Implementing the Lexicographic Maxmin Bargaining Solution

TL;DR

This paper introduces a novel mechanism that successfully implements the lexicographic maxmin bargaining solution as the unique subgame perfect equilibrium in an n-player setting, filling a notable gap in bargaining theory.

Contribution

The paper constructs the first known mechanism for implementing the lexicographic maxmin solution, using a binary game tree and novel combinatorial properties.

Findings

Mechanism achieves unique subgame perfect equilibrium outcome.

Characterizes key combinatorial properties of the lexicographic maxmin solution.

Fills a gap in implementation of bargaining solutions.

Abstract

There has been much work on exhibiting mechanisms that implement various bargaining solutions, in particular, the Kalai-Smorodinsky solution \cite{moulin1984implementing} and the Nash Bargaining solution. Another well-known and axiomatically well-studied solution is the lexicographic maxmin solution. However, there is no mechanism known for its implementation. To fill this gap, we construct a mechanism that implements the lexicographic maxmin solution as the unique subgame perfect equilibrium outcome in the n-player setting. As is standard in the literature on implementation of bargaining solutions, we use the assumption that any player can grab the entire surplus. Our mechanism consists of a binary game tree, with each node corresponding to a subgame where the players are allowed to choose between two outcomes. We characterize novel combinatorial properties of the lexicographic maxmin solution which are crucial to the design of our mechanism.