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When agreement-accepting free-riders are a necessary evil for the evolution of cooperation - PubMed

  • ️Sun Jan 01 2017

When agreement-accepting free-riders are a necessary evil for the evolution of cooperation

Luis A Martinez-Vaquero et al. Sci Rep. 2017.

Abstract

Agreements and commitments have provided a novel mechanism to promote cooperation in social dilemmas in both one-shot and repeated games. Individuals requesting others to commit to cooperate (proposers) incur a cost, while their co-players are not necessarily required to pay any, allowing them to free-ride on the proposal investment cost (acceptors). Although there is a clear complementarity in these behaviours, no dynamic evidence is currently available that proves that they coexist in different forms of commitment creation. Using a stochastic evolutionary model allowing for mixed population states, we identify non-trivial roles of acceptors as well as the importance of intention recognition in commitments. In the one-shot prisoner's dilemma, alliances between proposers and acceptors are necessary to isolate defectors when proposers do not know the acceptance intentions of the others. However, when the intentions are clear beforehand, the proposers can emerge by themselves. In repeated games with noise, the incapacity of proposers and acceptors to set up alliances makes the emergence of the first harder whenever the latter are present. As a result, acceptors will exploit proposers and take over the population when an apology-forgiveness mechanism with too low apology cost is introduced, and hence reduce the overall cooperation level.

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Conflict of interest statement

The authors declare that they have no competing interests.

Figures

Figure 1
Figure 1

Proposers avoid pure defectors without the need of acceptors in RC games. Composition of the recurrent sets for RC one-shot games. Composition of the recurrent sets for each studied RC type game when all the strategies are included (left) and when acceptors are removed (right). Each pie-chart is associated with a specific (ϵ, δ) pair. Each strategy is assigned a different colour. The pieces of the pie-chart separated by thick lines correspond to different nodes of the recurrent set. Their sizes are proportional to their probabilities in the stationary state. If a sector is of a single colour it means that the node corresponds to a pure state; if it is subdivided in smaller sectors with different colours it means that the node corresponds to a mixed state, the different colours representing the coexisting strategies. The sizes of these sub-sectors are proportional to their fraction within the mixed state. On the top-right of this figure, a visual explanation through an illustration in the figure has been included.

Figure 2
Figure 2

Mixed states between proposers and acceptors are necessary to maintain pure defectors at bay in PC games. Composition of the recurrent sets for PC one-shot games. Same as Fig. 1 but for PC type games.

Figure 3
Figure 3

Pure defection is stable in PC games but not in RC games. A mixed state between cooperative proposers and acceptors is stable in both scenarios. Replicator dynamics for a subset of strategies in one-shot games. Replicator dynamics for an infinite population with (P, C, D), (A, C, D), and (N, D) strategies for RC (left) and PC (right) games. Small filled (empty) circles represent stable (unstable) rest points. We assumed ϵ = 0.25 and δ = 4. Several filled (empty) circles along the ACD-ND axis reveal that the whole interval is stable (unstable) in the axis; the grey filled circle represents the change of stability in the axis for the PC game. Regions where the dynamics lead to a mixed state of (A, C, D) with (P, C, D) or (N, D) in the PC game are separated by a black curve. Figures obtained with Dynamo.

Figure 4
Figure 4

No mixed states are formed in repeated games. Acceptors reduce cooperation in the absence of noise. Composition of the recurrent sets for repeated games without forgiveness. Strategies represented as in Fig. 1 for RC and PC repeated games for different values of α. Different pies for a given game stand for different attractors. We assumed ϵ = 0.25 and δ = 4.

Figure 5
Figure 5

Acceptors reduce cooperation when an apology-forgiveness mechanism is introduced; defective acceptors take over the population when the apology cost is too low in RC games. Composition of the recurrent sets for RC repeated games with forgiveness. Same as Fig. 1 but for RC repeated games as a function of α and γ. Dotted (plain) sectors represent strategies that (do not) include apology. We assumed ϵ = 0.25 and δ = 4.

Figure 6
Figure 6

Similar to PC games, in RC games acceptors reduce cooperation in general and take over the population whenever the apology cost is too low when an apology-forgiveness mechanism is introduced. Composition of the recurrent sets for PC repeated games with forgiveness. Same as Fig. 5 but for PC repeated games.

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