Cost of institutional incentives for promoting cooperation in $2\times2$ games and collective risk games

Prosocial behaviours, which appear to contradict Darwinian principles of individual payoff maximisation, have been extensively studied across multiple disciplines. Cooperation, requiring a personal cost for collective benefits, is widespread in nature and has been explained through mechanisms such as kin selection, direct and indirect reciprocity, and network reciprocity. Institutional incentives, which reward cooperation and punish anti-social behaviour, offer a promising approach to fostering cooperation in groups of self-interested individuals. This study investigates the behaviour of the cost functions associated with these types of interventions, using both analytical and numerical methods. Focusing on reward, punishment, and hybrid schemes, we analyse their associated cost functions under evolutionary dynamics governed by Fermi's rule, exploring their asymptotic behaviour in the limits of neutral drift and strong selection. Our analysis focuses on two game types: General $2\times2$ Games (with particular attention paid to the cooperative and defective Prisoner's Dilemma) and the Collective Risk Game. In addition to deriving key analytical results, we use numerical simulations to study how parameters such as the intensity of selection affect the behaviour of the aforementioned incentive cost functions.