Fairness in the 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 behaviour experimentally observed in the context of one vs. one Ultimatum Game. 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.

Fig 1. Graphical example of the MPMR UG 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.

Fig 2. Numeric values of the proposers’ Nash equilibria and expected payoffs.

(A) Numeric values of offers from Eq (12). (B) Expected payoffs of the proposers. (C) Expected payoff of the responders. See Eq (13) for the formulas. The values are shown for varying numbers of responders and proposers.

Fig 3. The expected payoffs and offer levels for varying proposer-responder ratio c .

Here we show the expected payoff for proposers (blue), for responders (red dashed), the offer level (black) and 1-the offer level (black dashed) in subgame-perfect Nash equilibrium with respect to the proposer-responder ratio c, i.e. K = cL, in the large L limit (compare Eq (14) and Eq (15)). 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.

Fig 4. Comparison of theoretical predictions of proposers’ offers in the subgame-perfect Nash equilibrium and evolutionary simulation results.

On the left, we show the theoretical predictions derived in the previous sections, on the right the results of the evolutionary simulations for 9 different scenarios with varying numbers of proposers and responders.