Stochastic ultimatum game: Spite-driven resource feedback fosters fairness

Resource scarcity can fundamentally encourage antisocial behaviour, whereas resource abundance can promote fair behaviour. Experimental evidence indeed suggests that scarcity induces spiteful behaviour, while repeated interactions enhance fairness. However, existing studies of game--environment feedback systems are largely confined to the evolution of cooperation and they overlook the interplay between resources, spite, and fairness. To address this lacuna, we develop a stochastic ultimatum game framework in which an offerer and an accepter repeatedly interact to negotiate exploitation of a self-renewable resource under the ownership of the offerer. Successful agreements deplete the resource, whereas unsuccessful agreements inhibit exploitation and facilitate replenishment. The mutation--selection driven two-species stochastic evolutionary dynamics reveal that the emergence of spite and fairness strongly depends on the resource growth rate. Fairness predominantly prevails for resources with high growth rates. Intriguingly, low resource growth rates give rise to a resource feedback loop driven by spite: spiteful behaviour dominates in the depleted state, facilitating transition of the resource state to replete state which, in turn, promotes fairness through repeated interactions.