategy is a best reply and thus each player is doing the best he or she can, given what the others players
FIGURE 4.18 Best-Reply Method Applied to the Voting Game. There Are Five Nash Equilibria
3 votes for A
1
2
A
B
C
BA C
2 , 0 , 0
2 ,0,0
2 ,0,0
1, 2 , 1
2 , 0 ,0
2 ,0,0
2 , 0 ,0
2 ,0,0 0, 1 , 2
3 votes for B
1
2
A
B
C
BA
2 ,0, 0
1, 2 ,1
1, 2 ,1
1 , 2 , 1
1 , 2 ,1
1 , 2 ,1
C
1 , 2 ,1
1 , 2 ,1
0,1, 2
3 votes for C
1
2
A
B
C
BA C
2 ,0, 0
0,1, 2
0, 1 , 2
1 , 2 , 1
0, 1 , 2
0, 1 , 2
0 ,1, 2
0 , 1 , 2
0 , 1 , 2
4.4 Three-Player Games 105
are doing; in other words, it is a Nash equilibrium. Inspecting Figure 4.18, we see that there are five strategy profiles for which all three players are using best replies and thus are Nash equilibria: (A, A, A), (B, B, B), (C, C, C), (A, C, C), and (B, B, C). Note that the equilibria lead to different outcomes: (A, A, A) results in offer A’s being accepted, since all are voting for A. (B, B, B) and (B, B, C) result in offer B’s being accepted, and (C, C, C) and (A, C, C) lead to C’s being chosen.
We have rather robotically derived the set of Nash equilibria. Although use- ful, it is more important to understand what makes them equilibria. Consider equilibrium (A, A, A). Why is it optimal for shareholders 2 and 3 to vote for their least preferred alternative? The answer is that neither shareholder is piv- otal, in that the outcome is the same—alternative A wins—regardless of how each votes. Now consider shareholder 2. If she votes for A, then A wins with 100 votes; if she votes for B, then A still wins (though now with only 65 votes); and if she votes for C, then A still wins (again with 65 votes). It is true that share- holders 2 and 3 could work together to achieve higher payoffs: if they both vote for B, then B wins and shareholders 2 and 3 get payoffs of 2 and 1, respectively, which is better than 0 (which is what they get when A wins). But such coordi- nation among players is not permitted. Nash equilibrium requires only that each player, acting independently of others, can do no better.*
Equilibrium (A, A, A) has another interesting property: shareholders 2 and 3 are using a weakly dominated strategy in voting for A. As shown in TABLE 4.3, vot- ing for A is weakly dominated in voting for B for shareholder 2: all of the payoffs in the column “2 votes for B” are at least as great as those in the column “2 votes for A,” and in some of the rows the payoff is strictly greater. So, regardless of how shareholders 1 and 3 vote, voting for B gives shareholder 2 at least as high a pay- off as does voting for A. Of course, when shareholders 1 and 3 vote for A—as they do at Nash equilibrium (A, A, A)—a vote for A and a vote for B result in the same payoff of 0 for shareholder 2, so she is acting optimally by voting for A. However,
*Note that, in each of the five Nash equilibria, at least one shareholder is not pivotal. With (B, B, B) and (C, C, C), all three players are not pivotal, just as with (A, A, A). With (A, C, C), shareholder 1 is not piv- otal, although shareholders 2 and 3 are pivotal, since each could result in A’s winning if they voted for A. With (B, B, C), shareholder 3 is not pivotal, although shareholders 1 and 2 are.
TABLE 4.3 PAYOFFS TO PLAYER 2
1’s Strategy 3’s Strategy 2 Votes for B 2 Votes for A
A A 0 � 0
A B 2 0
A C 1 � 0
B A 2 0
B B 2 � 2
B C 2 1
C A 0 � 0
C B 2 � 2
C C 1 � 1
106 CHAPTER 4: STABLE PLAY: NASH EQUILIBRIA IN DISCRETE GAMES WITH TWO OR THREE PLAYERS
there are other votes by shareholders 1 and 3 (e.g., when one of them votes for A and the other for B) for which shareholder 2 does strictly better by voting for B rather than A.
We then find that a player using a weakly dominated strategy is not ruled out by Nash equilibrium. Though voting for B always generates at least as high a payoff for shareholder 2 as does voting for A (and, in some cases, a strictly higher payoff), as long as A gives the same payoff that voting for B does for the strategies that shareholders 1 and 3 are actually using, then A is a best reply and thereby consistent with Nash equilibrium.
A Nash equilibrium does not preclude a player’s using a weakly dominated strategy.