A Nonequilibrium Analysis of Unanimity Rule, Majority Rule, and Pareto
Dougherty, Keith L., Edward, Julian, Economic Inquiry
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. …