“Vote vs. Utility-maximizing Candidates: Comparing Dynamic Models of Elections.” Quality and...

Quality and Quantit.v, 16 (1982) 507-526 Elsevier Scientific Publishing Company, Amsterdam - Printed in The Netherlands

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Vote-Maximizing versus Utility-Maximizing Candidates: Comparing Dynamic Models of

Bi-Party Competition * FRANCOK PETRY

Istituto Universitario Europeo, Badia Fiesolana. San Domenico di Fiesole 50016

Background: Some Fundamental Results in Spatial Voting Games

Building on Hotelling’s famous unidimensional metaphor, Downs developed a spatial model of party competition where admissible policies are located in a one-dimensional Euclidian space so that Black’s single-peaked preference theorem applies. Indeed, this assumption is mentioned explicitly in Downs’ (1957) An Economic Theory of Democracy (p. 116). In addition, Downs’ model postulates that candidates are vote-maximizing (or better, plurality-maximizing). They are interested solely in winning governmental position, and the choice of a particular policy platform is solely a means to that end.

When two parties compete for votes by advocating particular policies and when they have single-peaked preferences for these policies, there is a winning policy-the median-and the candidate who picks that policy is the lucky winner. Downs further showed that the two candidates will eventually both take positions near the median optimum and remain there. This is so because, given the two assumptions in the model, it would not pay either candidate to move away from the median, and if he or she did, there would be an incentive to move back.

Downs’ overarching postulates are sufficient for the existence of a stable equilibrium outcome. The postulates also supply a coherent explanation of bi-party elections. Unfortunately, the conditions that would prompt candidates and voters to accept behavior consistent with Downs’ postulates do not seem to bear sufficiently on the issue of party competition.

The Downsian model lacks plausibility in two main respects. First, it

* A preliminary version of this paper was presented at the European Section of the Public Choice Society Meeting, Badia Fiesolana. Florence, 17 April 1980.

0033-5 177/82/0000-0000/$02.75 0 1982 Elsevier Scientific Publishing Company

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ignores factors such as policy considerations made by the candidates. In- deed, it is straightforward to verify that if candidates have policy preferences, they will not automatically converge to the median optimum. To demonstrate this, assume a one-dimensional party competition where all voters’ preferred positions (designated S,-S,) are situated on one side of one candidate’s preferred position sl,, while the other candidate’s preferred position L$ is situated on the other side of &:

E

51 S2 S3 3 S Y

The utility that each player associates with policy proposals other than those of his or her preferred position decreases the further he or she moves from that position.

Assuming that candidates seek to maximize their own utility rather than to maximize votes, the challenger at each election will seek to win that election only if the winning policy platform is not farther away from his or her preferred position than is the incumbent’s platform. It follows that the candidate whose preferred position is at Sv has no incentive to move further to the left than the point S,, which constitutes the equilibrium outcome in the above example, rather than S,, the median voter’s optimum.

The specious character of the median-voter result has been further high- lighted by recent theorems in Arrovian (Arrow, 1963) social choice. Kramer (1973) has shown that, when the policy space is at least two-dimensional, Black’s theorem almost always fails and intransitivity of majority rule is more than likely. Majority-rule transitivity is only a sufficient condition for the existence of a unique equilibrium outcome (Condorcet winner, core, etc.) in more than one dimension. This leaves the possibility that such an equilibrium point may still exist even though the voting process is intransitive over an unconstrained feasible set of alternatives. Plott (1967), however, showed that the necessary condition for a core to exist is very restrictive. Essentially, Plott’s condition requires that all voters’ most preferred points that are not majority choices can be put into pairs such that for each pair, the circular indifference curves through the majority-winning point are separated by hyperplanes through that point.

The above results indicate that a unique equilibrium outcome is not to be expected in spatial voting games in general. Hence various attempts have been made to characterize plural equilibrium outcomes that do not necessi- tate restrictions on voter’s preference patterns. Buchanan (1968) argued that, although majority rule exhibits cyclic behavior, it does so in an area which is constrained to the Pareto set for all voters. Tullock (1967) conjectured that the cycles tend to move toward the central area in the interior of the Pareto set-the area of voters’ greatest agreement-and remain there.

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Tullock’s elaboration is interesting because it suggests a way out of Arrow’s impossibility theorem. Unfortunately, as shown by McKelvey (1975, 1979), in multidimensional choice problems in general there will be a single majority-rule cycle extending over the entire agenda of social alternatives. Roughly this result states that if voters’ most preferred policies are represented by points in some Euclidian space, then for any pair of policies x and y, a sequence of points (x, x’, x”, . . ., I’) can be found such that each point in the sequence is preferred to the preceding one by a majority of voters. More important, however, the majority-rule trajectory thus generated can reach any point in the feasible space. In other words, if the sequence does reach pointy, it would be by chance only and there would be no a priori reason to suppose that it would stay there. An illustration of a typical majority-rule cycle is given in two dimensions in Fig. 1, from which it can be verified that, beginning at an arbitrary point x, in the space, each subsequent point x, is majority-preferred to the preceding point in the sequence. The majority-rule trajectory soon moves outside the Pareto set itself and could eventually reach any point in the space.

McKelvey’s result applies to unconstrained alternative spaces. This leaves the possibility that, while there is a cyclic sequence over the entire space, majority rule is transitive in some subset of that space (e.g., as defined by budget constraint). Schofield (1978) showed that such conjecture is unwar- ranted. Schofield’s result states that, if preferences are continuous and if for any proposal x in the issue space, all points in some arbitrary neighborhood of x can defeat x through some sequential majority-rule path, then not only x is involved in a cyclic preference-relation but so also is the whole agenda of alternatives.

Suppose that two candidates compete for votes repeatedly over a series of elections by advocating particular policy platforms. The space of competition is multidimensional, so that the “voter paradox” is incorporated. At each election, whichever candidate receives a majority vote is therefore elected to office and enacts the policy advocated in his or her campaign. In the next election, whatever policy has been advocated by the incumbent, the challenger will pick some alternative policy which will be majority-winning. Consequently, the challenger will always win and the two candidates will alternate in office. More important, however, and in view of the above theorems, the majority-rule sequence thus generated will eventually fill the whole feasible space.

The results reported above provide a rather special characterization of the competitive majority-voting process, where no equilibrium can be found. The diagnosis is obtained not so much from actual lack of voters’ consensus as from the impossibility of finding one. The social-choice “system”, which can be characterized as “chaotic” (Schofield, 1980), constitutes an anti-the-

Fig. 1. Typical Majority-Rule Cycling Trajectory.

ory, so to speak. If it is true that any responsive (democratic) election process can result in any outcome, then the view might be taken that it is indeed a lost cause to attempt to develop causal models-or theories-of such a process.

This of course would contradict the common-sense observation that real-world elections do reach some stable outcome, as well as our legitimate desire to explain how such an outcome is reached. Yet while it is true that competitive majority elections display some coherence, the voter-paradox result shows that this cannot be attributed to equilibrium properties that are inherent to the process itself.

In the light of this impossibility result, one direction of scholarship has been to focus on the candidates in elections, seen as independent actors in spatial voting games. Within this context, candidates are conceived as ideal types whose physical existence is only incidental. Candidates are characterized only by some abstract principle of behavior-or by some goal-that will account for the existence of some equilibrium outcome, or at least provide a coherent explanation of the election process. More precisely, the aim is to show that the majority-rule trajectory generated by the competitive election process is “forced” to converge to some equilibrium set in the alternative space.

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At this stage, several points are worth mentioning. The models that will be investigated each produce a set of predicted outcomes, any one of which may form and none of which is more likely to emerge than another. The problem is therefore to make the equilibrium set as small as possible. It is also required that the models provide an understanding of the process by which an outcome actually emerges. In other words, predictions must be motivated by plausible behavioral assumptions. Finally, since Arrovian social choice hardly constitutes a theory, the burden of proof must rely on the proposi- tions of equilibrium models. In other words, equilibrium phenomena must be accounted for and deduced from the assumptions of these models. This implies that the formulation will be axiomatic. By axiom is meant a necessary assumption or minimal expectation about process behavior. Any process that allows a behavior that does not satisfy an axiom violates that axiom.

This paper attempts to compare and contrast two “axiomatic” models of bi-party competition. One (Kramer, 1977) postulates, in the Downsian tradition, that candidates are vote-maximizers, while the other (Wittman, 1973, 1977) assumes that candidates seek to maximize their own utility. Kramer’s model is more complete than Wittman’s. One of our tasks will therefore be to generalize and complement Wittman’s conclusions. Both of the models, although neither is completely axiomatized, do catch some significant and distinctive features of bi-party competition. This is why we have thought that contrasting them would be warranted.

Primitive Notions

Models of party competition must account for an equilibrium at two distinct levels. These are the levels of voters’ choice for candidates, and of party positioning in the competition space. Assuming that in the bi-party case voters vote for the candidate whose platform is closer to their respective preferred position, the question of finding an equilibrium will depend solely upon the spatial positioning of the two candidates.

Let N= { 1, 2,..., n} be the set of voters, where n is odd, and candidates vote for themselves. The platforms are points in A, a subset of Euclidian k-dimensional space A k, where k < n. The Euclidian distance between two points x and y is denoted d( x, y).

Each voter i has a preference ordering Ri over the points in A, representa- ble by a utility function ui. For all i E N and for any alternative x in A, there is a point S, E Rk such that U,(X) =f(d( S,, x)), where f: R --) R is a linear function. S, represents i’s saturation point-the position in Rk he or she most prefers-while the utility associated by i with any alternative in A

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decreases the farther that alternative is from S,. For two dimensions, i’s indifference curves are concentric circles around S,, where the smaller the circle, the higher i’s utility level.

Assume that voter i has a choice between two alternative platforms x and y. Write xR,y if i prefers x to Y or is indifferent between the two alternatives. Write xP,v if xR,y and not yR,x. Let x,( y,) be candidate X’s ( Y’s) platform at election t. The incumbent at each election offers the same platform that he or she offered at the preceding election. The challenger’s platform is winning if a majority of voters strictly prefer that platform to the incumbent’s. Since in general there will be no Condorcet winner, and in view of the formal results above, the incumbent’s platform at any election is likely to be defeated by the challenger at the next election. Voter i votes for candidate X at election t if x,P,y,, votes for candidate Y if JJ,P,~,, and votes for the incumbent if he or she is indifferent between x, and y,.

Vote-Maximizing Candidates

Contrary to the “chaotic” characterization given by social-choice theory, Kramer (1977) showed that the majority-rule trajectory generated by two- party competition will not extend over the entire issue space when candidates maximize votes. To obtain “complete” intransitivity of majority rule, it is necessary to assume that the challenger at any election may choose any policy platform which defeats the incumbent’s platform, irrespective of the margin by which the first defeats the second. This would imply that candidates have zero marginal utility for votes once a bare majority is achieved.

In contrast, the vote-maximizing assumption implies that a challenger seeks to win an election with the greatest possible margin. While there is still unavoidable rotation in office, the assumption generates majority-rule trajectories where each platform is vote-maximizing against the preceding. The set of vote-maximizing trajectories constitutes a proper subset of the initial set of majority-rule sequences. Kramer demonstrated that the vote-maximizing trajectories display some convergence properties and yield a dynamic equilibrium in intransitive party competition. The demonstration was done in two steps: (1) proof of the existence of a minimax set; and (2) characterization of the vote-maximizing trajectory. Kramer’s characterization of the vote-maximizing trajectory, although ingenious, is of technical interest only. Essentially it was shown that the trajectory must approach the minimax set because at any election t + 1, the Euclidian distance between the winning platform x,+ i ( yt+ ,) and the minimax set must be smaller than the distance between y, (x,) and the minimax set at election t.

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The determination of the minimax set is of more interest here. Let point .v represent a platform such that the number of voters who prefer any rival point x is as small as possible. Call y the minimax point and the number of voters who prefer a rival point x the minimax number, n*. If the choice problem is one-dimensional, y is the median and n* < ( n + 1)/2. If )I* = II - 1, then a majority-preference cycle extends to the Pareto set for all voters. Since in the typical case II * > (n + 1)/2, majority rule will be cyclic and there will be no Condorcet winner.

Call the set of points which can be defeated by no more than n* votes the minimax set, M(n*). To find M(n*), it suffices to take the non-empty intersection for the smallest majorities. Thus, in Fig. 2, all points situated in the Pareto set for all the voters, M(4), are vote-maximizing against any point situated outside that set. Consider now M(3), which is constituted by the intersection of the Pareto sets for all four-membered coalitions. M(3) is the smallest non-empty intersection that can be obtained by this method and it constitutes the minimax set toward which the vote-maximizing trajectory converges.

Kramer’s dynamic-equilibrium model can be represented in the form of a two-person, zero-sum, non-cooperative game, for which an equilibrium strategy pair always exists. To demonstrate this, denote candidate X’s admissible platform choices as xi, x2,. . . , and candidate Y’s admissible choices as y, , y,, . . . A pay-off (vote) function u~( x,, yi), k being one or the

k

Fig. 2. Minimax Set for Five Voters.

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other candidate, represents the expected vote for that candidate when he or she chooses xi( y,), as a function of the particularyj (x,) selected by the other candidate at each election. Call a strategy the set of instructions which tell a candidate which admissible choice to select at every election. Let X choose x* from his or her admissible set xi, x2, . . . , and Y choose y* from the set y,, Y,, ... The pair (x*, v*) is said to be an equilibrium strategy pair if

n,(x*,y*) a~~(x~,y*) for all x, E {xi, x2, . ..}

U,(X*,Y*)~u,.(x*,Y,) forally,E {YlrJ$, --.} 0)

Assuming that no abstention is allowed, then the game is constant-sum, and U, = 1 - Ok. Hence, an equilibrium outcome exists if

f-&(x*,uj) -Jx*, v*) 2 n,( xi, v*) for all xi and yj (2)

The game for five voters can be depicted in a matrix where each cell contains the pay-off that accrues to each candidate depending upon the platforms they both select:

Candidate Y

\ y2 y3

Candidate X Xl

x2

X3

Key :

W = Win

L =Lose

For example, choices x2 and y2 by candidates X and Y respectively could be platforms in the Pareto set for all voters, M(4), and both would beat a platform outside that set (say choice xi). Choices x3 and y, could represent platforms that are in the minimax set M(3). They constitute the equilibrium strategy pair, the only pair to satisfy eqn. (2) above.

To be sure, policy preferences may be introduced in the vote-maximizing model without undermining the minimax result. To achieve that result, however, it is necessary to assume that the process can be divided into two independent, sequential stages. In the first stage, candidates maximize votes as if they were indifferent between possible alternative policy proposals. In the second stage, policy preferences are revealed, and as long as the challenger can achieve a margin of victory that is greater than n* in the first

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stage, a vote-policy trade-off can take place whereby the winner sacrifices extra votes so as to offer a policy proposal that he or she prefers.

However, it does not seem plausible to assume that candidates in elections will always offer policy proposals that will ensure them more than n* votes. There is considerable empirical evidence against the assumption that candidates are interested only in winning office, regardless of the content of the policies they offer. Candidates in elections often appear to be seeking to win in order to enact a given policy, rather than offering a policy platform for the sole purpose of winning office. As shown by the long-celebrated example of the unsuccessful Goldwater candidacy in the 1964 U.S. presidential election, the choice of a particular platform by a party need not be the best for winning an election.

The office-seeking assumption also lacks plausibility because its implications do not seem to be verified empirically. For example, one implication of the vote-maximizing assumption is that the candidates’ proposals will converge. This can be taken to mean that the candidates will take up the same positions on policy issues over the long run. This, however, fails to be confirmed by empirical observation. Candidates in recent elections in Eng- land and the U.S. have not proposed similar positions on policy issues.

Finally, the vote-maximizing assumption does not seem to follow from the rationality postulate in all cases. For example, candidates might increase their chances of winning office by not offering vote-maximizing platforms but instead, by maintaining strong ideological links with particular sub-cultures in society. Another variant would be that a rational candidate will prefer to advocate the views of his or her financial backers in a policy platform even if that might entail a loss of votes.

Utility-Maximizing Candidates

Wittman (1973, 1977) introduced a dynamic model where the candidates are concerned above all with the utility they derive from their respective platforms, and where winning is solely a means to that end. To justify this postulate, Wittman used an analogy with the business world. Down’s vote- maximizing model is conceptually similar to a sales-maximizing model, where the winner is the firm which has sold the greatest number of units of a particular good. In contrast, the utility-maximizing model is similar to a profit-maximization model, where winning depends primarily on the price level of each unit of the good that is sold.

In the Downsian model, candidates’ preferences for policies are either unconsequential or they can be considered independently of vote considerations. This, in a sense, ensures that the vote-maximizing majority-rule

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sequence will converge to the minimax set. The class of models introduced by Wittman is of a different nature, however. Here, a candidate’s problem is to select, among the set of admissible platforms, the one which brings him or her the greatest utility. Vote considerations therefore interfere with candidates’ policy preferences at the same stage. The problem then is to characterize this interference in a general, plausible fashion, and to determine whether the majority-rule trajectory thus generated will follow some coherent pattern.

Wittman’s utility-maximizing model assumes that the challenger at each election chooses a winning platform which is preferred by himself or herself to all others. To formalize this, let xi, x2,. . . be the set of winning platforms for the challenger at election t. This set is obviously a function of the particular v, selected by the challenger’s opponent at the preceding election, t-l. Let uk( xjr JQ) represent candidate k’s utility when he or she chooses xi, as a function of the particular yJ selected by the opponent. The largest utility pay-off that candidate k can expect is designated by maxi uk( xi, JJ~). Call the platform that generates this maximum pay-off, utility-maximizing.

What characterizes this model is its myopia. In this respect, Wittman’s utility-maximizing model is comparable to Kramer’s vote-maximizing model. In both models, the challenger is assumed to maximize at each election his or her immediate return (whether it is vote or utility) without regard to the possible consequences in subsequent elections.

It is well known that in spatial voting games where utilities are assumed to be transferable, the “size principle” applies (see for example, McKelvey et al., 1978). At each election, the challenger will seek the support of a minimum winning coalition, i.e., of a winning coalition that would cease to be winning if some voter were taken from it. To impose the transferable-utility condition, the view must be taken that the players compete for some divisible and freely transferable commodity, over which they all possess a linear utility function. A similar approach would be to assume that the players compete for some private good, i.e., a good that is supplied to individual consumers on an exclusive basis, such that the consumption by one consumer of a portion of the good deprives other customers of the benefit of that portion.

Wittman (1973) developed an earlier model in which policy income was represented as a fixed amount of transferable commodity to be divided among the players and where, consequently, the utility-maximizing candidates sought to form minimum winning coalitions. Moreover, since the game was constant-sum, inequality (2) applied and an equilibrium strategy pair existed in that model. It was shown that the resulting income distribution was such that the candidates monopolize most-if not all-of policy income. This version of the utility-maximizing hypothesis is in marked contrast

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with the vote-maximizing thesis. Although both lead to an equilibrium outcome, the two models differ in their implications. In the vote-maximizing model, the voters always end up receiving something from the candidates, so that abstention from voting is a meaningful threat-strategy to be considered by dissatisfied voters. In contrast, the implication of the pay-off distribution version of Wittman’s model is that the voters will get nothing-or very little -no matter what candidate they vote for. Abstention from voting is therefore ineffective. This is a case of “no exit, no voice” (Hirschman, 1970).

Clearly, in the class of spatial games that are considered here, the transferable-utility assumption cannot hold. A more appropriate view would be to conceive of the pay-off values that are received by the players as resource allocations toward preferred policy bundles. A similar approach would be to consider the pay-offs distributed to the players as analogous to public goods. A public good is defined here as having the property that making a portion of it available to one consumer does not necessarily restrict the availability of that portion to other consumers.

If the policies advocated by the candidates are public goods, then the relative utility that the players receive from one policy proposal rather than another is not necessarily related to the size of the coalition that supports that proposal. Instead, players’ utilities are a function of their relative policy-preferences. In general, when two candidates compete for votes in a non-transferable environment, there is no a priori reason to suppose that they will seek the support of minimal winning coalitions of voters. Utility- maximizing elections are a special case, however, where the candidates will indeed seek the support of no more than (n + 1)/2 voters at each election. To see why this is so. consider an initial utility-maximizing election with S,y the preferred position of the majority-winning candidate. The set of points that are strictly preferred by a majority of voters to S, is delineated by (n + 1)/2 voters’ indifference curves with respect to S,. At the next election, the challenger Y, if he or she is to have a chance to win against S,X, will need to move from his or her preferred position &, which is by definition outside of the set of points that are majority-preferred to S,, to the point inside that set cloijest to s,,. In the general case, the distance from the challenger’s preferred position S, to a non-minimally winning intersection will necessarily be larger than the distance to the minimal winning intersection. Two utility-maximizing candidates will therefore seek the support of no more than (n + 1)/2 voters at each election, whether utility is transferable or not.

The contrasting natures of the vote-maximizing and utility-maximizing models are summarized in Fig. 3. For any pair of platforms x and y, define n( x, JJ) as the number of voters who prefer x to y. By varying x over the set of alternatives A, a vote function f( y) is obtained whose value indicates the number of voters who prefer x to y, when v is fixed. A vote-maximizing

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”

n/2

MINIMUM UTILITY

LOSS

Fig. 3. Vote-Maximizing versus Utility-Maximizing Strategies.

candidate will select the platform x which maximizes that value, regardless of the policy distance (utility loss) that separates x from his or her preferred position S,. By contrast, a utility-maximizing candidate will select that platform x’ which is winning against y and such that the distance between S, and x’ is minimized.

As was shown in the introduction, if (in one dimension) the incumbent’s preferred position is situated between all voter’s preferred positions on one side and the challenger’s position on the other, then the incumbent’s preferred position is in the core. This observation can be generalized to more than one dimension. If the incumbent’s preferred position S, is closer in Euclidian distance to the challenger’s preferred position than to any minimum winning intersection of voters’ indifference contours with respect to S,, then S, is a core equilibrium.

As shown by Wittman (1977) when there is a total median in the competition space, that point is also in the core. This is a very special condition, which is equivalent to Plott’s requirement of symmetry of voters’ preferences (see earlier). Given the severity of the restrictions that must be imposed on voters’ preferences in order to guarantee the existence of a total median, it is interesting to investigate the path followed by utility-maximizing platforms over a series of elections, when a total median does not exist.

We first introduce two definitions. Call m the set of (n + 1)/2 most preferred positions in R”. MI” C m is a

minimax set for the challenger (say X) at election t if for every M, #MI”

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there exists a voter j E M, such that d( S,, xr) a d( S,, y,- ,). CallS,EM~pivotalifforeveryS,EM~,d(S,,y,_,)-d(S,,x,)~d(S,,

r?,-,) - d( S,, x,). In other words, a pivotal voter’s preferred position S, is in the minimax set at election t and the distance-differential between the challenger’s and the incumbent’s platforms is not greater for S, than for any other voter’s preferred position in the minimax set MF.

Consider the point S, which represents the pivotal voter’s preferred position at the initial election, in Fig. 4. At the next election, the challenger Y will offer a platform inside voter i’s indifference contour with respect to S,. Note that, although there may be more than one pivotal voter from the initial election to the next, for the subsequent election there will be a unique pivotal voter. As long as S, is pivotal in the game of Fig. 4, the utility-maximizing strategy for the challenger at any election will be to offer a platform along the straight line joining S, (S,) and S,, when the sum of utilities for the challenger and the pivotal voter is at its maximum: u,(x,) = 1 - U-J xl) or u,( y,) = 1 - u.“( y,). As a result, the utility-maximizing sequence starts to converge (if S, were the unique pivot in the whole sequence, there would be a core equilibrium at S, and the sequence would be a Cauchy sequence).

However, a series of winning utility-maximizing platforms will stop converging at election t, when the challenger (say Y) finds that he or she can

s, ‘Xt-i

Fig. 4. Realigning Utility-Maximizing Election.

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maximize his or her utility by enacting a platform ,v~ which is majority-winning against x,-, = x, and such that S, is no longer pivotal between y, and x,: d(Si9 .Yt) ad(Si9 x,). Thus in Fig. 4, candidate Y at election t has a choice between yt and the platform y which is situated in the interior of a circle centered on S,, the initial pivot, with a radius of d(S,, xr). Platform y, although winning against x,, is not Y’s best choice, however, because platform y, is also winning and Y prefers y, to y:

In Fig. 4, the initial converging majority-rule sequence is interrupted at election t, when candidate Y seeks the support of the new pivotal voter j, while candidate X has no incentive to shift from the initial platform path. The shift of trajectory may be a drastic one, depending on the location of the new pivotal voter.

The next question is whether the utility-maximizing sequence will con- tinue to move from one pivotal voter to another, thus generating an endless cycle. In view of the McKelvey and Schofield results above, it is established that in general a cycling trajectory may end up anywhere in the alternative space. Another cycling pattern would be for the trajectory to “orbit” around a subset of Rk. Neither of these conjectures seems warranted, however, because, as is now shown with the help of a simple three-voter, two-candidate example, the utility-maximizing trajectory will move back to its initial path immediately after the realigning election.

First, we show that at election t + 1. there is a platform w = x,+, winning against y, and such that w is situated along the line segment joining Sdy, the challenger’s preferred position, and x,- i.

Consider election t + 1, when X is the challenger in Fig. 4. X’s indifference curve at x,-, is represented by a circle centered on S, with a radius

4L x,-,) < d( S],, S,). Consistent with our observation concerning the strategy of a utility-maximizing candidate, xl-i is located along the line joining S, and S,, so that u,(x~-,) = 1 - u,(x~~,). Call z the intersection of i’s indifference curve at xI-, with the line joining S, and S,, so that u,(z) = 1 - U,,(Z). By construction, for any point on voter i’s indifference curve at x,-, other than z, there exists a pointy on the line joining TV and S, such that yP,z. Thus Y’s utility-maximizing platform at election t must be situated on or inside a circle centered on Sy with radius d(&, z), so that d($ y,) G d(S,, z). By assumption, an indifferent voter votes for the incumbent, therefore it must be the case that y, # z. However, since uI( z) = 1 - U&Z), yI will be situated farther away from S, than is x,-,, i.e., d( S,,

y,) > d( Si, x,_ ,). Consequently, some point in the neighborhood of x,-i along the segment joining S, and x,-i is in the common intersection of i’s indifference curve with respect to y, and X’s indifference curve with respect

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to x,-,9 so that the point w = x,+, will be strictly preferred toy, by i and to X I~, by candidate X.

Consider next j’s indifference curve at x,- ,. Suppose that d( S,, w) < d( S,, x,-,) while d(S,, w) < d(S,, x,~,), in other words, the trajectory that is followed by candidate X’s platforms, including w, prior to the platform .x,-, cuts across j’s indifference curve at x,- ,. By definition, at election t - 1, voter k supports Y’s platform,y,-, =y-,, so that d(Sk,yri,_,) c&S,, x,~,). Now, if S,, were inside the circle centered on S, with radius d( S,, x,- ,), the platform x,- , could not be winning at election cP 1 because this would imply the inequalities d(S,, q,,)<d(S,, x,~,) and d(S,, y,p2)=d(Sk, $,.)<d(S,, x tP,), so that both j and k would support ,v,-* = Sy against .x,-,. S,. is therefore outside the circle centered on S’ with radius d( S’, x,- ,).

By assumption, the platform y, is situated on the line joining SV and S,. To show that the point w = x,+, is inside j’s indifference curve with respect to y,, suppose that j’s preference order for the points x,- ,, -v~ and w is as follows: ytPjWPjXt- ,. A contradiction may now be deduced. Call W’ the intersection of j’s indifference curve at w with the line joining S,, and S,. Since S, is outsidej’s indifference curve with respect to x,- ,, Y’s preference order for the three points is just the opposite ofj’s: x,-, Pvw’Pv y, (see Fig. 5). Surely then, Y could maximize utility further by offering-y,, such that his or her preference order be x,- , Pv y, PYw’, since Y’s goal is only to beat x,- , with the support of voterj. Y being utility-maximizing, it follows that he or she will offer a platform y, at election t such that wP, y, P,x,- ,, the only preference order admissible if x,- ,P,,y,P,.w’. Hence, the point w = x,+, which is already preferred to y, by X and i will also gain the support of j.

Suppose now that d( S,., w) > d( S,, x,- ,), i.e., there is no platform offered by X prior to x,-, which 1s preferred byj to x,- ,. It may be shown that voter k will prefer the point w = x,+ , to yt at election t + 1. If x,- , is preferred by j to any point w on the line joining S, and x,_ ,, S, must be situated outside of the cone formed by the utility-maximizing trajectory converging toward S,,

Fig. 5. y,P,w - wf’,.Y(.

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the initial pivotal voter’s preferred position. This implies that -y( will be situated on the same side of the cone as Sj, since otherwise y, would not be utility-maximizing for candidate Y. Consider now the median line cutting the cone through S, as well as the median line perpendicular to the line joining yf and x,- ,. Sk is necessarily in the intersection of two planes, one being delineated by FV’s side of the first median, the other being delineated by yI’s side of the second median. Consider any point in the intersection of these two planes. That point’s indifference curve at y, has a common intersection with i’s indifference curve with respect to yr such that a portion of the line segment joining S, and x,-, is inside the intersection (see Fig. 6).

Thus, the challenger at election t + 1 will offer a utility-maximizing platform x,+, such that d(S],, x,+i) < d( S,, x,~,). Now since x,+, occupies the same location as-or is in the immediate neighborhood of-a platform which was previously offered by X, it follows that Y will maximize his or her utility at election t + 2 by offering a platform yr+2 such that uu( Y~+~) = 1 - ui( JJ!+~). This completes the cycle (see Fig. 7).

In the game of Fig. 7, x,,, does not quite reach S,. Particular cases can be constructed where the platform which immediately follows the realigning platform is indeed the challenger’s preferred position. In such cases, the initial incumbent is always assured of seeing his or her preferred position

Fig. 6. d(S,, w)<d(S,, xl-,), d(S,, w)c d(S,, Y() and d(h, W)<d(%, A).

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‘i

Fig. 7. Typical Utility-Maximizing Trajectory.

enacted at regular intervals. The initial challenger will also get a chance to see his or her preferred position enacted if the initial incumbent is the challenger at the realigning election.

This result offers an interesting contrast with the vote-maximizing hypothesis. If candidates are interested solely in winning office, their platforms are subjected to a “centripetal” force, i.e., they converge toward the minimax set. Utility-maximizing platforms, by contrast, are prevented from converging because the trajectory that is generated inevitably goes back toward its points of origin.

This centrifugal drive applies generally in the absence of a core or a strict solution. Recall our definition (eqn. (1)) of an equilibrium strategy pair. Once at (x*, y*), neither candidate has an incentive to move to another point if

u,(x*,y*> au,(x,,y*) and u,(x*,y*) )u,(x*,Y~)

If a set of utility-maximizing platforms is to be in equilibrium, each pair of platforms must be such that there is no other strategy pair outside of that set which is strictly preferred to it by both candidates. Formally, an equilibrium

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strategy pair (x, v) is said to be “jointly admissible” if, for any other strategy pair (x’, v’) it is not the case that

In addition, all jointly admissible pairs must be equivalent in the sense that both candidates must be indifferent as to which equilibrium pair prevails. Formally, (x, y) and (x’, y’) are equivalent strategy pairs if

ux(x’, y’) = ux(x, v> and uy(x’, y’) = u,(x, y>

If all jointly admissible equilibrium pairs are equivalent, they constitute a strict solution. If a pair of proposals is in the core then it is a strict solution.

Consider the three-voter game represented in Fig. 8. In this game, the pairs (x, x’) and ( y, JJ’) are not jointly admissible. One candidate will always prefer another pair of outcomes to (x, x’) or (y, v’). In other words, candidates X and Y will offer only platforms x or x’ and y or y’, respectively.

The equivalence requirement of the strict solution severely restricts the voters’ preference patterns. It is also clear that the equilibrium in this game is not robust. Any slight move by one voter (or candidate) will make the equilibrium collapse to an unstable situation similar to that of Fig. 7.

Another interesting contrast may be drawn between the vote-maximizing and utility-maximizing models. Consider the utility-maximizing elections in Figs. 4 and 6. In the case of Fig. 4, platform x,+ r attracts the vote ofj so that voter k supports both platforms yt and Y~+~. As a result, voter i is the unique pivotal voter between platforms xt+i and JJ,+~. In the case of Fig. 6, in contrast, both i and k support platform x,+r and they are both pivotal between that platform and the next, Y,+~. Consequently, two coalition structures are possible:

Election Coalitions Coalitions (Fig. 4) (Fig. 6)

t-l {X, i,j> {X, i,j> t { Y,j, k} { Y,j, k} t-t1 {X id {X, i, k} t+2 {Y,i,k} {Y,i,k}

t-t3 {X, i,.i> {X, i,j>

Note that only four out of the six possible minimal winning coalitions are admissible in the example. The coalitions {X, j, k} and {Y, i, j} are outside the prediction set.

Consider now a five-voter, two-candidate election with the voters dis- played in the usual five-branch star fashion of Fig. 2. The grand coalition is {X, Y, i, j, k, 1, m}. Suppose that X and Y are utility-maximizing candidates

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and that the winning coalitions in the two initial elections are {X, i, j, f} and {Y, i, k, m}, respectively. Note that { i,j, I} and {i, k, m} are not necessarily connected, i.e., the two sets may split the opposition geometrically. Suppose,

sx SY

Fig. 8. The Strong Solution

however, that { i,j, f} and {i, k, m} are connected. Then it is straightforward to verify that the winning coalition at the realigning election (say (j, k, m}) will indeed split the opposition geometrically.

Consider now the same voter configuration with X and Y maximizing votes. Suppose that the vote-maximizing trajectory has converged inside the minimax set so that only minimal winning coalitions will form. Then it is easy to verify that any point y inside the minimax set M(3) can be beaten through majority rule by a point x only if x is supported by a connected coalition.

Interpretation and Further Conjecture

The motivational premise of the utility-maximizing model that we have analyzed is that candidates are concerned above all about their respective policy preferences. The smaller the distance between a candidate’s platform and his or her preferred position, the greater his or her utility. We consider this assumption to be reasonable in various electoral contexts, e.g., when the candidates are more concerned about the symbolic content of their respective platforms than with the policies that can or will actually be enacted. Such “irresponsible” party competition has often been observed, and criti- cized, by political scientists, when parties fail to control policy implementa- tion which is then entrusted to an independent bureaucracy.

The main result in our model is that the trajectory is prevented from converging because there is always a platform along the path of convergence which is beaten through majority vote by a platform enacted previously. This result implies in particular that a policy-oriented candidate always runs the

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risk of losing a large amount of utility, when his or her utility-maximizing platform is beaten by the opponent’s preferred position or by a point in the neighborhood of that position. However, insofar as these utility losses are largely symbolic, they do not undermine the prospect of further utility gains by the candidate in subsequent elections. Hence candidates still have an incentive to offer platforms that minimize the distance to their preferred positions.

The utility-maximizing assumption may also be justified on the grounds of party ideology or special appeal to particular sub-cultures in society. “Doctrinaire” parties, it is often explained, have a tendency to “stay put” because they care more about policy than votes. Our characterization of the utility-maximizing trajectory over a series of elections has been based on a three-voter example only. With many voters, it seems likely that the first realigning election will occur fairly soon in the sequence, so that successive platforms will tend to cluster in the neighborhood of either candidate’s preferred position. The greater the number of voters, the more clustered each candidate’s platforms. This conjecture, if proved true, would provide a nice theoretical confirmation of our intuition-and much empirical evidence- about party competition in a variety of contexts.

References

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285-297. Kramer, G. (1977). “A dynamical model of political equilibrium”, Journal of Economic Theory

16: 310-334. McKelvey, R. (1975). “Intransitivities in spatial voting games”, Mimeo, Carnegie Mellon

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Econometrica 47: 1085-l 113. McKelvey, R., Ordeshook. P. and Weiner, M. (1978). “The competitive solution for N-person

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