Grand challenges in social physics: In pursuit of moral behavior

10/12/2018 ∙ by Valerio Capraro, et al. ∙ Middlesex University London 0

Methods of statistical physics have proven valuable for studying the evolution of cooperation in social dilemma games. However, recent empirical research shows that cooperative behavior in social dilemmas is only one kind of a more general class of behavior, namely moral behavior, which includes reciprocity, respecting others' property, honesty, equity, efficiency, as well as many others. Inspired by these experimental works, we here open up the path towards studying other forms of moral behavior with methods of statistical physics. We argue that this is a far-reaching direction for future research that can help us answer fundamental questions about human sociality. Why did our societies evolve as they did? What moral principles are more likely to emerge? What happens when different moral principles clash? Can we predict the break out of moral conflicts in advance and contribute to their solution? These are amongst the most important questions of our time, and methods of statistical physics could lead to new insights and contribute towards finding answers.



There are no comments yet.


This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.


Our time now is unique and special in that we are arguably richer, safer, and healthier than ever before pinker2011better ; pinker2018enlightenment , but simultaneously, we are also facing some of the greatest challenges of our evolution. Climate change, the depletion of natural resources, staggering inequality, the spread of misinformation, persistent armed conflicts, just to name a few examples, all require our best efforts to act together and to renounce part of our individual interests for the greater good. Understanding when, why, and how people deviate from their best self-interest to act pro-socially, benefitting other people and the society as a whole, is thus amongst the most important aims of contemporary scientific research.

Pro-social behavior can come in many forms, the most studied of which is cooperation. Indeed, cooperation is so important that many have contended that our capacity to cooperate at large scales with unrelated others is what makes human societies so successful ostrom2000collective ; fehr2002altruistic ; milinski2002reputation ; gintis2003explaining ; fehr2004social ; henrich2006costly ; nowak2006five ; herrmann2008antisocial ; bowles2011cooperative ; capraro2013model ; rand2013human ; perc2017statistical . Moreover, the psychological basis of cooperation, shared intentionality, that is, ‘the ability and motivation to engage with others in collaborative, co-operative activities with joint goals and intentions’ is what makes humans uniquely human, as it is possessed by children, but not by great apes tomasello2005search .

Although human cooperation is believed to originate from our evolutionary struggles for survival hrdy_11 , it is clear that the challenges that pressured our ancestors into cooperation today are gone. Nevertheless, we are still cooperating, and on ever larger scales, to the point that we may deserve being called ‘SuperCooperators’ nowak_11 . Taking nothing away from the immense importance of cooperation for our evolutionary success and for the wellbeing of our societies, recent empirical research shows, however, that to cooperate is just a particular manifestation of moral behavior capraro2018right . And while methods of statistical physics have been used prolifically to study cooperation perc2017statistical , other forms of moral behavior have not. Our goal here is to outline the many possibilities for future research at the interface between physics and moral behavior, beyond the traditional framework of cooperation in social dilemmas.


To study cooperative behavior, scientists use social dilemma games, such as the prisoner’s dilemma rapoport1965prisoner , the stag hunt skyrms_04 , or the public goods game hardin2009tragedy . In these games, players have to decide whether to cooperate or to defect: cooperation maximizes the payoff of the group, while defection maximizes the payoff of an individual. This leads to a conflict between individual and group interests, which is at the heart of each social dilemma, and in particular at the heart of the cooperation problem.

Since cooperating is not individually optimal, cooperative behavior cannot evolve among self-interested individuals, unless other mechanisms are at play. Several mechanisms for the evolution of cooperation have been identified and studied, including kin selection hamilton1964genetical , direct reciprocity trivers1971evolution , indirect reciprocity nowak1998evolution , social preferences fehr1999theory ; bolton2000erc ; charness2002understanding ; engelmann2004inequality

, the internalization of social heuristics

rand2012spontaneous , translucency capraro2015translucent , cooperative equilibria halpern2010cooperative ; capraro2013model ; barcelo2015group , as well as many others.

One realistic mechanism for the evolution of cooperation is network reciprocity. Everyday interactions among humans do not happen in a vacuum. We are more likely to interact and cooperate within our network of family members, friends, and coworkers, and we rarely interact, let alone cooperate, with strangers. One can formalize this situation by assuming that individuals occupy the vertices of a graph and interact only with their neighbors. Can this spatial structure promote the evolution of cooperation? The answer is yes rand_pnas14 . And the intuition is that, in this setting, cooperators can form clusters and protect themselves from the invasion of defectors nowak1992evolutionary ; lieberman2005evolutionary ; ohtsuki2006simple . These ‘games on graphs are difficult to analyze mathematically’, but ‘are easy to study by computer simulations’ nowak2006five . Games on networks present the natural setting in which one can apply the techniques and methods of statistical physics and network science to study cooperation perc_bs10 ; wang_epjb15 , as well as other forms of moral behavior.

Statistical physics of human cooperation

Methods of statistical physics have come a long way in improving our understanding of the emergence of cooperation and the phase transitions leading to other counterintuitive evolutionary outcomes. Research has revealed that such outcomes depend on the structure of the social network, the type and strength of interactions, and on the complexity and number of competing strategies

santos_prl05 ; pacheco_prl06 ; gomez-gardenes_prl07 ; ohtsuki_prl07 ; lee_s_prl11 ; mathiesen_prl11 ; szolnoki2012defense ; assaf_prl12 ; gomez_prl13 ; knebel_prl13 ; pinheiro_prl14 . Aspects particularly relevant to human cooperation have also been studied in much detail perc2017statistical . The workhorse behind this research has been the spatial public goods game szolnoki_pre09c ; perc_ejp17 , with extensions towards different forms of punishment helbing_njp10 ; szolnoki_pre11a ; perc_njp12 ; wang2013impact ; chen_njp14 ; chen_pre15 ; szolnoki_prx17 , rewarding szolnoki_epl10 ; hilbe_prsb10 ; szolnoki_njp12 ; szolnoki_prsb15 , and tolerance szolnoki_njp16 , to name just some examples. The Monte Carlo method is thereby typically used perc_ejp18 , which ensures that the research is aligned with statistical physics methodology. This in turn enables a comparison of simulation results with generalized mean-field approximations dickman_pre01 ; szolnoki_pre02 ; dickman_pre02 , and a proper determination of phase transitions between different stable strategy configurations szolnoki2012defense . Ultimately, the goal is to identify and understand pattern formation, the spatiotemporal dynamics of solutions, and the principles of self-organization that may lead to socially favorable evolutionary outcomes.

As an example of an evolutionary game that yields an impressively intricate phase diagram, we mention an -strategy public goods game with diverse tolerance levels. The phase diagram is presented in Fig. 1 of szolnoki_njp16 , based on which several observations can be made. In the first place, we can observe that higher tolerance levels are supported by higher multiplication factors in the public goods game, and vice versa. This is in agreement with experience, in that overly tolerant strategies cannot survive in the presence of other less tolerant strategies. From the viewpoint of the considered -strategy public goods game this is also not surprising, because players adopting the most tolerant strategy act as loners only if everybody else in the group is a defector. And evidently, such naive tolerance can not compete with other less tolerant strategies that also compete in the game. Secondly, the phase diagram also reveals that if the cost of inspection is too high, or if the value of the multiplication factor is either very low or very high, tolerance is not viable at all. Even if one tries out different tolerance levels, the evolutionary pressure from other strategies is simply too strong.

While it is beyond the scope of this work to go further into details, it should be noted that phase diagrams as the one presented in Fig. 1 of szolnoki_njp16 provide an in-depth understanding of the evolutionary dynamics and of the phase transitions that lead from one stable strategy configuration to the other. The key for obtaining accurate locations of phase transition points and the correct phases is the application of the stability analysis of competing subsystem solutions perc_ejp18 . A subsystem solution can be formed by any subset of all the competing strategies, and on their own (if separated from other strategies) these subsystems solutions are stable. If the subsystem solution is formed by a single strategy this is trivially true, but it can be likewise true if more than one strategy forms a subsystem solution. The dominant subsystem solution, and hence the phase that is ultimately depicted in the phase diagram as the stable solution of the whole system, can only be determined by letting all the subsystem solutions compete against each other.

By means of this approach, several important insights have been obtained. By peer-based strategies, for example, we note the importance of indirect territorial competition in peer punishment helbing_ploscb10 , the spontaneous emergence of cyclic dominance in rewarding szolnoki_epl10 , and an exotic first-order phase transition observed with correlated strategies szolnoki_prx13 . By institutionalized strategies, we have observed the fascinating spatiotemporal complexity that is due to pool punishment szolnoki_pre11a , while in the realm of self-organization of incentives for cooperation, we have demonstrated the elevated effectiveness of adaptive punishment perc_njp12 , the possibility of probabilistic sharing to solve the problem of costly punishment chen_njp14 , and the many evolutionary advantages of adaptive rewarding szolnoki_njp12 . With antisocial strategies, we have shown the restoration of the effectiveness of prosocial punishment when accounting for second-order free-riding on antisocial punishment szolnoki_prx17 , and the rather surprising lack of adverse effects with antisocial rewarding szolnoki_prsb15 .

While this is just a short snippet of statistical physics research concerning human cooperation, it hopefully showcases successfully the potency of the approach for studying complex mathematical models that describe human behavior, thus recommending itself also for relevantly studying other types of moral behavior to which we attend to in what follows.

Moral behavior

Empirical research has indeed shown that cooperation in social dilemmas is only one facet of a more general class of behavior – moral behavior. When subjects are asked to report what they think is the morally right thing to do in social dilemmas, they typically answer: ‘to cooperate’ capraro2018right .

Morality is universal across human societies. Virtually all societies adopt behavioral rules that are presented to the people as moral principles. But where do these rules come from? A classical non-scientific explanation, still adopted by many societies and religious thinkers, is that they are emanated directly from God. However, in recent years, social scientists have been developing a scientific theory of morality, according to which morality evolved as a a mechanism ‘to promote and sustain cooperation’ greene2015rise . As psychology-star Michael Tomasello put it: ‘human morality arose evolutionarily as a set of skills and motives for cooperating with others’ tomasello2013origins . Similar positions have also been put forward in rawls2009theory ; mackie1990ethics ; wong1984moral ; rai2011moral ; curry2016morality ; curry2018good . However, the word ‘cooperation’ in these statements does not refer only to cooperation in social dilemmas. How does this general form of cooperation translates into specific behaviors?

A recent study exploring morality in societies across the world found that seven moral rules are universal: love your family, help your group, return favors, be brave, defer to authority, be fair, and respect others’ property. Although, what is not universal is how they are ranked curry2018good . Of course, not all these rules are easy to study using simple games on networks, but some are. For example, ‘returning favors’ can be studied using a sequential prisoner’s dilemma, in which the players do not choose their strategy simultaneously, but sequentially. Alternatively, it can be studied using the trust game berg1995trust . In the trust game, player starts with a sum of money and has to decide how much of it, if any, to transfer to player 2. Any amount transferred gets multiplied by a factor larger than 1 and handed to player 2. Then player 2 has to decide how much of it to keep and how much of it to return to player 1.

Similarly, ‘help your group’ can be studied using games with labeled players, in which agents come with a label representing the group(s) they belong to tajfel1970experiments . ‘Fairness’ can be studied using the ultimatum game guth1982experimental , as has already been done along these lines page_prsb00 ; kuperman_epjb08 ; eguiluz_acs09 ; da-silva_r_jtb09 ; deng_ll_pa11 ; gao_j_epl11 ; szolnoki2012defense ; szolnoki2012accuracy ; deng_ll_jsm12 ; iranzo_pone12 ; miyaji_csf13 , or the dictator game forsythe1994fairness . ‘Respect others’ property’ can be studied using games with special frames, as, for example, the Dictator game in the Take frame, for which it is known that taking is considered to be more morally wrong than giving krupka2013identifying .

Beyond these seven rules, there are other forms of moral behavior that are worth studying, as, for example, ‘honesty’. A common game theoretic paradigm to study honest behavior is the sender-receiver game gneezy2005deception . In this game, player 1 is given a private information (for example, the outcome of a die) and is asked to communicate this piece of information to player 2. Player 1 can either communicate the truthful piece of information, or can lie. The role of player 2 is to guess the original piece of information. If player 2 guesses the original piece of information, then players 1 and 2 are both paid according to some option A. Conversely, if player 2 does not guess the original piece of information, then players 1 and 2 are both paid according to option B. Crucially, only player 1 knows the payoffs associated with options A and B. A variant of this game in which player 2 makes no choice has also been introduced and studied biziou2015does ; capraro2017does , in order to avoid the confound of sophisticated deception, that is, players who tell the truth because they believe that player 2 will not believe them sutter2009deception .

Other important forms of moral behavior that ought to be investigated are ‘equity’, that is, a desire to minimize payoff differences among players; ‘efficiency’, that is, a desire to maximize the total welfare; and ‘maximin’, that is, a desire to maximize the worse off payoff. These types of behavior are usually studied using simple distribution experiments, in which people have to decide between two or more allocations of money charness2002understanding ; engelmann2004inequality ; capraro2014benevolent ; capraro2018right ; tappin2018doing .


Methods of statistical physics and network science have proven to be very valuable for successfully studying the evolution of cooperation in social dilemma games. However, empirical research shows that this kind of behavior is only one form of a more general class of moral behavior. The later includes love your family, help your group, return favors, be brave, defer to authority, be fair, respect others’ property, honesty, equity, and efficiency, as well as many others. We have outlined a set of games and mathematical models that could be used efficiently to study particular aspects of some of these forms of moral behavior.

Taken together, the application of statistical physics to study the evolution of moral behavior has the potential to become a flourishing and vibrant avenue of future research. We believe so for two reasons. In the first place, it would allow us to understand why our societies evolved as they did and which moral principles are more likely to evolve. Secondly, since many social conflicts are ultimately conflicts between different moral positions nagel1987moral ; pearce1997moral ; bartos2002using , exploring the evolution of moral behavior could allow us to predict in advance the consequences of a moral conflict, and suggest strategies to avoid it or, in case it is unavoidable, strategies to minimize its costs. We hope that at least parts of our vision will be put to practice in the near future.

This work was supported by the Slovenian Research Agency (Grants J1-7009, J4-9302, J1-9112 and P5-0027).


  • (1) Pinker, S. The better angels of our nature: The decline of violence in history and its causes. Penguin uk, (2011).
  • (2) Pinker, S. Enlightenment now: the case for reason, science, humanism, and progress. Penguin, (2018).
  • (3) Ostrom, E. Collective action and the evolution of social norms. Journal of Economic Perspectives 14, 137–158 (2000).
  • (4) Fehr, E. and Gächter, S. Altruistic punishment in humans. Nature 415, 137 (2002).
  • (5) Milinski, M., Semmann, D., and Krambeck, H.-J. Reputation helps solve the ?tragedy of the commons? Nature 415, 424 (2002).
  • (6) Gintis, H., Bowles, S., Boyd, R., and Fehr, E. Explaining altruistic behavior in humans. Evolution and Human Behavior 24, 153–172 (2003).
  • (7) Fehr, E. and Fischbacher, U. Social norms and human cooperation. Trends in Cognitive Sciences 8, 185–190 (2004).
  • (8) Henrich, J., McElreath, R., Barr, A., Ensminger, J., Barrett, C., Bolyanatz, A., Cardenas, J. C., Gurven, M., Gwako, E., Henrich, N., et al. Costly punishment across human societies. Science 312, 1767–1770 (2006).
  • (9) Nowak, M. A. Five rules for the evolution of cooperation. Science 314, 1560–1563 (2006).
  • (10) Herrmann, B., Thöni, C., and Gächter, S. Antisocial punishment across societies. Science 319, 1362–1367 (2008).
  • (11) Bowles, S. and Gintis, H. A cooperative species: Human reciprocity and its evolution. Princeton University Press, (2011).
  • (12) Capraro, V. A model of human cooperation in social dilemmas. PLoS ONE 8, e72427 (2013).
  • (13) Rand, D. G. and Nowak, M. A. Human cooperation. Trends in Cognitive Sciences 17, 413–425 (2013).
  • (14) Perc, M., Jordan, J. J., Rand, D. G., Wang, Z., Boccaletti, S., and Szolnoki, A. Statistical physics of human cooperation. Physics Reports 687, 1–51 (2017).
  • (15) Tomasello, M., Carpenter, M., Call, J., Behne, T., and Moll, H. In search of the uniquely human. Behavioral and Brain Sciences 28, 721–727 (2005).
  • (16) Hrdy, S. B. Mothers and Others: The Evolutionary Origins of Mutual Understanding. Harvard University Press, Cambridge, MA, (2011).
  • (17) Nowak, M. A. and Highfield, R. SuperCooperators: Altruism, Evolution, and Why We Need Each Other to Succeed. Free Press, New York, (2011).
  • (18) Capraro, V. and Rand, D. G. Do the right thing: Experimental evidence that preferences for moral behavior, rather than equity or efficiency per se, drive human prosociality. Judgment and Decision Making 13, 99–111 (2018).
  • (19) Rapoport, A. and Chammah, A. M. Prisoner’s dilemma: A study in conflict and cooperation, volume 165. University of Michigan Press, (1965).
  • (20) Skyrms, B. The Stag Hunt and the Evolution of Social Structure. Cambridge University Press, Cambridge, U.K., (2004).
  • (21) Hardin, G. The tragedy of the commons. Journal of Natural Resources Policy Research 1, 243–253 (2009).
  • (22) Hamilton, W. D. The genetical evolution of social behaviour. i. Journal of Theoretical Biology 7, 1–16 (1964).
  • (23) Trivers, R. L. The evolution of reciprocal altruism. The Quarterly Review of Biology 46, 35–57 (1971).
  • (24) Nowak, M. A. and Sigmund, K. Evolution of indirect reciprocity by image scoring. Nature 393, 573 (1998).
  • (25) Fehr, E. and Schmidt, K. M. A theory of fairness, competition, and cooperation. The Quarterly Journal of Economics 114, 817–868 (1999).
  • (26) Bolton, G. E. and Ockenfels, A. Erc: A theory of equity, reciprocity, and competition. The American Economic Review 90, 166–193 (2000).
  • (27) Charness, G. and Rabin, M. Understanding social preferences with simple tests. The Quarterly Journal of Economics 117, 817–869 (2002).
  • (28) Engelmann, D. and Strobel, M. Inequality aversion, efficiency, and maximin preferences in simple distribution experiments. The American Economic Review 94, 857–869 (2004).
  • (29) Rand, D. G., Greene, J. D., and Nowak, M. A. Spontaneous giving and calculated greed. Nature 489, 427 (2012).
  • (30) Capraro, V. and Halpern, J. Y. Translucent players: Explaining cooperative behavior in social dilemmas. Proceedings of the 15th conference on Theoretical Aspects of Rationality and Knowledge 215, 114–126 (2016).
  • (31) Halpern, J. Y. and Rong, N. Cooperative equilibrium. In Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems: volume 1-Volume 1, 1465–1466. International Foundation for Autonomous Agents and Multiagent Systems, (2010).
  • (32) Barcelo, H. and Capraro, V. Group size effect on cooperation in one-shot social dilemmas. Scientific Reports 5, 7937 (2015).
  • (33) Rand, D. G., Nowak, M. A., Fowler, J. H., and Christakis, N. A. Static network structure can stabilize human cooperation. Proc. Natl. Acad. Sci. USA 111, 17093–17098 (2014).
  • (34) Nowak, M. A. and May, R. M. Evolutionary games and spatial chaos. Nature 359, 826 (1992).
  • (35) Lieberman, E., Hauert, C., and Nowak, M. A. Evolutionary dynamics on graphs. Nature 433, 312 (2005).
  • (36) Ohtsuki, H., Hauert, C., Lieberman, E., and Nowak, M. A. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502 (2006).
  • (37) Perc, M. and Szolnoki, A. Coevolutionary games – A mini review. BioSystems 99, 109–125 (2010).
  • (38) Wang, Z., Wang, L., Szolnoki, A., and Perc, M. Evolutionary games on multilayer networks: A colloquium. European Physical Journal B 88, 124 (2015).
  • (39) Santos, F. C. and Pacheco, J. M. Scale-free networks provide a unifying framework for the emergence of cooperation. Phys. Rev. Lett. 95, 098104 (2005).
  • (40) Pacheco, J. M., Traulsen, A., and Nowak, M. A. Coevolution of strategy and structure in complex networks with dynamical linking. Phys. Rev. Lett. 97, 258103 (2006).
  • (41) Gómez-Gardeñes, J., Campillo, M., Floría, L. M., and Moreno, Y. Dynamical organization of cooperation in complex networks. Phys. Rev. Lett. 98, 108103 (2007).
  • (42) Ohtsuki, H., Nowak, M. A., and Pacheco, J. M. Breaking the symmetry between interaction and replacement in evolutionary dynamics on graphs. Phys. Rev. Lett. 98, 108106 (2007).
  • (43) Lee, S., Holme, P., and Wu, Z.-X. Emergent hierarchical structures in multiadaptive games. Phys. Rev. Lett. 106, 028702 (2011).
  • (44) Mathiesen, J., Mitarai, N., Sneppen, K., and Trusina, A. Ecosystems with mutually exclusive interactions self-organize to a state of high diversity. Phys. Rev. Lett. 107, 188101 (2011).
  • (45) Szolnoki, A., Perc, M., and Szabo, G. Defense mechanisms of empathetic players in the spatial ultimatum game. Physical Review Letters 109, 078701 (2012).
  • (46) Assaf, M. and Mobilia, M. Metastability and anomalous fixation in evolutionary games on scale-free networks. Phys. Rev. Lett. 109, 188701 (2012).
  • (47) Gómez, S., Díaz-Guilera, A., Gómez-Gardeñes, J., Pérez-Vicente, C., Moreno, Y., and Arenas, A. Diffusion dynamics on multiplex networks. Phys. Rev. Lett. 110, 028701 (2013).
  • (48) Knebel, J., Krüger, T., Weber, M., and Frey, E. Coexistence and survival in conservative Lotka-Volterra networks. Phys. Rev. Lett. 110, 168106 (2013).
  • (49) Pinheiro, F., Santos, M. D., Santos, F., and Pacheco, J. Origin of peer influence in social networks. Phys. Rev. Lett. 112, 098702 (2014).
  • (50) Szolnoki, A., Perc, M., and Szabó, G. Topology-independent impact of noise on cooperation in spatial public goods games. Physical Review E 80, 056109 (2009).
  • (51) Perc, M. High-performance parallel computing in the classroom using the public goods game as an example. European Journal of Physics 38, 045801 (2017).
  • (52) Helbing, D., Szolnoki, A., Perc, M., and Szabó, G. Punish, but not too hard: How costly punishment spreads in the spatial public goods game. New Journal of Physics 12, 083005 (2010).
  • (53) Szolnoki, A., Szabó, G., and Perc, M. Phase diagrams for the spatial public goods game with pool punishment. Physical Review E 83, 036101 (2011).
  • (54) Perc, M. and Szolnoki, A. Self-organization of punishment in structured populations. New Journal of Physics 14, 043013 (2012).
  • (55) Wang, Z., Xia, C.-Y., Meloni, S., Zhou, C.-S., and Moreno, Y. Impact of social punishment on cooperative behavior in complex networks. Scientific Reports 3, 3055 (2013).
  • (56) Chen, X., Szolnoki, A., and Perc, M. Probabilistic sharing solves the problem of costly punishment. New Journal of Physics 16, 083016 (2014).
  • (57) Chen, X., Szolnoki, A., and Perc, M. Competition and cooperation among different punishing strategies in the spatial public goods game. Physical Review E 92, 012819 (2015).
  • (58) Szolnoki, A. and Perc, M. Second-order free-riding on antisocial punishment restores the effectiveness of prosocial punishment. Physical Review X 7, 041027 (2017).
  • (59) Szolnoki, A. and Perc, M. Reward and cooperation in the spatial public goods game. EPL (Europhysics Letters) 92, 38003 (2010).
  • (60) Hilbe, C. and Sigmund, K. Incentives and opportunism: from the carrot to the stick. Proc. R. Soc. B 277, 2427–2433 (2010).
  • (61) Szolnoki, A. and Perc, M. Evolutionary advantages of adaptive rewarding. New Journal of Physics 14, 093016 (2012).
  • (62) Szolnoki, A. and Perc, M. Antisocial pool rewarding does not deter public cooperation. Proceedings of the Royal Society B 282, 20151975 (2015).
  • (63) Szolnoki, A. and Perc, M. Competition of tolerant strategies in the spatial public goods game. New Journal of Physics 18, 083021 (2016).
  • (64) Perc, M. Stability of subsystem solutions in agent-based models. European Journal of Physics 39, 014001 (2018).
  • (65) Dickman, R. First- and second-order phase transitions in a driven lattice gas with nearest-neighbor exclusion. Phys. Rev. E 64, 016124 (2001).
  • (66) Szolnoki, A. Dynamical mean-field approximation for a pair contact process with a particle source. Phys. Rev. E 66, 057102 (2002).
  • (67) Dickman, R. n-site approximations and coherent-anomaly-method analysis for a stochastic sandpile. Phys. Rev. E 66, 036122 (2002).
  • (68) Helbing, D., Szolnoki, A., Perc, M., and Szabó, G. Evolutionary establishment of moral and double moral standards through spatial interactions. PLoS Computational Biology 6, e1000758 (2010).
  • (69) Szolnoki, A. and Perc, M. Correlation of positive and negative reciprocity fails to confer an evolutionary advantage: Phase transitions to elementary strategies. Physical Review X 3, 041021 (2013).
  • (70) Greene, J. D. The rise of moral cognition. Cognition 135, 39–42 (2015).
  • (71) Tomasello, M. and Vaish, A. Origins of human cooperation and morality. Annual review of psychology 64, 231–255 (2013).
  • (72) Rawls, J. A theory of justice: Revised edition. Harvard university press, (2009).
  • (73) Mackie, J. Ethics: Inventing right and wrong. Penguin UK, (1990).
  • (74) Wong, D. B. Moral relativity. Univ of California Press, (1984).
  • (75) Rai, T. S. and Fiske, A. P. Moral psychology is relationship regulation: moral motives for unity, hierarchy, equality, and proportionality. Psychological Review 118, 57 (2011).
  • (76) Curry, O. S. Morality as cooperation: A problem-centred approach. In The evolution of morality, 27–51. Springer (2016).
  • (77) Curry, O., Mullins, D., and Whitehouse, H. Is it good to cooperate? testing the theory of morality-as-cooperation in 60 societies. Current Anthropology 1, (2018).
  • (78) Berg, J., Dickhaut, J., and McCabe, K. Trust, reciprocity, and social history. Games and Economic Behavior 10, 122–142 (1995).
  • (79) Tajfel, H. Experiments in intergroup discrimination. Scientific American 223, 96–103 (1970).
  • (80) Güth, W., Schmittberger, R., and Schwarze, B. An experimental analysis of ultimatum bargaining. Journal of Economic Behavior & Organization 3, 367–388 (1982).
  • (81) Page, K. M., Nowak, M. A., and Sigmund, K. The spatial ultimatum game. Proc. R. Soc. Lond. B 267, 2177–2182 (2000).
  • (82) Kuperman, M. N. and Risau-Gusman, S. The effect of topology on the spatial ultimatum game. Eur. Phys. J. B 62, 233–238 (2008).
  • (83) Equíluz, V. M. and Tessone, C. Critical behavior in an evolutionary ultimatum game with social structure. Adv. Complex Systems 12, 221–232 (2009).
  • (84) da Silva, R., Kellerman, G. A., and Lamb, L. C. Statistical fluctuations in population bargaining in the ultimatum game: Static and evolutionary aspects. J. Theor. Biol. 258, 208–218 (2009).
  • (85) Deng, L., Tang, W., and Zhang, J. The coevolutionay ultimatum game on different network topologies. Physica A 390, 4227–4235 (2011).
  • (86) Gao, J., Li, Z., Wu, T., and Wang, L. The coevolutionary ultimatum game. EPL 93, 48003 (2011).
  • (87) Szolnoki, A., Perc, M., and Szabó, G. Accuracy in strategy imitations promotes the evolution of fairness in the spatial ultimatum game. EPL (Europhysics Letters) 100, 28005 (2012).
  • (88) Deng, L., Wang, C., Tang, W., Zhou, G., and Cai, J. A network growth model based on the evolutionary ultimatum game. J. Stat. Mech. 2012, P11013 (2012).
  • (89) Iranzo, J., Floría, L., Moreno, Y., and Sánchez, A. Empathy emerges spontaneously in the ultimatum game: Small groups and networks. PLoS ONE 7, e43781 (2011).
  • (90) Miyaji, K., Wang, Z., Tanimoto, J., Hagishima, A., and Kokubo, S. The evolution of fairness in the coevolutionary ultimatum games. Chaos, Solitons & Fractals 56, 13–18 (2013).
  • (91) Forsythe, R., Horowitz, J. L., Savin, N. E., and Sefton, M. Fairness in simple bargaining experiments. Games and Economic Behavior 6, 347–369 (1994).
  • (92) Krupka, E. L. and Weber, R. A. Identifying social norms using coordination games: Why does dictator game sharing vary? Journal of the European Economic Association 11, 495–524 (2013).
  • (93) Gneezy, U. Deception: The role of consequences. The American Economic Review 95, 384–394 (2005).
  • (94) Biziou-van Pol, L., Haenen, J., Novaro, A., Liberman, A. O., and Capraro, V. Does telling white lies signal pro-social preferences? Judgment and Decision Making 10, 538–548 (2015).
  • (95) Capraro, V. Does the truth come naturally? time pressure increases honesty in one-shot deception games. Economics Letters 158, 54–57 (2017).
  • (96) Sutter, M. Deception through telling the truth?! experimental evidence from individuals and teams. The Economic Journal 119, 47–60 (2009).
  • (97) Capraro, V., Smyth, C., Mylona, K., and Niblo, G. A. Benevolent characteristics promote cooperative behaviour among humans. PLoS ONE 9, e102881 (2014).
  • (98) Tappin, B. M. and Capraro, V. Doing good vs. avoiding bad in prosocial choice: A refined test and extension of the morality preference hypothesis. Journal of Experimental Social Psychology 79, 64–70 (2018).
  • (99) Nagel, T. Moral conflict and political legitimacy. Philosophy & Public Affairs 16, 215–240 (1987).
  • (100) Pearce, W. B. and Littlejohn, S. W. Moral conflict: When social worlds collide. Sage, (1997).
  • (101) Bartos, O. J. and Wehr, P. Using conflict theory. Cambridge University Press, (2002).