SITUATION: COMPETING FOR ELECTED OFFICE
Earlier this week, I watched the last of the presidential debates, and by the end of it I was ready to change the channel in disgust. Neither George Bush nor John Kerry seemed to be saying anything original, or anything that had- n’t been heard before. By the end of it, I couldn’t help but feel like I was watching Tweedle-Dum and Tweedle-Dee dancing around in Alice & Wonderland, except here, maybe we’d be better off calling them Tweedle- Dumb and Tweedle-Dumber.6
Candidates running for elected office compete in many ways—advertising on television, gaining endorsements
, giving stump speeches—but perhaps the most significant method of competition lies in the positions they take on the issues. To model this form of competition, suppose there is just one issue and a position on that issue is represented by a number in the interval [0,1]. For example, the issue could be taxes, where a higher number indicates a higher tax rate. Or it could be how funds are allocated between welfare programs and defense expenditures, with a higher number corresponding to more defense
6.2 Solving for Nash Equilibria Without Calculus 155
and less welfare. Or we could imagine that candidates aren’t so much taking positions, but rather conveying their ideology. A position of 0 could corre- spond to the “far left” (very liberal) and of 1 to the “far right” (very conserva- tive), with a moderate position being represented by
In describing the strategic form of this game, the players are candidates D(emocratic) and R(epublican), and each has a strategy set of [0,1]. Let de- note the position (or strategy) of candidate Taking a rather cynical view, let us assume that candidates care only about being elected and not about the positions they take. To write down a candidate’s payoff function, we then need to describe how strategies (i.e., positions of the two candidates) de- termine the electoral outcome. Doing this will take a bit of work.
Suppose each voter has an ideal position from the position space [0,1]. Furthermore, suppose there are many voters evenly distributed over [0,1] in terms of their ideal positions. In other words, the number of voters whose most preferred position is .3 is the same as the number whose most preferred position is .72, and so forth. Assume that all voters vote and cast their votes for the candidate whose announced position is closest to their ideal one. There is then no question about the credibility of a candidate’s announced position: voters believe what is said. Perhaps this idealistic assumption offsets the cyn- ical one about the candidates’ preferences!
Now, consider the case when candidate D’s position is to the left of candidate R’s: (See FIGURE 6.7.) Voters who lie to the left of can- didate D’s position (i.e., their ideal position is lower than vote for her, since is closer to their ideal than is Analogously, those voters who lie to the right of vote for candidate R. What about the voters between and (Pundits refer to them as “swing voters.”) Consider the voter who is smack-dab in the mid- dle—the one located at Since the voter at is equidis- tant between the two candidates’ positions, he’s indifferent between them. Thus, swing voters who lie to the left of are closer to and vote for candidate D. Analogously, those who are to the right of vote for candidate R.*
With this information, we can figure out how positions determine an elec- toral outcome. If then candidate D’s share of the vote is as all voters to the left of vote for D. Hence, if then less than half of the voters vote for candidate D, and candidate R wins. If, instead, then more than half of the voters vote for candi- date D, and candidate D wins. If , then the preceding argument applies if one switches around the two candidates. Finally, if then, since the candidates have taken identical positions, the vote is split equally.
In writing down candidates’ payoff functions, assume that the payoff to winning the election is 2 while the loser’s payoff is 0. What if both candidates receive the same number of votes? This happens when either both candi- dates take the same position or In those situations, we’ll(xD � xR)/2 �
1 2.
xD � xR, xR 6 xD
1 2 6 (xD � xR)/2,
(xD � xR)/2 6 12,(xD � xR)/2 (xD � xR)/2,xD 6 xR,
(xD � xR)/2 xD(xD � xR)/2
(xD � xR)/2(xD � xR)/2.
xR?xD
xR
xR. xDxD)
xD 6 xR.
i � D, R. xi
1 2.
*Although the voter located at is indifferent, it doesn’t matter what he does, since there is an infi- nite number of voters.
xD � xR 2
FIGURE 6.7 The Determination of a Winner in the Campaign
Vote for D Vote for R
xRxD0 1xD � xR 2
156 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
assume that each candidate has an equal chance of winning, so the associ- ated payoff is 1.
Using the preceding analysis that describes how positions affect vote totals, we can now write down a payoff function. If then candidate D’s payoff is
and if then
The payoff function for candidate R is analogously defined. As an initial step toward understanding the incentives of candidates, con-
sider the pair of positions shown in FIGURE 6.8. The two candidates shown are relatively liberal—both are located to the left of —but candidate D is more liberal than candidate R: Voters vote as described at the bottom of Figure 6.8, with all voters whose ideal position is to the right of
voting for candidate R. Candidate R then wins by garnering more than half of the vote. If candidate D instead took a position between and (say, in the figure), she would gather more than half of the votes and thus raise her payoff from 0 to 2; this is the voting outcome described at the top of Figure 6.8. Thus, is not a Nash equilibrium, since candidate D can do better than to espouse x¿D.
(x¿D, x¿R)
x0D
1 2x¿R
(x¿D � x¿R)/2
x¿D 6 x¿R 6 12.
1 2
(x¿D, x¿R)
μ 2 if 12 6 xD � xR2 and xR 6 xD1 if xD � xR2 = 12 or xD = xR. 0 if
xD � xR 2 6 12 and xR 6 xD
xR � xD,
μ 0 if xD � xR2 6 12 and xD 6 xR1 if xD � xR2 = 12 or xD = xR, 2 if 12 6
xD � xR 2 and xD 6 xR
xD � xR,
FIGURE 6.8 A Candidate Does Better by Locating Between the Other Candidate’s Position and When Candidate D Chooses Voting Is Described at the Bottom. When Candidate D Chooses Voting Is Described at the Topx0D,
x�D, 1 2.
Vote for D Vote for R
x�D x�R x0D
0 1x�D � x�R 2
1 2
Vote for DVote for R
More generally, if, say, candidate D is taking a position different from then candidate R can ensure victory by locating between and candidate D’s position. But this means that candidate D loses for sure. However, a strategy that results in certain loss is clearly not optimal for candidate D, because she can always gain at least a tie (with a payoff of 1) by taking the exact same po- sition as candidate R. We then conclude that if either or both candidates are located away from the pair of positions is not a Nash equilibrium.12,
1 2
1 2,
6.3 Solving for Nash Equilibria with Calculus (Optional) 157
We have managed to eliminate all strategy pairs but one: In that case, the candidates split the vote and each receives a payoff of 1. Now, consider candidate D’s choosing a different position, say, in FIGURE 6.9. This results in her share of the vote drop- ping from to which means that her payoff declines from 1 to 0. This argu- ment works as well for candidate R, so both candidates locating at constitutes a unique Nash equilibrium.
Even though there are two candidates, electoral competition results in vot- ers having no choice: both candidates support the same position! What at- tracts candidates to that moderate position is that it is the most preferred po- sition of the median voter; half of the voters are more liberal than the median voter and half are more conservative. A candidate who strays from the median voter is taking an unpopular stance that the other candidate can capitalize on to ensure victory. This result is known as policy convergence, since the candi- dates converge on the same platform.
As is typically the case with a simple model, the result it delivers is extreme. In reality, candidates do not offer identical or even near-identical positions. The departure of reality from theory is at least partly due to some important ele- ments that are absent from the model. Most candidates don’t just care about winning; they also care about the policies they would implement if elected. Also, we assumed that all voters vote but, in fact, voter turnout can well depend on the positions taken. A candidate may take an extreme position to induce vot- ers with similar views to turn out. In spite of these weaknesses of the model, still it delivers the insight that, in trying to win an election, there is a force drawing candidates closer together in their positions. That general tendency is well apt to be true even if it doesn’t result in full policy convergence.
6.3 Solving for Nash Equilibria with Calculus (Optional) IN THIS SECTION, GAMES are considered in which calculus can be used to solve for Nash equilibria. We’ll start with a general treatment of the subject and then move on to a few examples. Some students will prefer this sequence, while others may find it more user friendly to first have a concrete example before taking on the more abstract. My suggestion is to read through this section, but don’t fret if it doesn’t all make sense. If, indeed, that is the case, then, after reading the first example, come back and reread this material.
Recall that a player’s best reply function describes a strategy that maxi- mizes his payoff, given the strategies of the other players. In formally defining a best-reply function, let represent player i’s payoff function in an n-player game. Once you plug a strategy profile into out pops a number which is the payoff that player i assigns to that strategy pro- file. A best reply for player i to other players using is a strategy that maximizes
A player can have more than one best reply to a particular configuration of other players’ strategies; it just means that there is more than one strategy that
Vi(s1, . . . , sn). (s1, . . . , si�1, si�1, . . . , sn)
Vi(s1, . . . , sn), Vi(s1, . . . , sn)
1 2
(x¿D � 12)/2,12
x¿D
xD � 1 2 � xR. FIGURE 6.9 Both Candidates Choosing Is the
Unique Nash Equilibrium
1 2
Vote for D Vote for R
x�D0 1� xR 1 2
x�D �
2
1 2
158 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
generates the highest payoff. In the games analyzed in this section, a player al- ways has a unique best reply. So let
denote the unique best reply for player i, given the strategies chosen by the other players. If we let denote the strategy set of player i, then sat- isfies the following condition:*
BRiSin � 1
BRi(s1, . . . , si�1, si�1, . . . , sn)
*Recall that is read as “ is a member of ”.Sisisi � Si
[6.1]for all si � Si.Vi(s1, . . . , si�1, BRi, si�1, . . . , sn) � Vi(s1, . . . , si�1, si, si�1, . . . , sn)
The strategy profile is a Nash equilibrium when each player’s strategy is a best reply to the strategies of the other players:
A Nash equilibrium is then a strategy profile that satisfies the preceding n equations. Solving for a Nash equilibrium means solving these n equations for n unknowns, There may be no solution, one solution, a finite num- ber of solutions, or even an infinite number of solutions.
To pursue this solution method, each player’s best-reply function must first be derived by solving Equation (6.1) for Suppose, then, that [0,1], so that choosing a strategy means choosing a number from the interval [0,1]. Suppose further that takes the shape shown in FIGURE 6.10. Here we are fix- ing the strategies of all players but i at
and then plotting how player i’s payoff depends on It is not difficult to see that player i’s best reply is
A key property of in Figure 6.10 is that it is hill shaped, with being its acme. Now, here’s the critical observation: The slope of the payoff function at
is zero, as shown by the flat tangent. Generally, when the payoff func- tion is hill shaped, the optimal strategy is the point at which the slope of the payoff function is zero. Let’s convince ourselves of this claim before showing how we can use this property.
Consider a strategy, such as where the slope is positive. A positive slope means that the payoff function is increasing in Thus, cannot be a best reply, because player i earns a higher payoff with a slightly bigger strategy. Now consider a strategy where the slope is negative, such as ins–i
s¿isi. s¿i,
si � s 0 i
s0iVi
s0i � BRi(s¿1, . . . , s¿i�1, s¿i�1, . . . , s¿n).
si.
(s1, . . . , si�1, si�1, . . . , sn) � (s¿1, . . . , s¿i�1, s¿i�1, . . . , s¿n)
Vi
Si �BRi.
s*1, . . . , s * n.
s*n � BRn(s * 1, . . . , s
* n�1).
o
s*2 � BR2(s * 1, s
* 3, . . . , s
* n)
s*1 � BR1(s * 2, . . . , s
* n)
(s*1, . . . , s * n)
6.3 Solving for Nash Equilibria with Calculus (Optional) 159
Figure 6.10. Since the payoff function is decreasing in i’s strategy, cannot be a best reply, because player i earns a higher payoff with a slightly smaller strategy. It is only at the point where the slope is zero that the payoff is maximized.
It is at this stage that we introduce the wonder developed by Gottfried Wilhelm Leibniz and Sir Issac Newton in the late 17th century. Calculus pro- vides an easy way in which to derive the slope of a function. That slope is sim- ply the function’s first derivative, which is denoted symbolically as
By the previous argument, the best reply for player i is defined as the strategy that makes the derivative equal to zero:
More generally, for any strategies of the other players, the best-reply func- tion, is the solution of the following equation:
Remember that this equation determines the best reply if the payoff function for player i is hill shaped. For those who can’t get enough of calculus, a con- dition which ensures that the payoff function is hill shaped is that the second derivative of the payoff function is always negative:
02Vi(s1, . . . , sn) 0s2i
6 0.
0Vi(s1, . . . , si�1, BRi, si�1, . . . , sn) 0si
� 0.
BRi(s1, . . . , si�1, si�1, . . . , sn),
0Vi(s¿1, . . . , s¿i�1, s0i , s¿i�1, . . . , s¿n) 0si
� 0.
0Vi(s1, . . . , sn) 0si
.
s–i
Pa yo
ff
0 1s�i
Vi(s�i, . . . , s�i�1, si, s�i�1, . . . , s�n)
s0i si
s�i�
FIGURE 6.10 The Relationship Between Player i ’s Payoff and when the Strategies of All Players But i Are Fixed at (s1, . . . , si�1, si�1, . . . , sn) � (s�1 . . . , s�i�1, s�i�1, . . . , s�n)
si
160 CHAPTER 6: STABLE PLAY: NASH EQUILIBRIA IN CONTINUOUS GAMES
When this last condition holds, the payoff function is said to be strictly con- cave. If all this is a bit too abstract for you, it ought to become clearer as we look at two examples.
But before moving to these examples, it is worth noting that, with infinite strategy sets, a Nash equilibrium is assured of existing under certain condi- tions. Assume that each player’s strategy set is an interval of real numbers with a lowest value and a highest value—for example, the interval [0,10]. Assume also that players’ payoff functions are smooth (the curves representing them have no kinks), continuous (the curves have no jumps), and hill shaped (tech- nically speaking, the curves are strictly concave). The first two conditions en- sure that the derivative we are seeking exists. Then, under the stated assump- tions, there always exists a Nash equilibrium. Furthermore, if the game is symmetric, then a symmetric Nash equilibrium exists.
Note that the method described here will not work with the games in Section 6.2. To take a derivative, a function has to be differentiable, which requires that it be continuous and have no kinks. The payoff function for the situation of price competition with identical products is not continu- ous: there is a jump when a shop matches the price of its competitor. Although there are no jumps in the payoff functions in the Price-Matching Guarantees game, there is a kink when one shop matches the other’s price. (See Figure 6.6.) Nevertheless, calculus can still be handy even in those games. We’ll observe the truth of the latter statement in the last example in this section, which explores the power of matching grants in generating charitable donations.