An Experimental Study of the Efficiency of UnanimityRule and Majority RuleKeith Dougherty*Brian PittsJustin Moeller Robi RaganVersion 0.4March 24, 2009Abstract. We test several claims about the relationship between unanimity rule andPareto optimality. Buchanan and Tullock (1962), Mueller (2003), and other scholarsargue that unanimity rule is more capable of producing Pareto optimal outcomes thanother voting rules, such as majority rule, because unanimity rule passes an alternativeonly if it makes everyone better off. Majority rule can pass alternatives that make someindividuals worse off. Dougherty and Edward (2008), in contrast, claim that majorityrule is at least as likely to select Pareto optimal outcomes as unanimity rule in finitegames if proposals are random, sincere, or strategic. We test the two sets ofconjectures in a two dimensional framework using laboratory experiments. Our resultssuggest: 1) majority rule enters the Pareto set more quickly than unanimity rule, 2)majority rule leaves the Pareto set at the same rate as unanimity rule, and 3) majorityrule is more likely to select a Pareto optimal outcome than unanimity rule in the finalround of play. Our results also suggest that proposers do not behave observationally-rational in the final round and complete information does not affect the primary result.* All authors: Department of Political Science, University of Georgia, Athens, GA 30602. Corresponding author: (706) 542-2989, <[email protected]>.Throughout the paper, we use the term “majority rule” to refer to simple majority rule1(i.e. a proposal passes if and only if the yeas exceed the nays) and “unanimity rule” to refer tosimple unanimity rule (i.e. a proposal passes if and only if the yeas > 0 and the nays = 0). SeeDougherty and Edward (2004) and Riker (1982) for more details. Pareto optimality, also known as Pareto efficiency, is the most widely accepted criteria2for evaluating efficiency. An alternative is Pareto optimal if no alternative is Pareto preferred toit. That is, no other alternative can make at least one individual better off without makinganother individual worse off. A k-majority rule requires at least k individuals to vote in favor of a proposal in order for3the proposal to pass, where N/2 < k # N and N is the number of individuals; otherwise thestatus quo is chosen. Two special cases of k-majority rule are majority rule (k = N/2) andunanimity rule (k = N). See Dougherty and Edward (2004) for more precise definitions whennon-voters and “votes to abstain” are permitted.11. IntroductionDecision making bodies often adjust, and sometimes re-adjust, their voting thresholdsbetween majority rule and unanimity rule . For example, the Treaty establishing a1Constitution for Europe recently proposed reducing the unanimity rule requirement inthe Council of Ministers to a qualified majority in 26 issue areas. The FinancialAccounting Standards Board (FASB) reduced its voting threshold from a supermajorityto a majority in 1977 only to increase it back to another supermajority rule in 1991 (King1994). Each of these changes were an attempt to establish a more desirable threshold.A common criterion for evaluating such voting rules is Pareto optimality. Most2theoretical work concludes that unanimity rule is the best k-majority rule for promoting3Pareto optimality (Buchanan and Tullock 1962; Chen and Ordeshook 1998; Brennanand Buchanan 2000, 151-55; Mueller 2003, 1996). Buchanan and Tullock make it thecornerstone of their seminal work, The Calculus of Consent, and it has been widelyaccepted ever since. Traditionally viewed, unanimity rule guarantees more efficientoutcomes than other voting rules because unanimity rule only passes proposals thatThe “chaos theorem” states that in n-dimensional space unless ideal points are radially4symmetric, no alternative is in equilibrium using pairwise majority rule. In other words, for everyalternative in the space, there is another alternative that a majority of individuals prefer to it. Ideal points are radially symmetric around an alternative x if there is an ordinal, pairwisesymmetry of all the ideal points along any vector passing through x – a rare condition (Hinich &Munger 1997, 65). The Pareto set is the set of all Pareto optimal alternatives.5 Complete information refers to situations in which voters know the preferences (in this6case ideal points) of the other members of the group.2make everyone better off. Other k-majority rules, such as majority rule, can passproposals that make some individuals worse off.Dougherty and Edward (2005, 2008) challenge this claim. They analyze a seriesof two dimensional spatial voting models where the so-called “chaos” theorem applies4(McKelvey 1976; Schofield 1978). They conclude that majority rule is usually morelikely to select a Pareto-optimal outcome than unanimity rule if proposals are random. They also show that majority rule is at least as likely to select Pareto optimal outcomesif proposals are sincere or strategic. We test these competing sets of theories using laboratory experiments ongroups voting in a two dimensional issue space. In particular, we test three primaryhypotheses: 1) groups using majority rule are more likely to enter the Pareto set in5fewer rounds than groups using unanimity rule; 2) groups using majority rule are morelikely to leave the Pareto set than groups using unanimity rule; and 3) groups usingmajority rule are at least as likely to select Pareto optimal outcomes at the end of thegame as groups using unanimity rule. We also determine whether completeinformation affects the relative performance of the two voting rules, and we investigate6whether subjects propose rationally in the final round. The final round provides a3particularly clean period for testing such behavior because rational subjects shouldalways make a proposal that moves the outcome closer to their ideal point in the finalround (if such an alternative is available).We find that groups using majority rule enter the Pareto set in fewer rounds thangroups using unanimity rule and that groups using majority rule are more likely to selectPareto optimal outcomes than groups using unanimity rule. Such results provideevidence in favor of the theoretical claims of Dougherty and Edward (2008). Contraryto both sets of theories, we find no evidence that groups using majority rule are morelikely to leave the Pareto set than groups using unanimity