Dr. Kraines contributed to the theory of homology and cohomology operations, particularly to Massey products and loop operations. Among the applications of his work has been his construction of the counterexample to the transfer conjecture of Quillen. He has also studied the variational bicomplex of Vinogradov and introduced the cohomology of quantum electrodynamics. Dr. Kraines has applied game theoretical techniques to study the evolution of cooperation. With Dr. Vivian Kraines, he introduced a stochastic learning approach, dubbed the Pavlov strategy, for the iterated Prisoner's Dilemma. They show that, in a noisy environment, agents using the Pavlov strategy may achieve a higher level of cooperation than those using Tit for Tat type strategies. Using computer simulations, dynamic systems and Markov chains, they extend their analysis to the evolution of the rate of learning in a society of Pavlov type agents. Recently, they have explored the natural selection of stochastic strategies in the simultaneous and the alternating Prisoner's Dilemma and identified several evolutionarily stable strategies.

Office Location: | 120 Science Drive, Durham, NC 27708, Durham, NC 27708 |

Office Phone: | (919) 660-2849 |

Email Address: | |

Web Page: | http://www.math.duke.edu/~dkrain |

**Teaching (Fall 2023):**

- MATH 283S.01,
*ADVANCED PROBLEM SOLVING*Synopsis- Physics 119, Tu 06:15 PM-07:45 PM

**Office Hours:**- send message to dkrain@math.duke.edu

**Education:**Ph.D. University of California - Berkeley 1965 M.A. University of California - Berkeley 1963 B.A. Oberlin College 1961

**Research Interests:***Algebraic Topology and Game Theory*Dr. Kraines contributed to the theory of homology and cohomology operations, particularly to Massey products and loop operations. Among the applications of his work has been his construction of the counterexample to the transfer conjecture of Quillen. He has also studied the variational bicomplex of Vinogradov and introduced the cohomology of quantum electrodynamics. Dr. Kraines has applied game theoretical techniques to study the evolution of cooperation. With Dr. Vivian Kraines, he introduced a stochastic learning approach, dubbed the Pavlov strategy, for the iterated Prisoner's Dilemma. They show that, in a noisy environment, agents using the Pavlov strategy may achieve a higher level of cooperation than those using Tit for Tat type strategies. Using computer simulations, dynamic systems and Markov chains, they extend their analysis to the evolution of the rate of learning in a society of Pavlov type agents. Recently, they have explored the natural selection of stochastic strategies in the simultaneous and the alternating Prisoner's Dilemma and identified several evolutionarily stable strategies.

**Keywords:**Animals • Cooperative Behavior • Game Theory • Humans • Mathematics • Memory • Models, Genetic • Population Dynamics • Selection, Genetic

**Undergraduate Research Supervised**- Theodore Freylinghuysen (2010 - 2012)
- Hans Kist (2009 - 2011)

**Recent Publications**- Kraines, D; Kraines, V,
*The threshold of cooperation among adaptive agents: Pavlov and the stag hunt*, Lecture Notes in Computer Science (Including Subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1193 (January, 2015), pp. 219-231, ISSN 0302-9743 [abs] - Kraines, DP; Kraines, VY,
*Natural selection of memory-one strategies for the iterated prisoner's dilemma.*, Journal of Theoretical Biology, vol. 203 no. 4 (April, 2000), pp. 335-355, ISSN 0022-5193 [10736212], [doi] [abs] - Kraines, D; Kraines, V,
*Evolution of Learning among Pavlov Strategies in a Competitive Environment with Noise*, Journal of Conflict Resolution, vol. 39 no. 3 (January, 1995), pp. 439-466, SAGE Publications [doi] [abs] - Kraines, D; Kraines, V,
*Learning to cooperate with Pavlov an adaptive strategy for the iterated Prisoner's Dilemma with noise*, Theory and Decision, vol. 35 no. 2 (September, 1993), pp. 107-150, Springer Nature, ISSN 0040-5833 [doi] [abs] - Kraines, D; Kraines, V,
*Pavlov and the prisoner's dilemma*, Theory and Decision, vol. 26 no. 1 (January, 1989), pp. 47-79, Springer Nature, ISSN 0040-5833 [doi] [abs]

- Kraines, D; Kraines, V,