Fairness in Multi-Proposer-Multi-Responder Ultimatum Game

TL;DR

This paper extends the classic Ultimatum Game to multiple proposers and responders, deriving equilibria and exploring implications for fairness thresholds in community interactions.

Contribution

It introduces a multi-proposer-multi-responder model, deriving equilibria and proposing new fairness thresholds based on asymptotic player numbers.

Findings

Derived subgame-perfect Nash equilibria for the extended game.

Proposed two estimates (31.8% and 36.8%) for fair share thresholds.

Connected theoretical results to experimental observations in one-on-one games.

Abstract

The Ultimatum Game is conventionally formulated in the context of two players. Nonetheless, real-life scenarios often entail community interactions among numerous individuals. To address this, we introduce an extended version of the Ultimatum Game, called the Multi-Proposer-Multi-Responder Ultimatum Game. In this model, multiple responders and proposers simultaneously interact in a one-shot game, introducing competition both within proposers and within responders. We derive subgame-perfect Nash equilibria for all scenarios and explore how these non-trivial values might provide insight into proposal and rejection behavior experimentally observed in the context of one vs. one Ultimatum Game scenarios. Additionally, by considering the asymptotic numbers of players, we propose two potential estimates for a "fair" threshold: either 31.8% or 36.8% of the pie (share) for the responder.