Abstract
Counterfactual Thinking is a human cognitive ability studied in a wide variety of domains. It captures the process of reasoning about a past event that did not occur, namely what would have happened had this event occurred, or, otherwise, to reason about an event that did occur but what would ensue had it not. Given the wide cognitive empowerment of counterfactual reasoning in the human individual, the question arises of how the presence of individuals with this capability may improve cooperation in populations of self-regarding individuals. Here we propose a mathematical model, grounded on Evolutionary Game Theory, to examine the population dynamics emerging from the interplay between counterfactual thinking and social learning (i.e., individuals that learn from the actions and success of others) whenever the individuals in the population face a collective dilemma. Our results suggest that counterfactual reasoning fosters coordination in collective action problems occurring in large populations, and has a limited impact on cooperation dilemmas in which coordination is not required. Moreover, we show that a small prevalence of individuals resorting to counterfactual thinking is enough to nudge an entire population towards highly cooperative standards.
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Notes
- 1.
Formally, one can write this process as an average over a hyper-geometric sampling in a population of size Z and k cooperators. This gives the probability of an agent to interact with N-1 other players, where, among those, j are cooperators. In this case, the average fitness fD and fC of Ds and Cs, respectively, in a population with k Cs, is given by [17] \( f_{D} (k) = \left( {\begin{array}{*{20}c} {Z - 1} \\ {N - 1} \\ \end{array} } \right)^{ - 1} \sum\limits_{j = 0}^{N - 1} {\,\left( {\begin{array}{*{20}c} k \\ j \\ \end{array} } \right){\kern 1pt} } \;\left( {\begin{array}{*{20}c} {Z - k - 1} \\ {N - j - 1} \\ \end{array} } \right)\,P_{D} (j) \) and \( f_{C} (k) = \left( {\begin{array}{*{20}c} {Z - 1} \\ {N - 1} \\ \end{array} } \right)^{ - 1} \sum\limits_{j = 0}^{N - 1} {\,\left( {\begin{array}{*{20}c} {k - 1} \\ j \\ \end{array} } \right){\kern 1pt} } \;\left( {\begin{array}{*{20}c} {Z - k} \\ {N - j - 1} \\ \end{array} } \right)\,P_{C} (j + 1) \), where PC and PD are given by Eq. (1).
- 2.
For simplicity we assume that the population is large enough such that \( Z \approx Z - 1 \).
- 3.
Strictly speaking, by the finite population analogues of the internal fixed points in infinite populations.
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Acknowledgements
We are grateful to The Anh Han and Tom Lenaerts for comments. We are also grateful to the anonymous reviewers for their improvement recommendations This work was supported by FCT-Portugal/MEC, grants NOVA-LINCS UID/CEC/04516/2013, INESC-ID UID/CEC/50021/2013, PTDC/EEI-SII/5081/2014, and PTDC/MAT/STA/3358/2014.
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Pereira, L.M., Santos, F.C. (2019). Counterfactual Thinking in Cooperation Dynamics. In: Nepomuceno-Fernández, Á., Magnani, L., Salguero-Lamillar, F., Barés-Gómez, C., Fontaine, M. (eds) Model-Based Reasoning in Science and Technology. MBR 2018. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-030-32722-4_5
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