Fairness in Multi-Proposer-Multi-Responder Ultimatum Game

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.

Figures (3)

• Figure 1: Graphical representation of the game for the case of $L=3$ responders and $K=2$ proposers. In the first stage, proposers announce their offers, prompting each responder to determine their selection strategy. In this scenario, responder $1$ chooses a mixed strategy while the other two responders play pure strategies. In the second stage, it is probabilistically decided that responder $1$ chooses proposer $1$. Since two responders chose proposer $2,$ another probabilistic realisation determines who gets paired with the proposer. In this case, responder $2$ is unpaired, resulting in a zero payoff. Similarly, if one of the proposers (or both) would not be selected they would receive a zero payoff.
• Figure 2: Numeric values of the proposers' Nash equilibria offers from \ref{['eq:final_Nash']} (left) and the expected payoffs for the proposers (middle) and the responders (right) under these equilibria (see \ref{['eq:exp_payoffs_final']}). These are shown for varying number of responders and proposers, under the assumption that responders play their evolutionarily stable strategy and thus select each proposer with the same probability when the offers are identical.
• Figure 3: The expected payoff for proposers (blue), for responders (red dotted), the offer level (black) and 1-the offer level (black dotted) in subgame-perfect Nash equilibrium with respect to the proposer-responder ratio $c$, i.e. $K=cL$, in the large $L$ limit (compare \ref{['eq:limit_offer']} and \ref{['eq:limit_payoffs']}). The differences between the offer and expected payoffs arise from "inefficiencies". That is, with some probability some responders choose the same proposer and some proposers may not get chosen by any responder.