## Evolving Aspirations and Cooperation (1998)

Venue: | Journal of Economic Theory |

Citations: | 42 - 2 self |

### BibTeX

@ARTICLE{Karandikar98evolvingaspirations,

author = {Rajeeva Karandikar and Dilip Mookherjee and Debraj Ray and Fernando Vega-redondo},

title = {Evolving Aspirations and Cooperation},

journal = {Journal of Economic Theory},

year = {1998},

volume = {80},

pages = {292--331}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper therefore builds on [3], in which a model of consistent aspirations-based learning was introduced

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Citation Context ...ges weakly to Then P ' has a unique invariant measure + ' , which converges weakly to +* as ' a 0. Proof. Given any '>0, properties [b] and [c] imply that P ' is a T-chain. Applying Theorem 16.2.5 in =-=[14]-=-, P ' is uniformly ergodic, and has a unique invariant measure + ' . Then implying that ASPIRATIONS AND COOPERATION + ' [(1&,(')) P+,(') Q ' ]=+ ' , + ' [,(') Q ' ]=+ ' &(1&,(')) + ' P. 311s312 KARAND... |

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Citation Context ...her than ``belief'' learning. They originated in the mathematical psychology literature [7], and have received a certain degree of support in laboratory experiment situations involving human subjects =-=[15, 16, 20, 22, 23]-=-. Other recent explorations of such models of learning include [4] and [6], both of which explore the relationship with ``replicator dynamics'' models, and [10], which develops an axiomatic ``case-bas... |

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Citation Context ... It should be mentioned, nevertheless, that we obtain this similarity despite the fact that players are only ``learning'' to play the stage game. This is in contrast to some recent literature such as =-=[11]-=-, in which convergence to an equilibrium of the repeated game is considered. 5. RELATED LITERATURE The model of adaptive behavior considered in this paper presumes a limited form of rationality, where... |

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Citation Context ... and v=_, and therefore the probability of such convergence tends to zero as well. An analogue of Lemma 3 applied to player 2 similarly takes care of the (D, C) pss, and the proof of [1] is complete. =-=[2]-=- There exists a positive measure of states s under q(.|(C, D) pss) under which and lim R * 1 * ((C, D) pss | s)=0, lim R * 1 * ((D, C) pss | s)=0, and the same is true starting from the (D, C) pss. Th... |

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Citation Context ...n the terminology of [21], such models represent ``stimulus'' 305s306 KARANDIKAR ET AL. or ``reinforcement'' rather than ``belief'' learning. They originated in the mathematical psychology literature =-=[7]-=-, and have received a certain degree of support in laboratory experiment situations involving human subjects [15, 16, 20, 22, 23]. Other recent explorations of such models of learning include [4] and ... |

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Citation Context ..., and have received a certain degree of support in laboratory experiment situations involving human subjects [15, 16, 20, 22, 23]. Other recent explorations of such models of learning include [4] and =-=[6]-=-, both of which explore the relationship with ``replicator dynamics'' models, and [10], which develops an axiomatic ``case-based decision theory'' where players satisfice relative to aspiration levels... |

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Citation Context ...her than ``belief'' learning. They originated in the mathematical psychology literature [7], and have received a certain degree of support in laboratory experiment situations involving human subjects =-=[15, 16, 20, 22, 23]-=-. Other recent explorations of such models of learning include [4] and [6], both of which explore the relationship with ``replicator dynamics'' models, and [10], which develops an axiomatic ``case-bas... |

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Citation Context ...depicted in Fig. 4. In analyzing this, the following lemma will be useful. Lemma 4. Let T(*) be a sequence of positive integers such that * , Z t ] be a finite horizon stochastic takes values only in =-=[0, 1]-=- T(*) as* 1. For each *, let [X * t process with terminal date T(*), such that X * t (there is no restriction on Z * t ). Suppose that for each *, X * 0 is equal to a constant i(*) (which can take the... |

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Citation Context ...her than ``belief'' learning. They originated in the mathematical psychology literature [7], and have received a certain degree of support in laboratory experiment situations involving human subjects =-=[15, 16, 20, 22, 23]-=-. Other recent explorations of such models of learning include [4] and [6], both of which explore the relationship with ``replicator dynamics'' models, and [10], which develops an axiomatic ``case-bas... |

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Citation Context ...Q ' { : T (1&,(')) t=0 t P t f =+ = ' f &[1&,(')] T+1 + ' P T+1 f. Taking T , and using sup x |+ ' P T+1 f(x)| sup x | f(x)|, we get where Note that lim T g ' (x)#,(') : (1 (T+1)) + ' Q ' g ' =+ ' f, =-=(5)-=- t=0 [1&,(')] t P t f(x). (6) T t=0 Ptf =Rf implies g ' (x) Rf(x), for each x # E, as' a 0. Moreover, | g ' (x)| sup y | f( y)|#M< for all '. We may therefore apply Lemma 1 to conclude that lim ' 0 su... |

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Citation Context ...air of agents plays the game repeatedly over time. This also stands in contrast with some of the well-known models whose concern is to provide an evolutionary basis for the rise of cooperation, e.g., =-=[4, 8, 19]-=-. These papers consider situations in which the repeated Prisoner's Dilemma is recurrently being played between pairs of individuals randomly selected from the general population. In a sense, the obje... |

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Citation Context ... instead it concerns the adjustment of aspirations or payoff expectations on the basis of past experience, which in turn shapes agent behavior in the face of current experience. In the terminology of =-=[21]-=-, such models represent ``stimulus'' 305s306 KARANDIKAR ET AL. or ``reinforcement'' rather than ``belief'' learning. They originated in the mathematical psychology literature [7], and have received a ... |

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Citation Context ...rk of previous authors? The reason is that the long run distribution will not generally be concentrated on a single pss. This is in sharp contrast with random matching contexts considered by [12] and =-=[24]-=-, where the corresponding process singles out a unique limit state. In the informal discussion, which we postpone to Section 4, we attempt to provide a clearer intuitive explanation of this observatio... |

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Citation Context ... but in the context of a specific class of 2_2 games. Aspirations do turn out to converge, and the long run outcome is essentially cooperative, thus complementing the analysis in that paper. [13] and =-=[18]-=- both apply the Gilboa Schmeidler case-based theory to games of coordination or the Prisoners Dilemma. They allow aspirations to evolve simultaneously with the strategies selected by players in a cont... |

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Citation Context ...air of agents plays the game repeatedly over time. This also stands in contrast with some of the well-known models whose concern is to provide an evolutionary basis for the rise of cooperation, e.g., =-=[4, 8, 19]-=-. These papers consider situations in which the repeated Prisoner's Dilemma is recurrently being played between pairs of individuals randomly selected from the general population. In a sense, the obje... |

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Citation Context ...us preventing them from being perpetually satisfied with any given action. We examine the long run outcomes that are induced by vanishingly small tremble probabilities. This paper therefore builds on =-=[3]-=-, in which a model of consistent aspirations-based learning was introduced. 1 We make precise and prove the following result (Theorem 1 below): If the speed of updating aspiration levels is sufficient... |

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Citation Context ...ture [7], and have received a certain degree of support in laboratory experiment situations involving human subjects [15, 16, 20, 22, 23]. Other recent explorations of such models of learning include =-=[4]-=- and [6], both of which explore the relationship with ``replicator dynamics'' models, and [10], which develops an axiomatic ``case-based decision theory'' where players satisfice relative to aspiratio... |

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Citation Context ...actual experienced payoffs. Hence the theory in these papers imparts a certain additional degree of ``ambitiousness'' to players, which helps in ensuring convergence to cooperative outcomes. Finally, =-=[9]-=- and [17] consider models in which numerous pairs of players simultaneously play a given bilateral game, their aspirations being formed on the basis of the average payoff earned across the whole popul... |

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Citation Context ...ke the work of previous authors? The reason is that the long run distribution will not generally be concentrated on a single pss. This is in sharp contrast with random matching contexts considered by =-=[12]-=- and [24], where the corresponding process singles out a unique limit state. In the informal discussion, which we postpone to Section 4, we attempt to provide a clearer intuitive explanation of this o... |

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Citation Context ...s evolve, but in the context of a specific class of 2_2 games. Aspirations do turn out to converge, and the long run outcome is essentially cooperative, thus complementing the analysis in that paper. =-=[13]-=- and [18] both apply the Gilboa Schmeidler case-based theory to games of coordination or the Prisoners Dilemma. They allow aspirations to evolve simultaneously with the strategies selected by players ... |

2 |
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Citation Context ...xperienced payoffs. Hence the theory in these papers imparts a certain additional degree of ``ambitiousness'' to players, which helps in ensuring convergence to cooperative outcomes. Finally, [9] and =-=[17]-=- consider models in which numerous pairs of players simultaneously play a given bilateral game, their aspirations being formed on the basis of the average payoff earned across the whole population. [9... |

1 |
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Citation Context ...nvolving human subjects [15, 16, 20, 22, 23]. Other recent explorations of such models of learning include [4] and [6], both of which explore the relationship with ``replicator dynamics'' models, and =-=[10]-=-, which develops an axiomatic ``case-based decision theory'' where players satisfice relative to aspiration levels that are based on past experience. The structure of interaction between players in ou... |