Abstract
We study the success of generalised trigger strategies in the evolution of cooperation in international environmental negotiations where the performance of these strategies is derived from asymmetric \(n\)-player prisoners’ dilemmas. Our results suggest that there exist regions in the relevant parameter space—i.e. costs and benefits, low and high tit-for-tat thresholds, probability of continued interaction—such that (partial) cooperation may emerge as long-run attractor of the evolutionary dynamics in these asymmetric social dilemmas.
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Notes
The random matching assumption is relaxed in literature on assortative matching, homophily or preferential attachment (see e.g. Boyd and Richerson 1988; Bergstrom 2003; Alger and Weibull 2013). In all these models cooperators interact with a disproportionately higher probability with similar players (i.e. cooperators), and much higher levels of cooperation can be sustained vis-a-vis the random matching. As our goal is to isolate the role of direct reciprocity, we leave the extension of the model to correlated matching probabilities for future research.
We could model reciprocity in the coalitional, internal–external stability model of environmental agreements but we believe that (generalized) tit-for-tat is a natural framework to think about countries’ conditioning behavior in revising emission-reduction strategies in the context of the protracted COP framework, which is a long-run process that is punctuated occasionally (Kyoto, Copenhagen, Durban, etc.).
One could, of course, assume full rationality in the game \(\{{ TFT}_{\alpha _{1},\beta _{1}}^{i},AllD_{i}\}\) vs. \(\{{ TFT}_{\alpha _{2},\beta _{2}}^{j},AllD_{j}\}\) and compute Nash equilibria accordingly. Replicator dynamics offers, arguably, a more realistic description of human interaction as it is grounded in simple and intuitive heuristics (e.g. imitate the better-performing strategy, see Sandholm 2010).
For evolutionary games with three or more strategies, discrete-time replicator dynamics may generate more complicated long-run attractors than its continuous-time counterpart; the phenomenon is caused by overshooting of the interior attractors due to arbitrarily large time steps in the discretization of the continous-time process (Weibull 1997).
We thank an anonymous referee for raising the important point of convergence time to the (cooperative) equilibrium. This issue is particularly acute in stochastic evolutionary dynamics where a specific sequence of mutations needs to occur for the system to escape from the basin of attraction of one equilibrium (see Nisan et al. 2008; Feldman and Tamir 2012 for examples of games and stochastic adjustments with convergence in polynomial time), but less so for continuous-time deterministic systems where adjustment takes place almost instantaneously once the repeated game expected payoffs are known.
A similar sensitivity test can be performed with respect to type \(J\) triggers \((\alpha _{2},\beta _{2})\) but with fewer degrees of freedom, given the constraint on thresholds \(\alpha _{1}\le \alpha _{2},\beta _{1}\le \beta _{2}\) which effectively restricts \(\alpha _{2}\in [\alpha _{1}, \frac{n}{2}-1]\) and \(\beta _{2}\in [\beta _{1},\frac{n}{2}-1].\)
We assume that type \(j\) player continues to cooperate unless both \( k=\alpha _{2}\) and \(l=\beta _{2}\) hold. This assumption is not innocuous as a more stringent strategy (i.e. start cooperate and revert to perpetual defection if either \(k=\alpha _{2}\) or \(l=\beta _{2}\)) would lead to a different payoff structure for the \(AllD\) strategy. A similar assumption applies when the second corner point is hit: \(k=\alpha _{1}\) and \(l=\beta _{1}.\)
The incomplete beta function is given by: \(B(x;a,b)= \int _{0}^{x}t^{a-1}(1-t)^{b-1}dt\) which can be normalized by the (complete) beta function \(B(a,b)=\int _{0}^{1}t^{a-1}(1-t)^{b-1}dt\) to obtain the regularized incomplete beta function \(I_{x}(a,b)=\frac{B(x;a,b)}{B(a,b)}.\) For a binomially distributed random variable \(X\sim B(p,n)\) the regularized beta function \(I\) characterizes the cumulative probability distribution, i.e. \(P(X\le \alpha )=I_{1-p}(n-\alpha ,\alpha +1)=1-I_{p}(\alpha +1,n-\alpha ).\)
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We are very grateful for the comments of two anonymous reviewers and the participants of a workshop at ZEW, Mannheim, Germany, in 2014. Part of Marius Ochea’s work on the paper was financed by the project “SENSE”, funded by the TECT program of the European Science Foundation, and part of Aart de Zeeuw’s work on the paper was financed by the project “Improving international cooperation on emission abatement”, funded by the NORKLIMA program of the Norwegian Research Council and by the project “Thalis-Optimal Management of Dynamical Systems of the Economy and the Environment” co-funded by the European Social Fund and Greek national funds.
Appendix: Repeated Game Payoff Differentials
Appendix: Repeated Game Payoff Differentials
The replicator dynamics (2)–(3) is driven by the payoff differentials \(\Delta \Pi _{i}=\Pi _{{ TFT}_{\alpha _{1},\beta _{1}}^{i}}\) \(-\Pi _{AllD^{i}}\). We focus on case (i) in Sect. 2.1, where the conditioning behavior of a type \(J\) country is more demanding with respect to the cooperation of both types of countries: \(\{\alpha _{1}\le \alpha _{2},\beta _{1}\le \beta _{2}\}\). Figure 1 displays the possible paths of actions of the countries using reciprocal strategies.
Given the fractions \((\rho _{1},\rho _{2})\) of reciprocators in the groups of countries \(I,J\), we assume that the number of reciprocal strategies \({ TFT}\) in a sample of size \(\frac{n}{2}\) is binomially distributed:
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\(\#\) of \({ TFT}_{\alpha _{1},\beta _{1}}^{i}\) in a sample of \(\frac{n }{2}\equiv k\sim B_{1}\left( \frac{n}{2}\rho _{1},\frac{n}{2}\rho _{1}(1-\rho _{1})\right) \),
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\(\#\) of \({ TFT}_{\alpha _{2},\beta _{2}}^{j}\) in a sample of \(\frac{n }{2}\equiv l\sim B_{2}\left( \frac{n}{2}\rho _{2},\frac{n}{2}\rho _{2}(1-\rho _{2})\right) \).
Figure 1 shows that the space \((k,l)\) can be split in three payoff-equivalent regions that we denote by \(R_{1},R_{2},R_{3}\). We first compute \(\Pi _{{ TFT}_{\alpha _{1},\beta _{1}}^{i}},\) \(\Pi _{AllD^{i}}\) and \( \Delta \Pi _{i}\) for each region.
1.1 A.1 \(R_{1}\equiv \{(k,l),k\ge \alpha _{2},l\ge \beta _{2}\}\)
In region \(1\), both types \(I,J\) start with and continue cooperation \(C\). For type \(I\), the expected net benefit of the reciprocal strategy \({ TFT}\) is
whereas the \(AllD\) strategy yields
with
Combining \(\Pi _{{ TFT}_{\alpha _{1},\beta _{1}}^{i}}^{1}\) and \(\Pi _{AllD^{i}}^{1}\) we can preserve both lower bounds of the summation operators as long as we adjust for the bordercaseFootnote 7 payoffs at (\(k=\alpha _{2})\) and \( (l=\beta _{2})\):
Finally, by using the expressions for \(V(C_{i}\mid k+1\mid l),V(D_{i}\mid k\mid l)\) and \(V(D_{i}\mid k\mid 0)\), we obtain region \(1\)’s payoff difference:
where \(I_{\rho _{1}}(\cdot ,\cdot ),I_{\rho _{2}}(\cdot ,\cdot )\) stand for the regularized incomplete beta functions.Footnote 8
1.2 A.2 \(R_{2}\equiv \{(k,l),(k\ge \alpha _{1}\wedge l\ge \beta _{1})\backslash R_{1}\}\)
In region \(2\), type \(I\) starts with and continues cooperation \(C\), while type \(J\) starts with cooperation \(C\) but switches to defection \(D\). The expected net benefits of a reciprocal strategy \({ TFT}\) from type \(I\) are
with
whereas the expected net benefits accruing to the \(AllD\) strategy are given by
with
Combining \(\Pi _{{ TFT}_{\alpha _{1},\beta _{1}}^{i}}^{2}\) and \(\Pi _{AllD^{i}}^{2}\), and accounting for the bordercase \((k=\alpha _{1})\wedge (l=\beta _{1})\), we obtain for \(\Delta \Pi _{i}^{2}=\Pi _{{ TFT}_{\alpha _{1},\beta _{1}}^{i}}^{2}-\Pi _{AllD^{i}}^{2}:\)
Using (5), (6) for \(V(C_{i}\mid k+1\mid l),V(C_{i}\mid k+1\mid 0),V(D_{i}\mid k\mid l)\) and \(V(D_{i}\mid 0\mid 0)\) we can simplify, with the help of the incomplete beta function to
1.3 A.3 \(R_{3}\equiv \{(k,l),(k\ge 0\wedge l\ge 0)\backslash \{R_{1}\cup R_{2}\}\}\)
In region \(3\), both types \(I,J\) start with cooperation \(C\) and switch to defection \(D\). The expected net benefits of the reciprocal strategy \({ TFT}\) from type \(I\) are
with
whereas \(AllD\) strategies from type \(I\) get
with
Combining \(\Pi _{{ TFT}_{\alpha _{1},\beta _{1}}^{i}}^{3}\) and \(\Pi _{AllD^{i}}^{3}\), we can write the repeated game’s payoff difference in region \(3\) as
1.4 A.4 Payoff Differentials
Summing up the results for the three regions (9), (10) and (11), the expected payoff difference between a type \(I\) country using reciprocal strategy \({ TFT}_{\alpha _{1},\beta _{1}}^{i}\) and an unconditional defector \(AllD^{i}\) yields
A similar computation yields type \(J\)’s payoff differential between \( { TFT}_{\alpha _{2},\beta _{2}}^{j}\) and \(AllD^{j}\):
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Ochea, M.I., de Zeeuw, A. Evolution of Reciprocity in Asymmetric International Environmental Negotiations. Environ Resource Econ 62, 837–854 (2015). https://doi.org/10.1007/s10640-014-9841-5
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DOI: https://doi.org/10.1007/s10640-014-9841-5