Tournament games and condorcet voting Original Research Article

A Condorcet voting procedure asks each voter to rank candidates in order of preference. Then for each pair of candidates, a tally determines which candidate is preferred by a majority. The results can be modeled by a tournament where nodes represent the candidates and arcs point toward the loser of each two way race. If one candidate beats every other candidate, he/she is the winner. If there is no such winner, Felsenthal and Machover proposed that the “winner” be a probabilistic combination of candidates that win at least as often as they lose against every candidate. A winner is then picked according to the probabilities. This is the unique optimal strategy to a generalized “scissors, paper, and stone” game played on the tournament. What is the maximum number of lottery balls (the least common denominator of the probabilities) needed to pick a winner from n candidates?