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The paradox of grading systems


We distinguish between (i) voting systems in which voters can rank candidates and (ii) those in which they can grade candidates, using two or more grades. In approval voting, voters can assign two grades only—approve (1) or not approve (0)—to candidates. While two grades rule out a discrepancy between the average-grade winners, who receive the highest average grade, and the superior-grade winners, who receive more superior grades in pairwise comparisons (akin to Condorcet winners), more than two grades allow it. We call this discrepancy between the two kinds of winners the paradox of grading systems, which we illustrate with several examples and whose probability we estimate for sincere and strategic voters through a Monte Carlo simulation. We discuss the tradeoff between (i) allowing more than two grades, but risking the paradox, and (ii) precluding the paradox, but restricting voters to two grades.

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  1. Because ties for highest median grade are common when relatively few grades can be assigned, Balinski and Laraki (2011) propose a method for breaking ties in a system they call “majority judgment.” Cumulative voting, in which voters can allocate a fixed number of votes to one or more candidates, effectively allows voters to grade candidates, though it is not usually thought of as a grading system. It is used to elect multiple winners, unlike the systems analyzed here, and affords parties or factions the opportunity to elect candidates in proportion to their share in the electorate. For analyses of its properties and experience with it, see Brams (2003 [1975]) and Bowler et al. (2003).

  2. Merrill and Nagel (1987) distinguish a balloting method, such as one that allows two grades, from a method for aggregating the ballots (what they call “decision rules”), such as one that elects the candidate with more superior grades in pairwise comparisons—or, equivalently, in the case of AV the highest average grade. As we will show, using a different aggregation method can produce a winner different from the AG-SG winner, even when only two grades are possible (the Hare system is an example). However, systems that choose as winners the candidates with the highest median grade or the highest Borda score duplicate AV in always selecting AG–SG winners for the case of two grades.

  3. For any g, the method breaks ties as follows: Order the grades for each candidate from low to high; for an even number of grades, treat the grade just below the middle, rather than the average of the two middle grades, as the median; delete each candidate’s (tied) median grade; obtain the median for each candidate from the new (smaller) sets of grades; and if a tie still exists, continue to delete grades successively, one at a time, in the same way as before, until the tie is broken. As an example for g = 2, suppose the ordered grades from six voters are 0 0 1 1 1 1 for candidate X and 0 1 1 1 1 1 for Y, with the remaining candidates receiving more 0s than 1s (the medians, as just defined, are underscored). Deleting both underscored medians yields 0 0 1 1 1 for X and 0 1 1 1 1 for Y. The new medians are again tied, so they are deleted, giving 0 0 1 1 for X and 0 1 1 1 for Y. The tie is now broken in favor of Y as the median-grade winner, who indeed did receive more 1s.

  4. The Borda count gives rise to the same paradox in Example 1 if the grades of 2, 1, and 0 are interpreted as ranks, because the Borda winner, C, is different from the three candidates in the Condorcet cycle. In our later examples, however, the grades do not necessarily correspond to Borda scores, including tied scores, and the Borda and AG winners can differ, illustrating that ranking systems like Borda are categorically different from grading systems.

  5. Taft, also, would have benefited from the approval of both his supporters and Roosevelt supporters, but probably not to the extent of Roosevelt for two reasons: Roosevelt (i) edged him out in the plurality vote and (ii) probably would have drawn more support from Wilson voters, and even supporters of Eugene Debs, the Socialist candidate who received 6 % of the vote, because he generally was perceived to be less conservative than Taft.

  6. If a voter’s ranking is not strict, the usual convention is to give all tied candidates the mean of the ranks as if the ranking were strict. To illustrate in Example 2, with ranks of 2, 1, and 0 for the three candidates, the Borda scores of the first voter would be (2, ½, ½), of the second voter (0, 1½, 1½), and of the third voter (0, 1½, 1½). In this example, B and C would be the tied winners, and no paradox of grading systems would occur, because these candidates are preferred by two of the three voters. In general, when translating ranks into grades or grades into ranks, ranking systems and grading systems may yield different outcomes.

  7. Such a dichotomization is effectively what AV forces, and not just for two candidates. In the presence of two candidates only, plurality voting suffices to enable a voter to vote for just his or her preferred candidate.

  8. This was also true in Example 2, but “all except one” meant only 2 of the 3 voters rather than, as in Example 4, 4 of the 5 voters.

  9. We are grateful to Sean J. Vasquez for writing the computer program to do the simulation.

  10. The sampling is carried out with replacement, so in general it may include duplicate combinations, but that is unlikely for this example.

  11. For n Bernoulli trials each with success probability p (the binomial distribution), the standard error of the sample proportion of successes is \( \sqrt {p(1 - p)/n} \), which is maximized at p = ½.

  12. We did not calculate the probability of a strong paradox. It seemed to us that the possibility, not the certainty, of different AG and SG winners was the first question to address in inquiring whether a discrepancy posed a serious problem.

  13. Regenwetter et al. (2006) make a similar point about the probability of the Condorcet paradox, showing that the paradox turns up much less often than its theoretical probability, based on the impartial-culture assumption.

  14. The first column of Table 3 when g = 3 repeats probabilities from Table  2 .

  15. Some uses of, and experience with, AV in elections are discussed in Brams and Fishburn (2005), Laslier and Sanver (2010), Felsenthal and Machover (2012), Baujard et al. (2014), and references therein. Brams (2008), Brams and Sanver (2009), Sanver (2010), and Camps et al. (2014) suggest ways in which ranking and grading systems can be combined or otherwise reconciled.


  • Alcantud, J. C. R., & Laruelle, A. (2014). Dis&Approval voting: A characterization. Social Choice and Welfare, 43(1), 1–10.

    Article  Google Scholar 

  • Arrow, K. J. (1963 [1951]). Social choice and individual values. New Haven, CT: Yale University Press.

  • Balinski, M. L., & Laraki, R. (2011). Majority judgment. Cambridge, MA: MIT Press.

    Book  Google Scholar 

  • Baujard, A., Gavrel, F., Igersheim, H., Laslier, J. F., & Lebon, I. (2014). Who’s favored by evaluative voting? An experiment conducted during the 2012 French presidential election. Electoral Studies, 34, 131–145.

    Article  Google Scholar 

  • Bowler, S., Donovan, T., & Brockington, D. (2003). Electoral reform and minority representation: Local experiments with alternative elections. Columbus, OH: Ohio State University Press.

    Google Scholar 

  • Brams, S. J. (2003 [1975]). Game theory and politics. Mineola, NY: Dover.

  • Brams, S. J. (2008). Mathematics and democracy: Designing better voting and fair-division procedures. Princeton, NJ: Princeton University Press.

    Book  Google Scholar 

  • Brams, S. J., & Fishburn, P. C. (1978). Approval voting. American Political Science Review, 72(3), 831–847.

    Article  Google Scholar 

  • Brams, S. J., & Fishburn, P. C. (2005). Going from theory to practice: The mixed success of approval voting. Social Choice and Welfare, 25(2), 457–474.

    Article  Google Scholar 

  • Brams, S. J., & Fishburn, P. C. (2007 [1983]). Approval voting. New York: Springer.

  • Brams, S. J., Fishburn, P. C., & Merrill, S. III. (1988). The responsiveness of approval voting: Comments on Saari and Van Newenhizen; Rejoinder to Saari and Van Newenhizen. Public Choice, 59(2), 121–131; 149.

    Article  Google Scholar 

  • Brams, S. J., & Sanver, M. R. (2009). Voting systems that combine approval and preference. In S. J. Brams, W. V. Gehrlein, & F. S. Roberts (Eds.), The mathematics of preference, choice, and order: Essays in honor of Peter C. Fishburn (pp. 215–237). Berlin: Springer.

    Chapter  Google Scholar 

  • Camps, R., Mora, X., & Saumell, L. (2014). Choosing by means of approval-preferential voting: The revised approval choice. Preprint,

  • Center for Election Science (2015).

  • Center for Range Voting (2015).

  • Felsenthal, D. S. (1989). On combining approval with disapproval voting. Behavioral Science, 34(1), 53–60.

    Article  Google Scholar 

  • Felsenthal, D. S., & Machover, M. (Eds.). (2012). Electoral systems: Paradoxes, assumptions, and procedures. Berlin: Springer.

    Google Scholar 

  • Hillinger, C. (2005). The case for utilitarian voting. Homo Oeconomicus, 22(3), 295–321.

    Google Scholar 

  • Laslier, J.-F., & Sanver, M. R. (Eds.). (2010). Handbook on approval voting. Berlin: Springer.

    Google Scholar 

  • Merrill, S. III., & Nagel, J. (1987). The effect of approval balloting on strategic voting under alternative decision rules. American Political Science Review, 81(2), 509–524.

    Article  Google Scholar 

  • Potthoff, R. F. (2013). Simple manipulation-resistant voting systems designed to elect Condorcet candidates and suitable for large-scale public elections. Social Choice and Welfare, 40(1), 101–122.

    Article  Google Scholar 

  • Potthoff, R. F. (2014). Condorcet completion methods that inhibit manipulation through exploiting knowledge of electorate preferences. Games, 5(4), 204–233.

    Article  Google Scholar 

  • Regenwetter, M., Grofman, B., Marley, A. A. J., & Tsetlin, I. (2006). Behavioral social choice: Probabilistic models, statistical inference, and applications. Cambridge, UK: Cambridge University Press.

    Google Scholar 

  • Saari, D. G., & Van Newenhizen, J. (1988). Is approval voting an ‘unmitigated evil’: A response to Brams, Fishburn, and Merrill. Public Choice, 59(2), 133–147.

    Article  Google Scholar 

  • Sanver, M. R. (2010). Approval as an intrinsic part of preference. In J.-F. Laslier & M. R. Sanver (Eds.), Handbook on approval voting (pp. 469–481). Berlin: Springer.

    Chapter  Google Scholar 

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We thank two reviewers and the editor for valuable comments.

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Correspondence to Steven J. Brams.

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Brams, S.J., Potthoff, R.F. The paradox of grading systems. Public Choice 165, 193–210 (2015).

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  • Voting and elections
  • Grading systems
  • Ranking systems
  • Approval voting
  • Condorcet winner
  • Monte Carlo simulation