Recent Empirical Examples and Theoretical Extensions



A number of elections have already been analyzed to try to ascertain the probable effects of approval voting: the 1980 New Hampshire presidential primaries (Section 1.3); the 1976 House Majority Leader contest (Chapter 4); the 1959 Boston mayoral race (Section 6.6); and the 1970 United States Senate election in New York (Section 7.5), in which we also looked at possible polling effects. In this chapter, we shall examine what seems almost a reprise of the Senate race in New York, ten years later, and then turn our attention to presidential elections. We shall report on a reconstruction of the first of two recent three-way presidential elections (1968), reserving the second (1980) for a more detailed statistical analysis in Chapter 9.


Nash Equilibrium Presidential Election Vote System Approval Vote Electoral College 
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Footnotes to Chapter 8

  1. 1.
    Gerald De Maio and Douglas Muzzio, “The 1980 Elections and Approval Voting,” Presidential Studies Quarterly 9,3 (Summer 1981), 341–363.Google Scholar
  2. 2.
    Quoted in Irvin Molotsky, “Javits Says He Would Consider Any Offer of 1. a Position if Reagan Sought Him,” New York Times, November 14, 1980, p. Bl.Google Scholar
  3. 3.
    Material in this section is based in part on Steven J. Brams and Peter C. Fishburn, “Approval Voting,” American Political Science Review 72,3 (September 1978), 840–842; S. J. Brams, The Presidential Election Game (New Haven: Yale University Press, 1978), pp. 221–228; and S. J. Brams, Comparison Voting, Innovative Instructional Unit (Washington, DC: American Political Science Association, 1978), pp. 51–66, which will appear in slightly revised form in Modules in Applied Mathematics: Political and Related Models, Vol. 2, edited by Steven J. Brams, William F. Lucas, and Philip D. Straffin, Jr. (New York: Springer-Verlag, 1982). However, we also report on studies published since the aforementioned work was done.Google Scholar
  4. 4.
    John Kellett and Kenneth Mott, “Presidential Primaries: Measuring Popular Choice,” Polity 9,4 (Summer 1977), 528–537. Similar results were found for gubernatorial primaries held in New Jersey in June 1981, in which the average voter would have approved of 2.1 candidates. See Steven J. Brams, Arnold Urken, George Sharrard, and Douglas Muzzio, “Results of Exit Poll of New Jersey Voters in Democratic and Republican Gubernatorial Primaries, June 2, 1981” (press release, June 17, 1981).CrossRefGoogle Scholar
  5. 5.
    William R. Keech and Donald R. Matthews, The Party’s Choice (Washington, DC: Brookings Institution, 1976), p. 212.Google Scholar
  6. 6.
    Richard A. Joslyn, “The Impact of Decision Rules in Multi-Candidate Campaigns: The Case of the 1972 Democratic Presidential Nomination,” Public Choice 25 (Spring 1976), 1–18. The effects of different rules for allocating delegates in the 1972 Democratic presidential primaries (winner-take-all, proportional, and districted) is explored in James I. Lengle and Byron Shafer, “Primary Rules, Political Power, and Social Change,” American Political Science Review 70, 1 (March 1976), 25–40; Louis Maisel and Gerald J. Lieberman, “The Impact of Electoral Rules on Primary Elections: The Democratic Presidential Primaries in 1976,” in The Impact of the Electoral Process, edited by Louis Maisel and Joseph Cooper (Beverly Hills, CA: Sage, 1977), pp. 39–80; and Thomas H. Hammond, “Another Look at the Role of ‘The Rules’ in the 1972 Democratic Presidential Primaries,” Western Political Quarterly 33, 1 (March 1980), 50–72. The effects of allocating delegates in presidential primaries, based on approval votes rather than plurality votes, has not been studied. A potential problem, which Philip D. Straffin, Jr., pointed out to S. J. Brams in conversation in 1980, is that candidates would find it in their interest to set up temporary Doppelgängers to run, who would pick up essentially the same approval votes as they would. Then, after the Doppelgängers dropped out, a candidate would presumably inherit their delegate votes, gaining an edge over candidates without Doppelgängers. This would be a clever strategy but, we think, would soon become so transparent that it would undermine the attractiveness of the Doppelgdngers, and maybe the candidate himself who set them up. With a high risk of backfiring, we doubt such a strategy would be tried.CrossRefGoogle Scholar
  7. 7.
    Samuel Merrill, III, “Strategic Decisions under One-Stage Multi-Candidate Voting Systems,” Public Choice 36,1 (1981), 127, Table 1.CrossRefGoogle Scholar
  8. 8.
    See Daniel A. Mazmanian, Third Parties in Presidential Elections (Washington, DC: Brookings Institution, 1974).Google Scholar
  9. 9.
    D. Roderik Kiewiet, “Approval Voting: The Case of the 1968 Presidential Election,” Polity 12,1 (Fall 1979), 170–181.CrossRefGoogle Scholar
  10. 10.
    Richard M. Scammon and Ben J. Wattenberg, The Real Majority: An Extraordinary Examination of the American Electorate (New York: Coward, McCann and Geoghegan, 1970), pp. 171–172.Google Scholar
  11. 11.
    D. R. Kiewiet, “Approval Voting: The Case of the 1968 Election,” p. 178.Google Scholar
  12. 12.
    On the debate over the Electoral College and an analysis of its biases, see S. J. Brams, Presidential Election Game, Ch. 3. The case for direct popular vote is made in Neal R. Peirce and Lawrence D. Longley, The People’s President: The Electoral College in American History and the Direct Vote Alternative, Rev. Ed. (New Haven: Yale University Press, 1981).Google Scholar
  13. 13.
    A median divides the area under a distribution curve exactly in half, which means in this example that half the voters have attitudes to the left of the point where the median line intersects the horizontal axis and half have attitudes to the right of this point. Moreover, because the distribution is symmetric—the curve to the left of the median is a mirror image of the curve to the right—the same numbers of voters have attitudes equal distances to the left and right of the median. For an elementary analysis of spatial models, and references to the literature, see S. J. Brams, Presidential Election Game, Ch. 1.Google Scholar
  14. 14.
    For reasons why voters support third-party candidates under plurality voting, see William H. Riker, “The Number of Political Parties: A Reexamination of Duverger’s Law,” Comparative Politics 9,1 (October 1976), 93–106; W. H. Riker, “Duverger’s Law: Plurality Voting and Party Systems,” American Political Science Review 76, 4 (December 1982), 753–766; and Steven J. Brams and Philip D. Straffin, Jr., “The Entry Problem in a Political Race,” in Political Equilibrium, edited by Peter C. Ordeshook and Kenneth A. Shepsle (Boston: Kluwer-Nijhoff, 1982), pp. 181–195.CrossRefGoogle Scholar
  15. 15.
    Steven J. Brams, Paradoxes in Politics: An Introduction to the Nonobvious in Political Science (New York: Free Press, 1976), pp. 34–37.Google Scholar
  16. 16.
    A more formal development of these ideas is given in Steven J. Brams, Game Theory and Politics (New York: Free Press, 1975), Ch. 2. Equilibrium results for approval voting when voters are sincere are given in K. H. Kim and Fred W. Roush, Introduction to Mathematical Consensus Theory, Lecture Notes in Pure and Applied Mathematics, Vol. 59 (New York: Marcel Dekker, 1980), pp. 100–102.Google Scholar
  17. 17.
    John Nash, “Non-cooperative Games,” Annals of Mathematics 54,2 (September 1951), 286–295.CrossRefMathSciNetGoogle Scholar
  18. 18.
    Gideon Doron and Richard Kronick, “Single Transferrable Vote: An Example of a Perverse Social Choice Function,” American Journal of Political Science 21,2 (May 1977), 303–311.CrossRefGoogle Scholar
  19. 19.
    Report of the Royal Commission Appointed to Enquire into Electoral Systems (London: HMSO, 1910), Cd. 5163, p. 21; and James Creed Meredith, Proportional Representation in Ireland (Dublin, 1913), p. 93. We are grateful to Duff Spafford for these references.Google Scholar
  20. 20.
    John H. Smith, “Aggregation of Preferences with Variable Electorate,” Econometrica 41,6 (November 1973), 1027–1041.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Peter C. Fishburn, “Monotonicity Paradoxes in the Theory of Elections,” Discrete Applied Mathematics 4,2 (April 1982), 119–134. A related paradox, in which some voters, by truncating their preferences (e.g., indicating only a first choice rather than an ordering over all candidates), can induce a better outcome under the Hare system, is given in Steven J. Brams, “The AMS Nomination Procedure is Vulnerable to ‘Truncation of Preferences,’” Notices of the American Mathematical Society 29, 2 (February 1982), 136–138, where an extreme case of nonmonotonicity is also given: a candidate loses when some voters raise him from last to first place in their preference orders. The truncation problem is also shown to affect other preferential voting systems in Hannu Nurmi, “On Taking Preferences Seriously” (mimeographed, 1982). For a catalogue of paradoxes to which preferential voting is vulnerable, see P. C. Fishburn and S. J. Brams, “Paradoxes of Preferential Voting,” Mathematics Magazine (forthcoming).zbMATHCrossRefMathSciNetGoogle Scholar

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