Divide the Dollar: Three solutions and extensions

Abstract

Divide the Dollar (DD) is a game in which two players independently bid up to 100 cents for a dollar. Each player receives his or her bid if the sum of the bids does not exceed a dollar; otherwise they receive nothing. This game has multiple Nash equilibria, including the egalitarian division of (50, 50), but this division is not compelling except for its symmetry and presumed fairness.

This division is easy to induce, however, by punishing — more severely than does DD — deviations from it, but these solutions are not ‘reasonable’. By altering the rules of DD, however, one can induce an egalitarian division (by successive elimination of weakly dominated strategies), but no reasonable payoff scheme produces this division with egalitarian bids of 50.

Three alternatives to DD are analyzed. DD1, which rewards lowest bidders first, shows how an egalitarian outcome can be induced with equal but nonegalitarian bids. DD2, which adds a second stage that provides the players with new information yet restricts their choices at the same time, is used to introduce ‘dominance inducibility’. DD3 combines the features of DD1 and DD2, is reasonable (like DD1), makes calculations transparent (like DD2), and induces egalitarian bids as well as the egalitarian outcome. The possible application of the different procedures to a real-world allocation problem (setting of salaries by a team), in which there may be entitlements, is described.