Playing Divide-and-Choose Given Uncertain Preferences

TL;DR

This paper analyzes the divide-and-choose method for fair division when players have uncertain, Bayesian preferences, revealing strategic differences and optimal division structures under various information and correlation scenarios.

Contribution

It characterizes optimal strategies and equilibria in divide-and-choose with uncertain preferences, providing algorithms and insights into strategic behavior and utility outcomes.

Findings

Optimal divisions depend on preference knowledge and correlation.

Diversification incentives lead to multiple goods being divided at equilibrium.

Utilities vary with the number of goods and information levels.

Abstract

We study the classic divide-and-choose method for equitably allocating divisible goods between two players who are rational, self-interested Bayesian agents. The players have additive values for the goods. The prior distributions on those values are common knowledge. We consider both the cases of independent values and values that are correlated across players (as occurs when there is a common-value component). We describe the structure of optimal divisions in the divide-and-choose game and identify several cases where it is possible to efficiently compute equilibria. An approximation algorithm is presented for the case when the distribution over the chooser's value for each good follows a normal distribution, along with a randomized approximation algorithm for the case of uniform distributions over intervals. A mixture of analytic results and computational simulations illuminates several striking differences between optimal strategies in the cases of known versus unknown preferences. Most notably, given unknown preferences, the divider has a compelling "diversification" incentive in creating the chooser's two options. This incentive leads to multiple goods being divided at equilibrium, quite contrary to the divider's optimal strategy when preferences are known. In many contexts, such as buy-and-sell provisions between partners, or in judging fairness, it is important to assess the relative expected utilities of the divider and chooser. Those utilities, we show, depend on the players' levels of knowledge about each other's values, the correlations between the players' values, and the number of goods being divided. Under fairly mild assumptions, we show that the chooser is strictly better off for a small number of goods, while the divider is strictly better off for a large number of goods.