Prejudice driven spite: A discontinuous phase transition in ultimatum game

In a mix of prejudiced and unprejudiced individuals engaged in strategic interactions, the individual intensity of prejudice is expected to have effect on overall level of societal prejudice. High level of prejudice should lead to discrimination that may manifest as unfairness and, perhaps, even spite. In this paper, we investigate this idea in the classical paradigm of the ultimatum game which we theoretically modify to introduce prejudice at the level of players, terming its intensity as prejudicity. The stochastic evolutionary game dynamics, in the regime of replication-selection, reveals the emergence of spiteful behaviour as a dominant behaviour via a first order phase transition -- a discontinuous jump in the frequency of spiteful individuals at a threshold value of prejudicity. The phase transition is quite robust and becomes progressively conspicuous in the limit of large population size where deterministic evolutionary game dynamics, viz., replicator dynamics, approximates the system closely. The emergence of spite driven by prejudice is also found to persist when one considers long-term evolutionary dynamics in the mutation-selection dominated regime.