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Avoiding or restricting defectors in public goods games? - PubMed

  • ️Thu Jan 01 2015

Avoiding or restricting defectors in public goods games?

The Anh Han et al. J R Soc Interface. 2015.

Abstract

When creating a public good, strategies or mechanisms are required to handle defectors. We first show mathematically and numerically that prior agreements with posterior compensations provide a strategic solution that leads to substantial levels of cooperation in the context of public goods games, results that are corroborated by available experimental data. Notwithstanding this success, one cannot, as with other approaches, fully exclude the presence of defectors, raising the question of how they can be dealt with to avoid the demise of the common good. We show that both avoiding creation of the common good, whenever full agreement is not reached, and limiting the benefit that disagreeing defectors can acquire, using costly restriction mechanisms, are relevant choices. Nonetheless, restriction mechanisms are found the more favourable, especially in larger group interactions. Given decreasing restriction costs, introducing restraining measures to cope with public goods free-riding issues is the ultimate advantageous solution for all participants, rather than avoiding its creation.

Keywords: commitment; cooperation; evolutionary games; public goods.

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

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Figures

Figure 1.
Figure 1.

(a) Stationary distribution and fixation probabilities. The population spends most of the time in the homogeneous state of AVOID. The black arrows identify the advantageous transitions, where ρN = 1/Z denotes the neutral fixation probability. The dashed lines denote neutral transitions. Note the cyclic pattern from cooperation to defection to commitment strategies and back. (b) Contour plot of the frequency of AVOID as a function of εP and δ. For a small enough cost of arranging the commitment, AVOID is abundant whenever a sufficient compensation is associated with the commitment deal. Parameters: N = 5, Z = 100, r = 3; β = 0.1; in panel (a), εP = 0.25, δ = 2.

Figure 2.
Figure 2.

(a) Transition probabilities and stationary distributions in case of RESTRICT. For an efficient restriction (εR = 0.5 and ψ = 0.25), the population spends most of the time in the homogeneous state of RESTRICT. Notations are the same as in figure 1a. (b) Frequencies of each strategy for varying ψ, in case of RESTRICT. For a given cost of restriction (εR = 0.5), in general the better the effect of restriction on non-committers (i.e. the smaller ψ), the greater the frequency of RESTRICT. (c) Frequency of RESTRICT as a function of εR and ψ, in a population with C, D, FREE and FAKE strategies. For a large range of cost for restricting the access of non-committers, εR, and the restriction, ψ, RESTRICT is highly frequent, having a higher frequency than AVOID. The double-stroke line corresponds to the part having the same frequency as AVOID (i.e. 0.64, with the same parameter values), and the area below this line identifies the area in which RESTRICT is more frequent than AVOID. In general, the larger εR, the smaller ψ is required for RESTRICT to be advantageous to AVOID. (d) Frequencies of each strategy as a function of the group size, N. RESTRICT becomes more frequent when the group size increases, even for a rather high cost of restriction (εR = 2.0). Parameters: in panels (a–c): N = 5; in all cases, Z = 100, r = 3; εP = 0.25, δ = 2; β = 0.1.

Figure 3.
Figure 3.

(a) Range of parameters ψ, εR and εP, generated from the analytical formula in equation (2.10), in which RESTRICT is better than AVOID. For a large range of cost for restricting, the access of non-committers, εR, and the effect of restriction, ψ, RESTRICT is better than AVOID. In general, the larger εR, the smaller ψ is required for RESTRICT to be advantageous to AVOID. (b) Group size is an important factor for making RESTRICT more viable than AVOID. We compute, as a function of the group size, N, the frequencies of RESTRICT for different values of restriction cost εR (the curves without markers), in comparison to the frequency of AVOID (the red curve with circled markers). In general, the lower the cost of restriction, the higher the frequency of RESTRICT. Also, the threshold of N above which RESTRICT is more frequent than AVOID is smaller. Parameters: in panel (b), Z = 100, εP = 0,25, ψ = 0.25, β = 0.1; in both panels, N = 5, r = 3.

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