# On the fair division of a random object

## TL;DR

This paper studies fair division rules for a random object based on limited utility information, proposing optimal rules that ensure fairness and efficiency in expectation despite unknown utility distributions.

## Contribution

It characterizes the optimal division rules using only realized utilities and mean utility, achieving near-optimal performance without full distribution knowledge.

## Key findings

- Rules perform close to the best with full distribution info.
- Full characterization of division rules under limited utility info.
- Ensures fairness and efficiency in expectation.

## Abstract

Ann likes oranges much more than apples; Bob likes apples much more than oranges. Tomorrow they will receive one fruit that will be an orange or an apple with equal probability. Giving one half to each agent is fair for each realization of the fruit. However, agreeing that whatever fruit appears will go to the agent who likes it more gives a higher expected utility to each agent and is fair in the average sense: in expectation, each agent prefers his allocation to the equal division of the fruit, i.e., he gets a fair share. We turn this familiar observation into an economic design problem: upon drawing a random object (the fruit), we learn the realized utility of each agent and can compare it to the mean of his distribution of utilities; no other statistical information about the distribution is available. We fully characterize the division rules using only this sparse information in the most efficient possible way, while giving everyone a fair share. Although the probability distribution of individual utilities is arbitrary and mostly unknown to the manager, these rules perform in the same range as the best rule when the manager has full access to this distribution.