Optimization of institutional incentives for cooperation in structured populations

TL;DR

This paper investigates how to optimally implement positive and negative incentives to promote cooperation in structured populations, minimizing costs while ensuring collective goals, supported by theoretical derivations and extensive simulations across various network types.

Contribution

It derives the optimal, time-invariant incentive protocols for cooperation in structured populations and compares their costs, providing a rigorous framework for incentive design in game theory.

Findings

Optimal incentive protocols are identical and time-invariant.

Both rewarding and punishing schemes have comparable implementation costs.

Results are robust across different network structures.

Abstract

The application of incentives, such as reward and punishment, is a frequently applied way for promoting cooperation among interacting individuals in structured populations. However, how to properly use the incentives is still a challenging problem for incentive-providing institutions. In particular, since the implementation of incentive is costly, to explore the optimal incentive protocol, which ensures the desired collective goal at a minimal cost, is worthy of study. In this work, we consider the positive and negative incentives respectively for a structured population of individuals whose conflicting interactions are characterized by a prisoner's dilemma game. We establish an index function for quantifying the cumulative cost during the process of incentive implementation, and theoretically derive the optimal positive and negative incentive protocols for cooperation on regular networks. We find that both types of optimal incentive protocols are identical and time-invariant. Moreover, we compare the optimal rewarding and punishing schemes concerning implementation cost and provide a rigorous basis for the usage of incentives in the game-theoretical framework. We further perform computer simulations to support our theoretical results and explore their robustness for different types of population structures, including regular, random, small-world, and scale-free networks.