We outline a model to study the evolution of cooperation in a population of
agents playing the prisoner's dilemma in signed networks. We highlight that if
only dyadic interactions are taken into account, cooperation never evolves.
However, when triadic considerations are introduced, a window of opportunity
for emergence of cooperation as a stable behaviour emerges.Comment: In Proceedings Wivace 2013, arXiv:1309.712

Righi, Simone

Takács, Károly

Electronic Proceedings in Theoretical Computer Science

English

arXiv

Simone Righi

Károly Takács

Crossref

Signed Networks, Triadic Interactions and the Evolution of Cooperation

We outline a model to study the evolution of cooperation in a population of agents playing the prisoner's dilemma in signed networks. We highlight that if only dyadic interactions are taken into account, cooperation never evolves. However, when triadic considerations are introduced, a window of opportunity for emergence of cooperation as a stable behaviour emerges

RIGHI, SIMONE

Takács, Károly

Archivio istituzionale della ricerca - Università di Modena e Reggio Emilia

14/09/2025 16:38Signed Networks, Triadic Interactions and the Evolution of Cooperation / Righi, Simone; Takács, Károly. -In: ELECTRONIC PROCEEDINGS IN THEORETICAL COMPUTER SCIENCE. - ISSN 2075-2180. - 130:(2013), pp.104-107. (Intervento presentato al  convegno 2013 Italian Workshop on Artificial Life and EvolutionaryComputation, Wivace 2013 tenutosi a Milano nel 1-2 Luglio 2013) [10.4204/EPTCS.130.17].Terms of use:The terms and conditions for the reuse of this version of the manuscript are specified in the publishingpolicy. For all terms of use and more information see the publisher's website.(Article begins on next page)Electronic Proceedings in Theoretical Computer ScienceThis is the peer reviewd version of the followng article:A. Graudenzi, G. Caravagna, G. Mauri, M. Antoniotti (Eds.):Wivace 2013 - Italian Workshop onArtificial Life and Evolutionary ComputationEPTCS 130, 2013, pp. 104–107, doi:10.4204/EPTCS.130.17c© S. Righi, K. TakácsSigned Networks, Triadic Interactions and the Evolution ofCooperationSimone Righi Károly TakácsMTA TK ”Lendület” Research Center for Educationaland Network Studies (RECENS).Budapest (Hungary)simone.righi@tk.mta.hu karoly.takacs@tk.mta.huWe outline a model to study the evolution of cooperation in a population of agents playing the pris-oner’s dilemma in signed networks. We highlight that if only dyadic interactions are taken intoaccount, cooperation never evolves. However, when triadic considerations are introduced, a windowof opportunity for emergence of cooperation as a stable behaviour emerges.Introduction. In this paper we examine how cooperation can be supported by negative ties and triadicinteractions. Previous research has demonstrated that cooperation is more likely to evolve in socialdilemma games, such as the Prisoners Dilemma, if the game is played in networks [1, 4, 5, 12]. Besidessparseness [3, 4, 5], the structure of the network is also important because on top of direct ties, indirectrelations also control behavior and contribute to the establishment of cooperation through reputationmechanisms, such as image scoring [2]. These ideas have been fruitful in explaining why and underwhich conditions can cooperation evolve and be stable among humans. In these studies however, onlypositive relations between players are analyzed.Negative ties could potentially be better enforcers of cooperation than positive ones. At the dyadiclevel, social psychological mechanisms of vengeance and anger are manifested through the use of pow-erful trigger strategies in the repeated Prisoners Dilemma game [7, 8, 9, 10]. Another mechanism isselectivity that prescribes cooperation with those who are liked and defection with those who are dis-liked. Experiments demonstrate that subjects play differently in social dilemma games with related andunrelated others (e.g., [6]). There is also evidence of homophilous selection of partners that can stabi-lize cooperative regimes [11]. Furthermore, we hypothesize that when interactions happen on a triadicbasis, cooperation can increase. For instance, if A and B are friends, and C is their common enemy, thepresence of the negative ties towards C could strengthen cooperation between A and B. Signed relationscould reliably signal cooperative or non-cooperative intentions, and hence could guarantee the spread ofappropriate reputational information that result in confirming expectations about cooperation.Model. We incorporate both dyadic and triadic interaction mechanisms in an agent-based model inwhich the (random) signed network and cooperation co-evolve. Our model is inspired by the tradition ofcomputational evolutionary game theory models where the more successful strategies tend to diffuse inthe population. We analyze the evolutionary success of different strategies that are defecting or cooper-ating and sensitive to the sign of links from their partners to others to a different way and extent. Indeed,each agent in our setup can be of one of the following three types:• Unconditional Defector (UD): which will always defect.• Conditional Player(CO): whose strategy choice is influenced by the sign of his ties. He cooperatesif he likes the opponent (positive tie) and he defects if he does not like the opponent (negative tie).S. Righi, K. Takács 105• Unconditional Cooperator (UC): It cooperates always regardless to the sign of his ties.Starting with a population composed of a similar number of agents for each type we iterate the followingsteps until an equilibrium is reached:1. Two (or three in the case of triadic interactions) connected agents are selected randomly.2. The prisoner’s dilemma is played between the two (or three, playing the game between each coupleof the triad) agents selected and the total payoff is calculated for each of them1.3. The signs of the network ties involved in the interactions evolve as consequence of the actions ofthe agents in step 3. In particular:• if both agents played the Defection strategy then the sign of their tie is turned negative.• if both agents played the Cooperation strategy then the sign between them is turned positive.• if one of the agent Cooperates while the other Defects then the tie is turned positive withprobability Ppos and negative with probability Pneg.4. The agent with higher payoff has a probability Pinv of invading the agent with lower payoff (in caseof equality each agent has the same probability of invasion2).Studying this setup in the two different cases in which the Prisoner’s dilemma is played in couples andin triads we obtain the preliminary results outlined in the next paragraph3Outline of the results. In the presence of negative ties, cooperation cannot be sustained in a randomnetwork where only dyadic considerations affect the agents’ behavior. The reason is described by the firstpanel of Figure 1. Here we consider the stability, in isolation, of each possible pair of agents and sign ofthe tie that links them. If a pair is unstable then the dynamics of the model will transform it in some otherpair. In presence of only dyadic interactions only the pairs circled in blue4 are stable. However, sincethe prisoner’s dilemma is not played in isolation but in a population of agents, one must also considerthe stability of the dyads to invasion by other types of agents. Indeed, as UD always dominates UC ina dyadic comparison, the dyad UC+UC is only stable in isolation. Moreover, UD dominates CO in adyadic comparison if they are linked by a positive tie, they are equivalent otherwise. So if Pneg > 0 thenCO+CO and CO−CO are also only stable in isolation. Generalizing this result to the whole networkwe can conclude that the only stable configuration for the case of dyadic interactions is a population ofDefectors linked by a network of negative signs.Introducing triadic interactions enlarges the window of opportunity for the diffusion of cooperativebehavior allowing for the emergence of stable cooperative clusters. Indeed, with the same strategy usedabove5, we obtain the updating dynamics of the lower left panel of Figure 1: all triads formed of agentsof the same types are stable in isolation. Moreover, as can be seen on the right panel of Figure 1 none ofthis triads dominates any other one. This allows us to conclude that if interactions happen in triads it ispossible to observe the emergence of some degree of cooperation in an evolutionary prisoner’s dilemmaplayed on signed networks. The actual emergence of cooperation depends on the values assumed by thetransition probabilities and from the network topology assumed. We are currently designing simulationsto corroborate the conceptual analysis provided here.1If a triad is selected all three agents need to have connections with each other so that everyone will play exactly twice.2If there is a payoffs order a > b > c, then only a can invade one of the other agents. However, if the payoffs order is of thetype a = b > c than either a or b can be selected. The selected agent will then randomly choose to invade one of the remainingplayers of the triad randomly.3For reasons of space, in this paper we limit our analysis to a conceptual analysis.4Namely: UC+UC, CO+CO, UD-UD and CO-CO.5Signs are not reported in the image so to simplify the drawing.106 Signed Networks, Triadic Interactions and the Evolution of CooperationFigure 1: The first panel shows the effects of model’s dynamics on each signed pair for the case in whichonly dyadic interactions are considered. The arrows between dyads indicate the possible transformationsfrom one pair to another allowed by the model’s dynamics. Only the four pairs circled in blue (thefour left ones) are stable. The lower left panel repeat the same exercise for triadic interactions. Thelower right panel shows that none of the triads which are stable in isolation is dominated by any other.The signs are omitted from the figure and each triad represent all possible triads with the combinationof agents’ types indicated, regardless of the signs that link them. The arrows between triads indicatepossible transformations from one triad to another, accounting for the possibilities given by differentnetwork signs, allowed by the model’s dynamics..References[1] Hauert C & Doebeli M (2004): Spatial structure often inhibits the evolution of cooperation in the snowdriftgame. Nature 428(6983), pp. 643–646. doi:10.1038/nature02360[2] Wedekind C & Milinski M (2000): Cooperation through image scoring in humans. Science 288(5467), pp.850–852. doi:10.1126/science.288.5467.850[3] Nowak MA (2006): Five rules for the evolution of cooperation. Science 314(5805), pp. 1560–1563.doi:10.1126/science.1133755[4] Lieberman E Ohtsuki H, Hauert C & Nowak MA (2006): A simple rule for the evolution of cooperation ongraphs and social networks. Nature 441(7092), pp. 502–505. doi:10.1038/nature04605[5] Pinheiro FL Pacheco JM & Santos FC (2009): Population structure induces a symmetry break-ing favoring the emergence of cooperation. PLoS computational biology 5(12), p. e1000596.doi:10.1371/journal.pcbi.1000596.g004[6] Ferraro PJ & Cummings RG: Cultural diversity, discrimination, and economic outcomes: an experimentalanalysis. Economic Inquiry 45(2).[7] Axelrod R (1984): The Evolution of Cooperation. New York, Basic Books.S. Righi, K. Takács 107[8] Axelrod R (1997): The Complexity of Cooperation. Agent-Based Models of Competition and Collaboration.Princeton NJ, Princeton University Press.[9] Axelrod R & Hamilton WD (1981): The evolution of cooperation. Science 211(4489), pp. 1390–1396.doi:10.1126/science.7466396.[10] Trivers RL (1971): The evolution of reciprocal altruism. Quarterly review of biology, pp. 35–57.[11] Yamagishi T & Hayashi N (1996): Selective Play: Social Embeddedness of Social Dilemmas. In LiebrandWBG & Messick DM, editors: Frontiers in Social Dilemma Research, Springer, Berlin.[12] Zhang C Wang S, Szalay MS & Csermely P (2008): Learning and innovative elements ofstrategy adoption rules expand cooperative network topologies. PLoS One 3(4), p. e1917.doi:10.1371/journal.pone.0001917.s001[13] Hayashi N Yamagishi T & Jin N (1994): Prisoners Dilemma Networks: Selection Strategy Versus ActionStrategy. In Albers W Schulz U & Mueller U, editors: Social Dilemmas and Cooperation, Springer Verlag,Berlin.

Directory of Open Access Journals

A. Graudenzi, G. Caravagna, G. Mauri, M. Antoniotti (Eds.):Wivace 2013 - Italian Workshop onArtificial Life and Evolutionary ComputationEPTCS 130, 2013, pp. 104–107, doi:10.4204/EPTCS.130.17c© S. Righi, K. TakácsSigned Networks, Triadic Interactions and the Evolution ofCooperationSimone Righi Károly TakácsMTA TK ”Lendület” Research Center for Educationaland Network Studies (RECENS).Budapest (Hungary)simone.righi@tk.mta.hu karoly.takacs@tk.mta.huWe outline a model to study the evolution of cooperation in a population of agents playing the pris-oner’s dilemma in signed networks. We highlight that if only dyadic interactions are taken intoaccount, cooperation never evolves. However, when triadic considerations are introduced, a windowof opportunity for emergence of cooperation as a stable behaviour emerges.Introduction. In this paper we examine how cooperation can be supported by negative ties and triadicinteractions. Previous research has demonstrated that cooperation is more likely to evolve in socialdilemma games, such as the Prisoners Dilemma, if the game is played in networks [1, 4, 5, 12]. Besidessparseness [3, 4, 5], the structure of the network is also important because on top of direct ties, indirectrelations also control behavior and contribute to the establishment of cooperation through reputationmechanisms, such as image scoring [2]. These ideas have been fruitful in explaining why and underwhich conditions can cooperation evolve and be stable among humans. In these studies however, onlypositive relations between players are analyzed.Negative ties could potentially be better enforcers of cooperation than positive ones. At the dyadiclevel, social psychological mechanisms of vengeance and anger are manifested through the use of pow-erful trigger strategies in the repeated Prisoners Dilemma game [7, 8, 9, 10]. Another mechanism isselectivity that prescribes cooperation with those who are liked and defection with those who are dis-liked. Experiments demonstrate that subjects play differently in social dilemma games with related andunrelated others (e.g., [6]). There is also evidence of homophilous selection of partners that can stabi-lize cooperative regimes [11]. Furthermore, we hypothesize that when interactions happen on a triadicbasis, cooperation can increase. For instance, if A and B are friends, and C is their common enemy, thepresence of the negative ties towards C could strengthen cooperation between A and B. Signed relationscould reliably signal cooperative or non-cooperative intentions, and hence could guarantee the spread ofappropriate reputational information that result in confirming expectations about cooperation.Model. We incorporate both dyadic and triadic interaction mechanisms in an agent-based model inwhich the (random) signed network and cooperation co-evolve. Our model is inspired by the tradition ofcomputational evolutionary game theory models where the more successful strategies tend to diffuse inthe population. We analyze the evolutionary success of different strategies that are defecting or cooper-ating and sensitive to the sign of links from their partners to others to a different way and extent. Indeed,each agent in our setup can be of one of the following three types:• Unconditional Defector (UD): which will always defect.• Conditional Player(CO): whose strategy choice is influenced by the sign of his ties. He cooperatesif he likes the opponent (positive tie) and he defects if he does not like the opponent (negative tie).S. Righi, K. Takács 105• Unconditional Cooperator (UC): It cooperates always regardless to the sign of his ties.Starting with a population composed of a similar number of agents for each type we iterate the followingsteps until an equilibrium is reached:1. Two (or three in the case of triadic interactions) connected agents are selected randomly.2. The prisoner’s dilemma is played between the two (or three, playing the game between each coupleof the triad) agents selected and the total payoff is calculated for each of them1.3. The signs of the network ties involved in the interactions evolve as consequence of the actions ofthe agents in step 3. In particular:• if both agents played the Defection strategy then the sign of their tie is turned negative.• if both agents played the Cooperation strategy then the sign between them is turned positive.• if one of the agent Cooperates while the other Defects then the tie is turned positive withprobability Ppos and negative with probability Pneg.4. The agent with higher payoff has a probability Pinv of invading the agent with lower payoff (in caseof equality each agent has the same probability of invasion2).Studying this setup in the two different cases in which the Prisoner’s dilemma is played in couples andin triads we obtain the preliminary results outlined in the next paragraph3Outline of the results. In the presence of negative ties, cooperation cannot be sustained in a randomnetwork where only dyadic considerations affect the agents’ behavior. The reason is described by the firstpanel of Figure 1. Here we consider the stability, in isolation, of each possible pair of agents and sign ofthe tie that links them. If a pair is unstable then the dynamics of the model will transform it in some otherpair. In presence of only dyadic interactions only the pairs circled in blue4 are stable. However, sincethe prisoner’s dilemma is not played in isolation but in a population of agents, one must also considerthe stability of the dyads to invasion by other types of agents. Indeed, as UD always dominates UC ina dyadic comparison, the dyad UC+UC is only stable in isolation. Moreover, UD dominates CO in adyadic comparison if they are linked by a positive tie, they are equivalent otherwise. So if Pneg > 0 thenCO+CO and CO−CO are also only stable in isolation. Generalizing this result to the whole networkwe can conclude that the only stable configuration for the case of dyadic interactions is a population ofDefectors linked by a network of negative signs.Introducing triadic interactions enlarges the window of opportunity for the diffusion of cooperativebehavior allowing for the emergence of stable cooperative clusters. Indeed, with the same strategy usedabove5, we obtain the updating dynamics of the lower left panel of Figure 1: all triads formed of agentsof the same types are stable in isolation. Moreover, as can be seen on the right panel of Figure 1 none ofthis triads dominates any other one. This allows us to conclude that if interactions happen in triads it ispossible to observe the emergence of some degree of cooperation in an evolutionary prisoner’s dilemmaplayed on signed networks. The actual emergence of cooperation depends on the values assumed by thetransition probabilities and from the network topology assumed. We are currently designing simulationsto corroborate the conceptual analysis provided here.1If a triad is selected all three agents need to have connections with each other so that everyone will play exactly twice.2If there is a payoffs order a > b > c, then only a can invade one of the other agents. However, if the payoffs order is of thetype a = b > c than either a or b can be selected. The selected agent will then randomly choose to invade one of the remainingplayers of the triad randomly.3For reasons of space, in this paper we limit our analysis to a conceptual analysis.4Namely: UC+UC, CO+CO, UD-UD and CO-CO.5Signs are not reported in the image so to simplify the drawing.106 Signed Networks, Triadic Interactions and the Evolution of CooperationFigure 1: The first panel shows the effects of model’s dynamics on each signed pair for the case in whichonly dyadic interactions are considered. The arrows between dyads indicate the possible transformationsfrom one pair to another allowed by the model’s dynamics. Only the four pairs circled in blue (thefour left ones) are stable. The lower left panel repeat the same exercise for triadic interactions. Thelower right panel shows that none of the triads which are stable in isolation is dominated by any other.The signs are omitted from the figure and each triad represent all possible triads with the combinationof agents’ types indicated, regardless of the signs that link them. The arrows between triads indicatepossible transformations from one triad to another, accounting for the possibilities given by differentnetwork signs, allowed by the model’s dynamics..References[1] Hauert C & Doebeli M (2004): Spatial structure often inhibits the evolution of cooperation in the snowdriftgame. Nature 428(6983), pp. 643–646. doi:10.1038/nature02360[2] Wedekind C & Milinski M (2000): Cooperation through image scoring in humans. Science 288(5467), pp.850–852. doi:10.1126/science.288.5467.850[3] Nowak MA (2006): Five rules for the evolution of cooperation. Science 314(5805), pp. 1560–1563.doi:10.1126/science.1133755[4] Lieberman E Ohtsuki H, Hauert C & Nowak MA (2006): A simple rule for the evolution of cooperation ongraphs and social networks. Nature 441(7092), pp. 502–505. doi:10.1038/nature04605[5] Pinheiro FL Pacheco JM & Santos FC (2009): Population structure induces a symmetry break-ing favoring the emergence of cooperation. PLoS computational biology 5(12), p. e1000596.doi:10.1371/journal.pcbi.1000596.g004[6] Ferraro PJ & Cummings RG: Cultural diversity, discrimination, and economic outcomes: an experimentalanalysis. Economic Inquiry 45(2).[7] Axelrod R (1984): The Evolution of Cooperation. New York, Basic Books.S. Righi, K. Takács 107[8] Axelrod R (1997): The Complexity of Cooperation. Agent-Based Models of Competition and Collaboration.Princeton NJ, Princeton University Press.[9] Axelrod R & Hamilton WD (1981): The evolution of cooperation. Science 211(4489), pp. 1390–1396.doi:10.1126/science.7466396.[10] Trivers RL (1971): The evolution of reciprocal altruism. Quarterly review of biology, pp. 35–57.[11] Yamagishi T & Hayashi N (1996): Selective Play: Social Embeddedness of Social Dilemmas. In LiebrandWBG & Messick DM, editors: Frontiers in Social Dilemma Research, Springer, Berlin.[12] Zhang C Wang S, Szalay MS & Csermely P (2008): Learning and innovative elements ofstrategy adoption rules expand cooperative network topologies. PLoS One 3(4), p. e1917.doi:10.1371/journal.pone.0001917.s001[13] Hayashi N Yamagishi T & Jin N (1994): Prisoners Dilemma Networks: Selection Strategy Versus ActionStrategy. In Albers W Schulz U & Mueller U, editors: Social Dilemmas and Cooperation, Springer Verlag,Berlin.

Signed Networks, Triadic Interactions and the Evolution of Cooperation

Abstract

Similar works

Full text

Available Versions

Crossref

Archivio istituzionale della ricerca - Università di Modena e Reggio Emilia

Directory of Open Access Journals

Archivio istituzionale della ricerca - Alma Mater Studiorum Università di Bologna