1. For example, suppose the US progressive-income-tax code, defined by previously enacted laws, has no expiration date. Suppose next that the Congress takes up legislation to institute a flat-tax. Because this new proposal is dramatic and controversial, it fails to obtain a majority of votes in the Senate and dies. The status quo policy—namely, the existing progressive income tax—therefore remains in effect.Google Scholar

2. For example, any voter can consider a set of policies {a, b, c} and say: ‘I like c best, a next best, and b least’.Google Scholar

3. An example of rules of procedure in a corporate setting is a nominating convention that has a right to propose changes to the company's board of directors, subject to approval by an up-or-down share-weighted majority vote of shareholders. An example in a legislature is a standing committee that has a right to make a proposal but, thereafter, any member of the entire chamber (e.g. House or Senate) may propose amendments to the committee's proposal which, if adopted by a majority vote, replace or become part of the committee's proposal.Google Scholar

4. For example, a simple majority rule says that if more than half of the voters vote for alternative x over alternative y, then x shall be the collective choice. A two-thirds over-ride rule says that if the president vetoes a bill, the Congress can enact it into law anyway if at least two-thirds of the House and two-thirds of the Senate vote to over-ride the veto.Google Scholar

5. Examples of optimality criteria include utility maximization and expected utility maximization at the individual level, and Nash equilibrium at the level of players in a formally specified game.Google Scholar

6. For example, if, in an n-person committee that votes under a simple-majority rule and an open amendment procedure, there is a single policy x* that at least a majority of committee members prefer to all other policies, then x* is said to be stable.Google Scholar

7. Black, Duncan, The Theory of Committees and Elections (Cambridge University Press, London, 1958).Google Scholar

8. Collective choice can also be represented in a multidimensional choice space, however such extensions add considerable complexity and tend to be much more difficult to apply.Google Scholar

9. Left and right arc not intended to imply ideological content here, since liberals usually favor increases (rightward shifts) on the minimum wage while conservatives tend to oppose such changes.Google Scholar

10. For example, a consumer's utility for goods is usually assumed to be increasing in the good (albeit at a decreasing rate), a firm's utility for profits is usually assumed to be increasing in profits (albeit at a decreasing rate), and so on.Google Scholar

11. Technically, a single-peaked preference (or utility) function is defined as a mapping from a set of policy alternatives into a utility measure such that the graph of utility ‘changes its direction at most once, from up to down’ (Black, The Theory of Committees and Elections, p. 7). So, for example, if a decision-maker had a maximum at the boundary of the space (say a pacifist at zero in Figure 2) and her utility curve never rose thereafter, then her preferences would be single-peaked.Google Scholar

12. This can be confirmed by inspecting whether U_L (q) is greater than or less than U_L (X_R ), and likewise for U_M (q) and U_m (X_R ).Google Scholar

13. Formally, for any pair of bills y to the right of the median voter's ideal point x_m , at least 2 of the 3 voters prefer x to y. Google Scholar

14. Notice that this implies that any proposal between b and c = x_M would pass. M, however, receives her maximum payoff with c = x_M. Google Scholar

15. The open amendment procedure does not limit voters to a single amendment.Google Scholar

16. If any such voter favored such a change, his or her utility function would be increasing in this portion of the policy space, thereby contradicting the single-peakedness assumption.Google Scholar

17. With the exception of Nebraska, all US states as well as the Congress are bicameral, i.e. have both a lower and upper chamber. Therefore, the assumption of unicameralism obviously is not realistic. To the extent that upper and lower chamber preferences are similar, however, the simplifying assumption is defensible.Google Scholar

18. For a recent study of the filibuster, see Binder, Sarah A. and Smith, Steven S. (1997), Politics or Principle: Filibustering in the United States Senate (Brookings Institution, Washington, DC). Although institutions such as the veto and filibuster appear to be unique to the US political system and the pivotal politics theory was developed as a theory of US lawmaking, most democratic systems have supermajoritarian decision rules at some stages of their lawmaking processes and many presidential systems have some form of executive veto. In Japan, for example, a supermajority in the lower house of parliament can pass legislation over the opposition of the upper house. Likewise, many measures taken in the European Union require ‘qualified majorities’ in which the requisite number of country-weighted votes exceeds one-half.Google Scholar

19. Technically, there are filibuster pivots on both sides of the median voter. On the president's side, however, most anything that can be blocked by a filibuster can also be blocked more easily by a presidential veto (assuming the president is more extreme than the filibuster pivot). We therefore simplify the analysis by dispensing with the president-side filibuster pivot.Google Scholar

20. In retrospect (and as Republicans claimed during the election), the economy was probably well on the way to recovery even before this initiative was launched, and some Democrats, too, conceded as much. Nonetheless, the initiative was launched.Google Scholar

21. Congressional Quarterly Almanac (1993), p. 706.Google Scholar

22. The congressional budget resolution is passed early in the annual budget cycle. It sets targets for income and spending across several functional categories, after which more specific taxation and appropriations legislation is passed that is supposed to conform with the overall targets but, as a practical matter, often doesn't.Google Scholar

23. Technically, the 60^th senator, who is pivotal, needs only to be indifferent between the status quo and the bill.Google Scholar

24. Exempted now were also the highest-paid 10% of companies’ work forces. Eligibility was further restricted to employees who had worked at least 25 hours per week for the previous 12 months. As testimony to the weakness of the bill as well as Democratic frustrations regarding its dilution, House Speaker Tom Foley said, ‘This is not a generous bill. It would not require even one day of paid leave’ (Congressional Quarterly Almanac (1992), p. 355). (The bill would require only unpaid leave.)Google Scholar

25. Senator Christopher Dodd of Connecticut had sought a roll-call vote but Republican leaders objected and threatened delaying tactics that would have resulted in passage only after the convention recess.Google Scholar

26. Still, the bill had attracted 37 Republican supporters, adding credibility to supporters’ claims that it was a bipartisan coalition.Google Scholar

27. Groseclose, Timothy and Snyder, James M., ‘Buying Supermajorities’, American Political Science Review, Vol. 90 (1996), pp. 303–15; Timothy Groseclose, ‘An Examination of the Market for Favors and Votes in Congress’, Economic Inquiry, Vol. 34 (1995), pp. 320–40; Snyder (1991); and Alan Wiseman, Votes for Smokes: An Analysis of the Tobacco Lobbying, unpublished manuscript, Stanford University, Stanford, CA (1998). See also Baron, and Krehbiel's, case, Federal Express, in David P. Baron, Business and Its Environment, 3^rd edition (Prentice-Hall, New York, forthcoming).Google Scholar

28. Austen-Smith, David and Wright, John R., ‘Counteractive Lobbying’, American Journal of Political Science, Vol. 38 (1994), pp. 25–44; Thomas W. Gilligan and Keith Krehbiel, ‘Asymmetric Information and Legislative Rules with a Heterogeneous Committee’, American Journal of Political Science, Vol. 33 (1988), pp. 459–90.Google Scholar

29. Baron, David P., Summertime in the European Union, unpublished manuscript, Stanford University, Stanford, CA (1997); Keith Krehbiel, Institutional Analysis for Summertime in the European Union, unpublished manuscript, Stanford University, Stanford, CA (1998); Keith Krehbiel, Pivotal Politics: A Theory of US Lawmaking (University of Chicago Press, Chicago, 1998).Google Scholar