The Condorcet Criterion: Which System Best Finds the Majority Candidate?


We showed in Chapter 2 that approval voting promotes sincere voting and discourages strategic or manipulative voting more than other nonranked voting systems. This is in part due to the fact that, unlike other nonranked systems, approval voting imposes no restrictions on the number of candidates for whom an individual can vote. Thus, voters can express their approval for all candidates they consider acceptable, without worrying about abandoning their favorite candidates when such candidates have only a slim chance of winning.


Vote System Preference Order American Political Science Review Approval Vote Voter Preference 


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Footnotes to Chapter 3

  1. 1.
    Marquis de Condorcet, Essai sur l’application de l’analyse à la probabilité dés decisions rendues á la pluralité des voix (Paris, 1785); and Jean-Charles de Borda, “Mémoire sur les election au scrutin,” Histoire de l’Académie Royale des Sciences (Paris, 1781). A translation, with commentary, of Borda’s original work can be found in Alfred de Grazia, “Mathematical Derivation of an Election System,” ISIS 44, 135–136 (June 1953), 42–51.Google Scholar
  2. 2.
    Duncan Black, The Theory of Committees and Elections (Cambridge: Cambridge University Press, 1958).MATHGoogle Scholar
  3. 3.
    For a review of the literature on the paradox of voting, and Arrow’s Impossibility Theorem that describes conditions underlying it, see Jerry S. Kelly, Arrow Impossibility Theorems (New York: Academic Press, 1978); Steven J. Brains, Paradoxes in Politics: An Introduction to the Nonobvious in Political Science (New York: Free Press, 1976), Ch. 2; Peter C. Fishburn, The Theory of Social Choice (Princeton: Princeton University Press, 1973); and P. C. Fishburn, “Paradoxes of Voting,” American Political Science Review 68, 2 (June 1974), 537–546.MATHGoogle Scholar
  4. 4.
    Peter C. Fishburn, “Condorcet Social Choice Functions,” SIAM Journal on Applied Mathematics 33,3 (November 1977), 469–489; P. C. Fishburn, “Dimensions of Election Procedures: Analyses and Comparisons,” Theory and Decision (forthcoming); Jeffrey T. Richelson, “A Comparative Analysis of Social Choice Functions,” Behavioral Science 20, 5 (September 1975), 331–337; J. T. Richelson, “A Comparative Analysis of Social Choice Functions, II,” Behavioral Science 23, 1 (January 1978), 38–44; J. T. Richelson, “A Comparative Analysis of Social Choice Functions, III,” Behavioral Science 23, 3 (May 1978), 169–178; J. T. Richelson, “A Comparative Analysis of Social Choice Functions, I, II, III: A Summary,” Behavioral Science 24, 5 (September 1979), 355; J. T. Richelson, “A Comparative Analysis of Social Choice Functions, IV,” Behavioral Science 26, 4 (October 1981), 346–353; Philip D. Straffin, Jr., Topics in the Theory of Voting, UMAP Expository Monograph Series (Boston: Birkhäuser Boston, 1980); Hannu Nurmi, “Majority Rule: Second Thoughts and Refutations,” Quality and Quantity 14, 6 (December 1980), 743–765; H. Nurmi, “On the Properties of Voting Systems,” Scandinavian Political Studies 4 (1981), 19–32; H. Nurmi, “Review Article: Voting Procedures,” British Journal of Political Science (forthcoming); Prakash P. Shenoy and David B. Smith, “Voting Schemes for the Financial Accounting Standards Board” (mimeographed, 1981); and 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). Properties of different runoff systems are compared in Jeffrey T. Richelson, “Running off Empty: Run-off Point Systems,” Public Choice 35, 4 (1980), 457–468; and the effects of different spatial modeling assumptions are considered in John Chamberlin and Michael D. Cohen, “Toward Applicable Social Choice Theory: A Comparison of Social Choice Functions under Spatial Model Assumptions,” American Political Science Review 72, 4 (December 1978), 1341–1356. At a less technical level, good comparisons of different voting systems, with illustrations from recent United States presidential elections, can be found in Richard G. Niemi and William H. Riker, “The Choice of Voting Systems,” Scientific American (June 1976), pp. 21ff; and Martin Gardner (written by Lynn Arthur Steen), “Mathematical Games (From Counting Votes to Making Votes Count: The Mathematics of Elections),” Scientific American (October 1980), pp. 16ff.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Ken-ichi Inada, “A Note on the Simple Majority Decision Rule,” Econometrica 32,4 (October 1964), 525–531.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Peter C. Fishburn, “Simple Voting Systems and Majority Rule,” Behavioral Science 19,3 (May 1974), 166–176; P. C. Fishburn and William V. Gehrlein, “An Analysis of Simple Two-stage Voting Systems,” Behavioral Science 21, 1 (January 1976), 1–23; P. C. Fishburn and W. V. Gehrlein, “An Analysis of Voting Procedures with Nonranked Voting,” Behavioral Science 22, 3 (May 1977), 178–185; W. V. Gehrlein and P. C. Fishburn, “Coincidence Probabilities for Simple Majority and Positional Voting Rules,” Social Science Research 7, 3 (September 1978), 272–283; W. V. Gehrlein and P. C. Fishburn, “Effects of Abstentions on Voting Procedures in Three-Candidate Elections,” Behavioral Science 24, 5 (September 1979), 346–354; W. V. Gehrlein and P. C. Fishburn, “Constant Scoring Rules for Choosing One among Many Alternatives,” Quality and Quantity 15, 2 (April 1981), 203–210; P. C. Fishburn and W. V. Gehrlein, “Majority Efficiencies for Simple Voting Procedures: Summary and Interpretation,” Theory and Decision 14, 2 (June 1982), 141–153; Raphael Gillett, “The Asymptotic Likelihood of Agreement between Plurality and Condorcet Outcomes,” Behavioral Science 25, 1 (January 1980), 23–32; and W. V. Gehrlein, “Condorcet Efficiency and Constant Scoring Rules,” Mathematical Social Sciences 2, 2 (March 1982), 123–130.CrossRefGoogle Scholar
  7. 7.
    Robert J. Weber, “Comparison of Voting Systems” (mimeographed, 1977); R. J. Weber, “Multiply-Weighted Voting Systems” (mimeographed, 1977); R. J. Weber, “Reproducing Voting Systems” (mimeographed, 1977); Robert F. Bordley, “A Pragmatic Scheme for Evaluating Election Schemes,” American Political Science Review (forthcoming); and Samuel Merrill, III, “A Comparison of Multicandidate Electoral Systems in Terms of Optimal Voting Strategies,” American Journal of Political Science (forthcoming).Google Scholar
  8. 11.
    Peter C. Fishburn and Steven J. Brams, “Approval Voting, Condorcet’s Principle, and Runoff Elections,” Public Choice 36,1 (1981), 89–114.CrossRefGoogle Scholar
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    William V. Gehrlein and Peter C. Fishburn, “The Effects of Abstentions on Election Outcomes,” Public Choice 33,2 (1978), 69–82.CrossRefGoogle Scholar
  10. 14.
    The modern record for number of candidates running in a congressional election is 31, which occurred in the primary races for Maryland’s Fifth Congressional District after the incumbent seat was declared vacant because of the incumbent’s illness. Ben A. Franklin, “31 on Ballot Tomorrow for House Seat in Maryland,” New York Times, April 6, 1981, p. B15. The Republican winner easily triumphed against 12 opponents with 63 percent of the vote, but the Democratic winner won with only 30 percent against 19 opponents. “Scott and Hoyer Nominated in Maryland 5th District,” Congressional Quarterly Weekly Report 39, 5 (April 11, 1981), 645.Google Scholar
  11. 15.
    These results have been extended to the election of committees in which more than one person is to be elected to office. See Peter C. Fishburn, “An Analysis of Simple Voting Systems for Electing Committees,” SIAM Journal on Applied Mathematics 41,3 (December 1981), 499–502; and Jeffrey B. Sidney, “Single Ballot Non-Ranked Voting Systems for Committee Selection” (mimeographed, 1981). Expected-utility calculations that a voter might make in the election of a committee are illustrated in Dale T. Hoffman, “A Model for Strategic Voting,” SIAM Journal on Applied Mathematics 42, 4 (August 1982), 751–761. The use of approval voting in the selection of multiple propositions—such as those that might come before a zoning board where majority support is required for passage—is analyzed in Roger L. Faith and James M. Buchanan, “Towards a Theory of Yes-No Voting,” Public Choice 37, 2 (1981), 231–245.MATHCrossRefMathSciNetGoogle Scholar

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