Francisco C. Santos’s research while affiliated with Inesc-ID and other places

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Publications (214)


Average cooperation ratio (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta$$\end{document}) (and respective standard deviation) promoted by the emotion-based image scoring norm (blue circles) and the baseline image scoring (orange squares), for the different probabilities of accounting for emotion (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}), when all errors—execution, assessment and assignment—are present (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha =\chi =\varepsilon =0.002$$\end{document}). While the (low) performance of Image Scoring remains consistent with the literature, by looking at the performance of emotion-based image scoring we obtain clear benefits by using emotional expressions as part of the moral evaluation process—higher \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} values lead to higher cooperation levels under the emotion-based norm, while having no impact on the baseline norm. Despite this, such behaviour is non-monotonous: when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =1$$\end{document}, cooperation levels under the emotion-based norm plummet. Each data point averages the results of 300 runs with the same parameter configuration. Other parameters: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=5, c=1, z=50, \mu =1/z, \beta =1$$\end{document}.
Contourplots of cooperation by error magnitude, for each of the studied errors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon , \alpha , \chi$$\end{document}, respectively. Encoded by colour, according to the colour bar on the right-hand side, we plot the cooperation index difference between emotion-based image scoring and basic image scoring, by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} value on the y-axis and by error magnitude on the x-axis, from among the following values: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon , \chi , \alpha \in \{\frac{10^{-3}}{z}, \frac{10^{-2}}{z}, \frac{10^{-1}}{z}, \frac{1}{z}, \frac{10}{z}\}$$\end{document}, where z is the population size. For high \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} values, we can see that the greater the magnitude of each error, the greater the impact of using an emotion-based social norm, with two notable exceptions: when errors become too frequent (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =\chi = \alpha = \frac{10}{z} = 0.2$$\end{document}—one error occurring on average in every 5 interactions), and when \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma =1$$\end{document}—once again showing a phase transition occurring in the interval of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0.9 \le \gamma \le 1$$\end{document}. Each data point averages the results of 100 simulations. Other parameters are set to the same values as the previous figure.
Average relative frequencies of emotional profiles, reputations and strategies by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document}, for the emotion-based image scoring norm. Blue circles represent the average frequency of cooperative emotional profiles (complemented by the omitted frequency of competitive emotional profiles). Lines represent the frequency of the four possible strategies—always cooperate, always defect, discriminate and paradoxically discriminate. The green shaded area represents the average frequency of good reputations (complemented by the omitted frequency of bad reputations). One can observe how the dominance of the discriminate strategy, the cooperative emotional profile and the prevalence of good reputations for high \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma$$\end{document} values steers the system towards the cooperation levels observed in Fig. 1. Error probabilities are set to values of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon =\chi =\alpha =\frac{10^{-2}}{z}=0.002$$\end{document}, and all other parameters are set to the same values as the previous figure.
Comparison of the normal version and an emotion-based version of the social norm.
Evolution of indirect reciprocity under emotion expression
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March 2025

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45 Reads

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Celso M. de Melo

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Francisco C. Santos

Do emotion expressions impact the evolution of cooperation? Indirect Reciprocity offers a solution to the cooperation dilemma with prior work focusing on the role of social norms in propagating others’ reputations and contributing to evolutionarily stable cooperation. Recent experimental studies, however, show that emotion expressions shape pro-social behaviour, communicate one’s intentions to others, and serve an error-correcting function; yet, the role of emotion signals in the evolution of cooperation remains unexplored. We present the first model of IR based on evolutionary game theory that exposes how emotion expressions positively influence the evolution of cooperation, particularly in scenarios of frequent errors. Our findings provide evolutionary support for the existence of emotion-based social norms, which help foster cooperation among unrelated individuals.

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Schematic representation of the Bottom-Up institutional model developed here to address the sustainable governance of GRC making use of the CRD dilemma. There are three pro-social strategies: C (Cooperators) P (Punishers) and R (rewarders) that contribute with a fraction c of their endowment to the CRD (left-pointing arrows); and one anti-social strategy, D (Defectors), that does not contribute. P and R also pay an additional tax \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _t$$\end{document} to fund an Electoral institution (right-pointing arrows) which decides, based on a majority rule, how to use its revenue \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Delta _{RP}$$\end{document}: Either to Punish the Ds, to reward the pro-social strategies: C, P and R or to reward the pro-social and Punish the anti-social in case of a tie. In the well-mixed approximation, the configuration of the population can be specified by the state vector \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{i}=\{i_C,i_P,i_R,i_D\}$$\end{document}, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$i_S$$\end{document} is the number of individuals using strategy S in the population. We use a corresponding notation to specify the composition of each group: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{j}=\{j_C,j_P,j_R,j_D\}$$\end{document} where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j_S$$\end{document} is the number of individuals using strategy S in the group—see Methods for full details of the model.
Stochastic Evolutionary dynamics of the Bottom-Up Electoral model developed here, showing that the overall dynamics is dominated, for most parameter values, by 2 interior attractors, both depicted with solid orange spheres: One close to the ALL-D configuration, and another (“cooperative”) at a configuration where Cs clearly dominate. The black and blue arrows (to the left of the dashed triangle) illustrate the most likely paths (in a stochastic sense) that converge to the ALL-D attractor, whereas the blue, green and red arrows (to the right of the dashed triangle) illustrate those paths that converge to the cooperative attractor. Whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r < 0.5$$\end{document} (for the model parameters chosen) the population remains, most of the time, near the ALL-D configuration, well below the fitness barrier illustrated by black dashed lines that qualitatively represent the intersection of the (quasi-planar) fitness barrier top surface with the surface of the simplex. Whenever \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r > 0.7$$\end{document} (see Fig. 3) the population is now able both to tunnel through the barrier and to evolve towards the cooperative attractor where (in a total of 70 individuals) there are 52 Cs, 7 Rs, 6 Ps and 5 Ds. Finally, among the plethora of trajectories (in blue) converging to the “cooperative” attractor, we distinguish 2 types belonging to different “classes” of evolutionary paths: 1) In green we illustrate paths where convergence evolves mostly due to an initial rise of Rs without any significant participation of Ps. 2) In red, paths where convergence evolves mostly through a significant increase of Ps without any significant participation of Rs. As discussed in detail in the main text, the first class of trajectories is more abundant than the second class. Parameters used: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z=70$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =1/Z$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =5$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=8$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{pg}=6$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_I=2$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=1$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.1$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =2$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _t=0.03$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$r=0.8$$\end{document}.
The population average group achievement \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta _G$$\end{document} (defined in Methods) is plotted as a function of risk for the case of i) No institutions (black dashed line with open circles); ii) Local sanctioning (punishing) institutions (red solid line with solid circles); iii) Local rewarding institutions (green dashed line with open circles) and iv) Local Electoral institutions (blue solid line with solid circles). Clearly, the present bottom-up approach leads to higher overall cooperation, for all values of risk, compared to other models developed previously. Parameters used: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z=140$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu =1/Z$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\beta =2$$\end{document}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N=8$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_{pg}=6$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n_I=2$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b=1$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$c=0.1$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi _t=0.03$$\end{document}, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =2$$\end{document}.
A new electoral bottom-up model of institutional governance

January 2025

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28 Reads

The sustainable governance of Global Risky Commons (GRC)—global commons in the presence of a sizable risk of overall failure—is ubiquitous and requires a global solution. A prominent example is the mitigation of the adverse effects of global warming. In this context, the Collective Risk Dilemma (CRD) provides a convenient baseline model which captures many important features associated with GRC type problems by formulating them as problems of cooperation. Here we make use of the CRD to develop, for the first time, a bottom-up institutional governance framework of GRC. We find that the endogenous creation of local institutions that require a minimum consensus amongst group members—who, in turn, decide the nature of the institution (reward/punishment) via an electoral process—leads to higher overall cooperation than previously proposed designs, especially at low risk, proving that carrots and sticks implemented through local voting processes are more powerful than other designs. The stochastic evolutionary game theoretical model framework developed here further allows us to directly compare our results with those stemming from previous models of institutional governance. The model and the methods employed here are relevant and general enough to be applied to a variety of contemporary interdisciplinary problems.


Co-evolution of risk and cooperation in climate policies under wealth inequality

December 2024

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9 Reads

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1 Citation

PNAS Nexus

Worldwide cooperation is necessary to mitigate the effects of climate change. Many previous investigations employed the so-called collective risk dilemma, where the risk of losing everything whenever a target is not met was fixed from the outset, rendering predictions dependent on snapshot values assumed for this parameter, whose importance was found to be paramount. Here we couple risk with the overall success of mitigation, investigating the co-evolution of risk and cooperation in a world where countries are partitioned in two different wealth classes, allowing us to further assess the impact of wealth inequality and homophily on the co-evolutionary dynamics. We show that the stochastic dynamics is dominated by a global attractor, typically located in a region of low risk, where most developed countries cooperate most of the time while developing countries cooperate to a lesser extent. This scenario assumes no homophily which, when moderate, can contribute to increase overall cooperation, more so when combined with the presence of a small fraction of developing countries that opt for an unconditional cooperative behavior.


Counterfactual Thinking in Stochastic Dynamics of Cooperation

October 2024

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8 Reads

Understanding the conundrum of human cooperation has been declared one of the century’s grand challenges. Traditionally, the evolution of cooperative action in nature is analyzed through the lens of Evolutionary Game Theory, specifically, using the social learning framework, a model for Darwinian competition. However, more complex individuals may resort to more sophisticated learning rules such as Counterfactual Thinking (CT). Given these individuals’ cognitive empowerment, the question of how the presence of counterfactuals influences the evolution of cooperation in a hybrid population of these complex agents and social learners. Here we explore how cooperation emerges from the interplay of different strategy revision paradigms by analyzing large-scale Markov processes. We find that increasing the prevalence of CT individuals can promote cooperation, but such an increase is non-monotonous. Moreover, whereas counterfactual reasoning generally fosters cooperation, it fails to promote such behaviour among counterfactuals. Lastly, we find that increasing the population’s heterogeneity level enhances cooperation among social learners, but again not among counterfactuals. This indicates that, under certain circumstances, the presence of more sophisticated agents may help promote cooperation in hybrid populations. The proposed study may come as a starting point for a more profound understanding of agents’ counterfactual rationality impact on hybrid populations.




The art of compensation: How hybrid teams solve collective-risk dilemmas

February 2024

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91 Reads

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4 Citations

It is widely known how the human ability to cooperate has influenced the thriving of our species. However, as we move towards a hybrid human-machine future, it is still unclear how the introduction of artificial agents in our social interactions affect this cooperative capacity. In a one-shot collective risk dilemma, where enough members of a group must cooperate in order to avoid a collective disaster, we study the evolutionary dynamics of cooperation in a hybrid population. In our model, we consider a hybrid population composed of both adaptive and fixed behavior agents. The latter serve as proxies for the machine-like behavior of artificially intelligent agents who implement stochastic strategies previously learned offline. We observe that the adaptive individuals adjust their behavior in function of the presence of artificial agents in their groups to compensate their cooperative (or lack of thereof) efforts. We also find that risk plays a determinant role when assessing whether or not we should form hybrid teams to tackle a collective risk dilemma. When the risk of collective disaster is high, cooperation in the adaptive population falls dramatically in the presence of cooperative artificial agents. A story of compensation, rather than cooperation, where adaptive agents have to secure group success when the artificial agents are not cooperative enough, but will rather not cooperate if the others do so. On the contrary, when risk of collective disaster is low, success is highly improved while cooperation levels within the adaptive population remain the same. Artificial agents can improve the collective success of hybrid teams. However, their application requires a true risk assessment of the situation in order to actually benefit the adaptive population (i.e. the humans) in the long-term.


Figure 1. Number of clusters of opinions for two extreme social tensions (fear in SH and greed in SD)
Figure 2. Increase in number of clusters of opinions when comparing different social tensions (S and T values of the game) Panel of sensitivity analysis on ðe; bÞ showing increase in fragmentation when comparing two pairs of games (PD w.r.t. SH, and SD w.r.t. PD). Both heat-maps show a clear increase in the number of opinion clusters when values of S and T are higher and, therefore, a greedy and fearless population is defined.
Figure 3. Evolution of opinions for different social tensions
Figure 4. Snapshots of the same social network structure at time-steps t = 0 and t = 500 showing the evolution of opinions for each node for SD (top) and SH (bottom)
Success-driven opinion formation determines social tensions

February 2024

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50 Reads

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7 Citations

iScience

Polarization is common in politics and public opinion. It is believed to be shaped by media as well as ideologies, and often incited by misinformation. However, little is known about the microscopic dynamics behind polarization and the resulting social tensions. By coupling opinion formation with the strategy selection in different social dilemmas, we reveal how success at an individual level transforms to global consensus or lack thereof. When defection carries with it the fear of punishment in the absence of greed, as in the stag-hunt game, opinion fragmentation is the smallest. Conversely, if defection promises a higher payoff and also evokes greed, like in the prisoner’s dilemma and snowdrift game, consensus is more difficult to attain. Our research thus challenges the top-down narrative of social tensions, showing they might originate from fundamental principles at individual level, like the desire to prevail in pairwise evolutionary comparisons.


Evolution of a Theory of Mind

February 2024

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49 Reads

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4 Citations

iScience

Even though the Theory of Mind in upper primates has been under investigation for decades, how it may evolve remains an open problem. We propose here an evolutionary game theoretical model where a finite population of individuals may use reasoning strategies to infer a response to the anticipated behavior of others within the context of a sequential dilemma, i.e., the Centipede Game. We show that strategies with bounded reasoning evolve and flourish under natural selection, provided they are allowed to make reasoning mistakes and a temptation for higher future gains is in place. We further show that non-deterministic reasoning co-evolves with an optimism bias that may lead to the selection of new equilibria, closely associated with average behavior observed in experimental data. This work reveals both a novel perspective on the evolution of bounded rationality and a co-evolutionary link between the evolution of Theory of Mind and the emergence of misbeliefs.


Fig. 1 (a) Left panel: Learning gradients for social learners (SL, black line) and counterfactual learners (CT, red line) for the N-person SH game. If the learning gradient is positive (negative), the fraction of cooperators will tend to increase (decrease). Empty and full circles represent the finite population analogue of unstable and stable fixed points, respectively. Right panel: Stationary distribution of the Markov processes created by the transition probabilities pictured in the left panel; it characterizes the prevalence in time of each fraction of cooperators in finite populations. (b) Right panel: Overall cooperation as a function of the prevalence of individuals resorting to social learning (SL, χ) and counterfactual reasoning (CT, 1-χ). It shows that only a relatively small prevalence of counterfactual thinking is required to nudge cooperation in an entire population of self-regarding agents. Other parameters: Z = 50, N = 6, F = 5.5. M = N/2 (panel A), c = 1.0, μ = 0.01, β SL = β CT = 5.0
AI Modelling of Counterfactual Thinking for Judicial Reasoning and Governance of Law

December 2023

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40 Reads

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1 Citation

When speaking of moral judgment, we refer to a function of recognizing appropriate or condemnable actions and the possibility of choice between them by agents. Their ability to construct possible causal sequences enables them to devise alternatives in which choosing one implies setting aside others. This internal deliberation requires a cognitive ability, namely that of constructing counterfactual arguments. These serve not just to analyse possible futures, being prospective, but also to analyse past situations, by imagining the gains or losses resulting from alternatives to the actions actually carried out, given evaluative information subsequently known. Counterfactual thinking is in thus a prerequisite for AI agents concerned with Law cases, in order to pass judgement and, additionally, for evaluation of the ongoing governance of such AI agents. Moreover, given the wide cognitive empowerment of counterfactual reasoning in the human individual, namely in making judgments, the question arises of how the presence of individuals with this ability can improve cooperation and consensus in populations of otherwise self-regarding individuals. Our results, using Evolutionary Game Theory (EGT), suggest that counterfactual thinking fosters coordination in collective action problems occurring in large populations and has limited impact on cooperation dilemmas in which such coordination is not required.


Citations (60)


... This said, we did not address in detail other aspects of the present model which are also of relevance, namely the behavior of strategies in time. As is well known, evolutionary game models may lead to oscillatory behaviour, both under deterministic [27][28][29] and stochastic 30,31 dynamics. In the present model, the prevailing scenario portrayed in Fig. 2 does not lead to periodic oscillations, although one cannot rule out the occurrence of oscillatory behavior for particular parameter combinations or in situations where spatial effects are added to the model. ...

Reference:

A new electoral bottom-up model of institutional governance
Co-evolution of risk and cooperation in climate policies under wealth inequality
  • Citing Article
  • December 2024

PNAS Nexus

... In the future, KGs will become dynamic, evolving continuously to incorporate realtime data and changes. By including temporal aspects, Dynamic Knowledge Graphs (DKGs) [120] extend traditional static KGs. They gather information about entities, relations, and events that change over time. ...

Using dynamic knowledge graphs to detect emerging communities of knowledge
  • Citing Article
  • June 2024

Knowledge-Based Systems

... For example, studies have explored opinion propagation involving leader roles (Dong et al. 2017) and stubborn individuals (Han et al. 2019;Tian and Wang 2018), as well as the impact of individuals with heterogeneous bounded confidence thresholds on the opinion propagation process (Liang et al. 2013). Recently, some scholars have incorporated game payoffs into the opinion learning rules based on the continuous opinion model, exploring the impact of introducing games on opinion dynamics (Chica et al. 2024). Building upon their work, this paper introduces game payoffs into the HK model and investigates the effects of opinion dynamics on collective cooperative behavior by embedding three different game models. ...

Success-driven opinion formation determines social tensions

iScience

... The first focuses on developing algorithms to achieve human-level cooperation [22][23][24], exploring the phenomenon of human bias or machine penalty-the reluctance to cooperate with machines compared with playing with humans [25][26][27], and developing methods to mitigate or overcome this machine penalty [28,29]. The other perspective explores the role of AI as a scaffold for human cooperation, acting in various capacities such as planners structuring network interactions [17,30], independent decision-makers affecting population composition [31][32][33][34] and proxies making decisions on behalf of humans [35][36][37]. For a comprehensive understanding, Mu et al. [38] provide a thorough review. ...

The art of compensation: How hybrid teams solve collective-risk dilemmas

... Indicators of this have also been shown through agent modelling. ToM has been shown to be a successful strategy when modelling evolution of social behaviour, resembling human behaviour better than making fully rational decisions in the Incremental Centipede Game [44], and research into the negotiation game Coloured Trails has shown that ToM is a skill that grows in benefit as both the environment and resource dilemmas become more complicated [45]. ...

Evolution of a Theory of Mind
  • Citing Article
  • February 2024

iScience

... This susceptibility means that individuals may tend to provide socially desirable responses, which can compromise the reliability of the results [38]. To mitigate this, facial expression recognition has been employed to complement emotion measurement, given the established link between facial expressions and emotions [39][40][41]. ...

Emotion Expression and Cooperation under Collective Risks
  • Citing Article
  • September 2023

iScience

... These games enable us to study how collective cooperation may survive in a world where individual selfish actions produce better short-term outcomes. 17,[19][20][21] The set of applications of evolutionary games is immense, from collective disasters 22,23 to herding behavior, 24 sharing economy, 25 or tax fraud, 26 among others. ...

The evolution and social cost of herding mentality promote cooperation

iScience

... Modularity maximization [29] is arguably the most popular method for urban boundary delineation with community detection [20,26,[30][31][32][33]. However, this method is known to infer misleading partitions even for nonspatial networks [28,[34][35][36][37]. Modularity maximization can infer high-scoring partitions even in completely random graphs that by definition have no community structure [34], and suffers from a well-known "resolution limit" in which highly connected communities of nodes smaller than a certain size cannot be detected [35]. ...

Spatiotemporal trip profiles in public transportation reveal city modular structure
  • Citing Article
  • May 2023

Transportation Research Interdisciplinary Perspectives

... Adopting targeted incentive forms based on environmental changes can achieve the desired outcome more effectively. Furthermore, by contrasting the group-based incentive approach [27,28,39] with the stick-and-carrot institution incentive approach [31,[45][46][47], it becomes evident that information levels maintained by group participants and third-party institutions are different in the game, leading to diverse regulatory outcomes concerning incentives on the system. However, acquiring information often incurs costs. ...

Does Spending More Always Ensure Higher Cooperation? An Analysis of Institutional Incentives on Heterogeneous Networks

Dynamic Games and Applications