Bryan C. McCannon

Elmira College

bmccannon@elmira.edu

Anthony Downs' seminal work, An Economic Theory of Democracy (1957), laid the foundation for the study of the economics of political parties and candidates seeking office. Within his work, conditions are given for a rather interesting result. Under certain ideological conditions the incumbent always loses the upcoming election to the challenger. This is accomplished by the challenging candidate forming a coalition of minority groups. I refer to this result as Downs' Coalition of Minorities Proposition (CMP). Downs argues that under his standard framework, which has since been used extensively in the literature, along with a few additional assumptions, the challenger always wins the election.

This paper achieves three things. First, I provide a formalization of the CMP. I am able to show that Downs' result holds in an even less strict environment than he presented. So long as two of any arbitrarily large number of issues have the required properties then the incumbent is always beaten. I then turn to modifications of his environment to explain the re-election of incumbents. The second accomplishment is that I relax the assumption that the incumbent must announce its platform before the challenger. Under this modification, the incumbent retains power with a high probability. Thus, I provide a rationale for non-commitment of candidates running in an election. Specifically, the use of ambiguity in papers such as Alesina and Cukierman (1990) is an effective action of incumbents. Finally, I consider a restriction of the platform choices of the candidates. Due to the very nature of the issues there may exist a platform that cannot feasibly be implemented. In such a case, the incumbent can select a platform that guarantees the victory. Thus, there exist environments where the incumbent is unbeatable and other environments where the incumbent is beatable.

The turnover of incumbents is a widely studied phenomenon. Studies have shown that an incumbent gains an additional 5-10% of the vote (Gelman and King, 1990). A wide variety of explanations have been given in the literature that predominantly relies on exogenous differences between the incumbent and the challenger.¹  Downs' result stands in opposition to these papers. Only Mayhew (1975) postulates that the incumbency advantage may be due to the selection of platforms, saying that the decreased turnover in the U.S. House of Representatives may be due to the fact that "members have become more skilled at public position-taking on issues" (p. 311). Thus, I formalize Downs' proposition and test its robustness to changes in the environment to provide a rationale for why some incumbents are not re-elected and an additional rationale for why others are.

A Model of an Election with an Incumbent
There are two candidates and a continuum of voters so that no one voter ever determines the outcome of the election. Let X denote the incumbent and let Y denote the challenger. There is a set of binary issues. Index the issues i = 1, 2, ..., N. A stance on each binary issue takes the form of either L or R. A platform consists of a stance on each of the N issues. Consequently, there are 2^N possible platforms. Denote the set of platforms as P. The voters differ on their preferences on the issues. A voter's type is the platform that consists of his or her preferred stance on each issue. Thus, there are 2^N types of voters. Let U_k(p) denote the utility a voter of type k receives from platform p. For simplicity, I assume that voters of the same type receive the same utility from a platform. Each candidate has no preference for the issue but simply receives a payoff of W if elected and L if not (W > L).
 

The timing of events is as follows. First, the incumbent selects a platform. Next, the challenger observes the platform selected and chooses its own platform. Finally, each voter casts one vote for a candidate. Since there are only two candidates from which to select there is no strategic voting and each voter selects the one whose platform gives the highest utility. Thus, I suppress the voting game and consider only the platform selection game between X and Y.² The candidate with the greatest number of votes (the plurality of the votes) wins the election and implements the announced platform. Since Downs treats platform matching separately, I assume that if both candidates select the same platform the incumbent wins the election. I now solve for the Nash equilibrium.
 

Downs' Coalition of Minorities Proposition
The important characteristics of Anthony Downs' model, that I preserve, are that there are two candidates that commit to platforms in sequence, the issues are binary, and that the candidates do not have a preference for the issue that is implemented. Downs gave two conditions that guarantee the challenger always wins the election. They are

(1)           More than one-half of the citizens who vote are in the minority on at least one issue.

(2)           Each citizen who holds the minority view on some but not all issues has a stronger preference for those policies he or she favors when in the minority than for those he or she favors when in the majority.

To formalize Downs' claim let δ_k denote the fraction of the population of voters of type k. Initially, assume that there are two issues and thus four platforms and four types of voters. Let, for example, (LR) denote the platform (and hence type) that prefers L on the first issue and R on the second. Furthermore, without loss of generality, assume that R is the majority stance on each issue, or rather, ½ < δ_RR + δ_RL < 1 and ½ < δ_RR + δ_LR < 1. The first of Downs' statements requires

(1)           δ_LL + δ_LR + δ_RL > 1/2.                                                                                                         

The second condition implies that

(2)           U_LR(LL) > U_LR(RR) and U_RL(LL) > U_RL(RR).

Using this model and conditions (1) and (2) he argues that, in equilibrium, Y wins.

Proposition 1 (Coalition of Minorities Proposition) In all equilibria the challenger wins the election.

Regardless of which platform the incumbent selects there exists another platform that the challenger can select that wins. Consider Figure 1 where each box in the matrix represents a platform selection. If the incumbent selects (RR) then the challenger can announce (LL) and, from (2), voters of type (LR) and (RL) vote for the challenger. This is depicted in Panel (a). Since, from (1), more than one-half of the population prefer a minority stance on an issue the challenger wins. If the incumbent selects either (RL) or (LR), as depicted in Panel (b), the opponent can announce the majority stance on both issues so that the election is decided by voter's preferences for the issue disagreed upon by the candidates. The challenger, who has selected the majority's preferred stance, is victorious. Finally, if the incumbent selects the minority stance on both issues the challenger need only match it on one issue. The voters are split on one issue and the challenger, representing the majority on that issue, wins. This is depicted in Panel (c).

For notational purposes let a strategy for Y be a 4-tuple listing a platform if X selects (LL), (LR), (RL), and (RR) in order. For example, the strategy [(RR), (LL), (LL), (LR)] prescribes Y to announce R on both issues if X selects L on both issues. It calls for Y to select L on both issues if X announces either L on the first and R on the second or R on the first and L on the second. Finally, Y announces L on Issue 1 and R on Issue 2 if X selects R on both.³ The following proves the CMP.

Proof.   If X selects (LL) then Y needs to match on one issue. If Y selects (LR) (or (RL)) the type (RR) voters choose Y and Y receives δ_RR + δ_LR (or δ_RR + δ_RL) votes, which is greater than 1/2. If X selects (LR) (or (RL)) then Y can select (RR) and receive the votes of the type (RL) (or (LR)) voters. Again, this gives Y δ_RR + δ_RL > 1/2 (or δ_RR + δ_LR > 1/2) of the vote. Finally, if X selects (RR) then Y can select (LL). It follows from (2) that both types (LR) and (RL) voters prefer Y. Thus, Y receives δ_LR + δ_RL + δ_LL of the vote, which from (1), is greater than 1/2. Therefore, either [(LR), (RR), (RR), (LL)] or [(RL), (RR), (RR), (LL)] is a best response. Consequently, any profile where Y selects either strategy and X selects any of the four platforms is a Nash equilibrium. Furthermore, it is straightforward to see that any other strategy selected by Y allows for X's victory. Therefore, in all equilibria Y wins. QED

Now consider relaxing the assumption that N = 2. The CMP result not only holds with a greater number of issues but the two required properties need only hold for two of the N issues. The challenger can form a coalition of the minorities on these two issues and match the incumbent on all of the remaining N – 2 issues.

Proposition 2 Suppose N > 2 and there is two issues for which (1) and (2) hold. In all equilibria the challenger wins the election.

Proof. Y can announce either of the strategies from the proof of CMP for the two issues in which (1) and (2) hold and match X on the other N – 2 issues. Given this strategy X is indifferent between every platform making these Nash equilibrium. Furthermore, there is no equilibrium in which X wins the election since Y could deviate to this strategy and guarantee the victory. QED

Therefore, Downs' result guarantees that the challenging candidate is victorious if there exists any two issues for which more than one-half of the population prefers a minority stance on at least one of the two issues and these voters have a greater preference for the issue in which they have a minority stance than the issue in which they agree with the majority.
 

How Can the Incumbent Win?
The CMP is a rather strong result for races between incumbents and challengers. The natural question to ask is how robust is this result to modifications in the environment? I consider two modifications: the leader-follower game to platform selection and the unrestricted domain of platforms. Both extensions allow for the incumbent to retain its office.

Breaking the Leader-Follower Game

The assumption that the incumbent must first commit to a platform before the challenger follows from the belief that an incumbent has an observable record. The incumbent has had at least one term in office to take stances on the various issues. But in some situations the sequential order to the moves may not occur. An incumbent can withhold a stance and stay noncommittal on an issue. A Congressperson may not vote on a bill or act to keep a vote from occurring. Heckelman and Yates (2002) credit secret ballots as a way for incumbents to retain power. By having the vote cast in secret there is no record of individual voting, thus breaking the leader-follower game. Therefore, it is instructive to discuss the model without this requirement.

Consider a modified version of the model where the platforms are selected in a simultaneous-move game and there are two issues. Consequently, there are no pure strategy equilibria. For every announcement one candidate makes there always exists another that can form a coalition of minorities and win. Assume that  those who prefer (RR) have a stronger preference for R on the first issue than R on the second, or rather, U_RR(RL) > U_RR(LR). This is done to simplify the analysis and has no effect on the outcome. It simply labels the first issue as the one that satisfies the inequality for those who prefer (RR).

Proposition 3 In the mixed strategy equilibrium the challenger selects platforms (RR), (LR) and (LL) each with probability 1/3 and the incumbent chooses platforms (RR), (RL) and (LL) each with probability 1/3.

Proof. It follows from the proof of Proposition 1 that the platform (LL) wins an election when the opponent selects (RR), (LR) is victorious over (LL), (RL) beats both (LR) and (LL), and (RR) beats (LR) and (RL). If X selects (LL), (RL), and (RR) with probability 1/3 each then it follows that Y is indifferent between each of the four platforms and the described mixed strategy is a best response. Suppose Y selects (LL), (LR), and (RR) with probability 1/3 each. Since it is assumed that X wins when the two select the same platform it follows that X is indifferent between all four platforms and the described mixed strategy is a best response making this is a Nash equilibrium. QED

As a result, the incumbent wins the election with a probability of 2/3. The advantage to being the follower is the information gained at the time at which a decision made. The strategic advantage given this second-mover allows for an indefensible plan that cannot be beaten by the incumbent. No matter which political decisions the incumbent makes there exists a winning response for the challenger. Without this advantage the incumbent uses his or her well-documented incumbency advantage to retain its position.

Infeasible Platform

The previous work assumes that any combination of stances on the issues is a possible platform for a candidate to select. In practice, there may exist two issues where taking a particular stance on one requires that a stance on another issue cannot be taken. For example, suppose each candidate had to take a stance on two questions: "Would you increase the number of military personnel?" and "Would you institute a draft to force civilians to work in the military to increase the number of military personnel?". Presumably, it would be quite possible for one candidate to be opposed to increasing the size of the military and announce no for both. Another candidate may be willing to do both and answer yes to each. Also, a candidate could reasonably prefer to increase the number of people in the military but be opposed to having required service from civilians. This candidate could answer yes to the first and no to the second. But it is not reasonable to answer no to the first question and yes to the second.⁴ A response of yes to the second question requires that you want to increase the number of military personnel and therefore have a response of yes on the first question. This platform is not a feasible choice for a candidate.

Consider a modification to the original model. Suppose that N = 2 and let the set of platforms, P′, be all the possible combinations of a stance of L or R on each issue except for one particular platform, z, that cannot be chosen, or rather, P′ = P\{z}. The incumbent selects a platform in P′ and then the challenger selects after observing X's choice. In this new game the incumbent always wins the election. If a platform has the support of a majority of the population then the incumbent selects it. If not, the incumbent selects the "centrist" platform forcing the challenger to take an extreme position. Voters at the other extreme vote for the incumbent who wins the election.⁵

Proposition 4 Regardless of which platform is infeasible the incumbent wins the election.

Proof. Given the infeasibility of the platform z two of the platforms in P′ are in direct opposition to each other, that is, for each issue one platforms takes the stance of L and the other takes the stance of R. The third platform in P′ is centrist in that the platform is in agreement with each platform on one and only one issue. If a majority of the population supports a particular platform then the incumbent selects this platform. If no one platform has a majority support then the incumbent selects the centrist platform. The challenger selects another platform and the incumbent receives the votes of the platform in opposition to the challenger and wins. This holds for any platform z. Thus, in all equilibria, the incumbent wins. QED

The intuition behind this result is that, to form a coalition of minorities necessary to defeat the incumbent, a platform made up of stances, each of which appeals to a particular minority group, is essential. The incumbent, by strategically selecting its platform, can use the infeasibility of a platform to deter the formation of such a coalition.

It is important to point out the role the assumption that if both candidates select the same platform the incumbent wins, has on the result. If, instead, the challenger won such a tie with a positive probability then it would choose to match the incumbent and win with a positive probability. The result of Proposition 4, though, is that infeasibility of a platform leads to the incumbent winning. This continues to be possible, which is the point of this extension.
 

Conclusion
This paper formalizes the claim made by Anthony Downs that under certain ideological conditions the challenging candidate always defeats the incumbent by constructing a coalition of minority groups. This requires that there exists two issues for which more than one-half of the voters prefer a minority stance on at least one of the issues and that these voters have a stronger preference for this issue. I modify the environment to explain the re-election of incumbents. If the leader-follower game is altered to a simultaneous announcement of platforms the incumbent wins with a probability 2/3. Also, if there exists a platform that is technologically infeasible the incumbent can guarantee the victory.

There are quite a few avenues for future research. The model does not examine some possible objections to the results such as analyzing the effect of entry/exit in the political race or voter uncertainty. Candidate reputation and partisan bias, as illustrated by Van Houweling and Sniderman (2005), likely affect the behavior of voters. Also, the issues of abstention, strategic voting, and voter participation are not considered. The motivation of citizens to vote is an important research area, especially in Downsian models where voters are assumed to always vote for the most similar candidate and ignore any strategic considerations. The results here are found in an environment of complete information where voter preferences are fully known. Extending the work to incomplete information settings where the candidates might differ in their information may provide interesting insights. The model does provide a rationale for the ambiguity and non-commitment of politicians on certain issues while for other issues the politicians race to explain their stances early and thoroughly.

Notes

1.            There are many examples of explanations for the incumbency advantage in the literature. Votes received may be a function of previous election vote totals (Gelman and King, 1990), secret ballots alleviate vote buying opportunities for challengers (Heckelman and Yates, 2002), there are perceived quality differences between incumbents and challengers (Cox and Katz, 1996), there are differences in probability to winning that result in more campaign contributions (Baron, 1989), incumbents are able to generate more surplus for their constituents (Mao, 2001), and incumbents are able to offer more services (Snyder, 1990).

2.            This is equivalent to considering only strict Nash equilibrium in the voting stage, which is a common refinement in the literature.

3.            One may note that conditions (1) and (2) create a situation where there is no Condorcet Winner to the platforms selected.

4.            Another interpretation is that there is an issue with three possible choices: one centrist stance and two extreme stances.

5.            With two issues and the infeasibility of one platform this is, in effect, a unidimensional, spatial voting model where only three positions on the line can be selected. This is similar to the typical model associated with Downs and first introduced by Hotelling (1929).

References

Alesina, A. and A. Cukierman, 1990, The politics of ambiguity. Quarterly Journal of Economics 55, 829-850.

Baron, D., 1989, Service-induced campaign contributions and the electoral equilibrium. Quarterly Journal of Economics 104, 45-72.

Cox, G. and J. Katz, 1996, Why did the incumbency advantage is U.S. House elections grow? American Political Science Review 40, 478-497.

Downs, A., 1957, An economic theory of democracy (Harper and Row, New York).

Gelman, A. and G. King, 1990, Estimating incumbency advantage without bias. American Journal of Political Science 34, 1142-1164.

Heckelman, J. and A. Yates, 2002, Incumbency preservation through electoral legislation: the case of the secret ballot. Economics of Governance 3, 47-57.

Hotelling, H., 1929, Stability in competition. Economic Journal 39, 41-57.

Mao, W., 2001, On the inconsistent behavior in voting for incumbents and for term limitation. Economic Theory 17, 701-720.

Mayhew, D., 1975, Congressional elections: the case of the vanishing marginals. Polity 6, 295-317.

Snyder, J., 1990, Campaign contributions as investments: the U.S. House of Representatives, 1980-1986. Journal of Political Economy 98, 1195-1227.

Van Houweling, R. P. and P. M. Sniderman, 2005, The political logic of a Downsian Space. Working Paper 2005'44, Institute of Governmental Studies, University of California, Berkeley.