M. A. Nowak and K. Sigmund. Tit for tat in heterogeneous populations. Nature, 355:250--253, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Riolo Program for Study of Complex Systems The University of Michigan Ann Arbor, MI 48109 USA rlriolo umich.edu November, 1996 Abstract A simple model of tag mediated partner selection for agents playing the IPD is described. The agents strategies are represented as (i,q,q) triples as in [Nowak and May, 1992]. The tags are represented as arbitrary real numbers in [0, 1] and tag selection is biased toward agents with similar tags. Each agent plays a small sample of other agents each generation. The accumulated payoff from those plays is used as a fitness measure by an evolutionary algorithm which ....

.... individuals ( Stanley et al., 1994] Crowley et al. 1996] Individuals can be located in some spatial topology with a bias to inter act with nearby individuals, e.g. one dimensional grids ( Oliphant, 1994] Hoffmann and Waring, 1996] or two dimensional grids (pp.158 168 of [Axelrod, 1984] [Nowak and May, 1992], Grim, 1996] Individuals can use preferences for arbitrary tags (labels, surface markings) to bias selection of partners. Note that tag based selection of part ners in effect induces a kind of abstract topology in which the individuals are located near (prefer to interact) or far ....

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Nowak, Martin and May, R.M. "Tit for tat in hetero- geneous populations." Nature, 355, pp250-253.

....properties, shared by most high scoring behaviors in IPD experiments. It is good (it starts by cooperating) It is retaliating (it returns the opponent s defection) It is generous (it forgets the past if the defecting opponent cooperates again) Further strategies include stochastic ones ([11]) the Pavlov strategy ( 13] that cooperates when it has played the same move as its opponent etc. In the literature we may also find studies in an evolutionary perspective ( 6] theoretical or applied biological studies ( 3] 5] 10] and studies of modified IPD versions ( 14] We have designed ....

Nowak, M.A., and Sigmund, K.: Tit for tat in heterogeneous populations, Nature 355 (1992) 250-53

.... properties, shared by most high scoring behaviors in IPD experiments : It is good (it starts by cooperating) It is retaliating (it returns the opponent s defection) It is generous (it forgets the past if the defecting opponent cooperates again) Further strategies include stochastic ones (Nowak Sigmund 1992), the Pavlov strategy (Nowak Sigmund 1993) that cooperates when it has played the same move as its opponent etc. In the literature we may also find studies in an evolutionary perspective (Fogel 1993) theoretical or applied biological studies (Axelrod Dion 1988, Feldman Thomas 1987, Milinski ....

Nowak, M., and Sigmund, K. (1992) Tit for tat in heterogeneous populations. Nature 355:250-53.

.... such as in [1, 3, 4] The experimental studies of the IPD and its strategies need a lot of time computation and thus with the progress of computers, a lot of computer scientists and mathematicians have studied it as they have been able to use specific methods, like genetic algorithms, on it, see [2, 5, 9, 15, 16, 21, 22, 24, 25]. As cooperation is a topic of continuing interest for the social, zoological and biological sciences, a lot of works in those different fields have been made on the IPD: 6, 8, 12, 13, 14, 17, 18, 20, 23, 19] In this paper we study the consequences of the second equality inversion: S T ) 2 ....

M. Nowak and K. Sigmund. Tit for tat in heterogeneous populations. Nature, 355:250--253, 1992.

....imagine using a qualitative example nding technique to discover a range of qualitatively di erent patterns in a set of data. 2.7 Finding diverse behaviours in a simulated dynamical system. Another area of interest is suggested by the work of Nowak and Sigmund on the iterated prisoner s dilemma [80]. In their experiments they nd that the presence in the initial population of just one or two solutions from a particular part of the search space changes the long term dynamics of the game in a substantial way. In this case there are only a small number of qualitative dynamics producible within ....

M. A. Nowak and K. Sigmund. Tit for tat in heterogeneous populations. Nature, 355:250-253, 1992.

.... giving us clues as to when protean behavior or adaptive unpredictability, such as a rabbit s zig zag path when fleeing a fox can emerge [8] Game theoretic models of the strategic interactions between individuals demonstrate the complicated paths by which forgiveness and cooperation might evolve [9]. Simulations of large interacting populations where new species can form have been used to explore the evolution of communication for altruistic or mate attracting functions [10] And simulations of the mateselection process itself indicate how the perceptual mechanisms used in choosing sexual ....

M.A. Nowak and K. Sigmund, "Tit for Tat in Heterogeneous Populations," Nature, Vol. 355, no. 6357, January 1992, pp. 250-252.

....an eye , and shows that the evolution of such complex traits is possible through small adaptive steps. Simulations are also one of the best tools for elucidating the dynamics of an evolutionary process, showing what the course of evolution of a certain behavior could have looked like over time. Nowak and Sigmund s (1992) simulation of the evolution of forgiving or generous behavior in a population of individuals playing the Iterated Prisoner s Dilemma game is a good example of this kind of study. Their work showed that a possible evolutionary sequence for this behavior could have begun with individuals who ....

Nowak, M.A., and Sigmund, K. "Tit For Tat in Heterogeneous Populations." Nature, 355 (1992):250-252.

....The explanatory value of a game theoretic analysis appears to be robust to this simplification. When offense reward defense punishment and two lineages have tag genes that match well, the lineages engage in an iterated prisoner s dilemna (IPD) regarding the mutation of their defense genes. [3, 2, 34, 32, 7, 8, 35, 40] That is, mutual mutation will result in lower interaction payoffs than mutual lack of mutation. Yet, if one lineage mutates and the other doesn t, then the mutant lineage will gain a better payoff than mutual lack of mutation while the nonmutant lineage will score worse than mutual mutation. ....

M. A. Nowak and K. Sigmund. Tit for tat in heterogeneous populations. Nature, 355:250--253, 1992.

.... exchange and imitation processes in economy has already been described by Myerson (1990) Sharfstein and Stein (1990) Blume (1991) Banerjee (1992) Kirman (1992, 1993) and Ellison (1993) The studies on cooperation are often based on the iterated prisoner s dilemma, Lindgren (1991) and Nowak and Sigmund (1992), or on cost benefit analysis, Glance and Huberman (1993, 1994) A simple example that we use throughout the discussion is that of two brands of otherwise equivalent cars, except for the fact that one brand is polluting and the other one is non polluting thanks to its catalytic converter. The a ....

....cars. They represent metastable and stable configurations of nonpolluters and cheaters that benefit from the presence of the non polluters. This coexistence of cooperators and cheaters is similarly observed in many simulations of the iterated prisoner dilemma, as observed by Lindgren (1991) and Nowak and Sigmund (1992). Fig. 7. Slow evolution towards a coexistence distribution among polluting and non polluting cars. The initial conditions are a random distribution of polluters and non polluters. The conditions are a large difference in prior utilities ( 0 (0) 8) longer term memory (ff max=64) s = 10, all ....

NOWAK M. and SIGMUND K. (1992), " Tit-for-tat in heterogeneous populations", Nature 355, 250-253.

....The role of such a diversity for the evolution of cooperation is by now widely accepted. Diversity, which was initially attributed to stochasticities or errors (Boyd 1989, Nowak Sigmund 1989) is now being investigated as an intrinsic element and the motor of the evolutionary process (May 1987, Nowak Sigmund 1992, Glance Huberman 1994) The longer term ambition is the study of the relation between social autonomy and emergence of higher order organizations. The main hypothesis of this study is that higher order organizations emerge as a result of cooperation between lower level entities, and this ....

....of multiple satisfactions, or else governed by multiple hedonistic motivations, will unavoidably seek compromises between those motivations. Returning to the prisoner s dilemma and games theory that have benefited from a recent resurrection within the sciences of complexity and artificial life (Nowak May 1992), Sigmund 1992) Angeline 1994) Batali Kitcher 1994) the above presented quantitative tit for tat model may certainly be instantiated in such contexts by simply adding a special play maker agent who will set the rules of the interaction and will manipulate the rest of the agents by ....

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Nowak, M., K. Sigmund (1992). Tit-for-tat in heterogeneous populations, Nature, 355:250-253.

....this leads to a long vendetta. There are several ways to overcome this problem. One can, for instance, play Generous Tit For Tat (GTFT ) always cooperate if the other player cooperated in the previous round, but defect only with a certain probability if he defected (see Molander, 1985, and Nowak and Sigmund, 1992). Alternatively, one could use the strategy PAVLOV : cooperate if and only if you and your opponent used the same move in the previous round (see Kraines and Kraines (1988) Fudenberg and Maskin (1990) or Nowak and Sigmund (1993b) Both strategies are error proof: a mistaken defection is quickly ....

Nowak, M.A. and Sigmund, K. (1992), Tit for tat in heterogeneous populations, Nature 355, 250-2.

....depend on the behaviour of the partner and are a priori defined. Many extensions of the Prisoner s Dilemma have been studied, such as iterated versions (Axelrod and Hamilton, 1981) versions involving more individuals (Eshel and Cavalli Sforza, 1982) and games on a spatial grid (Axelrod, 1984; Nowak and May, 1992). A strategy that appeared to survive against a large array of alternatives is Tit for Tat, in which each of the two players copies the preceding behavioural act of its partner (Axelrod and Hamilton, 1981) Tit for Tat has become the prototypical co operative strategy in both theoretical (e.g. ....

.... A strategy that appeared to survive against a large array of alternatives is Tit for Tat, in which each of the two players copies the preceding behavioural act of its partner (Axelrod and Hamilton, 1981) Tit for Tat has become the prototypical co operative strategy in both theoretical (e.g. Nowak and Sigmund, 1992,1993) and empirical studies of animal behaviour (e.g. Milinski, 1987; Dugatkin, 1988) Evolutionary biologists often use game theory in the form of so called Evolutionarily Stable Strategies. This approach is illuminating, because it shows how the fitness of individuals adopting a given behaviour ....

Nowak, M. A. and K. Sigmund 1992. Tit for Tat in heterogeneous populations. Nature 355: 250253.

.... such as in [1, 3, 4] The experimental studies of the IPD and its strategies need a lot of time computation and thus with the progress of computers, a lot of computer scientists and mathematicians have studied it as they have been able to use specific methods, like genetic algorithms, on it, see [9, 2, 5, 8, 19, 21, 26, 27, 20, 31]. As cooperation is a topic of continuing interest for the social, zoological and biological sciences, a lot of works in those different fields have been made on the IPD: 6, 7, 16, 17, 18, 22, 23, 25, 29, 24] Although all people who have studied the IPD come from different research fields, it ....

M. Nowak and K. Sigmund. Tit for tat in heterogeneous populations. Nature, 355:250--253, 1992.

....= 1. When lineages 1 and 2 perfectly mismatch, a mutation in the offense gene produces these scores. shown in Table 3. This is a Prisoner s Dilemma payoff matrix. Axelrod and Hamilton, 1981, Axelrod, 1984, Nowak and Sigmund, 1993, Milinski, 1993, Boyd, 1989, Boyd and Lorberbaum, 1987, Nowak and Sigmund, 1992, Wilkinson, 1984] A lineage scores better by mutating, or defecting in the Prisoner s Dilemma terminology, regardless of what its opponent does. However, mutual cooperation scores slightly better for both players than mutual defection. The table for a mutation in the offense gene reverses each ....

M. A. Nowak and K. Sigmund. Tit for tat in heterogeneous populations. Nature, 355:250--253, 1992.

....one solution, even if the search space offers multiple solutions [12, p. 185] With co evolution, genetic drift makes the GA overspecialize to a single strategy of the game. In the case of the IPD, the population soon consists of slight variations of the same strategy, usually a cooperative one [13]. This stalls the co evolutionary arms race. Training against opponents which lack variety causes poor generalization [14] and makes those over specialized individuals vulnerable to novel opponents [6] 15, p. 325] Eventually, mutation produces a sufficiently novel opponent, which causes sudden ....

....than 50 Titfor Tat 2 strategies, then all individuals will eventually use Tit for Tat or All Cooperate, and remain there as long as a sufficient number of Tit for Tats are maintained [17, pp. 121 122] It has also been shown that Tit for Tat usually leads to even more cooperative strategies [13]. Figure 3 shows what can occur when this over specialization produces brittle strategies that can be invaded by defection. As with practically all non trivial games, the IPD does not offer a perfect 3 strategy [24] The challenge is to learn a strategy that is as general as possible. ....

M. A. Nowak and K. Sigmund, "Tit for tat in heterogeneous populations," Nature, vol. 355, pp. 250--253, 16 January 1992.

....matches both zero and one digits. This mechanism allows for generalisation. A label pattern containing all don t care (#) digits, would match all possible labels. 9.2 The Representation of Strategies Strategies are represented as pairs (p,q) of real values in the range [0. 1] as used in [28] [26]. These values represent the probabilities that an agent will co operate on a move proceeded by co operation from the last player to whom the rule was applied in a previous game (p) or co operate on a move proceeded by defection (q) This is a stochastic representation with a memory of one. It ....

Nowak, M. & Sigmund, K.: Tit for Tat in Heterogeneous populations. Nature, 355, (1992) 250-253

....in a non spatial population. First, simulations involving the prisoner s dilemma will be presented, and then the framework will be applied to the problem of communication. Evidence from prisoner s dilemma simulations The earliest work done on spatializing the prisoner s dilemma was done by Nowak and May (1992). They used the prisoner s dilemma as an update rule for two dimensional cellular automata. In their simulations, cells could be in one of two states (cooperate or defect) and at each time step, every cell was replaced by the cell bordering it that had the highest summed payoff from playing the ....

....on cooperation in the prisoner s dilemma. Further encouragement is given by analytic work in ethology showing that kin selection can provide a plausible account for the 6 The strategies that occur in Grim s simulations are a more generous version of the strategy called Generous Tit For Tat by Nowak and Sigmund (1992). 55 10 20 30 40 50 Generations (x100) 10 20 30 40 50 60 70 80 90 100 Population Figure IV.14: A contour map of a spatial population evolving over time. The vertical axis represents the 100 individuals in the population. Shaded areas represent areas containing defectors. The population begins ....

Nowak, M. and K. Sigmund (1992). Tit for tat in heterogeneous populations.

....given round to be proportional to its relative success in the previous round. The result obtained by R. Axelrod with the ecological simulation is that TIT FOR TAT quickly becomes the most common strategy. Other experiments and studies about this game, or some of its variants have been done (see [2, 5, 3, 6, 13, 14, 15, 20, 24, 25, 26, 28, 29]) This game has been put to use in various ways ( 22, 23, 7] 2 The Iterated Prisoner s Dilemma with Renunciation Now we are going to explore a new version of the Iterated Prisoner s Dilemma in which each strategy can give up the game. This choice of strategy is irreversible and each player ....

Nowak M., K. Sigmund. Tit for tat in heterogeneous populations. Nature, V. 355 16 January 1992. pp. 250-253.

....the cost (to the donor) is less than the benefit (to the recipient) and if the recipient is likely to return the favour. This principle was explored in many papers, we mention only Axelrod and Hamilton (1981) Axelrod (1984) Sugden (1986) Boyd and Lorberbaum (1987) May (1987) Lindgren (1991) Nowak and Sigmund (1992, 1993) Nowak, May and Sigmund (1995) Sigmund (1995) Leimar (1997) In his seminal paper of 1971, Trivers mentioned the further possibility of a generalised altruism , where the return is directed towards a third party. Individuals . may respond to an altruistic act that benefits themselves ....

Nowak, M.A. and Sigmund, K. (1992) Tit for tat in heterogeneous populations, Nature 355, 250-2.

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M. A. Nowak and K. Sigmund. Tit for tat in heterogeneous populations. Nature, 355:250--253, 1992.

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M. Nowak and K. Sigmund, "Tit for tat in heterogeneous populations," Nature, vol. 355, no. 6357, pp. 250--253, Jan. 1990.

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Nowak, M. A. and K. Sigmund, Tit for tat in heterogeneous populations, Nature (1992), 355:250-253 36

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-58. Nowak, M. A. & Sigmund, K. (1992). Tit for tat in heterogeneous populations. Nature, 355, 250-253. Ray, T. S. (1992). An approach to the synthesis of life. In: Articial Life II (Langton, C. G., Taylor, C., Farmer, J. D., & Rasmussen, S., eds), pp.

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Nowak,M. & Sigmund,K. (1992). Tit for tat in heterogeneous populations. Nature, 355, 250 - 253.

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