A duality principle for selection games

A dinner table seats k guests and holds n discrete morsels of food. Guests select morsels in turn until all are consumed. Each guest has a ranking of the morsels according to how much he would enjoy eating them; these rankings are commonly known. A gallant knight always prefers one food division over another if it provides strictly more enjoyable collections of food to one or more other players (without giving a less enjoyable collection to any other player) even if it makes his own collection less enjoyable. A boorish lout always selects the morsel that gives him the most enjoyment on the current turn, regardless of future consumption by himself and others. We show the way the food is divided when all guests are gallant knights is the same as when all guests are boorish louts but turn order is reversed. This implies and generalizes a classical result of Kohler and Chandrasekaran (1971) about two players strategically maximizing their own enjoyments. We also treat the case that the table contains a mixture of boorish louts and gallant knights. Our main result can also be formulated in terms of games in which selections are made by groups. In this formulation, the surprising fact is that a group can always find a selection that is simultaneously optimal for each member of the group. Suppose that k guests are seated at a table containing n discrete morsels of food (sushi rolls, say). Each player i has a strict ranking <i of the n morsels according to how enjoyable they are to eat. Players take turns selecting and consuming a morsel from the table, according to a fixed turn order P1, P2, . . . , Pm, where m ≤ n, and Pt ∈ {1, 2, . . . , k} is the player who selects a morsel on the t th turn. Our dinner guests do not simply play to maximize their own enjoyment. Instead, each guest conforms to one of two stereotypes: he is either a gallant knight or a boorish lout, as explained below. A plate is a subset of the morsels. Each player’s ranking of the morsels determines a corresponding partial ordering on plates of a fixed size according to pairwise comparison: Given two plates A and A′ of the same size, write A ≤i A′ if there is a bijection f : A→ A′ such that a ≤i f(a) for all a ∈ A. Given a sequence a = (a1, . . . , am) of morsel selections representing the play of a game, let Ai(a) = {at | Pt = i} be the plate eaten by player i. For each Date: February 13, 2012. 2010 Mathematics Subject Classification. 91A10, 91A18, 91A05, 91A50.