Vivian Y. Kraines

Pavlov and the prisoner's dilemma

Our Pavlov learns by conditioned response, through rewards and punishments, to cooperate or defect. We analyze the behavior of an extended play Prisoners Dilemma with Pavlov against various opponents and compute the time and cost to train Pavlov to cooperate. Among our results is that Pavlov and his clone would learn to cooperate more rapidly than if Pavlov played against the Tit for Tat strategy. This fact has implications for the evolution of cooperation.

Learning to cooperate with Pavlov. An adaptive strategy for the iterated prisoner's dilemma with noise

Conflict of interest may be modeled, heuristically, by the iterated Prisoners Dilemma game. Although several researchers have shown that the Tit-For-Tat strategy can encourage the evolution of cooperation, this strategy can never outscore any opponent and it does poorly against its clone in a noisy environment. Here we examine the family of Pavlovian strategies which adapts its play by positive and negative conditioning, much as many animals do. Mutual cooperation will evolve in a contest with Pavlov against a wide variety of opponents and in particular against its clone. And the strategy is quite stable in a noisy environment. Although this strategy cooperates and retaliates, as does Tit-For-Tat, it is not forgiving; Pavlov will exploit altruistic strategies until he is punished by mutual defection. Moreover, Pavlovian strategies are natural models for many real life conflict-of-interest encounters as well as human and computer simulations.

Evolution of Learning among Pavlov Strategies in a Competitive Environment with Noise.

Pavlov denotes a family of stochastic learning strategies that achieves the mutually cooperative outcome in the iterated prisoners dilemma against a wide variety of strategies, although it can be exploited to some extent by some. When restricted to an environment of only Pavlov-type strategies, slower learning mutants cannot invade an initial dominant population. More surprising, mutants who learn much faster than the current population tend to overreact and also cannot invade. In particular, the “immediate learning” version of Pavlov, sometimes called win-stay-lose-switch, often fares poorly in this environment. Only those strategies that learn marginally faster than the dominant variety will have greater fitness. Although faster learners will eventually dominate a given homogeneous Pavlov population, the process must proceed through a gradual increase in the rate of learning.

The Threshold of Cooperation Among Adaptive Agents: Pavlov and the Stag Hunt

Why is it that in an animal society, persistent selfishness is quite rare yet in human society, even strict laws and severe punishment do not eliminate selfish action against the interests of the whole? Stochastic learning agents called Pavlov strategies are used to model interactions in the multi-agent 2×2 Stag Hunt matrix game, a close relative of the Prisoners Dilemma. Markov chain methods and computer simulations establish a threshold learning rate for the stability of cooperation. A society of rapidly adapting agents may suffer strife and dissension while another society with slower learning agents will enjoy the benefits of virtually complete cooperation.

Recruitment and Retention of Students in Undergraduate Mathematics.

Pat Kenschaft is professor of mathematics at Montclair State College, where she has taught since earning her Ph.D. from the University of Pennsylvania, specializing in functional analysis. She is Chair of the MAA Committee on Participation of Women and was organizer, chair, and editor of the January, 1987 panel on which this article is based. Author of numerous research papers about women, blacks, and careers in mathematics, she has also written several college textbooks. She currently leads programs to develop mathematics education in elementary schools. Mother of two grown offspring and wife of Fred Chichester, she is an avid organic gardener.

Classroom Computer Capsules: Binary Operations

(1990). Classroom Computer Capsules: Binary Operations. The College Mathematics Journal: Vol. 21, No. 3, pp. 240-241.

INTEGRATED CALCULUS LABORATORY

ABSTRACT The author designed a one-credit, two-hour computer laboratory that integrated students from different Calculus classes. The students worked in teams pairing a Calculus I student with a Calculus II student and emphasized cooperative learning. The lab, being independent from a particular course, permitted the students to experiment freely without infringing on regular class time. Students worked on each topic for two lab sessions submitting both draft and final lab reports.