# A nonequilibrium analysis of unanimity rule, majority rule, and Pareto.

I. INTRODUCTION

Voting theory is important for understanding economics as well as politics. International organizations vote on trade regulations, legislatures vote on minimum wage, and governments vote on the provision of public goods. Voting also affects industrial organization. For instance, limited partnerships may vote on the allocation of resources to various units and shareholders may vote on investment decisions (Carrera and Richmond 1988). What voting rule best achieves Pareto-optimal outcomes in such cases?

In the absence of transactions costs, it is widely believed that unanimity rule is the best voting rule for promoting Pareto efficiency (Berggren 1996; Buchanan and Tullock 1962; Mueller 2003). Buchanan (1967) writes, "it is evident that [unanimous consent] is the political counterpart of the Pareto criterion for optimality" (p. 285). Traditionally viewed, unanimity rule guarantees more efficient outcomes than majority rule because unanimity rule only passes proposals that make everyone better off. Other voting rules, like majority rule, can pass proposals that make some individuals worse off.

One of the assumptions that is either explicit or implicit in the these studies is the assumption of rational proposals. We define a proposal as rational if it helps maximize the proposer's utility at the end of the game. In our analysis, proposals are not rational. Instead, they are generated either randomly or sincerely in a single-dimensional voting model and evaluated by the pareto criterion or pareto optimality. This produces two striking results with regard to Pareto optimality. First, if proposal generation is random, then majority rule is usually more likely to select a Pareto-optimal outcome than unanimity rule. Second, if individuals propose their ideal points, then majority rule always selects Pareto-optimal outcomes at least as well as unanimity rule.

Considering all possible proposals, as done in our random generation models, has a long tradition in social choice literature (Arrow 1951; Caplin and Nalebuff 1988; Niemi and Weisberg 1968; Riker 1982; Sen 1979). It also has the advantage of modeling certain types of incomplete information, modeling votes where bills are introduced for reasons other than making them pass (see Stewart 2001, pp. 338-41), modeling exogenously generated proposals, and for allowing mistakes by the proposer.

Such foundational work should be of interest to those who relate unanimity rule to Pareto principles in the study of public goods (Cornes and Sandler 1996; Lindahl 1967), political parties (Aldrich 1995), and legislative institutions (Colomer 2001; Niou and Ordeshook 1985; Tsebelis 2002), to name a few.

II. PREVIOUS STUDIES

Perhaps the first in-depth study of the relationship between unanimity rule and a Pareto principle was Buchanan and Tullock's Calculus of Consent (1962). Buchanan and Tullock argued that if decision-making costs were negligible, unanimity rule would always be the most desirable k-majority rule, simply because it preserved the Pareto principle (p. 88). Although the exact Pareto principle they had in mind is not entirely clear, it seems that Buchanan and Tullock were arguing for the criteria of accepting a change from the status quo only if it lead to a Pareto improvement; otherwise the status quo should be preferred (pp. 92-93; Sen 1979, p. 25). (1)

Mueller (2003, pp. 138 and 140-41) further compared unanimity rule to majority rule and concluded that unanimity rule typically performs better in terms of the Pareto criterion. To illustrate his reasoning, Mueller describes a community voting on both the purchase of a new firehouse and the taxes needed to pay for it in a single bill. If the decision was made under unanimity rule, the distribution of taxes would be redefined until the firehouse benefited all. Otherwise, the bill would not pass. If the decision was made under majority rule, a majority might distribute the taxes on a minority such that members of the majority would be net gainers and members of the minority would be net losers. Buchanan and Tullock (1962) and Tullock (1998, pp. 70-74) offer similar examples.

With regard to Pareto optimality, Mueller (2003, pp. 138-42), Brennan and Buchanan (2000, pp. 151-55), and Johnson (1991, pp. 157-61) claim that unanimity rule is better at preserving Pareto optimality than majority rule because unanimity rule only accepts changes that are Pareto improvements. Mueller writes, "Should an initial proposal fail to command a unanimous majority, it is redefined until it does, or until it is removed from the agenda" (2003, p. 138). Majority rule, in contrast, might redistribute resources and choose Pareto-suboptimal outcomes.

More recent studies have focused on the pareto optimality of unanimity rule in equilibrium (Chen and Ordeshook 1998; Tsebelis 2002), compared unanimity rule to Pareto improvements in free market exchange (Sobel and Holcombe 2001), analyzed the ability of unanimity rule to obtain a Pareto-optimal Lindahl equilibrium (Cornes and Sandler 1996), and analyzed the distance required for unanimity rule to select any alternative in the Pareto-optimal set (Colomer 1999).

For example, Chen and Ordeshook (1998) study the relationship between unanimity rule and Pareto-superior sets over a series of binary choices. They find that a Nash equilibrium always exists under unanimity rule that is at the intersection of the Pareto-superior set and the Pareto-optimal set. In contrast, we compare unanimity rule to majority rule and examine models in which equilibria are not necessarily obtained.

III. CLARIFYING CONCEPTS

To study the ability of unanimity rule and majority rule to select Pareto-preferred and Pareto-optimal alternatives, it will be technically convenient to use a spatial voting model--as done by Chen and Ordeshook (1998), Tsebelis (2002), and Colomer (1999). To focus the study, consider N individuals with ideal points [I.sub.i] in a single dimension of alternatives ranging from 0 to 1, inclusive. Ideal points reflect preferences such as spending on a public good, like defense. The bounds on this interval reflect the set of feasible alternatives. For a public good this might range from no expenditures at the minimum to the size of the budget constraint at the maximum.

Furthermore, assume that each individual has single peaked and symmetric utility. These assumptions imply that individuals prefer alternatives closer to their ideal points more than alternatives farther away. They also limit admissible preferences and eliminate the possibility of vote cycles. Although this setting may seem restrictive, recent empirical studies suggest that a single dimension may be more appropriate than multiple dimensions for the study of legislatures (Poole and Rosenthal 1997). Also assume no transactions costs, no vote trading, and sincere voting (unless otherwise specified).

Voting in our analysis proceeds by pairing the initial status quo q against a proposal [x.sub.1] in round 1. The winning alternative in round 1 is then paired against a new proposal [x.sub.2] in round 2, [x.sub.r] in round r, and so on, for a total of R rounds. Binary choices, like these, usually result from institutions such as legislative procedures or referenda.

Two voting rules can be compared in this context:

DEFINITION 1. Simple unanimity rule: alternative x passes by simple unanimity rule if and only if # {yeas} [greater than or equal to] 1 and # {nays} = 0; otherwise q is chosen. (2)

DEFINITION 2. Simple majority rule: alternative x passes by simple majority rule if and only if #{yeas} > #{nays}; otherwise q is chosen.

Because we introduce no other version of these voting rules, (3) we will refer to them to as unanimity rule and majority rule for the remainder of the work. They are evaluated by two Pareto concepts.

DEFINITION 3. The Pareto criterion. For any two alternatives y and z, y is Pareto preferred to z if and only if it makes at least one individual better off than z and no individual worse off than z (Sen 1979, p. 21).

For any two alternatives, we define Pareto preferred as in Definition 3; we refer to alternative z as Pareto dispreferred to y if y is Pareto preferred to z; and we refer to the two alternatives as Pareto indeterminate if neither y is Pareto preferred to z nor z is Pareto preferred to y.

DEFINITION 4. Pareto optimality. Alternative y is Pareto optimal if there does not exist an alternative z that is Pareto preferred to y (Sen 1979, p. 21).

In a single-dimensional model, the Pareto-optimal set consists of all the alternatives ranging from the smallest ideal point to the largest ideal point, inclusive.

IV. THE PARETO CRITERION

To put the literature in context, consider Mueller's (2003) description of a community voting on a firehouse. The same claims about majority rule and unanimity rule may also apply to limited partnerships voting on the allocation of resources, an international organization voting on trade regulations, and so on. Mueller (2003) claimed that under unanimity rule individuals would redefine the bill until no one objected to it. Otherwise the bill would not pass. An example of this can be illustrated in Figure 1, where there are three voters (a, b, and c), two proposals (x and y), and the status quo q. If someone proposed x, it would be defeated under unanimity rule. Hence, in Mueller's scenario it would be irrational to propose x. It would be better to propose another alternative, like y, because it would pass under unanimity rule. In this case, y is pareto preferred to q and unanimity rule has produced a Pareto-preferred choice.

[FIGURE 1 OMITTED]

If, in contrast, the decision was made under majority rule, a majority might pass a bill that made some individuals worse off, as in the move from q to x in Figure 1. This move would produce a Pareto-indeterminate outcome.

Now consider our framework. In our framework, the two voting rules consider the same proposal in each round, but they may consider different status quos in rounds r > 1. Again, unanimity rule typically outperforms majority rule in terms of the Pareto criterion:

THEOREM 1. For N [greater than or equal to] 3 and R > 1. Fix any q and any stream of proposals. If majority rule makes a Pareto-preferred choice after R rounds of voting, then so does unanimity rule. Furthermore, there exists a q and a stream of proposals for which unanimity rule makes a Pareto-preferred choice but majority rule does not. (4)

Proof. See the appendix.

The conclusion in Theorem 1 is fairly consistent with the literature, but for different reasons. Theorem 1 is based on the raw mechanics of majority rule and unanimity rule in multiple rounds of voting. Claims in the literature are largely based on an assumption of rational proposing given a particular voting rule. The different assumptions have little effect on the ability of majority rule and unanimity rule to adhere to the Pareto criterion. However, they do affect Pareto optimality.

V. PARETO OPTIMALITY

Comparing unanimity rule and majority rule in terms of Pareto optimality produces more surprising results. In a single-dimensional context, the Pareto optimal set is identical to the set of alternatives that cannot be beaten under pairwise unanimity rule (otherwise known as the unanimity rule core). In addition, the alternative that cannot be beaten under pairwise majority rule is the alternative at the ideal point of the median voter (Black 1948). Hence, in equilibrium, majority rule and unanimity rule both select a Pareto-optimal outcome with a probability of 1.

The comparison becomes more interesting, however, if we consider choices that may or may not be in equilibrium. Out of equilibrium behavior occurs when ideal points change due to an election, an environmental change alters preferences, or new information changes opinions. For example, terrorist acts, ecological disasters, the signing of a free trade accord, and the aging of a constituency can all change a legislator's preferences for public goods. Also note that out of equilibria status quos are widely analyzed in the literature (Hinich and Munger 1997; Stewart 2001; Tsebelis 2002).

There are two interesting phenomena within this framework. The first is a retentive force. We show this in Figure 2, where there are three voters: a, b, and c. If the choice is between q and x, then majority rule selects the proposal x, and unanimity rule selects the status quo q. Although neither alternative is Pareto preferred in this case, it is clear that unanimity rule has preserved Pareto optimality and majority rule has not. The second phenomena is an attractive force (see Figure 3). In this case, the choice is between a Pareto-suboptimal status quo q and a Pareto-optimal proposal x. Majority rule chooses the Pareto-optimal proposal, and unanimity rule chooses the Pareto-suboptimal status quo. Again, neither alternative is Pareto preferred, yet in this case majority rule has preserved Pareto optimality and unanimity rule has not.

[FIGURES 2-3 OMITTED]

The reasoning behind these two phenomena can be generalized in the following theorem. Assuming a randomly drawn status quo and a randomly drawn proposal in a single round of voting,

THEOREM 2. If q is Pareto optimal, then unanimity rule is at least as likely to select the Pareto-optimal alternative as majority rule. If q is Pareto suboptimal, then majority rule is more likely to select the Pareto-optimal alternative than unanimity rule.

Proof See the Appendix.

Because the status quo may or may not be an element of the Pareto-optimal set, the natural question is which voting rule is more likely to select a Pareto-optimal alternative when all possible ideal points, status quos, and proposals are considered. In other words, which is more important: the retentive force or the attractive force?

Experiment 1: A Uniform Distribution of Ideal Points and R = 1

To answer the question, consider the following probability experiment, conducted with the aid of a C program. In this experiment, ideal points are randomly drawn from a uniform probability distribution on [0, 1] such that [inverted A]i, [I.sub.i] [member of] [0, 1]. Alternatives q and x are randomly drawn from the same distribution, and R = 1.

For each trial in the experiment, the program randomly draws ideal points and alternatives, it calculates majority rule and unanimity rule winners, then it determines the Pareto optimality of each winner. If either voting rule chooses a Pareto-optimal outcome, it is assigned a success in the trial. The process is then repeated to determine the relative frequency that each voting rule chooses a Pareto-optimal alternative in a single round of voting. In a large number of trials, (5) this frequency should approximate the true probability in the population.

Two interesting findings result from this experiment (see Table 1). First, regardless of the number of individuals, majority rule is always more likely to select the Pareto-optimal outcome than unanimity rule. The reason for this is roughly as follows. Majority rule is more likely to change suboptimal outcomes to optimal ones than unanimity rule, as shown in Theorem 2. Furthermore, by the law of large numbers the ideal points will be fairly evenly distributed and the expected value of the median will be 0.5, which is typically far from the Pareto-suboptimal set. Under these circumstances, majority rule is unlikely to change Pareto-optimal alternatives to Pareto-suboptimal alternatives. Hence, majority rule outperforms unanimity rule.

Second, Table 1 suggests that the probability of each voting rule selecting the Pareto-optimal outcome increases as N increases. This occurs for at least two reasons: (1) as N increases, the probability that the Pareto-optimal set covers the entire interval approaches 1; and (2) as N increases, the standard error of the median decreases, making the drawn median closer to the expected value of 0.5. The latter makes majority rule less likely to choose a Pareto-suboptimal proposal over a Pareto-optimal status quo.

Experiment 2: Two Alternatives and a Skewed Population

To determine the robustness of these results and to analyze more realistic cases, we conduct a second series of probability experiments on several skewed distributions of ideal points. For each trial in these experiments, a status quo and a proposal are randomly drawn from the [0, 1] interval as before. However, ideal points are restricted to a prespecified range, such as [0.1, 0.2] [union] [0.7, 1.0]. This is done by setting an ideal point at each end of the range(s) and randomly drawing the remaining ideal points from within the range(s).

The results of these experiments are shown in Table 2. In each of these cases, majority rule is more likely to select a Pareto-optimal outcome than unanimity rule. Again, this is because majority rule benefits more from its attractive property than unanimity rule from its retentive property. In the first two cases, population 1 and population 2, the expected value of the median is at the midpoint of the ideal point range. Majority rule is unlikely to move from an optimal point to a suboptimal point in these cases. In the latter two cases, population 3 and population 4, the expected value of the median is not at the midpoint of the range. Surprisingly, however, it is close enough for majority rule still to be favored. If the distribution of ideal points were extremely skewed within the ideal point range, such as if the median voter was at the ideal point with the greatest value, then unanimity rule would usually outperform majority rule in terms of its ability to select Pareto-optimal outcomes. For example, suppose all ideal points lie within [0.0, 0.2] and [I.sub.med] = 0.2, where [I.sub.med] is the ideal point of the median voter. Then unanimity rule is more likely to select the Pareto-optimal outcome than majority rule. This is because majority rule is equally likely to bring proposals into the Pareto-optimal set as unanimity rule (because they have the same pivotal voter). At the same time, majority rule may move alternatives out of the Pareto-optimal set. Unanimity rule will not.

With small perturbations on the location of the median, however, majority rule again outperforms unanimity rule. For the range of ideal points [0.0, 0.2], a case that should favor unanimity rule, majority rule outperforms unanimity rule if [I.sub.med] [less than or equal to] 0.17 (see Table 3). Similar results were found for other populations where the median was skewed. This suggests that majority rule is more likely to select Pareto-optimal outcomes than unanimity rule even in fairly skewed populations.

Experiment 3." A Series of Binary Choices

To see if these results are an artifact of a single round of voting (two alternative choice) and to present a more realistic setting that better represents decision making in assemblies, we further considered what would occur if R > 1.

Mueller (2003, pp. 138 and 140-141) argued that if an assembly had enough time, unanimity rule would always be more capable of selecting Pareto-optimal outcomes than majority rule. However, the following series of probability experiments suggest a different conclusion if proposing is not rational.

In these experiments, a status quo and a proposal are randomly drawn in round 1. The winner of round I becomes the status quo in round 2 and is voted on against a new randomly drawn proposal in round 2, and so on. The ability to select Pareto-optimal outcomes is based on the Pareto optimality of the alternative chosen in round R. The results are summarized in Table 4.

Again, majority rule is more likely to select the Pareto-optimal alternative than unanimity rule, regardless of the population of ideal points considered or the number of rounds of voting. There two reasons for this. First, unanimity rule frequently chooses the initial status quo, which can be Pareto suboptimal. (6) Second, as a consequence of a corollary to the median voter theorem (Hinich and Munger 1997, p. 36), the distance between the outcome selected by majority rule and the median voter is nonincreasing with each subsequent round. In practice, this distance, when averaged over many trials, decreases fairly quickly (see Figure 4).

[FIGURE 4 OMITTED]

As the distance approaches zero, any outcome is increasingly likely to be Pareto optimal, particularly if the median is near the center of the ideal point range. Thus, for each round, the advantage of majority rule in terms of the attractive force exceeds the advantage of unanimity rule in terms of the retentive force. This advantage is enhanced for larger N because the median tends to converge to the center of the Pareto-optimal set, as N increases. (7)

Sincere Proposing and Restricted Status Quos

We make two additional assumptions for the next refinement of the probability experiment:

a. individuals propose their ideal points.

b. the status quo is typically in the Pareto-optimal set.

The first assumption is exactly what would occur if individuals proposed alternatives sincerely. The second assumption might occur if the status quo was randomly drawn from a normal distribution centered on the Pareto-optimal set. Note that the second assumption favors unanimity rule. Despite this we have Theorem 3.

THEOREM 3. Given assumptions a and b, majority rule is at least as likely to select the Pareto-optimal alternative as unanimity rule, in R rounds.

The proof of this theorem is quite simple. Because only ideal points are proposed, all proposals must be in the Pareto-optimal set. Hence the only cases that distinguish the performance of majority rule and unanimity rule are the cases where q is pareto suboptimal. By Theorem 2, these cases favor majority rule. The argument for Theorem 3 clearly carries over to multiple dimensions and also applies to cases with complete information.

Strategic Voting

Finally, we consider a theorem about rational proposing. Comparing this theorem to Theorem 3 highlights the differences between sincere and rational proposing. Assume (1) an arbitrary individual is designated the proposer, (2) the proposer and voters both behave strategically (i.e., players choose a set of actions that maximize their utility at the end of the game), (3) R [greater than or equal to] 1 rounds of voting, and (4) complete information.

THEOREM 4. Given assumptions 1-4, the outcomes in subgame perfect equilibrium are Pareto optimal for both majority and unanimity rule.

Proof. See the Appendix.

Theorem 4 also holds if individuals vote sincerely.

VI. CONCLUSION

This article shows that if there are at least two rounds of pairwise voting, then unanimity rule outperforms majority rule in terms of the Pareto criterion--consistent with the literature.

Less consistent with the literature are our findings with respect to Pareto optimality. We find that majority rule is more likely to select Pareto-optimal outcomes than unanimity rule when alternatives are randomly drawn from a single dimension. This is still true if the distribution of preferences are fairly skewed and if multiple alternatives are considered over a series of pairwise votes. Furthermore, we find that majority rule is at least as likely to select the Pareto-optimal outcome as unanimity rule if individuals propose their ideal points. In contrast, majority rule and unanimity rule perform equally as well if individuals propose rationally in a sequential equilibrium. All this suggests that the widely assumed relationship between unanimity rule and Pareto optimality needs to be treated more carefully.

For instance, unanimity rule is an ideal voting scheme for maintaining Pareto-optimal outcomes; standard game theory makes this clear (Colomer 2001, pp. 71-73). However, our nonequilibrium analysis shows that majority rule is better at obtaining Pareto-optimal outcomes than unanimity rule. Because the attractiveness property is more potent than the retentiveness property, majority rule outperforms unanimity rule in our model.

Institutional framers that want to obtain Pareto-optimal outcomes will want to consider these differences before they impose unanimity rule as a method of finding optimal quantities of public goods, the best allocation of resources in a limited partnership, or the most efficient rules for assemblies. After all, obtaining Pareto-optimal outcomes should be as much of a concern as maintaining them.

APPENDIX

THEOREM 1. For N [greater than or equal to] 3 and R > 1. Fix any q and any stream of proposals. If majority rule makes a Pareto-preferred choice after R rounds of voting, then so does unanimity rule. Furthermore, there exists a q and a stream of proposals for which unanimity rule makes a Pareto-preferred choice but majority rule does not. (8)

Proof. To see the first part, let [x.sub.j] be the first proposal that is Pareto preferred to q. Because unanimity rule passes a proposal if and only if it is Pareto preferred, unanimity rule passes [x.sub.j]. If unaminity rule passes any other proposals subsequently, then they must be Pareto preferred to [x.sub.j, hence Pareto preferred to q. Therefore, if a Paretopreferred alternative is proposed, unanimity rule must pass a Pareto-preferred alternative in R rounds of voting. The second part of Theorem 1 can be illustrated using Figure 1. If x is randomly drawn as the proposal in round 1, x will defeat q by majority rule, but it will not defeat q by unanimity rule. As a result, x is the new status quo for majority rule, and q is the new status quo for unanimity rule in round 2. If y is then proposed against the updated status quo, x will be selected by majority rule and y will be selected by unanimity rule. Because y is Pareto preferred to q, but x is not Pareto preferred to q, unanimity rule does a better job of selecting a Pareto-preferred alternative than majority rule in this case.

THEOREM 2. If q is Pareto optimal, then unanimity rule is at least as likely to select the Pareto-optimal alternative as majority rule. If q is Pareto suboptimal, then majority rule is more likely to select the Pareto-optimal alternative than unanimity rule.

Proof. The first statement is trivial. The second statement can be shown as follows. There are two cases where a Pareto-optimal x is offered against a Pareto-suboptimal q: one where q > [I.sub.max] and the other where q < [I.sub.min] (where [I.sub.max] and [I.sub.min] are the maximum and minimum ideal points, respectively). Consider the former. In this case, majority rule chooses a Pareto-optimal alternative iff [absolute value of x - [I.sub.med]] < [absolute value of q - [I.su b.med]]. Likewise, unanimity rule chooses the Pareto-optimal alternative iff [absolute value of x - [I.sub.max]] < [absolute value of q - [I.sub.max]]. In this context, two things can occur:

(A1.a) [I.sub.min] [less than or equal to] [I.sub.med] [less than or equal to] x [less than or equal to] [I.sub.max] < q, or

(A1.b) [I.sub.min] [less than or equal to] x < [I.sub.med] [less than or equal to] [I.sub.max] < q.

In the case of (A1.a), a majority will always choose x over q. However, unanimity rule will not choose x over q iff [absolute value of x - [I.sub.max]] > [absolute value of q - [I.sub.max]]. This implies that majority rule is more likely to select the Pareto-optimal alternative for this case. In the case of (A1.b), if majority rule chooses q, then unanimity rule must choose q. If majority rule chooses x, then unanimity rule chooses x only if [absolute value of x - [I.sub.max]] < [absolute value of q - [I.sub.max]], which may not occur. This again implies that majority rule is more likely to select the Pareto-optimal alternative than unanimity rule. By symmetry, the same holds true for q < [I.sub.min].

THEOREM 4. Given assumptions 1-4 the outcomes in subgame perfect equilibrium (SPE) are Pareto optimal for both majority and unanimity rule.

Proof. For an outcome to be chosen in SPE, it must be a Nash equilibrium in the final round. Hence, the proof focuses on all possible status quos in the final round. Here we adopt the following notation: [q.sub.R] is the status quo in round R, [I.sub.p] is the proposer's ideal point, [W.sub.m](q) is the winset of q under majority rule, [W.sub.u](q) is the winset of q under unanimity rule, and [x.sup.*] is the alternative proposed by the proposer.

First, consider majority rule. Assume [I.sub.p] [less than or equal to] [I.sub.med]. The case of [I.sub.p] > [I.sub.med] is shown similarly. There are three possibilities: (a) [q.sub.R] < [I.sub.p], (b) [I.sub.p] [less than or equal to] [q.sub.R] [less than or equal to] [I.sub.med], and (c) [q.sub.R] > [I.sub.med]. In case (a), we have [I.sub.p] [member of] [W.sub.m](q), so the proposer will propose [x.sup.*] = [I.sub.p]. In case (b) q is already Pareto optimal, and there exists no winning proposal that will improve the outcome for the proposer, hence the proposer will offer an unwinnable proposal. In case (c), the proposer will propose [x.sup.*] [member of] [W.sub.m](q) such that [absolute value of [x.sup.*] - [I.sub.p]] is minimized. Clearly, [x.sup.*] [member of] [[I.sub.p], [I.sub.med]], hence it is Pareto optimal.

Second, consider unanimity rule. In the final round, q is either: (a) in the Pareto-optimal set, (b) less than the Pareto-optimal set, or (c) greater than the Pareto-optimal set. If (a), then q prevails (which is pareto optimal). If (b), then two cases can occur: (i) [I.sub.p] [member of] [W.sub.u](q), in which case the proposer proposes [x.sup.*] = [I.sub.p]; or (ii) [I.sub.p] [not member of] [W.sub.u](q), in which case [I.sub.p] is further to the right than all of [W.sub.u](q). The proposer proposes [x.sup.*] [member of] [W.sub.u](q) such that [absolute value of [x.sup.*] - [I.sub.p]] is minimized. This proposal must be Pareto optimal because [I.sub.min] < [x.sup.*] [less than or equal to] [I.sub.p]. Case (c) follows by symmetry.

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(1.) Buchanan and Tullock (1962) wrote, "The welfare-political-economist approach indicates that a specific choice is Pareto-optimal only if all parties reach agreement" (p. 94). At the very least, this blurs the distinction between Pareto optimality and unanimity rule. Also see Brennan and Buchanan (2000, p. 151).

(2.) Simple unanimity rule is used for unanimous consent agreements in the U.S. Senate, for nonprocedural decisions among permanent members of the UN Security Council, and for taxation issues in the Council of the European Union, to name a few.

(3.) The simple definition of unanimity rule and majority rule implicitly drops abstention in the tally (Dougherty and Edward 2004a, 2004b; Riker 1982). With the exception of Theorem l, the same results apply for absolute unanimity rule and absolute majority rule, which count only affirmative votes and implicitly treat abstentions the same as votes against the proposal.

(4.) If R = 1, then majority rule and unanimity rule are equally likely to satisfy the Pareto criterion. This is because for R = 1 both voting rules consider the same pair of alternatives, and any Pareto-preferred alternative will be selected by both voting rules.

(5.) With 1 million trials, we are 95% confident that the true probability is within 0.001 of the relative frequency reported.

(6.) For example, in the case of ideal point range = [0.0, 0.2], N = 5, and R = 20, unanimity rule remains at the initial status quo with a probability of 0.224. Majority rule remains at the initial status quo with a probability of 0.047.

(7.) Preliminary analysis suggests that Experiment 3 applies to two dimensions as well.

(8.) Although majority rule can select a Pareto-dispreferred alternative in R > 2 rounds of voting, this possibility is ruled out in our model by the assumptions of single peaked and symmetric preferences. These assumptions imply that alternatives must be closer to the median voter for them to pass under majority rule. Hence, the median voter must prefer q [less than or equal to] [x.sub.1] [less than or equal to] [x.sub.2] ... [less than or equal to] [x.sub.r]. Because at least one individual has these preferences, it is impossible for [x.sub.r] to be Pareto dispreferred to q.

KEITH L. DOUGHERTY and JULIAN EDWARD *

* We thank an anonymous referee for useful comments. Dougherty: Assistant Professor, Department of Political Science, University of Georgia, Athens, GA 30602. Phone 1-706-542-2989, Fax 1-706-542-4421, E-mail dougherk@uga.edu

Edward: Associate Professor, Department of Mathematics, Florida International University, Miami, FL 33199. Phone 1-305-348-3050, Fax 1-305-348-6158, E-mail edwardj@fiu.edu

Voting theory is important for understanding economics as well as politics. International organizations vote on trade regulations, legislatures vote on minimum wage, and governments vote on the provision of public goods. Voting also affects industrial organization. For instance, limited partnerships may vote on the allocation of resources to various units and shareholders may vote on investment decisions (Carrera and Richmond 1988). What voting rule best achieves Pareto-optimal outcomes in such cases?

In the absence of transactions costs, it is widely believed that unanimity rule is the best voting rule for promoting Pareto efficiency (Berggren 1996; Buchanan and Tullock 1962; Mueller 2003). Buchanan (1967) writes, "it is evident that [unanimous consent] is the political counterpart of the Pareto criterion for optimality" (p. 285). Traditionally viewed, unanimity rule guarantees more efficient outcomes than majority rule because unanimity rule only passes proposals that make everyone better off. Other voting rules, like majority rule, can pass proposals that make some individuals worse off.

One of the assumptions that is either explicit or implicit in the these studies is the assumption of rational proposals. We define a proposal as rational if it helps maximize the proposer's utility at the end of the game. In our analysis, proposals are not rational. Instead, they are generated either randomly or sincerely in a single-dimensional voting model and evaluated by the pareto criterion or pareto optimality. This produces two striking results with regard to Pareto optimality. First, if proposal generation is random, then majority rule is usually more likely to select a Pareto-optimal outcome than unanimity rule. Second, if individuals propose their ideal points, then majority rule always selects Pareto-optimal outcomes at least as well as unanimity rule.

Considering all possible proposals, as done in our random generation models, has a long tradition in social choice literature (Arrow 1951; Caplin and Nalebuff 1988; Niemi and Weisberg 1968; Riker 1982; Sen 1979). It also has the advantage of modeling certain types of incomplete information, modeling votes where bills are introduced for reasons other than making them pass (see Stewart 2001, pp. 338-41), modeling exogenously generated proposals, and for allowing mistakes by the proposer.

Such foundational work should be of interest to those who relate unanimity rule to Pareto principles in the study of public goods (Cornes and Sandler 1996; Lindahl 1967), political parties (Aldrich 1995), and legislative institutions (Colomer 2001; Niou and Ordeshook 1985; Tsebelis 2002), to name a few.

II. PREVIOUS STUDIES

Perhaps the first in-depth study of the relationship between unanimity rule and a Pareto principle was Buchanan and Tullock's Calculus of Consent (1962). Buchanan and Tullock argued that if decision-making costs were negligible, unanimity rule would always be the most desirable k-majority rule, simply because it preserved the Pareto principle (p. 88). Although the exact Pareto principle they had in mind is not entirely clear, it seems that Buchanan and Tullock were arguing for the criteria of accepting a change from the status quo only if it lead to a Pareto improvement; otherwise the status quo should be preferred (pp. 92-93; Sen 1979, p. 25). (1)

Mueller (2003, pp. 138 and 140-41) further compared unanimity rule to majority rule and concluded that unanimity rule typically performs better in terms of the Pareto criterion. To illustrate his reasoning, Mueller describes a community voting on both the purchase of a new firehouse and the taxes needed to pay for it in a single bill. If the decision was made under unanimity rule, the distribution of taxes would be redefined until the firehouse benefited all. Otherwise, the bill would not pass. If the decision was made under majority rule, a majority might distribute the taxes on a minority such that members of the majority would be net gainers and members of the minority would be net losers. Buchanan and Tullock (1962) and Tullock (1998, pp. 70-74) offer similar examples.

With regard to Pareto optimality, Mueller (2003, pp. 138-42), Brennan and Buchanan (2000, pp. 151-55), and Johnson (1991, pp. 157-61) claim that unanimity rule is better at preserving Pareto optimality than majority rule because unanimity rule only accepts changes that are Pareto improvements. Mueller writes, "Should an initial proposal fail to command a unanimous majority, it is redefined until it does, or until it is removed from the agenda" (2003, p. 138). Majority rule, in contrast, might redistribute resources and choose Pareto-suboptimal outcomes.

More recent studies have focused on the pareto optimality of unanimity rule in equilibrium (Chen and Ordeshook 1998; Tsebelis 2002), compared unanimity rule to Pareto improvements in free market exchange (Sobel and Holcombe 2001), analyzed the ability of unanimity rule to obtain a Pareto-optimal Lindahl equilibrium (Cornes and Sandler 1996), and analyzed the distance required for unanimity rule to select any alternative in the Pareto-optimal set (Colomer 1999).

For example, Chen and Ordeshook (1998) study the relationship between unanimity rule and Pareto-superior sets over a series of binary choices. They find that a Nash equilibrium always exists under unanimity rule that is at the intersection of the Pareto-superior set and the Pareto-optimal set. In contrast, we compare unanimity rule to majority rule and examine models in which equilibria are not necessarily obtained.

III. CLARIFYING CONCEPTS

To study the ability of unanimity rule and majority rule to select Pareto-preferred and Pareto-optimal alternatives, it will be technically convenient to use a spatial voting model--as done by Chen and Ordeshook (1998), Tsebelis (2002), and Colomer (1999). To focus the study, consider N individuals with ideal points [I.sub.i] in a single dimension of alternatives ranging from 0 to 1, inclusive. Ideal points reflect preferences such as spending on a public good, like defense. The bounds on this interval reflect the set of feasible alternatives. For a public good this might range from no expenditures at the minimum to the size of the budget constraint at the maximum.

Furthermore, assume that each individual has single peaked and symmetric utility. These assumptions imply that individuals prefer alternatives closer to their ideal points more than alternatives farther away. They also limit admissible preferences and eliminate the possibility of vote cycles. Although this setting may seem restrictive, recent empirical studies suggest that a single dimension may be more appropriate than multiple dimensions for the study of legislatures (Poole and Rosenthal 1997). Also assume no transactions costs, no vote trading, and sincere voting (unless otherwise specified).

Voting in our analysis proceeds by pairing the initial status quo q against a proposal [x.sub.1] in round 1. The winning alternative in round 1 is then paired against a new proposal [x.sub.2] in round 2, [x.sub.r] in round r, and so on, for a total of R rounds. Binary choices, like these, usually result from institutions such as legislative procedures or referenda.

Two voting rules can be compared in this context:

DEFINITION 1. Simple unanimity rule: alternative x passes by simple unanimity rule if and only if # {yeas} [greater than or equal to] 1 and # {nays} = 0; otherwise q is chosen. (2)

DEFINITION 2. Simple majority rule: alternative x passes by simple majority rule if and only if #{yeas} > #{nays}; otherwise q is chosen.

Because we introduce no other version of these voting rules, (3) we will refer to them to as unanimity rule and majority rule for the remainder of the work. They are evaluated by two Pareto concepts.

DEFINITION 3. The Pareto criterion. For any two alternatives y and z, y is Pareto preferred to z if and only if it makes at least one individual better off than z and no individual worse off than z (Sen 1979, p. 21).

For any two alternatives, we define Pareto preferred as in Definition 3; we refer to alternative z as Pareto dispreferred to y if y is Pareto preferred to z; and we refer to the two alternatives as Pareto indeterminate if neither y is Pareto preferred to z nor z is Pareto preferred to y.

DEFINITION 4. Pareto optimality. Alternative y is Pareto optimal if there does not exist an alternative z that is Pareto preferred to y (Sen 1979, p. 21).

In a single-dimensional model, the Pareto-optimal set consists of all the alternatives ranging from the smallest ideal point to the largest ideal point, inclusive.

IV. THE PARETO CRITERION

To put the literature in context, consider Mueller's (2003) description of a community voting on a firehouse. The same claims about majority rule and unanimity rule may also apply to limited partnerships voting on the allocation of resources, an international organization voting on trade regulations, and so on. Mueller (2003) claimed that under unanimity rule individuals would redefine the bill until no one objected to it. Otherwise the bill would not pass. An example of this can be illustrated in Figure 1, where there are three voters (a, b, and c), two proposals (x and y), and the status quo q. If someone proposed x, it would be defeated under unanimity rule. Hence, in Mueller's scenario it would be irrational to propose x. It would be better to propose another alternative, like y, because it would pass under unanimity rule. In this case, y is pareto preferred to q and unanimity rule has produced a Pareto-preferred choice.

[FIGURE 1 OMITTED]

If, in contrast, the decision was made under majority rule, a majority might pass a bill that made some individuals worse off, as in the move from q to x in Figure 1. This move would produce a Pareto-indeterminate outcome.

Now consider our framework. In our framework, the two voting rules consider the same proposal in each round, but they may consider different status quos in rounds r > 1. Again, unanimity rule typically outperforms majority rule in terms of the Pareto criterion:

THEOREM 1. For N [greater than or equal to] 3 and R > 1. Fix any q and any stream of proposals. If majority rule makes a Pareto-preferred choice after R rounds of voting, then so does unanimity rule. Furthermore, there exists a q and a stream of proposals for which unanimity rule makes a Pareto-preferred choice but majority rule does not. (4)

Proof. See the appendix.

The conclusion in Theorem 1 is fairly consistent with the literature, but for different reasons. Theorem 1 is based on the raw mechanics of majority rule and unanimity rule in multiple rounds of voting. Claims in the literature are largely based on an assumption of rational proposing given a particular voting rule. The different assumptions have little effect on the ability of majority rule and unanimity rule to adhere to the Pareto criterion. However, they do affect Pareto optimality.

V. PARETO OPTIMALITY

Comparing unanimity rule and majority rule in terms of Pareto optimality produces more surprising results. In a single-dimensional context, the Pareto optimal set is identical to the set of alternatives that cannot be beaten under pairwise unanimity rule (otherwise known as the unanimity rule core). In addition, the alternative that cannot be beaten under pairwise majority rule is the alternative at the ideal point of the median voter (Black 1948). Hence, in equilibrium, majority rule and unanimity rule both select a Pareto-optimal outcome with a probability of 1.

The comparison becomes more interesting, however, if we consider choices that may or may not be in equilibrium. Out of equilibrium behavior occurs when ideal points change due to an election, an environmental change alters preferences, or new information changes opinions. For example, terrorist acts, ecological disasters, the signing of a free trade accord, and the aging of a constituency can all change a legislator's preferences for public goods. Also note that out of equilibria status quos are widely analyzed in the literature (Hinich and Munger 1997; Stewart 2001; Tsebelis 2002).

There are two interesting phenomena within this framework. The first is a retentive force. We show this in Figure 2, where there are three voters: a, b, and c. If the choice is between q and x, then majority rule selects the proposal x, and unanimity rule selects the status quo q. Although neither alternative is Pareto preferred in this case, it is clear that unanimity rule has preserved Pareto optimality and majority rule has not. The second phenomena is an attractive force (see Figure 3). In this case, the choice is between a Pareto-suboptimal status quo q and a Pareto-optimal proposal x. Majority rule chooses the Pareto-optimal proposal, and unanimity rule chooses the Pareto-suboptimal status quo. Again, neither alternative is Pareto preferred, yet in this case majority rule has preserved Pareto optimality and unanimity rule has not.

[FIGURES 2-3 OMITTED]

The reasoning behind these two phenomena can be generalized in the following theorem. Assuming a randomly drawn status quo and a randomly drawn proposal in a single round of voting,

THEOREM 2. If q is Pareto optimal, then unanimity rule is at least as likely to select the Pareto-optimal alternative as majority rule. If q is Pareto suboptimal, then majority rule is more likely to select the Pareto-optimal alternative than unanimity rule.

Proof See the Appendix.

Because the status quo may or may not be an element of the Pareto-optimal set, the natural question is which voting rule is more likely to select a Pareto-optimal alternative when all possible ideal points, status quos, and proposals are considered. In other words, which is more important: the retentive force or the attractive force?

Experiment 1: A Uniform Distribution of Ideal Points and R = 1

To answer the question, consider the following probability experiment, conducted with the aid of a C program. In this experiment, ideal points are randomly drawn from a uniform probability distribution on [0, 1] such that [inverted A]i, [I.sub.i] [member of] [0, 1]. Alternatives q and x are randomly drawn from the same distribution, and R = 1.

For each trial in the experiment, the program randomly draws ideal points and alternatives, it calculates majority rule and unanimity rule winners, then it determines the Pareto optimality of each winner. If either voting rule chooses a Pareto-optimal outcome, it is assigned a success in the trial. The process is then repeated to determine the relative frequency that each voting rule chooses a Pareto-optimal alternative in a single round of voting. In a large number of trials, (5) this frequency should approximate the true probability in the population.

Two interesting findings result from this experiment (see Table 1). First, regardless of the number of individuals, majority rule is always more likely to select the Pareto-optimal outcome than unanimity rule. The reason for this is roughly as follows. Majority rule is more likely to change suboptimal outcomes to optimal ones than unanimity rule, as shown in Theorem 2. Furthermore, by the law of large numbers the ideal points will be fairly evenly distributed and the expected value of the median will be 0.5, which is typically far from the Pareto-suboptimal set. Under these circumstances, majority rule is unlikely to change Pareto-optimal alternatives to Pareto-suboptimal alternatives. Hence, majority rule outperforms unanimity rule.

Second, Table 1 suggests that the probability of each voting rule selecting the Pareto-optimal outcome increases as N increases. This occurs for at least two reasons: (1) as N increases, the probability that the Pareto-optimal set covers the entire interval approaches 1; and (2) as N increases, the standard error of the median decreases, making the drawn median closer to the expected value of 0.5. The latter makes majority rule less likely to choose a Pareto-suboptimal proposal over a Pareto-optimal status quo.

Experiment 2: Two Alternatives and a Skewed Population

To determine the robustness of these results and to analyze more realistic cases, we conduct a second series of probability experiments on several skewed distributions of ideal points. For each trial in these experiments, a status quo and a proposal are randomly drawn from the [0, 1] interval as before. However, ideal points are restricted to a prespecified range, such as [0.1, 0.2] [union] [0.7, 1.0]. This is done by setting an ideal point at each end of the range(s) and randomly drawing the remaining ideal points from within the range(s).

The results of these experiments are shown in Table 2. In each of these cases, majority rule is more likely to select a Pareto-optimal outcome than unanimity rule. Again, this is because majority rule benefits more from its attractive property than unanimity rule from its retentive property. In the first two cases, population 1 and population 2, the expected value of the median is at the midpoint of the ideal point range. Majority rule is unlikely to move from an optimal point to a suboptimal point in these cases. In the latter two cases, population 3 and population 4, the expected value of the median is not at the midpoint of the range. Surprisingly, however, it is close enough for majority rule still to be favored. If the distribution of ideal points were extremely skewed within the ideal point range, such as if the median voter was at the ideal point with the greatest value, then unanimity rule would usually outperform majority rule in terms of its ability to select Pareto-optimal outcomes. For example, suppose all ideal points lie within [0.0, 0.2] and [I.sub.med] = 0.2, where [I.sub.med] is the ideal point of the median voter. Then unanimity rule is more likely to select the Pareto-optimal outcome than majority rule. This is because majority rule is equally likely to bring proposals into the Pareto-optimal set as unanimity rule (because they have the same pivotal voter). At the same time, majority rule may move alternatives out of the Pareto-optimal set. Unanimity rule will not.

With small perturbations on the location of the median, however, majority rule again outperforms unanimity rule. For the range of ideal points [0.0, 0.2], a case that should favor unanimity rule, majority rule outperforms unanimity rule if [I.sub.med] [less than or equal to] 0.17 (see Table 3). Similar results were found for other populations where the median was skewed. This suggests that majority rule is more likely to select Pareto-optimal outcomes than unanimity rule even in fairly skewed populations.

Experiment 3." A Series of Binary Choices

To see if these results are an artifact of a single round of voting (two alternative choice) and to present a more realistic setting that better represents decision making in assemblies, we further considered what would occur if R > 1.

Mueller (2003, pp. 138 and 140-141) argued that if an assembly had enough time, unanimity rule would always be more capable of selecting Pareto-optimal outcomes than majority rule. However, the following series of probability experiments suggest a different conclusion if proposing is not rational.

In these experiments, a status quo and a proposal are randomly drawn in round 1. The winner of round I becomes the status quo in round 2 and is voted on against a new randomly drawn proposal in round 2, and so on. The ability to select Pareto-optimal outcomes is based on the Pareto optimality of the alternative chosen in round R. The results are summarized in Table 4.

Again, majority rule is more likely to select the Pareto-optimal alternative than unanimity rule, regardless of the population of ideal points considered or the number of rounds of voting. There two reasons for this. First, unanimity rule frequently chooses the initial status quo, which can be Pareto suboptimal. (6) Second, as a consequence of a corollary to the median voter theorem (Hinich and Munger 1997, p. 36), the distance between the outcome selected by majority rule and the median voter is nonincreasing with each subsequent round. In practice, this distance, when averaged over many trials, decreases fairly quickly (see Figure 4).

[FIGURE 4 OMITTED]

As the distance approaches zero, any outcome is increasingly likely to be Pareto optimal, particularly if the median is near the center of the ideal point range. Thus, for each round, the advantage of majority rule in terms of the attractive force exceeds the advantage of unanimity rule in terms of the retentive force. This advantage is enhanced for larger N because the median tends to converge to the center of the Pareto-optimal set, as N increases. (7)

Sincere Proposing and Restricted Status Quos

We make two additional assumptions for the next refinement of the probability experiment:

a. individuals propose their ideal points.

b. the status quo is typically in the Pareto-optimal set.

The first assumption is exactly what would occur if individuals proposed alternatives sincerely. The second assumption might occur if the status quo was randomly drawn from a normal distribution centered on the Pareto-optimal set. Note that the second assumption favors unanimity rule. Despite this we have Theorem 3.

THEOREM 3. Given assumptions a and b, majority rule is at least as likely to select the Pareto-optimal alternative as unanimity rule, in R rounds.

The proof of this theorem is quite simple. Because only ideal points are proposed, all proposals must be in the Pareto-optimal set. Hence the only cases that distinguish the performance of majority rule and unanimity rule are the cases where q is pareto suboptimal. By Theorem 2, these cases favor majority rule. The argument for Theorem 3 clearly carries over to multiple dimensions and also applies to cases with complete information.

Strategic Voting

Finally, we consider a theorem about rational proposing. Comparing this theorem to Theorem 3 highlights the differences between sincere and rational proposing. Assume (1) an arbitrary individual is designated the proposer, (2) the proposer and voters both behave strategically (i.e., players choose a set of actions that maximize their utility at the end of the game), (3) R [greater than or equal to] 1 rounds of voting, and (4) complete information.

THEOREM 4. Given assumptions 1-4, the outcomes in subgame perfect equilibrium are Pareto optimal for both majority and unanimity rule.

Proof. See the Appendix.

Theorem 4 also holds if individuals vote sincerely.

VI. CONCLUSION

This article shows that if there are at least two rounds of pairwise voting, then unanimity rule outperforms majority rule in terms of the Pareto criterion--consistent with the literature.

Less consistent with the literature are our findings with respect to Pareto optimality. We find that majority rule is more likely to select Pareto-optimal outcomes than unanimity rule when alternatives are randomly drawn from a single dimension. This is still true if the distribution of preferences are fairly skewed and if multiple alternatives are considered over a series of pairwise votes. Furthermore, we find that majority rule is at least as likely to select the Pareto-optimal outcome as unanimity rule if individuals propose their ideal points. In contrast, majority rule and unanimity rule perform equally as well if individuals propose rationally in a sequential equilibrium. All this suggests that the widely assumed relationship between unanimity rule and Pareto optimality needs to be treated more carefully.

For instance, unanimity rule is an ideal voting scheme for maintaining Pareto-optimal outcomes; standard game theory makes this clear (Colomer 2001, pp. 71-73). However, our nonequilibrium analysis shows that majority rule is better at obtaining Pareto-optimal outcomes than unanimity rule. Because the attractiveness property is more potent than the retentiveness property, majority rule outperforms unanimity rule in our model.

Institutional framers that want to obtain Pareto-optimal outcomes will want to consider these differences before they impose unanimity rule as a method of finding optimal quantities of public goods, the best allocation of resources in a limited partnership, or the most efficient rules for assemblies. After all, obtaining Pareto-optimal outcomes should be as much of a concern as maintaining them.

APPENDIX

THEOREM 1. For N [greater than or equal to] 3 and R > 1. Fix any q and any stream of proposals. If majority rule makes a Pareto-preferred choice after R rounds of voting, then so does unanimity rule. Furthermore, there exists a q and a stream of proposals for which unanimity rule makes a Pareto-preferred choice but majority rule does not. (8)

Proof. To see the first part, let [x.sub.j] be the first proposal that is Pareto preferred to q. Because unanimity rule passes a proposal if and only if it is Pareto preferred, unanimity rule passes [x.sub.j]. If unaminity rule passes any other proposals subsequently, then they must be Pareto preferred to [x.sub.j, hence Pareto preferred to q. Therefore, if a Paretopreferred alternative is proposed, unanimity rule must pass a Pareto-preferred alternative in R rounds of voting. The second part of Theorem 1 can be illustrated using Figure 1. If x is randomly drawn as the proposal in round 1, x will defeat q by majority rule, but it will not defeat q by unanimity rule. As a result, x is the new status quo for majority rule, and q is the new status quo for unanimity rule in round 2. If y is then proposed against the updated status quo, x will be selected by majority rule and y will be selected by unanimity rule. Because y is Pareto preferred to q, but x is not Pareto preferred to q, unanimity rule does a better job of selecting a Pareto-preferred alternative than majority rule in this case.

THEOREM 2. If q is Pareto optimal, then unanimity rule is at least as likely to select the Pareto-optimal alternative as majority rule. If q is Pareto suboptimal, then majority rule is more likely to select the Pareto-optimal alternative than unanimity rule.

Proof. The first statement is trivial. The second statement can be shown as follows. There are two cases where a Pareto-optimal x is offered against a Pareto-suboptimal q: one where q > [I.sub.max] and the other where q < [I.sub.min] (where [I.sub.max] and [I.sub.min] are the maximum and minimum ideal points, respectively). Consider the former. In this case, majority rule chooses a Pareto-optimal alternative iff [absolute value of x - [I.sub.med]] < [absolute value of q - [I.su b.med]]. Likewise, unanimity rule chooses the Pareto-optimal alternative iff [absolute value of x - [I.sub.max]] < [absolute value of q - [I.sub.max]]. In this context, two things can occur:

(A1.a) [I.sub.min] [less than or equal to] [I.sub.med] [less than or equal to] x [less than or equal to] [I.sub.max] < q, or

(A1.b) [I.sub.min] [less than or equal to] x < [I.sub.med] [less than or equal to] [I.sub.max] < q.

In the case of (A1.a), a majority will always choose x over q. However, unanimity rule will not choose x over q iff [absolute value of x - [I.sub.max]] > [absolute value of q - [I.sub.max]]. This implies that majority rule is more likely to select the Pareto-optimal alternative for this case. In the case of (A1.b), if majority rule chooses q, then unanimity rule must choose q. If majority rule chooses x, then unanimity rule chooses x only if [absolute value of x - [I.sub.max]] < [absolute value of q - [I.sub.max]], which may not occur. This again implies that majority rule is more likely to select the Pareto-optimal alternative than unanimity rule. By symmetry, the same holds true for q < [I.sub.min].

THEOREM 4. Given assumptions 1-4 the outcomes in subgame perfect equilibrium (SPE) are Pareto optimal for both majority and unanimity rule.

Proof. For an outcome to be chosen in SPE, it must be a Nash equilibrium in the final round. Hence, the proof focuses on all possible status quos in the final round. Here we adopt the following notation: [q.sub.R] is the status quo in round R, [I.sub.p] is the proposer's ideal point, [W.sub.m](q) is the winset of q under majority rule, [W.sub.u](q) is the winset of q under unanimity rule, and [x.sup.*] is the alternative proposed by the proposer.

First, consider majority rule. Assume [I.sub.p] [less than or equal to] [I.sub.med]. The case of [I.sub.p] > [I.sub.med] is shown similarly. There are three possibilities: (a) [q.sub.R] < [I.sub.p], (b) [I.sub.p] [less than or equal to] [q.sub.R] [less than or equal to] [I.sub.med], and (c) [q.sub.R] > [I.sub.med]. In case (a), we have [I.sub.p] [member of] [W.sub.m](q), so the proposer will propose [x.sup.*] = [I.sub.p]. In case (b) q is already Pareto optimal, and there exists no winning proposal that will improve the outcome for the proposer, hence the proposer will offer an unwinnable proposal. In case (c), the proposer will propose [x.sup.*] [member of] [W.sub.m](q) such that [absolute value of [x.sup.*] - [I.sub.p]] is minimized. Clearly, [x.sup.*] [member of] [[I.sub.p], [I.sub.med]], hence it is Pareto optimal.

Second, consider unanimity rule. In the final round, q is either: (a) in the Pareto-optimal set, (b) less than the Pareto-optimal set, or (c) greater than the Pareto-optimal set. If (a), then q prevails (which is pareto optimal). If (b), then two cases can occur: (i) [I.sub.p] [member of] [W.sub.u](q), in which case the proposer proposes [x.sup.*] = [I.sub.p]; or (ii) [I.sub.p] [not member of] [W.sub.u](q), in which case [I.sub.p] is further to the right than all of [W.sub.u](q). The proposer proposes [x.sup.*] [member of] [W.sub.u](q) such that [absolute value of [x.sup.*] - [I.sub.p]] is minimized. This proposal must be Pareto optimal because [I.sub.min] < [x.sup.*] [less than or equal to] [I.sub.p]. Case (c) follows by symmetry.

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(1.) Buchanan and Tullock (1962) wrote, "The welfare-political-economist approach indicates that a specific choice is Pareto-optimal only if all parties reach agreement" (p. 94). At the very least, this blurs the distinction between Pareto optimality and unanimity rule. Also see Brennan and Buchanan (2000, p. 151).

(2.) Simple unanimity rule is used for unanimous consent agreements in the U.S. Senate, for nonprocedural decisions among permanent members of the UN Security Council, and for taxation issues in the Council of the European Union, to name a few.

(3.) The simple definition of unanimity rule and majority rule implicitly drops abstention in the tally (Dougherty and Edward 2004a, 2004b; Riker 1982). With the exception of Theorem l, the same results apply for absolute unanimity rule and absolute majority rule, which count only affirmative votes and implicitly treat abstentions the same as votes against the proposal.

(4.) If R = 1, then majority rule and unanimity rule are equally likely to satisfy the Pareto criterion. This is because for R = 1 both voting rules consider the same pair of alternatives, and any Pareto-preferred alternative will be selected by both voting rules.

(5.) With 1 million trials, we are 95% confident that the true probability is within 0.001 of the relative frequency reported.

(6.) For example, in the case of ideal point range = [0.0, 0.2], N = 5, and R = 20, unanimity rule remains at the initial status quo with a probability of 0.224. Majority rule remains at the initial status quo with a probability of 0.047.

(7.) Preliminary analysis suggests that Experiment 3 applies to two dimensions as well.

(8.) Although majority rule can select a Pareto-dispreferred alternative in R > 2 rounds of voting, this possibility is ruled out in our model by the assumptions of single peaked and symmetric preferences. These assumptions imply that alternatives must be closer to the median voter for them to pass under majority rule. Hence, the median voter must prefer q [less than or equal to] [x.sub.1] [less than or equal to] [x.sub.2] ... [less than or equal to] [x.sub.r]. Because at least one individual has these preferences, it is impossible for [x.sub.r] to be Pareto dispreferred to q.

KEITH L. DOUGHERTY and JULIAN EDWARD *

* We thank an anonymous referee for useful comments. Dougherty: Assistant Professor, Department of Political Science, University of Georgia, Athens, GA 30602. Phone 1-706-542-2989, Fax 1-706-542-4421, E-mail dougherk@uga.edu

Edward: Associate Professor, Department of Mathematics, Florida International University, Miami, FL 33199. Phone 1-305-348-3050, Fax 1-305-348-6158, E-mail edwardj@fiu.edu

TABLE 1 The Probability of Selecting the Pareto-Optimal Outcome: Two Alternatives and a Uniform Distribution of Ideal Points N Majority Rule Unanimity Rule 5 0.816 0.711 10 0.928 0.834 41 0.992 0.954 71 0.997 0.973 99 0.998 0.980 200 0.999 0.990 500 1.000 0.996 Notes: Trials = 1,000,000; R = 1. Regardless of N, majority rule is more likely to select the Pareto-optimal outcome when ideal points, the status quo, and a proposal are randomly drawn from a single dimension. TABLE 2 The Probability of Selecting the Pareto-Optimal Outcome: Two Alternatives and a Skewed Population Ideal Point Ranges Population 1 Population 2 [0.0, 0.2] [0.3, 0.5] N Maj. Rule Unan. Rule Maj. Rule Unan. Rule 5 0.356 0.340 0.353 0.321 10 0.358 0.340 0.355 0.320 41 0.358 0.339 0.358 0.320 71 0.360 0.340 0.359 0.320 99 0.359 0.340 0.359 0.320 Ideal Point Ranges Population 3 Population 4 [0.1, 0.2] [union] [0.0, 0.1] [union] [0.7, 1.0] [0.2, 0.4] N Maj. Rule Unan. Rule Maj. Rule Unan. Rule 5 0.957 0.905 0.640 0.560 10 0.976 0.905 0.632 0.560 41 0.990 0.905 0.631 0.560 71 0.990 0.905 0.631 0.560 99 0.990 0.905 0.630 0.560 99 0.990 0.905 0.630 0.560 Notes: Trials = 1,000,000; R = 1. In the cases presented, and all others analyzed, majority rule is more likely to select the Pareto-optimal outcome when a random proposal is compared against a random status quo. TABLE 3 The Probability of Selecting the Pareto-Optimal Outcome: For Various Medians and [inverted A]i, [I.sub.i] [member of] [0.0, 0.2] Median Majority Unanimity More Likely 0.20 0.320 0.341 Unan 0.18 0.345 0.341 Unan 0.17 0.341 0.340 Maj 0.16 0.346 0.340 Maj 0.14 0.354 0.340 Maj 0.12 0.359 0.340 Maj 0.10 0.360 0.340 Maj 0.08 0.360 0.340 Maj 0.06 0.360 0.340 Maj 0.04 0.360 0.340 Maj 0.02 0.360 0.340 Maj 0.00 0.360 0.340 Maj Notes. Trials = 5,000,000, N = 5. In heavily skewed distributions majority rule may still outperform unanimity rule in terms of Pareto optimality. TABLE 4 The Probability of Selecting the Pareto-Optimal Outcome in Single-Dimensional Voting and a Series of Binary Choices R = 3 R = 5 R = 20 Maj. Unan. Maj. Unan. Maj. Unan. N Rule Rule Rule Rule Rule Rule [for all]: [I.sub.i] [member of] [[0, 1].sup.a] 5 0.911 0.763 0.946 0.794 0.992 0.876 10 0.979 0.855 0.991 0.869 1.000 0.915 41 0.999 0.956 1.000 0.869 1.000 0.966 71 1.000 0.973 1.000 0.974 1.000 0.978 99 1.000 0.981 1.000 0.981 1.000 0.983 Ideal point range = [0.1, 0.2] and [0.7, 1.0] 5 0.965 0.914 0.973 0.921 0.999 0.948 10 0.988 0.913 0.992 0.920 1.000 0.948 41 1.000 0.914 1.000 0.921 1.000 0.948 71 1.000 0.914 1.000 0.921 1.000 0.948 99 1.000 0.914 1.000 0.920 1.000 0.949 R = 3 R = 5 R = 20 Maj. Unan. Maj. Unan. Maj. Unan. N Rule Rule Rule Rule Rule Rule Ideal point range = [0.0, 0.2] 5 0.575 0.513 0.713 0.610 0.966 0.803 10 0.582 0.513 0.724 0.609 0.980 0.803 41 0.589 0.513 0.735 0.609 0.989 0.803 71 0.590 0.513 0.736 0.610 0.990 0.803 99 0.590 0.513 0.737 0.609 0.990 0.803 Ideal point range = [0.0, 0.1] and [0.2, 0.4] 5 0.870 0.688 0.953 0.745 1.000 0.863 10 0.853 0.688 0.936 0.745 0.999 0.863 41 0.850 0.688 0.935 0.745 1.000 0.863 71 0.849 0.687 0.934 0.745 1.000 0.863 99 0.849 0.688 0.934 0.745 1.000 0.863 Note: R is the number of rounds in the series. Trials = 1,000,000. Majority rule is at least as likely to select the Pareto-optimal outcome in all cases depicted. (a) In the top left case, ideal points are not fixed at the extremes. Instead, they are randomly drawn from [0, 1] as described in experiment 1.

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Author: | Dougherty, Keith L.; Edward, Julian |
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Publication: | Economic Inquiry |

Geographic Code: | 1USA |

Date: | Oct 1, 2005 |

Words: | 6666 |

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