The evolution of anti-social rewarding and its countermeasures in public goods games - PubMed (original) (raw)

The evolution of anti-social rewarding and its countermeasures in public goods games

Miguel dos Santos. Proc Biol Sci. 2015.

Abstract

Cooperation in joint enterprises can easily break down when self-interests are in conflict with collective benefits, causing a tragedy of the commons. In such social dilemmas, the possibility for contributors to invest in a common pool-rewards fund, which will be shared exclusively among contributors, can be powerful for averting the tragedy, as long as the second-order dilemma (i.e. withdrawing contribution to reward funds) can be overcome (e.g. with second-order sanctions). However, the present paper reveals the vulnerability of such pool-rewarding mechanisms to the presence of reward funds raised by defectors and shared among them (i.e. anti-social rewarding), as it causes a cooperation breakdown, even when second-order sanctions are possible. I demonstrate that escaping this social trap requires the additional condition that coalitions of defectors fare poorly compared with pro-socials, with either (i) better rewarding abilities for the latter or (ii) reward funds that are contingent upon the public good produced beforehand, allowing groups of contributors to invest more in reward funds than groups of defectors. These results suggest that the establishment of cooperation through a collective positive incentive mechanism is highly vulnerable to anti-social rewarding and requires additional countermeasures to act in combination with second-order sanctions.

Keywords: cooperation; evolutionary games; pool rewards; social dilemmas.

© 2014 The Author(s) Published by the Royal Society. All rights reserved.

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Figures

Figure 1.

Figure 1.

Anti-social rewarding in the PGG with reward funds and exclusion. The presence of anti-social rewarding results in a cooperation breakdown, even though second-order free-riders (i.e. non-rewarding individuals) can be excluded from their respective reward fund with an exclusion rate α that exceeds a critical threshold _α_T = (N – _r_2)/[_r_2(N – 1)]. Each face of the state space S4 is represented by a triangle. The interior of the triangles (a) and S4 (b) are filled with trajectories representing the dynamics of the population. The basic model without anti-social rewarding (corresponding to the triangle D–C–RC) could lead to full cooperation as the vertex RC was a global attractor. There is an unstable equilibrium _Q_2 between C and RC and between the three strategies D–C–RC _Q_1, but small perturbations such as random mutations will eventually lead the dynamics on this face towards the vertex RC. However, the presence of anti-social rewarding makes the vertex RC unstable and RD will be the outcome, starting from any initial condition. A similar unstable equilibrium _Q_3 is possible between D and RD, but perturbations will eventually lead the system towards RD. Interior states of S4 also end up in the vertex RD. Parameters: N = 5, _c_1 = 1, _r_1 = 3, _c_2 = 1, _r_2 = 3 and α = 0.2. (Online version in colour.)

Figure 2.

Figure 2.

The interplay of exclusion and high rewarding efficacies for contributors in the PGG with reward funds. Full cooperation can be attained from any initial condition as rewarding players are able to sufficiently exclude second-order free-riders (i.e. non-rewarding individuals) from their corresponding reward fund (i.e. α > _α_T), and contributors are able to produce sufficient rewards to outcompete non-contributors (i.e. _c_3(_r_3 − 1) < _c_2(_r_2 − 1) − σ). (a) Unstable equilibria are possible between rewarding individuals and their corresponding non-rewarding type (i.e. _Q_1 between D and RD and _Q_2 between C and RC), but random perturbations will eventually lead the system to stable full cooperation. (b) Interior states of S4 also end up in the vertex RC. Parameters: N = 5, _c_1 = 1, _r_1 = 3, _c_2 = 1, _r_2 = 3, _c_3 = 1, _r_3 = 2 and α = 0.5. (Online version in colour.)

Figure 3.

Figure 3.

The interplay of exclusion and reward funds dependent on the public good produced. Full cooperation can be attained from any initial condition as rewarding players are able to sufficiently exclude second-order free-riders (i.e. non-rewarding individuals) from their corresponding reward fund (i.e. α > _α_T), and reward funds are raised with a fraction β of the public good produced. (a) An unstable equilibrium _Q_2 is possible between pro-social rewarders and their corresponding non-rewarding type. The whole edge D–RD is a line of fixed points because, in the absence of contributors, anti-social rewarders cannot invest in their reward fund. Random perturbations such as mutations will eventually lead the system to stable full cooperation. (b) Interior states of S4 also end up in the vertex RC. Parameters: N = 5, _c_1 = 1, _r_1 = 3, _r_2 = 3, β = 0.5 and α = 0.2. (Online version in colour.)

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