# Electoral Promises as a Commitment Device

## Introduction

This chapter extends the basic political-competition model in the Downsian tradition (Downs, 1957) by considering partially binding platforms, which suppose that although a candidate can choose any policy, there is a cost for betrayal.^{1} The policy to be implemented is affected by, but may be different from, the platform because of this cost, which increases with the degree of betrayal. In the model specified in this chapter, it is assumed all players have complete information; therefore, there no uncertainty.

The model of partially binding platforms presents the following two implications. First, although it has been difficult to show asymmetric electoral outcomes in previous frameworks, the model of partially binding platforms can show that candidates with asymmetric characteristics can and will choose different platforms and policies to be implemented. This is because, if their characteristics differ, one candidate may have a greater incentive to win - and would actually win - the election. As a result, an electoral outcome is asymmetric in equilibrium when two candidates have different characteristics.

Second, in existing frameworks, it has been difficult to explain why a candidate runs for an election even though he/she may lose in a two-candidate model. By contrast, the model of partially binding platforms shows that even though a candidate is aware that he/she will lose, he/she may not deviate by withdrawing and runs in order to induce the opponent to approach the median policy and thus, the loser’s ideal policy.

## The model

### Setting

The policy space is R. There is a continuum of voters, and their ideal policies are distributed on some interval of R. This distribution function is continuous and strictly increasing, which means there exists a unique median voter’s ideal policy (the median policy), *x,„.* Assume that this distribution is symmetric and single-peaked about *x,„.*

Suppose there are two potential candidates, and each decides whether to run for office.^{2} Denote *x,* as the ideal policy of potential candidate (or voter) *i.* If a candidate wins, he/she will obtain a benefit from holding office, *b >* 0, which is not related to the ideal policy. However, candidates do have to pay a cost for running, *к* > 0.

In the second period, each candidate announces a platform, denoted by *Zj* £ R. On observing the available platforms, voters can ascertain correctly the policy to be implemented by each candidate as they have complete information. Based on the (expected) policy to be implemented, all voters cast their votes according to the plurality rule; that is, the candidate with the most votes wins. Note that voting is sincere, and I rule out weakly dominated voting strategies. In the last period, the winning candidate, /, decides on the actual policy to be implemented, denoted by *Xi-*

The voter and candidate experience a disutility if the implemented policy differs from their ideal policy. In line with Calvert’s (1985) study, this disutility is represented by -/?н(|£-л_{(}|), where *x* represents the policy implemented by the winner. Assume that w(.) satisfies *u(0) = 0, u'(d)>* 0, and *u"(d)>0* when *d>* 0. The level of political motivation is *p* g (0,~), in which a higher or lower /3 means a candidate is more policy motivated or more office motivated, respectively. Without loss of generality, for now, I assume *ft =* 1 for both candidates. I discuss the case in which candidates have different *p* values later.

If the implemented policy is not the same as that of the platform, the winning candidate incurs a cost of betrayal. The function describing this cost is Яс(|г,- *- Xi)-* Assume that c(.) satisfies c(0) = 0, c'(0) = 0, c'(r/)>0, and c'(d)>0 when *d>* 0. Here, a>represents the relative importance of betrayal. In the last period, the winning candidate chooses a policy that maximizes —«(|^ — лу|)—Яс— _^|). Denote

*Xi* (c,) = argmax^ [-и (|^ - ,v,-1) - Яс (|г,- - )]. Therefore, if the candidate runs and wins, the utility is (г,-)-л-_{(}-|)-Ас(|г,- *- Xi* (-, )|)+— A:.

If the candidate runs but loses, the utility is 1 assume

*b>k.* In other words, potential candidates have an incentive to run if they will definitely win by announcing their ideal policy as their platform (i.e., *Zj =Xj = Xi* (-/) and -м(|луАс(|л,--a,|) = 0). Furthermore, assume that if no candidate enters the election, all obtain a payoff of-~, as in Osborne and Slivinski's (1996) study. As I assume *b > k,* even if a status-quo policy is introduced, at least one candidate will enter the race. Hence, the position of a status-quo policy does not matter.

The equilibrium concept is a subgame perfect equilibrium. I restrict the analysis to a pure strategy equilibrium. I also concentrate on the typical case in which one candidate’s ideal policy is to the left of the median policy, *x*_{m}, while that of the other candidate is to the right. Here, the candidate whose ideal policy is to the left of the median policy is denoted as candidate L, and the other is candidate *R* (i.e., *X]_ m <хц).* In summary, the timing of events is as follows.

- 1 Two potential candidates decide whether to run. If no candidate enters the election, all voters and potential candidates obtain a payoff of -M.
- 2 The candidates who decide to run announce their platforms.
- 3 Voters vote. The candidate with the most votes wins. If only one candidate runs, this candidate wins with a probability of 1.
- 4 The winning candidate chooses the policy to be implemented.

### Policy implemented by the winner

First, the scenario after period 2, that is, the no-entry model in which the two potential candidates have already decided to run, is analyzed. I ignore the cost of running because it is a sunk cost at this stage.

In the last period, the winning candidate implements the policy that maximizes the utility after a win, —^{М}(|ЛГ/ (-/)—Л71) — At*(| =_{x}- - *Xi* (-/)|), given _{=i}.

**Lemma 2.1**

*Consider that u"(d) >* 0, *for any d >* 0. *In equilibrium, Xi* (-;) *satisfies*

*given =j ** лу. *If X goes to infinity, Xj* ) *converges to zIfX goes to zero, Xi(=i) converges to Xj.*

The policy to be implemented will lie somewhere between the platform policy and the ideal policy, as shown in Figure 1.1 in Chapter 1. When *X* increases, the policy that the winning candidate chooses to implement approaches the platform policy. Similarly, when *X* decreases, the implemented policy approaches the ideal policy. If the policy a candidate chooses to implement lies closer to the median policy than that of the opponent, this candidate is certain to win.

There are three additional implications. First, if the disutility function is linear, given platform r,-, the winner may prefer to implement *Xj* rather than *Xi{ ^{:}i* which satisfies (2.1). Here, I denote

*ud) = Tt>*0, which is constant for all

*d*> 0 because и(.) is a linear function.

**Corollary 2.2**

*Consider that u"(d) =* 0, *for all d>* 0. *Then, given* г, *Ф* лу, *if X is sufficiently low such that X < U* / с'(|г,- - *the winner implements* лу. *Otherwise, the winner implements Xi{=i). which satisfies (2.1).*

**Proof: **See Appendix 2.А.1.

However, a candidate never chooses *z _{(}* ^лу and

*Xi(*hi equilibrium. This is because, in committing to лу, it is better to choose

^{=}i)^{= x}i*Zj = Xi{=i) = Xj,*as there is then no need to pay the cost of betrayal. Thus, if a decision on c, is included in the analyses, then in equilibrium, a candidate will either choose

*Xi{*which satisfies (2.1), or

^{:}i*Zj = Xi{=j) =*as Corollary 2.7 will show. Thus, this boundary case is trivial.

^{x}i,Second, if a candidate’s platform approaches his/her own ideal policy, the cost of betrayal and the disutility from winning decreases. In other words, if a candidate compromises more toward the median voter, his/her expected utility from winning decreases.

**Corollary 2.3**

*As Zj approaches* лу, w(| ДГ/ (-/) - лу|) *and* c(|r_{(}- - *Xi{=i* )|) *decrease.*

**Proof: **See Appendix 2.A.2.

Third, if the benefit from holding office *b* is very large, candidates are less concerned about the cost of betrayal, and hence, the policies they choose to implement converge to the median policy. That is, both candidates will implement the median policy, as in the basic Downsian model. Denote *=,(x) = ХГ ^{1}(х)>* such that candidate

*i*implements

*/*when he/she announces platform

*zfix)*where

*=j(x) * X-*

**Lemma 2.4**

*Ifb* > Яс(|с, *(x _{m}*) -

*for both i = L and R, both candidates announce Zj*(.v,„)

*and implement x*

_{m}in equilibrium.**Proof: **See Appendix 2.A.3.

This result is less interesting, and hence, I assume that at least one candidate has *b* < Я,с(|г,(л„,)-л„,|), in what follows. Note that with asymmetric characteristics, even if one candidate, /, has *b* > Я,с1|г, (л„_{)})- -л>,А he/she may not commit to implementing the median policy when the other candidate, *j,* has /> < Я_{;}- c (| ry(*x*_{m})-*x*_{m}| j because *i* can win even if Г s policy does not converge to the median policy.