Pareto optimality of policy proposals with probabilistic voting*

PETER C O U G H L I N * *

Abstract This paper studies the Pareto optimality properties of policy proposals and electoral outcomes when there is probabilistic voting. Theorem i proves that, when the position of one candidate is taken as fixed, the other candidate will propose a Pareto optimal alternative. This implies that whenever there is an electoral equilibrium (in pure strategies) the electoral outcome is Pareto optimal. It also implies that, even if there is no such equilibirum, the electoral outcomes from a sequence of elections will be Pareto optimal (except, possibly, for an initial status quo).

1. Introduction Economic and political theorists have long been concerned with whether or not outcomes from social processes are Pareto optimal. This has included analyses of electoral competition and majority rule (e.g., see Shubik, 1968, 1970; Ordeshook, 1971; or Riker and Ordeshook, 1973). In analyses of electoral competition and majority rule, there have been two types of electorate which have been of primary interest. First, there is the electorate with 'deterministic' voting. In this case, each individual always votes for the candidate whose policy proposal offers him the greater utility level. Then, there is the electorate with 'probabilistic' voting. In this case, there is a probability that a citizen, who is drawn from a collection of individuals with a common utility function or ideal point, will vote against the candidate whose proposed policy has the greater utility for him or will abstain. This second formulation of an electorate's behavior incorporates random factors, non-policy influences, indifference and alienation into voters' choices.1 Ordeshook (1971) has proven that outcomes from electoral competition with deterministic voting must be Pareto optimal when there is a static * This research was supported by the National Science Foundation and by Office of Naval Research Grant No. ONR-N00014-79-C-0685. I would also like to acknowledge helpful comments and suggestions from Ken Arrow, Mel Hinich, Shmuel Nitzan, Ken Shepsle, Kotaro Suzumura and two anonymous referees. ** Department of Economics, University of Maryland at College Park, College Park, M D 20742. Public Choice 39:427-433 (1982) 0048-5829/82/0393-0427501.05. © 1982 Martinus NijhoffPublishers, The Hague. Printed in the Netherlands.

428 equilibrium in pure strategies for the candidates. Subsequently, Kramer (1977) has proven that, when there is no equilibrium in pure strategies, electoral outcomes from a sequence of elections will tend to move toward a particular subset of the society's set of Pareto optimal alternatives (viz., the 'minmax set'2). Nonetheless, policy proposals and electoral outcomes in such a trajectory may well be outside the Pareto set - even after the minmax set is reached (this is easily shown by the example in Appendix A). This paper, alternatively, studies the Pareto properties of outcomes from electoral competition with probabilistic voting. In particular, it is concerned with probabilistic voting which is (1) utility-based and (2) minimally responsive to candidates' policy proposals (as in the references cited in note 1). Theorem 1 establishes that any expected-vote or expected-plurality maximizing candidate who is challenging a fixed opponent will always propose a Pareto optimal policy. This, in turn, implies (1) that every outcome is Pareto optimal when there is a static equilibrium (like Ordeshook, 1971), and (2) once an initial status quo is replaced, every outcome from a sequence of elections is Pareto optimal (unlike Kramer, 1977).

2. Elections with probabilistic voting S will denote the set of feasible social alternatives. 3 An electorate is specified by (i) an index set A such that, for each e E A, there is an associated utility function, U,: S ~ R+, and (ii) a probability measure space, 4 (A, d , #). A two-candidate contest with (utility-based) probabilistic voting is specified by (i) an electorate, (ii) two candidates, indexed by i e {1,2}, and (iii) probabilistic voting functions,

P~(OI, Oz) = P~(U~(O0, U~(02))

(1)

defined on S x S, for each ie{1,2} and s e A . 0, (with iE{1,2}) denotes a policy proposed by the ith candidate. Pia(O1,02) denotes the probability that a citizen, who is randomly drawn from the collection of individuals indexed by ~, will vote for candidate i when 0~ and 02 are proposed. P°(O~, 02)= 1 P~(O1, 02) - P2(01, 02) is the probability that the citizen abstains. Hinich, Ledyard and Ordeshook (1973) carefully discussed the behavioral heuristics which should be expressed in assumptions about voters' choices. Aggregate assumptions which correspond to the individualistic assumptions in their paper are -

(i) For each i ~ {i, 2} and ~ ~ A, P~ is a strictly monotone increasing function of U,(Oi), and (ii) For each i~ {1, 2} and ~ ~ A, P~ is a monotone decreasing function of U~(Oj) f o r j e {1,2},j :p i.

429 We also assume (with essentially no technical restriction) that (iii) For each i s {1, 2} and (0~, Oz) ~ S x S, P~(01, 02) is an integrable function of c~(with respect to (A, d , #)). When these three assumptions are satisfied, we say that there is responsive probabilistic voting. These assumptions express the idea that voters' behavior should be minimally responsive to changes in candidates' proposals which would lead to changes in voters' utilities. This response might be nothing more than a change in the likelihood of abstaining. It should be observed that if the strict monotonicity of (1) is replaced by simple monotonieity then deterministic voting is included. The example in Appendix A immediately rules out certain Pareto properties in this case. We therefore focus here on the remaining case with (i) satisfied.

3. Pareto optimality properties The following definitions are drawn from Hildenbrand (1974: 236): The Pareto relation, R, is defined by (x, y)~ R <:~ U~(x) > U~(y) a.e. 5 in A. An alternative, x e S, is said to be in the Pareto set if and only if x is a maximal element for R. Of course, any point that is in the Pareto set is also in the weak Pareto set (defined with (x, y)~ R w <:~ U~(x) > U~(y) a.e. in A).

Theorem: Suppose that there is a two-candidate contest with responsive probabilistic voting. Then, whenever the proposal of one candidate is taken as fixed, any expected-vote or expected-plurality policy proposed by the other candidate must be in the Pareto set. The proof of this theorem is in Appendix B. When there is a finite set of voters, this result has a close connection to traditional Welfare Economics. In this case, the 'challenger's' expected plurality or expected vote payoff function (see Appendix B) is a BergsonSamuelson Social Welfare Function. Therefore, since the maximal elements of such functions must be in the society's Pareto set, this fact can provide us with an alternative proof for this special case. 6 For expositional convenience, we will let FI(Oj, 02) and F2(Oa, 02) denote the payoff functions of candidates 1 and 2, respectively. These could be expected vote or expected plurality functions. Then, (0~,02)~S * * x S is an electoral equilibrium (in pure strategies) if and only if

FI(O~,O~) <_ FI(O~,O*),

VO~S,

V2(O*, 02) <_ F2(O~,0~), V 02 ~ S.

and

430

Corollary 1: Suppose that (0~',0") is an electoral equilibrium when the candidates maximize expected votes or expected pluralities. Then both 0* and 0* (and also the electoral outcome) are in the Pareto set. Such equilibria have been shown to exist only when special radial s y m m e t r y or concavity conditions have been included (see the references cited in note 1). W e therefore also consider outcomes from sequences of elections. K r a m e r (1977) developed a dynamical model for studying sequences of electoral outcomes when there is deterministic voting. The analogous d y n a m ical process of policy formation when there is an electorate with probabilistic voting is as follows, s o ~ S will denote a feasible initial status quo with some incumbent in office. In each election (in a sequence of elections), the incumbent defends the status quo, while a challenger proposes an expected-vote or plurality maximizing alternative. This leads to a trajectory of electoral outcomes, (st) with t = 0, 1,2 .... We say that so is replaced once the initial incumbent loses for the first time.

Corollary 2: Suppose that (st) is a trajectory of electoral outcomes when there is minimally responsive probabilistic voting. Then every electoral outcome which occurs after So has been replaced is in the Pareto set.

Appendix A The following example is drawn from Riker and Ordeshook (1973: 372): Consider a society with three voters (or three groups of voters of equal size) whose ideal points are given by a, b and c respectively (in Figure 1) and whose indifference contours are concentric circles around their ideal points.

X

Figure 1. Two-dimensional policy space.

431 Suppose that an incumbent is defending his policies at z. This point is in the society's minmax set. Now, it is possible for a challenger to maximize his plurality (at + 1) and his votes (at 2) by suggesting any alternative in one of the shaded regions. These positions are also all outside of the Pareto set. Furthermore, z may be followed by z~, then z2, then z3, then z4, all of which are outside of the Pareto set.

Appendix B Proof of Theorem: We begin by obtaining a useful property for points that are not in the Pareto set (PS), x e PS ¢~ gy e S: (y, x) ~ R or (x, y) ~ R. Therefore, x ~ PS ~ 3y ~ S: (y, x) ~ R and (x, y) ~ R, (x,y)~R ¢~ [3A1 e d : #(A1) > 0 and U~(x) < U~(y) a.e. in All. Therefore, x ~ P S if and only if 3y~S: [U,(y) _> U~(x) a.e. in A] and [3A 1 ~¢:/~(AI) > 0 and U~(y) > U~(x) a.e. in All. Now suppose, without any loss of generality, that candidate 2 has the fixed position, z ~ S. Then the expected votes which the two candidates could get at l's possible alternative proposals are given by

EV~(01, z) = fa P~(01' z)" dNa), i ~ C.

(1)

Suppose that the candidate proposes an x e X which is not in the Pareto set. Then there must be some y ~ S such that (y, x) E R. For this y we have U,(y) >_ U~(x) a.e. in A and ~A 1 e z¢ such that #(A0 > 0 and U~(y) > U,(x) a.e. in A1. Therefore by Assumption (i), P~(y, z) > P~(x, z) a.e. in A and P~(y,z) > P~(x,z) a.e. in At. Therefore,

EV~(y,z)= fA P~(y,z).d#(a) + fA\A P~(y,z)'dp(a)

(2)

= ~V~(x z).

I.e., candidate 1 did not maximize his expected vote. The expected pluralities which candidate 1 could get are given by

EPII(01, z) = EVt(01, z) -- EV2(01, z).

(3)

Now, Assumption (ii) implies p2(y, z) < p2(x, z) a.e. in A. Hence, EV2(y, z) < EV2(x, z). Therefore, by (2),

EPIl(y, z) = EVI(y, z) -- EV2(y, z)

(4)

> EVl(x,z) - eVz(x,z)

= EPII(x, z). I.e., candidate 1 did not maximize his expected plurality. Hence, the theorem follows by contradiction

Q.E.D.

432 It should be noticed that this theorem applies to all situations in which the mobile candidate is able to achieve a maximum for his objective (i.e., expected-vote or expected-plurality) function.

NOTES 1. For analyses of electoral competition with the second type of electorate (as well as references for the deterministic voting literature), see, for instance, Hinich and Ordeshook (1969, 1971), Hinich, Ledyard and Ordeshook (1972, 1973), Riker and Ordeshook (1973), McKelvey (1975), Comaner (1976), Denzau and Kats (1977), Hinich (1977), Kramer (1978), Coughlin (1979), Austen-Smith (1981) and Coughlin and Nitzan (1981 a, 198 lb). 2. Let the preferences be given for a society. Let N(x, y) be the number of voters who prefer y to x. Let v(x) = max n(x, y). yes

Then the number, v* = min v(x) is called the society's minmax number. The set {x E S: v(x) = v*} xeS

3. 4. 5.

6.

is the society's minmax set. This set may be finite or infinite. By using a probability measure space we are able to include both discrete and continuous distributions of indices (as well as many other cases). We say that a proposition holds almost everywhere (or a.e.) if there exists a subset A o E A with #(Ao) = 0 such that the proposition holds on the complement, A\Ao. I.e., the proposition fails to hold, at most, on a set of measure zero. The interpretation of(x, y) e R is that everyone in the society (except, possibly, those in a set with measure zero) thinks that x is at least as good as y. I would like to thank an anonymous referee for this observation. I have used the more general proof in Appendix B because the properties of Bergson-Samuelson Social Welfare Functions for societies with measure spaces of agents have not yet been determined.

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