The paradox of grading systems

Steven J. Brams, Richard F. Potthoff

    Research output: Contribution to journalArticlepeer-review


    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.

    Original languageEnglish (US)
    Pages (from-to)193-210
    Number of pages18
    JournalPublic Choice
    Issue number3-4
    StatePublished - Dec 1 2015


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

    ASJC Scopus subject areas

    • Sociology and Political Science
    • Economics and Econometrics


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