The all_d strategy itself also is never beaten by another strategy, and is known to be catastrophic because she gets angry with everyone (except stupid non-reactive strategies) and therefore does not earn nearly point, especially in evolutionary competitions where only survive efficient strategies after a few generations. ;f�,��2�G}��=k���=`�X��d�����Y���5C��0k���������:�5��Ը�&�p;i죩�3`;A>���Y3���Y3���:r�G;�[��ܩq
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It’s easy to find an explanation: all_d exploits strategies playing poorly (nonreactive for example); when they are gone, all_d is not able to win enough points to survive. The highly technical paper, " Iterated Prisoner’s Dilemma contains strategies that dominate any evolutionary opponent" by William H. Press and Freeman J. Dyson has now been published in PNAS (May 22, 2012), which was followed by a PNAS Commentary by Alexander Stewart and Joshua Plotkin of the Department of Biology, University of Pennsylvania, entitled " Extortion and cooperation in the … \(\texttt{Exp10}\) studies the 30 deterministic and probabilistic basic strategies like in \(\texttt{Exp2}\) ( Section 3.10) with the four new strategies discovered thanks to the complete classes experiments (Section 5.4 and 5.7). 2013). That means that if one increases the \(min(X,Y)\) of a memory class, not more than the max, then all the \(\textit{memory(X,Y)}\) are always in the increased class. if defecting were 1000x more profitable than cooperating, then δ would have to be very high in order to make cooperating still profitable). 12/18/2017 ∙ by Shiheng Wang, et al. The effect of memory size on the evolutionary stability of strategies in iterated pris-oner’s dilemma. The strategies were competed against each other, including itself. [doi:10.3998/mpub.20269], SIGMUND, K. (2010). [doi:10.1109/TEVC.2005.856203], HILBE, C., Nowak, M. A. uuid:a9704177-1dd1-11b2-0a00-aa00585095ff In each \(\textit{memory(X,Y)}\) complete class, all deterministic strategies can be completely described by their “genotype” i.e. 1 The Iterated Prisoner’s Dilemma The Prisoner’s Dilemma is a two person game that provides a simple model of a disturbing social phenomenon. To test the stability of these results, we have built a set of five experiments. Then we have used the systematic and objective complete classes method to evaluate them. Journal of Artificial Societies and Social Simulation, 3(4), 3: http://jasss.soc.surrey.ac.uk/3/4/3.html. O’RIORDAN, C. (2000). The results found are full of lessons. Journal of Theoretical Biology, 168(2), 117–130. Some strategies for this complete class are already among the 30 basic strategies that we have adopted. Human cooperation in the simultaneous and the alternating prisoner’s dilemma: Pavlov versus generous tit-for-tat. ]��s[ЁΤ�<5�Ѻ� ��Ii�}���冈Kl6�lI���q�%��Ca�N,��1��R�\���U�j8���+Y�����X���|u`�uz� In Section 4 we present the complete classes principal which is an objective frame to find and compare strategies: the main idea is to build a set of all the possible strategies using the same size of memory. Random: Random (=C.5 or R(.5,.5,.5) or S(.5,.5,.5,.5) below) Defects unconditionally. (1965). 1999; Mittal & Deb 2009; Poundstone 1992; Rapoport & Chammah 1965; Sigm… This had already been noted in several papers (Hilbe et al. In a 'one-shot' prisoner's dilemma game, the dominant strategy is always to defect, or confess. It is remarkable that the winner is winner21. It objectively shows that spiteful, tit_for_tat and pavlov are efficient strategies. According to the state of the art, in the first part of this paper we have collected the most well-known interesting strategies. AXELROD, R. M. (2006). Our Meeting with Gradual, A Good Strategy for the Iterated Prisoner's Dilemma. Each possible strategy has unique strengths and weaknesses that appear through the course of the game. Given that you have two choices, defect or cooperate, you’re really choosing to: 1) Defect, and get the best or third best result (0 years, or 5 years of prison) 2) Cooperate, and get the second best or … 202–209). When a set A of strategies is given, we can evaluate it in two ways to get a ranking. <> It appears here that mem2 is not a robust strategy. At this time, we consider, according with the final ranking in Section 7.10 that the best actual strategies in the IPD are in order. 10 0 obj [doi:10.1088/1674-1056/23/7/078905], FICICI, S. & Pollack, J. Evolutionnary Programming VII (EP'7), 1998, undef, France. Animal Behaviour, 85(1), 233 – 239. The PD TWO PERSON ITERATED model demonstrates an interesting concept: When interacting with someone over time in a prisoner's dilemma scenario, it is possible to tune your strategy to do well with theirs. This experiment shows that probabilistic strategies introduced by Press and Dyson are not good competitors (except forequalizerF, which is relatively efficient). INTRODUCTION In the iterated prisoner’s dilemma game, new successful strategies are regularly proposed especially outperforming the well-known tit for tat strategy. The \(\texttt{Exp8}\) experiment shows a confrontation including the two complete classes: memory(1,2) and memory(2,1). This experiment have been repeated fifty times with 1000 rounds meetings. <>stream
In this game, since winning against everyone is trivial (all_d does), it is obvious that “playing well” corresponds to earning a maximum of points, which in evolutionary competitions is equivalent to ending with the greatest population possible. *g�m�m����C:���닩u�[��a���!MܰẂ����n6����5q�T"��A�a7��OI&Eɤ��a�S�h:f��'婒����M��P� A���S"M�T��E�,)KџFzl2 �v4ƪٜf$=��"�D8RP.CF��#��ɞ7A�X�{�QD��R��G�QB�2�gG\��'��#l�AYM�f��Ԫ�R�>AA�ec1��E7�d�hr�9�N��{��Z��ɦ��W�08O������X6�QO��L6
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E6�(��O�,|��q� t��i\�Г�;[e��Y���5�;YVB�� �M?-L��W�-w��@p�`0�r~�Bi�m��C�4Bn�1�7��eJ�N����:q ��>��W��,�x����)�9��K���O��dQC^���i��? We can see that the first ten strategies are always the same. This results shows clearly that the qualities of \(\texttt{gradual}\) require a certain length of meeting. The Iterated Prisoners’ Dilemma: 20 Years on. It remains to be seen whether the 4 new strategies we have just introduced are really robust, and how they are ranked when confronted to the best previously identified strategies. We will name it winner12. [doi:10.1109/TCIAIG.2011.2166268]. In each case, all_d is quickly and totally eliminated. It is obvious that the best outcome for the group would be if both prisoners cooperated and stayed silent: six months for both prisoners. The collective interest is that everyone play \(\texttt{c}\), but a single logical analysis leads inevitably to \(\texttt{[d, d]}\) which is collectively the worst case! Arbortext Advanced Print Publisher 9.1.510/W Unicode 203–223). Keywords: Games, Prisoner’s dilemma, Strategies, Evolu-tionary algorithms I. In Genetic and Evolutionary Computation – GECCO 2003, (pp. LI, J. 2007; Li et al. We then changed the proportion (10,000 vs 10,000) by gradually decreasing the numbers of each of the strategies studied. of Computer Science and Engineering Indian Institute of Technology, Kanpur, India mshashi@iitk.ac.in Kalyanmoy Deb Dept. 2) Always Defect (AllD): Defects on every move. An agent using this strategy will first cooperate, then subsequently replicate an opponent's previous action. Abstract—The iterated prisoner’s dilemma game is a widely used tool for modelling and formalization of complex interactions within groups. This leads to a set of 66 strategies. The study examines the evolution of strategies for a single population of players pitted against a static environment, as well as the co-evolution of strategies for two distinct subpopulations of players competing against one another. A series of works (Beaufils et al. [doi:10.1016/j.anbehav.2012.10.030]. & Pollack, J. Extortion strategies ensure that an increase in one’s own payoff exceeds the increase in the other player’s payoff by a fixed percentage. It depends on how much you value future outcomes, and the relative benefits of various options (i.e. Scientific Reports, 4, 5496. Read (a draft of) the paper!. Game Theory, Group Strategy, Iterated Prisoner’s Dilemma (IPD), Agent Behaviour, Memory, Opponent Identification. Coded in Javascript by Wayne Davis. New York, NY: Doubleday. * Heuristic or Rule-based strategies: 24) APavlov: Plays TFT in the first six moves and identifies the opponent by means of a rule-based mechanism. Again, we find that among the 4 added strategies, 3 of them are really excellent. Tit-for-tat has been very successfully used as a strategy for the iterated prisoner's dilemma. This leads to a set of 49 strategies. The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely rational individuals might not cooperate, even if it appears that it is in their best interests to do so. It is generally assumed that there exists no simple ultimatum strategy whereby one player can en- force a unilateral claim to an unfair share of rewards. (1994). The winner is a strategy that plays tit_for_tat except that it starts with \(\texttt{d,c}\), and, when she betrayed twice and the other has nevertheless cooperated she reacts by a \(\texttt{d}\) (this is the only round that differentiates it from tit_for_tat). Evolution of extortion in Iterated Prisoner’s Dilemma games. 3) Random Player (RAND): Makes a random move. The two-player Iterated Prisoner’s Dilemma game is a model for both sentient and evolutionary behaviors, especially including the emergence of cooperation. We note that the only strategy that appears coming from Press and Dyson ideas is the \(\textit{equalizerF}\) strategy, that we will encounter often further. Princeton, NJ: Princeton University Press. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageC]/XObject<>>>/Rotate 0/Thumb 99 0 R/Type/Page>> We recall that in this game, we can-not play well against everyone. Note that in this table the extreme stability of the beginning of ranking. In Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems (AAMAS), (pp. Each time, discovered strategies have been justified by mathematical or experimental arguments trying to establish that we are dealing with better strategies than tit_for_tat. Notably, it was (on both occasions) both the simplest strategy and the most successful in direct competition. This winner12 makes us think to a mixture as simple as possible of tit_for_tat and spiteful: She plays tit_for_tat unless she has been betrayed two times consecutively, in which case she always betrays (plays all_d). Boston, MA: The MIT Press/Bradford Books. It was originally framed by Merrill Flood and Melvin Dresher while working at RAND in 1950. [doi:10.1007/3-540-45105-6_35], FICICI, S. G., Melnik, O. Invincible Strategies of Iterated Prisoner's Dilemma. Singapore: World Scientific Publishing Co. [doi:10.1142/6461], LI, J., Hingston, P. & Kendall, G. (2011). The char-acterization of these strategies turned out to be surprisingly concise, and we derive them by modeling iterated prisoner’s dilemma as a Markov process as in (Press and Dyson 2012). When we consider complete classes we note the first plays (which do not depend on the past) in lowercases, and the other plays in uppercases. pnas201206569 10409..10413 The result of the meetings (Tournament and Evolutionary competition) of this set of 17 classic deterministic strategies is a really good validity test of any IPD simulator. Indeed, for iterated games in general, but especially for the iterated prisoner’s dilemma, notions of Nash equilibrium, Pareto optimality or evolutionarily stable strategies Lorberbaum (1994); Lorberbaum et al. WELLMAN, M. P. (2006). Tournaments: each strategy meets each other (including itself) during a series of, if the payoff in the two moves is \(\texttt{2R}\) \(\texttt{[c,c]}\) and \(\texttt{[c,c]}\) then, if the payoff in the last move is \(\texttt{T+S}\) (\(\texttt{[c,d]}\) or \(\texttt{[d,c]}\) then, \(p_1\) if the last move is \(\texttt{[c,c]}\), \(p_2\) if the last move is \(\texttt{[c,d]}\), \(p_3\) if the last move is \(\texttt{[d,c]}\), \(p_4\) if the last move is \(\texttt{[d,d]}\). 23) Adaptive: Starts with C,C,C,C,C,C,D,D,D,D,D and then takes choices which have given the best average score re-calculated after every move. Cambridge, MA: MIT Press. This leads to a set of 53 strategies. Unconditional Defector: Du: Defects unconditionally. But moves being simultaneous, one cannot play optimally against these two strategies. Our aim in this paper is to identify new systematic, reproducible and objective experiments, suggesting several ways to design robust and efficient new strategies and more than that, a general scheme to identify new ones. The right strategies in the prisoner’s dilemma are not those who try to earn as many points than the opponent (such as equalizers) or require to earn more points than any other (as extortioners), these are the ones that encourage cooperation, know how to maintain it and even restore it if necessary after a sequence of unfortunate moves. & Delahaye, J.-P. (1999). This ranking confirms that the strategies we have adopted are effectively efficient strategies. Here, co-operation (neither player confessing) can be a Nash equilibrium. A game-theoretic memory mechanism for coevolution. Except from the run4, the first five strategies are always the same. There are a lot of reasons for you to want to play iterated prisoner’s dilemma, but the hard part can definitely be finding the right players as well as understanding the rules, setting as well as how all of the parties should act. 122 0 obj Published: 31-Oct-2017. Game theory, group strategy, iterated prisoner’s dilemma, IPD, agent’s behaviour, memory, opponent identi cation 1. In Proceedings of the Fi h International Workshop on the Synthesis and Simulation of Living Systems (ALIFE’5), (pp. Abstract We present tournament results and several powerful strategies for the Iterated Prisoner’s Dilemma created using reinforcement learning techniques (evolutionary and particle swarm algorithms). A new strategy for the iterated prisoner’s dilemma game September 4, 2012 September 4, 2012 Carson Chow Economics , Evolution , Mathematics , Pedagogy , Probablity , Sociology The game theory world was stunned recently when Bill Press and Freeman Dyson found a new strategy to the iterated prisoner’s dilemma (IPD) game. 1665–1666).International Foundation for Autonomous Agents and Multiagent Systems. Toward adaptive cooperative behavior. CAPSULE SUMMARY Executive summary: Robert Axelrod's 1980 tournaments of iterated prisoner's dilemma strategies have been condensed into the slogan, Don't be too clever, don't be unfair. Proceedings of the National Academy of Sciences, 93(7), 2686– 2689. Extortion subdues human players but is finally punished in the Prisoner’s Dilemma. 2000; Li & Kendall 2013; Li et al. & Plotkin, J. In Section 2 we recall the rules of the iterated prisoner’s dilemma and specially tournaments and evolutionary competitions used to evaluate strategies. 185–189). The general formula for the number of elements of a \(\textit{memory(X,Y)}\) complete class is \(2^{max(X,Y)}.2^{2^{X+Y}}\). In order to add a robustness test to the strategies identified, we conducted a series of experiments to test their stability against invasions of different types. Game theory, group strategy, iterated prisoner’s dilemma, IPD, agent’s behaviour, memory, opponent identi cation 1. This leads to a set of 2,048 strategies. Iterated Prisoner's Dilemma. This is also the case obviously for winner12. ∙ 0 ∙ share . 1996; Delahaye et al. Within the class of memory-one strategies for the iterated Prisoner's Dilemma, we characterize partner strategies, competitive strategies and zero-determinant strategies. We have begun to study the actual situation with the desire to reach clear and as unbiased as possible conclusions. We note that these are almost all mixtures of two basic strategies: tit_for_tat and spiteful. The iterated prisoner's dilemma is just like the regular game except you play it multiple times with an opponent and add up the scores. It is remarkable that three among the four new introduced strategies are in the four first evolutionary ranking. The third test verifies that the changes of coefficients in the payoff matrix have any effect. Taking an incremental approach , combining the results of several progressive series of massive confrontation experiments in order to be able to formulate, as closely as possible, robust conclusions. In turn, we take 10,000 copies of all_d and 10,000 copies of one of the 10 previously mentioned strategies that come together in an evolutionary competition. 2014; Szolnoki & Perc 2014). If we consider only deterministic strategies making their decision using the last move of each player, we can define a set of 32 strategies, each determined by a 5-choice genotype \(C_1 C_2 C_3 C_4 C_5\). The same experiment has been performed by replacing all_d by the random strategy. This paper reports results obtained with a strategy for the Iterated Prisoner's Dilemma. Some strategies with different genotypes yet still behave identically. New York, NY: Basic Books. Ref. This paper is a completed and extended version of the two page paper (Mathieu & Delahaye 2015). Traditionally the Prisoner’s Dilemma game has a dominant strategy of betrayal. \(\texttt{Exp12}\) is built with all the basic deterministic strategies obtained with the 17 initial basic strategies and the memory(1,1) complete class added with the 13 probabilistic strategies like in \(\texttt{Exp4}\) (Section 4.14) with the four new strategies discovered thanks to the complete classes experiments (Section 5.4 and Section 5.7). The winner was Anatol Rapoport who submitted the simple strategy (Tit-for- The two best strategies come from this paper. (2003). H��Wێ�}߯ �u ����Λc v���P����yٕ��9U=3��E����p `5�鮮:Uu���&�4�%��'�Mo/^�p�G߭.�u����˿�[��Y�G��ty��o��fs����G�����"��-�ʣ�*��F�D���n/.g__�"�L٥�M>b��ۋ�6u�1�Jk���mЧ-��V}h�P��i����Q�?~�o>��1�c A�~���f4��OF������c�!X��ϓ�Ρ�9�ؑ�T$W�LJ����a~���٫�t�V�l��[�tʥr`y2a����)�m: ��aX����=���=����cT�ވ+�����Y2����]�ӕ�%����KN Abstract and Figures The two-player Iterated Prisoner's Dilemma game is a model for both sentient and evolutionary behaviors, especially including the emergence of cooperation. This game theory is useful to demonstrate the evolution of co-operative behaviour. endobj One can see that, when the length of the meeting is greater than 10 rounds, then the first 10 strategies stay the same. Berlin/Heidelberg: Springer. EndNote
Nature Communications, 4, 2193. This result shows the robustness of these four strategies. Each \(\textit{memory(X,Y)}\) class contains a large number of memXY_… strategies. 33–41). The dilemma is iterated when we imagine that the situation of choice between \(\texttt{c}\) and \(\texttt{d}\) is presented periodically to the same two entities. For example, 10,000 all_d are eliminated by 100 winner12, but are not eliminated by 60. Prisoner’s Dilemma: A Study in Conflict and Cooperation. Defection and extortion as unexpected catalysts of unconditional cooperation in structured populations. all_ is always eliminated, except when the number of the strategy added is less than 75 copies. Berlin/Heidelberg: Springer. IEEE Transactions on Evolutionary Computation, 9(6), 580–602. The method of complete classes is clearly an efficient method to identify robust winners. In fact, when shopping for a big-ticket item such as a … B. In this game, there are two players, each of whom can make one of the two The following slightly simpler and less provocative strategy (which is usually a quality) seemed interesting to us: she plays \(\texttt{cc}\) at the beginning and then plays spiteful. endobj MATHIEU, P. & Delahaye, J.-P. (2015). Zero-determinant strategy: An underway revolution in game theory. Iterated Prisoner’s dilemma is a fun game to play as long as you’re with the right people as well as if you have the right setting and rules for this game. Here we try to list all strategies that have ever been studied in IPD literature. see that these strategies include many well-known ones such as TFT, Always Defect and extortionate strategies. Are there really no evolutionarily stable strategies in the iterated prisoner’s dilemma? On the other hand tit_for_tat finishes only 13th. We require that \(T>R>P>S\) and \(T+S<2R\) The classical chosen values are \(\texttt{T = 5, R = 3, P = 1, S = 0}\), which gives: \(\verb|[c, c] -> 3 + 3|, \verb|[d, d] -> 1 + 1| , \verb|[d, c] -> 5 + 0|\). The strategy gradual wins, and strangely, all_c is the seventh, but the three new introduced strategies (spiteful_cc, winner12, t _spiteful) are among the 10 best. (2013). ON "ITERATED PRISONER’S DILEMMA CONTAINS STRATEGIES THAT DOMINATE ANY EVOLUTIONARY OPPONENT" by William H. Press and Freeman J. Dyson By William Poundstone 1. 23 0 obj This page will look and function better with Javascript. Using sets of strategies coming from a particular class (eg using the last move of past of each player) are in competition. Playing well against all_d need to always betray (and in particular for the first move), and playing well against all_c need to always cooperate. No strategy is evolutionarily stable in the repeated prisoner’s dilemma. The strategy is compared to well-known strategies in the domain and results presented. Studies on dynamics in the classical iterated prisoner’s dilemma with few strategies: Is there any chaos in the pure dilemma ? Ann Arbor, MI: University of Michigan Press. [doi:10.1073/pnas.93.7.2686]. The lessons learned from these experiments generally concern many multiagent systems where strategies and behaviours are needed. The iterated prisoner’s dilemma is a game that allows to understand various basic truths about social behaviour and how cooperation between entities is established and evolves sharing same space: living organisms sharing an ecological niche, companies competitors fighting over a market, people with questions about the value of conducting a joint work, etc (Axelrod 2006; Beaufils & Mathieu 2006; Kendall et al. For example \(\textit{memory(0,3)}\) \(\subset\) \(\textit{memory (1,3)}\) \(\subset\) \(\textit{memory (2,3)}\) \(\subset\) \(\textit{memory (3,3)}\) but not in \(\textit{memory(0,4)}\). (2005). Equalizer (or dictator) SET-n (for P≤n≤R) A ZDstrategy These starting choices are written in lower case. Iterated Prisoner’s Dilemma in deceptive tournaments C.R. It is generally assumed that there exists no simple ultimatum strategy whereby one player can enforce a unilateral claim to an unfair share of rewards. The memory also seems useful to play well (confirming the results of Li & Kendall 2013; Moreira et al. Robert Axlerod wrote about Iterated Prisoners’ Dilemma in his book The Evolution of Cooperation (1984). Vreeswijk second reviewer dr. F.P.M. endobj The prisoner's dilemma is a standard example of a game analyzed in game theory that shows why two completely "rational" individuals might not cooperate, even if it appears that it is in their best interests to do so. 2020-12-08T12:33:34-08:00 colmor@geminga.nuigalway.ie Abstract Much debate has centered on the nature of social dilem-mas. This builds a set of 62 (= 17 + 13 + 32) strategies. A promising way to find other efficient strategies is probably to carefully study larger complete classes, to identify the best and check their robustness. We note that shorter the meetings are, more \(\texttt{mem2}\) is favoured and less gradual is disadvantaged. Questions on the Robot Game should be directed to fllrobotgame@usfirst.org <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageC]/XObject<>>>/Rotate 0/Thumb 137 0 R/Type/Page>> The optimal score against all strategies one-to-one meeting the National Academy of Sciences, (... 170 distinct opponents, including many well-known and classic strategies that spiteful, tit_for_tat and pavlov are efficient and! Living Systems ( AAMAS ), 155–169 our meeting with gradual, a good strategy for iterated! No evolutionarily stable in the iterated prisoner 's dilemma game, new successful strategies, where choices can depend chance... Past of each player ) are in competition compared with those of Exp12 ( see Section 6.11 ) are... ( 3 ), 3 of them are really excellent compete in tournament and evolutionary behaviors, especially the! International Workshop on the evolutionary stability of these four strategies theory is useful to the! 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Few strategies: is there any chaos in the prisoner ’ s dilemma with few strategies: tit_for_tat and are., 6913–6918 standard deviation meeting with gradual, a general scheme to find new ones of 17 strategies... Generous tit-for-tat this confirms the results of Li & Kendall 2013 ; Li Kendall..., even if winner21 seems less robust Transactions on, 13 ( ). Source for OFFICIAL answers iterated prisoner's' dilemma best strategy to work and never reveal any new robust strategy the outcome similar. E., Shastri, a gradual, a good strategy for the classical iterated pris-oner ’ dilemma... Several papers ( HILBE et al with these complete classes } \ ) strategy can enforce a fixed relationship. Results have to be compared with those of Exp12 ( see Section 6.11 which... ’ 06 ), agent behaviour, 85 ( 1 ) always cooperate ( AllC ) makes. The 4 added strategies, Evolu-tionary algorithms I notably, it was on! … Section 2 we recall the rules of the strategies defined during the previous sections together! One of the strategies gradual, a good strategy for the iterated 's... A source for OFFICIAL answers against any opponent in a karmic sense would be mutual cooperation of! Including itself { Exp12 } \ ) class contains a large number of iterated! Made with 1,000 rounds by meeting result shows the robustness of the meetings are, more there permutations. Game theory, and the relative benefits of various options ( i.e it objectively shows that strategies... Through the course of the game is played multiple times ( sometimes infinitely times... Fact, when shopping for a big-ticket item such as TFT, always defect ( AllD ) Cooperates... Rank of the obvious reasons is that it is a single best strategy... Evolutionarily stable in the first 10 unexpected catalysts of unconditional cooperation in structured.. Dilemma and the most successful in direct competition reveal any new interesting strategy random strategy dilemma, strategies Evolu-tionary. A review Colm O ’ Riordan Department of information Technology National University of Ireland, Galway Ireland strategy compared... The simplest strategy and the iterated Prisoners ’ dilemma: pavlov versus generous tit-for-tat successful direct!

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