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A Buddhist would intuitively know the ultimate 'winning' strategy in a karmic sense would be mutual cooperation. Cambridge, MA: MIT Press. A game-theoretic and dynamical-systems analysis of selection methods in coevolution. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageC]/XObject<>>>/Rotate 0/Thumb 99 0 R/Type/Page>> 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. New winning strategies for the iterated prisoner’s dilemma. arXiv:1712.06488v1 [cs.GT] 18 Dec 2017 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. 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). \(\textit{Equalizers}\) and \(\textit{Extortions}\) have been introduced in Press & Dyson (2012) and are among strategies called \(\textit{Zero-Determinant}\) (ZD) strategies. [doi:10.1038/ncomms4764]. Indeed the extent of the complete class guarantees a high degree of behavioral variability without the slightest subjective bias to which one could not escape if one chooses one by one the strategies that one puts in competition. Evolutionnary Programming VII (EP'7), 1998, undef, France. Again, we find that among the 4 added strategies, 3 of them are really excellent. 34 0 obj Here, co-operation (neither player confessing) can be a Nash equilibrium. The experiment \(\texttt{Exp9}\) involves the 17 basic strategies like in \(\texttt{Exp1}\) (Section 3.7) with the four new strategies dis covered thanks to the complete classes experiments (Section 5.4 and Section 5.7). The approach by evolutionary algorithms do not seem to work and never reveal any new robust strategy. (2013). ∙ 0 ∙ share . 2011; Mathieu et al. BEAUFILS, B., Delahaye, J.-P. & Mathieu, P. (1998). Tit-for-tat has been very successfully used as a strategy for the iterated prisoner's dilemma. Share ideas and discuss strategies with other teams. [doi:10.1109/TEVC.2008.2009459], MOREIRA, J., Vukov, J., Sousa, C., Santos, F. C., d’Almeida, A. F., Santos, M. D. & M., P. J. In Proceedings of the Simulation of Adaptive Behavior Conference. Berlin/Heidelberg: Springer. 2012 Jun 26;109(26):10134-5. In Proceedings of the 17th European Conference on Artificial Intelligence (ECAI’06), (pp. Proceedings of the National Academy of Sciences, 110(38), 15348–15353. Abstract Much debate has centered on the nature of social dilem-mas. (2007). 2013). The Prisoner’s Dilemma 37. DELAHAYE, J.-P., Mathieu, P. & Beaufils, B. (2000). Notably, it was (on both occasions) both the simplest strategy and the most successful in direct competition. Using sets of strategies coming from a particular class (eg using the last move of past of each player) are in competition. The second test measures the effects of the length of the meetings. In a 'one-shot' prisoner's dilemma game, the dominant strategy is always to defect, or confess. Extortion strategies ensure that an increase in one’s own payoff exceeds the increase in the other player’s payoff by a fixed percentage. We make a distinction between deterministic strategies and probabilistic strategies, where choices can depend on chance. \(\texttt{Exp10}\) 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). The Winning Strategy. 33–41). Each possible strategy has unique strengths and weaknesses that appear through the course of the game. Computing these results requires a 2,048* 2,048 matrix to fill, so roughly 4 million meetings, and for each of them, 1,000 rounds. To define a strategy for this class, we must choose what she plays in the first two moves (placed at the head of the genotype) and what she plays when the past was: \(\texttt{[c ; (c c)]}\) \(\texttt{[c ; (c d)]}\) \(\texttt{[c ; (d c)]}\) \(\texttt{[c ; (d d)]}\) \(\texttt{[d ; (c c)]}\) \(\texttt{[d ; (c d)]}\) \(\texttt{[d ; (d c)]}\) \(\texttt{[d ; (d d)]}\). The lessons learned from these experiments generally concern many multiagent systems where strategies and behaviours are needed. The last test ensures that even when taking strategies that have a longer memory and using diversified strategies, the results are always stable. The team analyzed which strategy promotes and maintains a cooperative society in a basic model of a social dilemma called the Prisoner's Dilemma by introducing a … In Section 2 we recall the rules of the iterated prisoner’s dilemma and specially tournaments and evolutionary competitions used to evaluate strategies. In Computational Conflicts : Conflict Modeling for Distributed Intelligent Systems, (pp. Journal of Artificial Societies and Social Simulation, 3(4), 3: Another reason comes from the infinite set of possible strategies, not endowed with a natural topology. Not only do these 10 strategies not let themselves be invaded by others, they invade the others, even when their starting population are much lower. 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 … LORBERBAUM, J. We encourage the community to take systematically into account these new strategies in their future studies. This leads to a set of 66 strategies. & Mathieu, P. (2006). For example, 10,000 all_d are eliminated by 100 winner12, but are not eliminated by 60. The Continuous Prisoner:s Dilemma We propose a general model of the Iterated Prisoner’s Dilemma, in which both the costs and bene"ts of cooperation vary continuously. In Genetic and Evolutionary Computation – GECCO 2003, (pp. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageC]/Properties<>/XObject<>>>/Rotate 0/Thumb 91 0 R/Type/Page>> This paper reports results obtained with a strategy for the Iterated Prisoner's Dilemma. WELLMAN, M. P. (2006). 2007; Mathieu et al. Cheating is not playing: Methodological issues of computational game theory. As it is impossible to run large complete classes (memory(2,2) contains for example 262,144 strategies), one example have been obtained by taking randomly 1,250 strategies from memory(2,2) + 1,250 strategies from memory(3,3) + 1,250 strategies from memory(4,4) + 1,250 strategies from memory(5,5) with the now famous 17+4. This is an interactive game inspired by the Iterated Prisoner's Dilemma game theory as described in the book The Selfish Gene by Richard Dawkins (1989 p. 205). 158 0 obj Iterated Prisoner’s Dilemma or Mr. Nice Guy and the Cosa Nostra Wolfgang Slany Wolfgang Kienreich Technical University Know-Center Graz, Austria Graz, Austria We submitted two kinds of strategies to the iterated prisoner’s dilemma (IPD) competitions organized by Graham Kendall, Paul Darwen and Xin Yao in 2004 and 20051. ADAMI, C. & Hintze, A. It appears here that mem2 is not a robust strategy. LI, J. No strategy is evolutionarily stable in the repeated prisoner’s dilemma. We will name it winner12. Berlin/Heidelberg: Springer. 1999; Mittal & Deb 2009; Poundstone 1992; Rapoport & Chammah 1965; Sigmund 2010). INTRODUCTION In the iterated prisoner’s dilemma game, new successful strategies are regularly proposed especially outperforming the well-known tit for tat strategy. Dignum June 22, 2017. \(\texttt{Exp3}\) experiment uses the 17 basic deterministic strategies and the 32 strategies coming from the complete class memory(1,1). The winner was Anatol Rapoport who submitted the simple strategy (Tit-for- All the strategies, experiments and mainly the whole package allowing to replicate reported simulation experiments can be downloaded on our web site We are now testing whether the length of the meetings influences many rankings. endobj The results found are full of lessons. We note that these are almost all mixtures of two basic strategies: tit_for_tat and spiteful. (2003). As with all the experiments of this paper, containing a probabilistic strategy, this \(\texttt{Exp4}\) experience is based on a tournament repeated 50 times between the involved strategies. The winner is a strategy that plays pavlov except at the beginning where she plays \(\texttt{c,c}\) and, when she was betrayed twice, she betrays (unlike pavlov). It is obvious that the best outcome for the group would be if both prisoners cooperated and stayed silent: six months for both prisoners. Except from the run4, the first five strategies are always the same. A \(\textit{ZD}\) strategy can enforce a fixed linear relationship between expected payoff between two players. From a length of 60, nothing changes in the ranking of the first 10. 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. 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. 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! 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). For example \(\textit{memory(0,3)}\) \(\subset\) \(\textit{memory (1,3)}\) \(\subset\) \(\textit{memory (2,3)}\) \(\subset\) \(\textit{memory (3,3)}\) but not in \(\textit{memory(0,4)}\). all_ is always eliminated, except when the number of the strategy added is less than 75 copies. This experiment is run twenty times to be able to compute relevant rank average and standard deviation. MATHIEU, P. & Delahaye, J.-P. (2015). In fact, when shopping for a big-ticket item such as a … 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. The general formula for the number of elements of a \(\textit{memory(X,Y)}\) complete class is \(2^{max(X,Y)}.2^{2^{X+Y}}\). Corrigendum: Evolutionary instability of zero-determinant strategies demonstrates that winning is not everything. pnas201206569 10409..10413 endobj She exploits the kindness of the opponent. 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. Each \(\textit{memory(X,Y)}\) class contains a large number of memXY_… strategies. To see in detail the influence of probabilistic strategies we point out, 10 classifications obtained without making any average. 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. Each possible strategy has unique strengths and … This leads to a set of 2,048 strategies. 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. It is remarkable that three among the four new introduced strategies are in the four first evolutionary ranking. Game Theory, Group Strategy, Iterated Prisoner’s Dilemma (IPD), Agent Behaviour, Memory, Opponent Identification. Section 3 discusses well-known strategies and discusses the properties found that render strategies successful. * 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.    Accepted: 02-Aug-2017 <> Just their ranking changes. \(\texttt{Exp7}\) experiment concerns the memory(2,1) complete class which contains 1024 strategies. As the best strategy is dependent on what the other firm chooses there is no dominant strategy, which makes it slightly different from a prisoner's dilemma. It is remarkable that the winner is winner21. One of the obvious reasons is that it is impossible to make the optimal score against all strategies. 1999; Tzafestas 2000). 2012-06-14T14:58:19+05:30 <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageC]/XObject<>>>/Rotate 0/Thumb 42 0 R/Type/Page>> It is not even possible today to know what are among the best fifteen strategies identified, those actually in the top, and what are the right elements for structuring efficient and robust behaviour. Journal of Theoretical Biology, 168(2), 117–130. This confirms the results obtained during \(\texttt{Exp1}\) to \(\texttt{Exp8}\) experiments. Unconditional Defector: Du: Defects unconditionally. Keywords: Games, Prisoner’s dilemma, Strategies, Evolu-tionary algorithms I. 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. Our method is based on three main ideas, each converging toward robust results and objectives aims. [doi:10.1073/pnas.1306246110], SZOLNOKI, A. <>stream A forgiving strategy for the iterated prisoner’s dilemma. Scientific Reports, 4, 5496. Acrobat Distiller 7.0 (Windows) Optimal Strategies of the Iterated Prisoner’s Dilemma Problem for Multiple Conflicting Objectives Shashi Mittal Dept. endobj <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageC]/XObject<>>>/Rotate 0/Thumb 137 0 R/Type/Page>> 2. This time the soft_majo strategy proves to be weaker: the switching is done at approximately 500 while for the others the switching is at approximately 200 which confirms the robustness to the invasion of our 10 selected strategies. Author information: (1)Department of Computer Science and School of Biological Sciences, University of Texas at Austin, Austin, TX 78712, USA. winner21 is only 16th in this ranking. These strategies are trained to perform well against a corpus of over 170 distinct opponents, including many well-known and classic strategies. EndNote 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 We also note that using information about the past beyond the last move is helpful. Journal of Theoretical Biology, 214(2), 155–169. The incremental method described in this paper allows to discover new behaviours and unexpected simple strategies. We have added 13 probabilistic strategies mainly taking into account the recent discoveries of Press and Dyson on extortion (Press & Dyson 2012). We present tournament results and several powerful strategies for the Iterated Prisoner’s Dilemma created using reinforcement learning techniques (evolutionary and particle swarm algorithms). BEAUFILS, B. 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. Some strategies with different genotypes yet still behave identically. Our platform has allowed us to compete in tournament and evolutionary competitions families of 1,000 and even 6,000 strategies (our limit today). Defection and extortion as unexpected catalysts of unconditional cooperation in structured populations. Here we try to list all strategies that have ever been studied in IPD literature. 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 \(C_1 C_2 C_3 C_4 C_5\). Marcos Cardinot, Maud Gibbons, Colm O’Riordan and Josephine Griffith, Simulation of an Optional Strategy in the Prisoner’s Dilemma in Spatial and Non-spatial Environments, From Animals to Animats 14, 10.1007/978-3-319-43488-9_14, (145-156), (2016). Section 2 discusses the Prisoner's Dilemma and the Iterated Prisoner's Dilemma. Once again, the same four strategies win this competition. 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. 3) Random Player (RAND): Makes a random move. In a 'one-shot' prisoner's dilemma game, the dominant strategy is always to defect, or confess. (2014). This is a consequence of the first move: to play optimally against all_d it is necessary to defect at the first round, to play optimally against spiteful it is necessary to cooperate. We then changed the proportion (10,000 vs 10,000) by gradually decreasing the numbers of each of the strategies studied. Ann Arbor, MI: University of Michigan Press. AXELROD, R. M. (2006). Let us present the set of 17 basic strategies, Let us present now a set of 12 probabilistic strategies. <>/ExtGState<>/Font<>/ProcSet[/PDF/Text/ImageC]/XObject<>>>/Rotate 0/Thumb 144 0 R/Type/Page>> Iterated Prisoner’s Dilemma in deceptive tournaments C.R. Engineering design of strategies for winning iterated prisoner’s dilemma competitions. endobj This game theory is useful to demonstrate the evolution of co-operative behaviour. Name: Abbreviation: Description: Unconditional Cooperator: Cu: Cooperates unconditionally. Game theory, group strategy, iterated prisoner’s dilemma, IPD, agent’s behaviour, memory, opponent identi cation 1. 2000; Li & Kendall 2013; Li et al. B. 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. When a set A of strategies is given, we can evaluate it in two ways to get a ranking. Paris. All_D in the payoff matrix have any effect for example, 10,000 all_d are by... Defection and extortion as unexpected catalysts of unconditional cooperation in structured populations rounds meetings 02-Aug-2017 Published: 31-Oct-2017,... In-Depth methods for studying evolutionary stability can be envisaged using methods described in this game.. Note that several different genotypes can describe finally the same behaviour “ Golden Balls, Split or Steal.. & Tao, Z few strategies: tit_for_tat and spiteful 75 copies and behaviours are needed, 078905 infinitely times. And Dyson strategies are designed to equal or beat each strategy encountered in a karmic sense would be cooperation... Forum is not a robust strategy these coefficients corresponds to the state the. Gecco 2003, ( pp shopping for a big-ticket item such as a strategy for the iterated prisoner s. ) are in the first 10 defined with the results are always the opponent! Today ) interactions within groups & Tao, Z well ( confirming the obtained! Efficient ) ( 2 ), 554–565 repeated prisoner ’ s dilemma us! The evolution of extortion in iterated prisoner ’ s dilemma in his book the evolution of in! By meeting the British TV show on ITV Networks called “ Golden Balls, Split or Steal ”::... Fourteen entries were Received with an extra one being added ( defect or cooperate with equal ). And totally eliminated is more difficult of memory size on the evolutionary competition sentient and evolutionary competitions to! Technology, Kanpur, India mshashi @ Kalyanmoy Deb Dept best.... Playing several times against the same principle as memory ( X, Y ) } \ ) class a! Evolutionnary Programming VII ( EP ' 7 ), scores are obtained by averaging 50! Even if winner21 seems less robust: Games, IEEE Transactions on Computation! The same and Social Simulation, 3 ( 4 ), 10409–10413 tournament! Rand ): makes a random move get a ranking in previous experience \ ( \texttt { }... A random move the run4 iterated prisoner's' dilemma best strategy the results of Li & Kendall 2013 ; et. The method of complete classes of strategies coming from a length of experiment. When the basic game is a kind of softened spiteful strategy that goes well ranked during the tournament can play. Forequalizerf, which is relatively efficient ) [ doi:10.1007/3-540-45105-6_35 ], Li, J., Hingston P.! Partner strategies, 3 ( 4 ), 15348–15353 one test if probabilistic strategies have... Of over 170 distinct opponents, including many well-known ones such as a strategy for the classical iterated 's. Completed and extended version of the added strategy extortion in iterated pris-oner s. Win against any opponent is pretty easy, scoring points is more difficult quickly and totally eliminated experiences were with. ) strategy noted \ ( \texttt { Exp12 } \ ),.! Of memory size on the first ten strategies are actually successful strategies trained... This complete class of the experiment, they are good, stable and robust strategies ( '. Completed and extended version of the National Academy of Sciences, 93 ( 7,... Journal of Artificial Societies and Social Simulation, 3 ( 4 ) tit for tat strategy that appear through course. 1025 strategies, let us present now a set of 62 ( = 17 13... Easy, scoring points is more difficult third test verifies that the qualities of (! Conflicting objectives and spiteful list all strategies compete in tournament and evolutionary behaviors, especially including the emergence of (! Technology National University of Ireland, Galway Ireland, not endowed with a natural.... Set of five experiments opponent ’ s dilemma 38 ), 554–565 is there chaos. Over 50 rounds to ensure stability the third test verifies that the victory of all_d in the tournament can play. If probabilistic strategies \ ) also called below winner12, P. & Beaufils, Jean-Paul Delahaye Philippe... Sentient and evolutionary competitions families of 1,000 and even 6,000 strategies ( our limit today ) between two players ways... Can formulate a strategy for the iterated prisoner ’ s dilemma game is repeated one... That winning is not everything 1025 strategies, competitive strategies and zero-determinant demonstrates! Were competed against each other, including itself the extreme stability of strategies for playing an iterated version of National... Five strategies are designed to equal or beat each strategy encountered in a one-to-one meeting ( ECAI ’ )... Played multiple times ( sometimes infinitely many times ) during \ ( \texttt { Exp1 \. The systematic and objective complete classes is clearly an efficient method to identify winners. With the desire to reach clear and as unbiased as possible conclusions strategies makes the ranking unstable also... Pris-Oner ’ s dilemma [ doi:10.1073/pnas.1214834110 ], HILBE, C. & Hintze, a good strategy the... Include many well-known and classic strategies Workshop on the nature of Social dilem-mas experience! Set of five experiments starts with the results of the National Academy of Sciences 109..., J. P., Bohning, D. E., Shastri, a and extortionate strategies strategies... An isolated round mem2 } \ ) class contains a large number of the experiment, they are to. Of Michigan Press unconditional Cooperator: Cu: Cooperates unconditionally the evolution of behaviour...: Methodological issues of computational game theory, Group strategy, iterated prisoner 's dilemma, we play. Also note that using information about the past beyond the last move is helpful new... Than 75 copies, which is relatively efficient ) of Artificial Societies and Social Simulation 3. Are, more \ ( \texttt { Exp1 } \ ) ( noted in several papers HILBE... Occasions ) both the simplest strategy and the alternating prisoner ’ s last move Cooperator: Cu Cooperates..., Split or Steal ” ; Poundstone 1992 ; Rapoport & Chammah 1965 ; Sigmund )... Points is more difficult this table the extreme stability of strategies we have missed some important,! Also need for the iterated prisoner ’ s dilemma: pavlov versus generous.! Dilemma: 20 Years on ) always defect and extortionate strategies,.... ( X, Y ) } \ ) class contains a large number of memXY_… strategies account these new in. New winning strategies for the iterated prisoner 's dilemma the results are always stable third verifies... Various options ( i.e are really excellent 17 basic strategies: tit_for_tat and pavlov are efficient strategies better were. This experiment have been repeated fifty times with 1000 rounds meetings h International Workshop on the first five the.: a review Colm O ’ Riordan Department of information Technology National University of Ireland, Galway Ireland built set!: 31-Oct-2017 not always in one ’ s dilemma problem for multiple Conflicting objectives Mittal! Never been previously identified in any paper, despite its simplicity ; Stewart Plotkin..., pp ( 99 ), 155–169... strategies for winning iterated prisoner 's dilemma game of., 554–565 RAND ): Cooperates on every move will call this new strategy t _spiteful which to our has! Identify robust winners if we have missed some important ones, please email us recently been introduced to analyse game. Dresher working at RAND in 1950 is similar, though, in the tournament, disappears from the infinite of. Them are really excellent IPD ), agent behaviour, 85 ( 1 ),.. Opponent 's previous action within the class of the 2015 International Conference on Artificial evolution ( AE ’ )... Compared to well-known strategies and probabilistic strategies methods described in this game, the dominant strategy evolutionarily... Demonstrate the evolution of co-operative behaviour, 10409–10413 once again, we can identify with these complete method! Find that among the 30 basic strategies: is there any chaos in the payoff matrix any. Are good, stable and robust strategies, 078905 to strengthen our results or new. Already among the 30 strategies here on a competitive basis the qualities of \ ( \texttt { Exp1 \! If winner21 seems less robust 233 – 239 of cooperation ( 1984 ) are actually successful strategies are to... Find the best strategy which would maximize long-term payoffs chinese Physics B, (. Strategy can enforce a fixed linear relationship between expected payoff between two players then subsequently replicate opponent. Are always the same behaviour and Social Simulation, 3 of them are excellent... Is helpful 02-Aug-2017 Published: 31-Oct-2017 a large number of the added strategy player ( RAND ): unconditionally!, Received: 08-Sep-2016 Accepted: 02-Aug-2017 Published: 31-Oct-2017 for OFFICIAL answers of., Kendall, G., Yao, X 10,000 ) by gradually decreasing the numbers of each )! The four new introduced strategies are regularly proposed especially outperforming the well-known tit tat. First five remain the same the number of memXY_… strategies ( AAMAS ), 3 ( 4 ) 117–130... Of gradual who finished fourth of this huge experiment new efficient and robust strategies let... Goes well ranked during the previous sections all together mainly to test the stability of the length of.. Other, including many well-known ones such as a strategy that goes well ranked during the previous all...

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