Cost efficiency of institutional incentives in finite populations

Abstract

Institutions can provide incentives to increase cooperation behaviour in a population where this behaviour is infrequent. This process is costly, and it is thus important to optimize the overall spending. This problem can be mathematically formulated as a multi-objective optimization problem where one wishes to minimize the cost of providing incentives while ensuring a desired level of cooperation within the population. In this paper, we provide a rigorous analysis for this problem. We study cooperation dilemmas in both the pairwise (the Donation game) and multi-player (the Public Goods game) settings. These are well-known models for studying cooperation since individually defection is always preferred over cooperation while mutual cooperation is the preferred collective outcome for the population as a whole. We prove the regularity of the (total incentive) cost function, characterize its asymptotic limits (infinite population, weak selection and large selection) and show exactly when reward or punishment is more efficient. We prove that the cost function exhibits a phase transition phenomena when the intensity of selection varies. We calculate the critical threshold in regards to the phase transition and study the optimization problem when the intensity of selection is under and above the critical value. It allows us to provide an exact calculation for the optimal cost of incentive, for a given intensity of selection. Finally, we provide numerical simulations to demonstrate the analytical results. Overall, our analysis provides for the first time a selection-dependent calculation of the optimal cost of institutional incentives (for both reward and punishment) that guarantees a minimum amount of cooperation. It is of crucial importance for real-world applications of institutional incentives since intensity of selection is specific to a given population and the underlying game payoff structure.

Institutional Incentives for the Evolution of Committed Cooperation: Ensuring Participation is as Important as Enhancing Compliance

“…Indeed, we have shown that the larger the cost of participation ( ) is, the greater fraction of this budget should to be used for encouraging participation to achieve an optimal level of cooperation. This observation confirms the importance of studying incentives in models considering an explicit process of commitment formation, which has been omitted in extant models of institutional incentives (Chen et al, 2015, Duong and Han, 2021, García and Traulsen, 2019, Góis et al, 2019, Wang et al, 2019. These works have not considered commitment-based interactions nor incentives for encouraging participation in the interaction (and its impact on the overall cooperation).…”

“…However, it might be more costly for the institutions to provide rewards when compliant behaviour is frequent. An interesting direction is to consider how to combine reward and punishment in a cost efficient way, as have been done in the context of social dilemmas (without considering commitment-based behaviours) (Chen et al, 2015, Duong and Han, 2021, Góis et al, 2019, Sasaki et al, 2012, Sun et al, 2021.…”

“…Evolutionary game theory (EGT) (Sigmund, 2010) provides an appropriate tool to study the evolution of cooperative behaviour in social dilemmas, as they are governed by institutional incentives (Chen et al, 2015, Cimpeanu et al, 2021, Duong and Han, 2021, Góis et al, 2019, Sasaki et al, 2012, Sun et al, 2021, Wang et al, 2019 and prior commitments (Akdeniz and van Veelen, 2021, Han et al, 2017. However, prior work has not addressed how incentives influence commitment-based behaviours and how they can be used to efficiently ensure high levels of compliance and cooperation.…”

Cost-efficient interventions for promoting fairness in the ultimatum game

“…Moreover, the target variable and the model score are unrelated, and we test for each possible set of targets using this unique measure of fairness. Similar to previous works (Chen et al, 2015, Cimpeanu et al, 2019, Duong and Han, 2021a, Wang et al, 2019, we measure cost-efficiency of an interference strategy by its total cost required over time. Furthermore, we will discuss explicitly the requirements for each strategy to be implementable, such as the availability of information about the population and local neighbourhoods, and budget constraints in each time step.…”

“…Hence, the external decision maker needs to take into consideration that they will have to repeatedly interfere in the system, in order to sustain the desired behaviour over time. Note however that, for simplicity, previous works have either omitted mutation (Han andTran-Thanh, 2018, Wang et al, 2019), or assumed that it is infinitely small (for analytical treatment) Han, 2021a, Han and. Mutation (behavioural exploration), where agents can freely experiment with new behaviours, is usually non-negligible in real populations and has been shown to play an important role in enabling cooperation in the context of social dilemmas (Antal et al, 2009, Duong and Han, 2019, 2021b, Han et al, 2012, Rand et al, 2013.…”