# Equilibria in Sequential Allocation

## TL;DR

This paper analyzes strategic behavior in sequential allocation, introducing a linear-time algorithm for pure Nash equilibria, and explores the properties of these equilibria including Pareto optimality and Stackelberg strategies.

## Contribution

It presents a novel linear-time algorithm for finding pure Nash equilibria called the 'bluff profile' and analyzes its properties in sequential allocation.

## Key findings

- The bluff profile is in pure Nash equilibrium for all utilities consistent with preferences.
- The outcome of the bluff profile is Pareto optimal with respect to pairwise comparisons.
- A dynamic program is provided for computing optimal Stackelberg strategies for two agents.

## Abstract

Sequential allocation is a simple mechanism for sharing multiple indivisible items. We study strategic behavior in sequential allocation. In particular, we consider Nash dynamics, as well as the computation and Pareto optimality of pure equilibria, and Stackelberg strategies. We first demonstrate that, even for two agents, better responses can cycle. We then present a linear-time algorithm that returns a profile (which we call the "bluff profile") that is in pure Nash equilibrium. Interestingly, the outcome of the bluff profile is the same as that of the truthful profile and the profile is in pure Nash equilibrium for \emph{all} cardinal utilities consistent with the ordinal preferences. We show that the outcome of the bluff profile is Pareto optimal with respect to pairwise comparisons. In contrast, we show that an assignment may not be Pareto optimal with respect to pairwise comparisons even if it is a result of a preference profile that is in pure Nash equilibrium for all utilities consistent with ordinal preferences. Finally, we present a dynamic program to compute an optimal Stackelberg strategy for two agents, where the second agent has a constant number of distinct values for the items.