Power and Equity: Which System Is Fairest?


In previous chapters, we compared approval voting to other single-ballot voting systems that may restrict the numbers of candidates one is allowed to vote for, and to runoff election systems in which the two candidates with the most first-ballot votes go onto the runoff ballot. Approval voting seems to be competitive and often superior to other systems in its ability to elicit honest or sincere responses from voters, its relative invulnerability to strategic manipulation, and its propensity to elect Condorcet or majority candidates. With respect to the last criterion, it was shown in Chapter 4 that approval voting would almost surely have found the probable Condorcet candidate in an important congressional leadership contest in 1976.


Utility Function Vote System Vote Strategy Average Utility Social Choice Function 
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Footnotes to Chapter 5

  1. 1.
    Peter C. Fishburn and Steven J. Brams, “Efficacy, Power, and Equity under Approval Voting,” Public Choice 37,3 (1981), 425–434.CrossRefGoogle Scholar
  2. 2.
    The most efficacious strategies also maximize the likelihood that a Condorcet candidate will be selected, linking efficacy to the Condorcet results of Chapter 3. See William V. Gehrlein and Peter C. Fishburn, “Constant Scoring Rules for Choosing One among Many Alternatives,” Quality and Quantity 15,2 (April 1981), 203–210.CrossRefMathSciNetGoogle Scholar
  3. 3.
    William H. Riker, Liberalism Against Populism: A Confrontation Between the Theory of Democracy and the Theory of Social Choice (San Francisco: W. H. Freeman, 1981), p. 110.Google Scholar
  4. 4.
    Peter C. Fishburn, “Symmetric and Consistent Aggregation with Dichotomous Voting,” in Aggregation and Revelation of Preferences, edited by Jean-Jacques Laffont (Amsterdam: North-Holland, 1979), pp. 201–218.Google Scholar
  5. 6.
    Dale T. Hoffman, “Relative Efficiency of Voting Systems: The Cost of Sincere Behavior” (mimeographed, 1979); and Robert J. Weber, “Comparison of Voting Systems” (mimeographed, 1977).Google Scholar
  6. 7.
    John F. Banzhaf, “Weighted Voting Doesn’t Work: A Mathematical Analysis,” Rutgers Law Review 19,2 (Winter 1965), 317–343; Peter C. Fishburn and William V. Gehrlein, “Collective Rationality versus Distribution of Power for Binary Social Choice Functions,” Journal of Economic Theory 15, 1 (June 1977), 72–91; William F. Riker, “Some Ambiguities in the Notion of Power,” American Political Science Review 48, 3 (September 1964), 787–792.Google Scholar
  7. 8.
    Jack H. Nagel, The Descriptive Analysis of Power (New Haven: Yale University Press, 1975).Google Scholar
  8. 9.
    John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, 3d Ed. (Princeton: Princeton University Press, 1953); and Peter C. Fishburn, Utility Theory for Decision Making (New York: John Wiley and Sons, 1970).MATHGoogle Scholar
  9. 10.
    D. T. Hoffman, “Relative Efficiency of Voting Systems: The Cost of Sincere Behavior (mimeographed, 1979); Samuel Merrill, III, “Strategic Voting in Multicandidate Elections under Uncertainty and Under Risk,” in Power, Voting, and Voting Power, edited by Manfred J. Holler, (Würzburg: Physica-Verlag, 1982), pp. 179–187; S. Merrill, III, “Decision Analysis of Multicandidate Voting Systems,” UMAP Module 384 (Newton, MA: Education Development Center, n.d.); S. Merrill, III, “Strategic Decisions under One-Stage Multicandidate Voting Systems,” Public Choice 36, 1 (1981), 115–134; and S. Merrill, III, “A Comparison of Multicandidate Electoral Systems in Terms of Optimal Voting Strategies,” American Journal of Political Science (forthcoming). Merrill compares a variety of voting systems (under the assumption of optimal voting strategies), including ranking systems like the Borda method, in the aforementioned papers and makes empirical estimates of the performance of candidates in the 1972 Democratic Party primaries under different systems in “Decision Analysis of Multicandidate Voting Systems,” and “Strategic Decisions under One-Stage Multicandidate Voting Systems” (see Section 8.3 for a discussion of these elections).Google Scholar
  10. 11.
    Samuel Merrill, III, “Approval Voting: A ‘Best Buy’ Method for Multicandidate Elections?” Mathematics Magazine 52,2 (March 1979), 98–102; R. J. Weber, “Comparison of Voting Systems” (mimeographed, 1977).MATHMathSciNetCrossRefGoogle Scholar
  11. 12.
    Peter C. Fishburn and Steven J. Brams, “Expected Utility and Approval Voting,” Behavioral Science 26,2 (April 1981), 136–142.CrossRefMathSciNetGoogle Scholar
  12. 13.
    P. C. Fishburn and S. J. Brams, “Expected Utility and Approval Voting.”Google Scholar
  13. 14.
    In a sense, all voters under approval voting vote in the same way, too: they vote for and against—by not voting-all the candidates. Under “direct approval voting,” voters would actually disapprove of (by voting “no”) those candidates they did not approve of (for whom they vote “yes”), which Morin argues would clarify the purpose of an approval voting election (personal communication to Brams, September 21, 1980), though it is formally equivalent to approval voting. See Richard A. Morin, Structural Reform: Ballots (New York: Vantage Press, 1980).Google Scholar
  14. 15.
    For other asymmetric examples, see S. Merrill, III, “Approval Voting: A ‘Best-Buy’ Method For Multicandidate Elections?”; Dale T. Hoffman, “A Model for Strategic Voting,” SIAM Journal on Applied Mathematics 42, 4 (August 1982), 751–761.Google Scholar

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