List of Selected Publications

Kertesz, J., Torok, J., Murase, Y., Jo, H. H., & Kaski, K. (2016). Multiplex Modeling of the Society. arXiv preprint arXiv:1609.08381v1

Abstract:

The society has a multi-layered structure, where the layers represent the different contexts. To model this structure we begin with a single-layer weighted social network (WSN) model showing the Granovetterian structure. We find that when merging such WSN models, a sufficient amount of inter-layer correlation is needed to maintain the relationship between topology and link weights, while these correlations destroy the enhancement in the community overlap due to multiple layers. To resolve this, we devise a geographic multi-layer WSN model, where the indirect inter-layer correlations due to the geographic constraints of individuals enhance the overlaps between the communities and, at the same time, the Granovetterian structure is preserved. Furthermore, the network of social interactions can be considered as a multiplex from another point of view too: each layer corresponds to one communication channel and the aggregate of all them constitutes the entire social network. However, usually one has information only about one of the channels, which should be considered as a sample of the whole. Here we show by simulations and analytical methods that this sampling may lead to bias. For example, while it is expected that the degree distribution of the whole social network has a maximum at a value larger than one, we get with reasonable assumptions about the sampling process a monotonously decreasing distribution as observed in empirical studies of single channel data. We analyse the far-reaching consequences of our findings.

Pereda, M. (2016). Evolution of cooperation under social pressure in multiplex networks. Physical Review E, 94(3), 032314. DOI: https://doi.org/10.1103/PhysRevE.94.032314

Abstract:

In this work, we aim to contribute to the understanding of human prosocial behavior by studying the influence that a particular form of social pressure, “being watched,” has on the evolution of cooperative behavior. We study how cooperation emerges in multiplex complex topologies by analyzing a particular bidirectionally coupled dynamics on top of a two-layer multiplex network (duplex). The coupled dynamics appears between the prisoner’s dilemma game in a network and a threshold cascade model in the other. The threshold model is intended to abstract the behavior of a network of vigilant nodes that impose the pressure of being observed altering hence the temptation to defect of the dilemma. Cooperation or defection in the game also affects the state of a node of being vigilant. We analyze these processes on different duplex networks structures and assess the influence of the topology, average degree and correlated multiplexity, on the outcome of cooperation. Interestingly, we find that the social pressure of vigilance may impact cooperation positively or negatively, depending on the duplex structure, specifically the degree correlations between layers is determinant. Our results give further quantitative insights in the promotion of cooperation under social pressure.

Abstract:

Studies of animal behavior consistently demonstrate that the social environment impacts cooperation, yet the effect of social dynamics has been largely excluded from studies of human cooperation. Here, we introduce a novel approach inspired by nonhuman primate research to address how social hierarchies impact human cooperation. Participants competed to earn hierarchy positions and then could cooperate with another individual in the hierarchy by investing in a common effort. Cooperation was achieved if the combined investments exceeded a threshold, and the higher ranked individual distributed the spoils unless control was contested by the partner. Compared to a condition lacking hierarchy, cooperation declined in the presence of a hierarchy due to a decrease in investment by lower ranked individuals. Furthermore, hierarchy was detrimental to cooperation regardless of whether it was earned or arbitrary. These findings mirror results from nonhuman primates and demonstrate that hierarchies are detrimental to cooperation. However, these results deviate from nonhuman primate findings by demonstrating that human behavior is responsive to changing hierarchical structures and suggests partnership dynamics that may improve cooperation. This work introduces a controlled way to investigate the social influences on human behavior, and demonstrates the evolutionary continuity of human behavior with other primate species.

Antonioni, A., Sánchez, A., & Tomassini, M. (2016). Cooperation Survives and Cheating Pays in a Dynamic Network Structure with Unreliable Reputation. Scientific reports, 6, 27160. DOI:10.1038/srep27160

Abstract:

In a networked society like ours, reputation is an indispensable tool to guide decisions about social or economic interactions with individuals otherwise unknown. Usually, information about prospective counterparts is incomplete, often being limited to an average success rate. Uncertainty on reputation is further increased by fraud, which is increasingly becoming a cause of concern. To address these issues, we have designed an experiment based on the Prisoner’s Dilemma as a model for social interactions. Participants could spend money to have their observable cooperativeness increased. We find that the aggregate cooperation level is practically unchanged, i.e., global behavior does not seem to be affected by unreliable reputations. However, at the individual level we find two distinct types of behavior, one of reliable subjects and one of cheaters, where the latter artificially fake their reputation in almost every interaction. Cheaters end up being better off than honest individuals, who not only keep their true reputation but are also more cooperative. In practice, this results in honest subjects paying the costs of fraud as cheaters earn the same as in a truthful environment. These findings point to the importance of ensuring the truthfulness of reputation for a more equitable and fair society.

Tamarit, I., & Sánchez, A. (2016). Emotions and strategic behaviour: The case of the ultimatum game. PloS one, 11(7), e0158733. http://dx.doi.org/10.1371/journal.pone.0158733

Abstract:

Human behaviour in economic interactions has attracted an increasing amount of attention over the last decades. The economic assumption that people would behave focusing on their own material self-interest was proved incomplete, once the empirical evidence consistently showed that many other motives may influence such behaviour. Therefore, models that can incorporate rational decision process as well as other intervening factors are a key issue to both understand the observations from economic experiments and to apply the lessons learned from them. In this paper, we incorporate the influence of emotions to the utility function in an explicit manner, using the Ultimatum Game as a case study. Our model is amenable to analytical study, and is connected with the Circumplex model of emotions and with Kahneman’s two-system theory. The simplicity of the model allows to obtain predictions for the offers and acceptance thresholds. We study two specific examples, when the model parameters are distributed uniformly or normally, and show that in the latter case the results are already qualitatively correct. Although this work can be considered as a first approach, it includes what we believe are the main stylized facts, is able to qualitatively reproduce experimental results in a very simple manner, and can be straightforwardly extended to other games.

Ellwardt, L., Hernández, P., Martínez-Canovas, G., & Muñoz-Herrera, M. (2014). Conflict and segregation in networks: An experiment on the interplay between individual preferences and social influence (No. 0114). University of Valencia, ERI-CES. DOI:10.3934/jdg.2016010

Abstract:

We examine the interplay between a person’s individual preference and the social influence others exert. We provide a model of network relationships with conflicting preferences, where individuals are better off coordinating with those around them, but where not all have a preference for the same action. We test our model in an experiment, varying the level of conflicting preferences between individuals. Our findings suggest that preferences are more salient than social influence, under conflicting preferences: subjects relate mainly with others who have the same preferences. This leads to two undesirable outcomes: network segregation and social inefficiency. The same force that helps people individually, hurts society.

Gossner, O., Hernández, P., & Peretz, R. (2016). The complexity of interacting automata. International Journal of Game Theory, 45(1-2), 461-496.DOI: 10.1007/s00182-015-0521-7

Abstract:

This paper studies the interaction of automata of size m. We characterise statistical properties satisfied by random plays generated by a correlated pair of automata with m states each. We show that in some respect the pair of automata can be identified with a more complex automaton of size comparable to mlogm. We investigate implications of these results on the correlated min–max value of repeated games played by automata.

Abstract:

A celebrated result of Abreu and Rubinstein states that in repeated games, when the players are restricted to playing strategies that can be implemented by finite automata and they have lexicographic preferences, the set of equilibrium payoffs is a strict subset of the set of feasible and individually rational payoffs. In this paper we explore the limitations of this result. We prove that if memory size is costly and players can use mixed automata, then a folk theorem obtains and the set of equilibrium payo is once again the set of feasible and individually rational payoffs. Our result emphasizes the role of memory cost and of mixing when players have bounded computational power.

Abstract:

New systems can be designed, developed, and managed as societies of agents that interact with each other by offering and providing services. These systems can be viewed as complex networks where nodes are bounded rational agents. In order to deal with complex goals, they require cooperation of the other agents to be able to locate the required services. The aim of this paper is formally and empirically analyze under which circumstances cooperation emerges in decentralized search of services. We propose a repeated game model that formalizes the interactions among agents in a search process where agents are free to choose between cooperate or not in the process. Agents make decisions based on the cost of their actions and the expected reward if they participate forwarding queries in a search process that ends successfully. We propose a strategy that is based on random-walks, and we study under what conditions the strategy is a Nash equilibrium. We performed several experiments in order to evaluate the model and the strategy and to analyze which network structures are more appropriate to promote cooperation.

Abstract:

In this paper we experimentally test Schelling’s (1971) segregation model and confirm the striking result of segregation. In addition, we extend Schelling’s model theoretically by adding strategic behaviour and moving costs. We obtain a unique subgame perfect equilibrium in which rational agents facing moving costs may find it optimal not to move (anticipating other participants’ movements). This equilibrium is far from full segregation. We run experiments for this extended Schelling model, and find that the percentage of full segregated societies notably decreases with the cost of moving and that the degree of segregation depends on the distribution of strategic subjects.

Bao, T., Hommes, C. H., & Makarewicz, T. (2015). Bubble formation and (in) efficient markets in learning-to-forecast and-optimise experiments. Economic Journal, Forthcoming, 15-107. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2657057

Abstract:

This experiment compares the price dynamics and bubble formation in an asset market with a price adjustment rule in three treatments where subjects (1) submit a price forecast only, (2) choose quantity to buy/sell and (3) perform both tasks. We find deviation of the market price from the fundamental price in all treatments, but to a larger degree in treatments (2) and (3). Mispricing is therefore a robust finding in markets with positive expectation feedback. Some very large, recurring bubbles arise, where the price is 3 times larger than the fundamental value, which were not seen in former experiments.

Barrio, R. A., Govezensky, T., Dunbar, R., Iñiguez, G., & Kaski, K. (2015). Dynamics of deceptive interactions in social networks. Journal of The Royal Society Interface, 12(112), 20150798. DOI: 10.1098/rsif.2015.0798

Abstract:

In this paper, we examine the role of lies in human social relations by implementing some salient characteristics of deceptive interactions into an opinion formation model, so as to describe the dynamical behaviour of a social network more realistically. In this model, we take into account such basic properties of social networks as the dynamics of the intensity of interactions, the influence of public opinion and the fact that in every human interaction it might be convenient to deceive or withhold information depending on the instantaneous situation of each individual in the network. We find that lies shape the topology of social networks, especially the formation of tightly linked, small communities with loose connections between them. We also find that agents with a larger proportion of deceptive interactions are the ones that connect communities of different opinion, and, in this sense, they have substantial centrality in the network. We then discuss the consequences of these results for the social behaviour of humans and predict the changes that could arise due to a varying tolerance for lies in society.

Cuesta, J. A., Gracia-Lázaro, C., Ferrer, A., Moreno, Y., & Sánchez, A. (2015). Reputation drives cooperative behaviour and network formation in human groups. Scientific reports, 5. doi:10.1038/srep07843

Abstract:

Cooperativeness is a defining feature of human nature. Theoreticians have suggested several mechanisms to explain this ubiquitous phenomenon, including reciprocity, reputation, and punishment, but the problem is still unsolved. Here we show, through experiments conducted with groups of people playing an iterated Prisoner’s Dilemma on a dynamic network, that it is reputation what really fosters cooperation. While this mechanism has already been observed in unstructured populations, we find that it acts equally when interactions are given by a network that players can reconfigure dynamically. Furthermore, our observations reveal that memory also drives the network formation process, and cooperators assort more, with longer link lifetimes, the longer the past actions record. Our analysis demonstrates, for the first time, that reputation can be very well quantified as a weighted mean of the fractions of past cooperative acts and the last action performed. This finding has potential applications in collaborative systems and e-commerce.

Hernández, P., Martínez-Cánovas, G., Muñoz-Herrera, M., & Sánchez, A. (2016). Equilibrium characterization of networks under conflicting preferences. Economics Letters. http://dx.doi.org/10.1016/j.econlet.2016.12.004

Abstract:

In this work we characterize equilibrium introduced in configurations for networks with conflicting preferences. We use the model Hernández et al. (2013) to study the effect of three main factors: the strength of individual preferences, the level of integration in the network, and the intensity of conflict in the population. Our aim is to understand how likely is it that social outcomes are either those in which preferences dominate choices or those in which some individuals sacrifice their preferences to achieve consensus with others. Our results show that, the stronger individual preferences, the harder to achieve consensus in choices. However, in cases where the payoff ratio is less extreme, full coordination (consensus) is always an equilibrium. Finally, if the level of conflict is low, individual preferences become less relevant and all players choosing what they prefer is not an equilibrium anymore.

Abstract:

Mechanisms supporting human ultra-cooperativeness are very much subject to debate. One psychological feature likely to be relevant is the formation of expectations, particularly about receiving cooperative or generous behavior from others. Without such expectations, social life will be seriously impeded and, in turn, expectations leading to satisfactory interactions can become norms and institutionalize cooperation. In this paper, we assess people’s expectations of generosity in a series of controlled experiments using the dictator game. Despite differences in respective roles, involvement in the game, degree of social distance or variation of stakes, the results are conclusive: subjects seldom predict that dictators will behave selfishly (by choosing the Nash equilibrium action, namely giving nothing). The majority of subjects expect that dictators will choose the equal split. This implies that generous behavior is not only observed in the lab, but also expected by subjects. In addition, expectations are accurate, matching closely the donations observed and showing that as a society we have a good grasp of how we interact. Finally, correlation between expectations and actual behavior suggests that expectations can be an important ingredient of generous or cooperative behavior.

Abstract:

Modern economies rely heavily on their infrastructure networks. These networks face threats ranging from natural disasters to human attacks. As networks are pervasive, the investments needed to protect them are very large; this motivates the study of targeted defence. What are the ‘key’ nodes to defend to maximize functionality of the network? What are the incentives of individual nodes to protect themselves in a networked environment and how do these incentives correspond to collective welfare? We provide a characterization of equilibrium attack and defence in terms of two classical concepts in graph theory – separators and transversals. We use this characterization to study the intensity of conflict (the resources spent on attack and defence) and the prospects of active conflict (when both adversary and defender target nodes for action) in networks. Finally, we show that welfare costs of decentralized defence can be very large.

Gagnon, Julien, and Sanjeev Goyal. 2017. “Networks, Markets, and Inequality.” American Economic Review, 107(1): 1-30. DOI: 10.1257/aer.20150635

Abstract:
The interaction between community and markets remains a central theme in the social sciences. The empirical evidence is rich: in some instances, markets strengthen social ties, while in others they undermine them. The impact of markets on inequality and welfare also varies widely. This paper develops a model where individuals in a social network choose whether to participate in their network and whether to participate in the market. We show that individual behavior is defined by the q-core of the network and the key to understanding the conflicting evidence is whether the market and the network are complements or substitutes.

Sanjeev Goyal, Penélope Hernández, Guillem Martínez-Cánovas, Frédéric Moisan, Manuel Muñoz-Herrera & Ángel Sánchez (2017). PDF

Abstract:

Individuals prefer to coordinate with others, but they dier on the preferred action. In theory, this can give rise to an integrated society with everyone conforming to the same action or a segregated society with members of dierent groups choosing diverse actions. Social welfare is maximum when society is integrated and everyone conforms on the majority’s action. In laboratory experiments, subjects with dierent preferences segregate into distinct groups and choose diverse actions. To understand the role of partner choice, we then consider an exogenous network of partners. Subjects in the experiment now choose to conform on the action preferred by the majority. Thus, there exists a tension between two deeply held values: social cohesion and freedom of association.

Jan E. Snellman, Gerardo Iñiguez, Tzipe Govezensky, R.A. Barrio, Kimmo K. Kaski. PDF

Abstract:

In human societies, people’s willingness to compete and strive for better social status, as well as being envious of those perceived in some way superior, lead to social structures that are intrinsically hierarchical. Here, we propose an agent-based, network model to mimic the ranking behaviour of individuals and its possible repercussions in human society. The main ingredient of the model is the assumption that the relevant feature of social interactions is each individual’s keenness to maximize his or her status relative to others. The social networks produced by the model are homophilous and assortative, as frequently observed in human communities, and most of the network properties seem quite independent of its size. However, we see that for a small number of agents the resulting network consists of disjoint weakly connected communities, while being highly assortative and homophilic. On the other hand, larger networks turn out to be more cohesive with larger communities but less homophilic.We find that the reason for these changes is that larger network size allows agents to use new strategies for maximizing their social status, allowing for more diverse links between themIn human societies, people’s willingness to compete and strive for better social status, as well as being envious of those perceived in some way superior, lead to social structures that are intrinsically hierarchical. Here, we propose an agent-based, network model to mimic the ranking behaviour of individuals and its possible repercussions in human society. The main ingredient of the model is the assumption that the relevant feature of social interactions is each individual’s keenness to maximize his or her status relative to others. The social networks produced by the model are homophilous and assortative, as frequently observed in human communities, and most of the network properties seem quite independent of its size. However, we see that for a small number of agents the resulting network consists of disjoint weakly connected communities, while being highly assortative and homophilic. On the other hand, larger networks turn out to be more cohesive with larger communities but less homophilic.We find that the reason for these changes is that larger network size allows agents to use new strategies for maximizing their social status, allowing for more diverse links between them.

Wei Chen, Carlos Gracia-Lázaro, Zhiwu Li, Long Wang  & Yamir Moreno.
Scientific Reports 7 (1), 4800. doi:10.1038/s41598-017-04284-6

Abstract:

In the animal world, the competition between individuals belonging to different species for a resource often requires the cooperation of several individuals in groups. This paper proposes a generalization of the Hawk-Dove Game for an arbitrary number of agents: the N-person Hawk-Dove Game. In this model, doves exemplify the cooperative behavior without intraspecies conflict, while hawks represent the aggressive behavior. In the absence of hawks, doves share the resource equally and avoid conflict, but having hawks around lead to doves escaping without fighting. Conversely, hawks fight for the resource at the cost of getting injured. Nevertheless, if doves are present in sufficient number to expel the hawks, they can aggregate to protect the resource, and thus avoid being plundered by hawks. We derive and numerically solve an exact equation for the evolution of the system in both finite and infinite well-mixed populations, finding the conditions for stable coexistence between both species. Furthermore, by varying the different parameters, we found a scenario of bifurcations that leads the system from dominating hawks and coexistence to bi-stability, multiple interior equilibria and dominating doves.

Monsivais, D., Bhattacharya, K., Ghosh, A., Dunbar, R. I., & Kaski, K. Scientific Reports 7, Article number: 10717 (2017). doi:10.1038/s41598-017-11125-z

Abstract:

We study the influence of seasonally and geographically related daily dynamics of daylight and ambient temperature on human resting or sleeping patterns using mobile phone data of a large number of individuals. We observe two daily inactivity periods in the people’s aggregated mobile phone calling patterns and infer these to represent the resting times of the population. We find that the nocturnal resting period is strongly influenced by the length of daylight, and that its seasonal variation depends on the latitude, such that for people living in two different cities separated by eight latitudinal degrees, the difference in the resting periods of people between the summer and winter in southern cities is almost twice that in the northern cities. We also observe that the duration of the afternoon resting period is influenced by the temperature, and that there is a threshold from which this influence sets in. Finally, we observe that the yearly dynamics of the afternoon and nocturnal resting periods appear to be counterbalancing each other. This also lends support to the notion that the total daily resting time of people is more or less conserved across the year.

Abstract:

Abstract:

Mental disorders have an enormous impact in our society, both in personal terms and in the economic costs associated with their treatment. In order to scale up services and bring down costs, administrations are starting to promote social interactions as key to care provision. We analyze quantitatively the importance of communities for effective mental health care, considering all community members involved. By means of citizen science practices, we have designed a suite of games that allow to probe into different behavioral traits of the role groups of the ecosystem. The evidence reinforces the idea of community social capital, with caregivers and professionals playing a leading role. Yet, the cost of collective action is mainly supported by individuals with a mental condition – which unveils their vulnerability. The results are in general agreement with previous findings but, since we broaden the perspective of previous studies, we are also able to find marked differences in the social behavior of certain groups of mental disorders. We finally point to the conditions under which cooperation among members of the ecosystem is better sustained, suggesting how virtuous cycles of inclusion and participation can be promoted in a ‘care in the community’ framework.

We introduce a general framework for exploring the problem of selecting a committee of representatives with the aim of studying a networked voting rule based on a decentralized large-scale platform, which can assure a strong accountability of the elected. The results of our simulations suggest that this algorithm-based approach is able to obtain a high representativeness for relatively small committees, performing even better than a classical voting rule based on a closed list of candidates. We show that a general relation between committee size and representatives exists in the form of an inverse square root law and that the normalized committee size approximately scales with the inverse of the community size, allowing the scalability to very large populations. These findings are not strongly influenced by the different networks used to describe the individuals’ interactions, except for the presence of few individuals with very high connectivity which can have a marginal negative effect in the committee selection process.

The interplay of social structure and cooperative behavior is under much scrutiny lately as behavior in social contexts becomes increasingly relevant for everyday life. Earlier experimental work showed that the existence of a social hierarchy, earned through competition, was detrimental for the evolution of cooperative behaviors. Here, we study the case in which individuals are ranked in a hierarchical structure based on their performance in a collective effort by having them play a Public Goods Game. In the first treatment, participants are ranked according to group earnings while, in the second treatment, their rankings are based on individual earnings. Subsequently, participants play asymmetric Prisoner’s Dilemma games where higher-ranked players gain more than lower ones. Our experiments show that there are no detrimental effects of the hierarchy formed based on group performance, yet when ranking is assigned individually we observe a decrease in cooperation. Our results show that different levels of cooperation arise from the fact that subjects are interpreting rankings as a reputation which carries information about which subjects were cooperators in the previous phase. Our results demonstrate that noting the manner in which a hierarchy is established is essential for understanding its effects on cooperation.

van Rooij, A. J., Ferguson, C. J., Colder Carras, M., Kardefelt-Winther, D., Shi, J., … Orben, A., … & Przybylski, A. K. (2018, February 8). Retrieved from psyarxiv.com/kc7r9

Pablo Lozano, Alberto Antonioni, Marco Tomassini & Angel Sánchez. Scientific Reports volume 8, Article number: 9093 (2018).  https://doi.org/10.1038/s41598-018-27544-5

Ignacio Tamarit, José A. Cuesta, Robin I. M. Dunbar, Angel Sánchez. Proceedings of the National Academy of Sciences Aug 2018, 115 (33) 8316-8321; DOI: 10.1073/pnas.1719233115

Hernández AR, Gracia-Lázaro C, Brigatti E, Moreno Y.  Physica A: Statistical Mechanics and its Applications. 2018 Jun 18.

Abstract:

We consider an open-ended set of cultural features in the Axelrod model of cultural dissemination. By replacing the features in which a high degree of consensus is achieved by new ones, we address here an essential ingredient of societies: the evolution of topics as a result of social dynamics and debate. Our results show that, once cultural clusters have been formed, the introduction of new topics into the social debate has little effect on them, but it does have a significant influence on the cultural overlap. Along with the Monte Carlo simulations, we derive and numerically solve an equation for the stationary cultural overlap based on a mean-field approach which reproduces the qualitative behavior of the model.

Vicens J, Bueno-Guerra N, Gutiérrez-Roig M, Gracia-Lázaro C, Gómez-Gardeñes J, Perelló J, Sánchez A, Moreno Y, Duch J. arXiv preprint arXiv:1709.02857. 2017 Sep 8. [PLoS One, accepted paper. (2018).]

Abstract:

Fighting against climate change is a global challenge shared by nations with heterogeneous economical resources and individuals with diverse propensity for cooperation. However, we lack a clear understanding of the role of key factors such as inequality of means when diverse agents interact together towards a common goal. Here, we report the results of a collective-risk dilemma experiment in which groups of subjects were initially given either equal or unequal endowments. We found that although the collective goal was always achieved regardless of the initial capital distribution, the effort distribution was highly inequitable. Specifically, participants with fewer resources contributed significantly more to the public goods than the richer -sometimes twice as much. An unsupervised learning algorithm clustered the subjects according to their individual behavior. We found that the poorest participants congregated within the two “generous clusters” whereas the richest were mostly classified into a “greedy cluster”. Our findings suggest that future policies would benefit both from reinforcing climate justice actions addressed to most vulnerable people and educating fairness instead of focusing on understanding of generic or global climate consequences, as the latter has not proven to drive equitable contributions.

Kopányi-Peuker,  Journal of Behavioral and Experimental Economics, Vol. 80., pp. 211-218, 2019.

Abstract:

This research investigates the effects of the group size and the magnitude of the volunteering cost in a controlled large-scale laboratory experiment, where subjects play the volunteer’s dilemma only once. The experiment varies group sizes ranging from groups of 3 to about 100, and 2 different cost/benefit ratios. Results show that high cost reduces volunteering probability only in the smallest groups, but not for other group sizes. Furthermore, I find non-monotonic group size effect on the individual volunteering decisions. These findings are compared to the predictions produced by symmetric mixed-strategy Nash equilibrium, procedural rationality and quantal response equilibrium.