We distinguish between voting systems in which voters can rank candidates and those in which they can grade candidates, such as approval voting (AV), in which voters can give two grades—approve (1) or not approve (0)—to candidates (Brams and Fishburn, 1978). Other theorists have proposed that voters be allowed to give more grades (some favor three grades, others more).
Only AV rules out a discrepancy between the average-grade winner, who receives the highest average grade, and a superior-grade winner, who receives more superior grades. When this discrepancy occurs, we say there is a paradox of grading systems. We give several examples of the paradox, describe one situation in which it cannot occur, and analyze its probability, which is surprising high. 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.
