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.

💡 Research Summary

The paper studies a class of sequential selection games in which k players take turns picking from a finite set of n discrete items (morsels). Each player i has a strict ranking of the items, which induces a partial order on “plates” (subsets of items of equal size) via a bijective comparison: a plate A is no worse than plate B for i if there exists a bijection f : A→B such that every a∈A satisfies a ≤_i f(a).

Two behavioral archetypes are defined. A “gallant knight” cares primarily about the welfare of the other players. Formally, player i prefers outcome a to b if either (K1) every other player receives a plate that is at least as good in the partial order and at least one other player receives a strictly better plate, or (K2) all other players receive identical plates and i’s own plate is strictly better. A “boorish lout” ignores future consequences and always selects, on his turn, the currently most preferred remaining item (rule (L)).

The game proceeds according to a fixed turn sequence P₁,…,P_m (m ≤ n). The authors model the game as a finite decision tree with perfect information. Each leaf corresponds to a complete allocation of items; each internal node is labeled by the player whose turn it is. Preferences are partial orders on leaves, and a sub‑game‑perfect Nash equilibrium (SPNE) is defined as a strategy profile that, at every node, selects a child whose contingent leaf is maximal with respect to the moving player’s order.

Theorem 1 (the core duality) states: the final allocation produced by any SPNE when all players are gallant knights and the turn order is P₁,…,P_m coincides exactly with the allocation obtained when all players are boorish louts but the turn order is reversed, P_m,…,P₁. The proof hinges on the observation that, after the last knight move, the knight behaves exactly like a lout (he simply takes his favorite remaining item). By repeatedly swapping a knight move with the following lout move (Lemma 5) and using the fact that lout moves are deterministic, the authors show that moving all knight turns to the end, reversing their order, and converting them into lout turns does not change the eventual partition of items.

The paper then generalizes this result to mixed‑nature players—agents who are forced to act as knights on some pre‑specified turns and as louts on others. Theorem 4 asserts that for any such mixed game, all SPNEs lead to the same allocation, which can be computed by the same “push‑knight‑to‑the‑end, reverse, convert‑to‑lout” transformation. The proof proceeds by induction on the number of knight turns, repeatedly applying Lemma 5.

An important corollary is computational: the lout game is trivial to solve in linear time (each player simply picks his current favorite), so the allocation for the original knight game can be obtained in O(nk) time by the transformation, despite the fact that computing equilibria in general sequential games is PSPACE‑complete.

The authors also discuss the relationship to earlier work. In the two‑player case where each player maximizes his total utility, the classic result of Kohler and Chandrasekaran (1971) is recovered as a special case of Theorem 1. The paper notes that for three or more players the analogous “self‑maximizing” game does not admit a known efficiently computable equilibrium, and exhibits pathological behavior as described by Brams and Straffin.

Finally, the duality principle is recast in a group‑selection framework: when a group makes a joint selection, there always exists a choice that is simultaneously optimal for every member of the group under the knight ordering. This highlights a surprising robustness of the knight preferences: even though they are only partially ordered, they uniquely determine the final division of items, though not the exact order of picks (multiple optimal plays may exist, as illustrated by Figure 1).

Overall, the paper provides a clean, elegant duality between altruistic and selfish sequential selection, offers an efficient algorithm for computing the altruistic outcome, and extends the insight to mixed‑behavior settings and group decision contexts, thereby contributing both to the theory of fair division and to algorithmic game theory.