Moderate Responder Committees Maximize Fairness in (NxM)-Person Ultimatum Games

We introduce and study a multiplayer version of the classical Ultimatum Game in which a group of N Proposers jointly offers a division of resources to a group of M Responders. In general, the proposal is rejected if the (average) proposed offer is lower than the (average) response threshold in the Responders group. A motivation for our work is the exchange of flexibilities between different smart energy communities, where the surplus of one community can be offered to meet the demand of a second community. We find that, in the absence of any mechanism, the co-evolving populations of Proposers and Responders converge to a state in which proposals and acceptance thresholds are low, as predicted by the rational choice theory. This is more evident if the Proposers' groups are larger (i.e., large N). Low proposals imply an unfair exchange that highly favors the Proposers. To circumvent this drawback, we test different committee selection rules which determine how Responders should be selected to form decision-making groups, contingent on their declared acceptance thresholds. We find that selecting the lowest-demanding Responders maintains unfairness. However, less trivially, selecting the highest-demanding individuals also fails to resolve this imbalance and yields a worse outcome for all due to a high fraction of rejected proposals. Selecting moderate Responders optimizes overall fitness. This result provides a practical message for institutional design and the model proposed allows testing policies and emergent behaviors on the intersection between social choice theory, committee selection and fairness elicitation.