- Associate Professor Emeritus of Mathematics

**External address:**233 Physics Bldg, Durham, NC 27708

**Internal office address:**Box 90320, Durham, NC 27708-0320

**Phone:**(919) 660-2849

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# David P. Kraines

**External address: **233 Physics Bldg, Durham, NC 27708**Internal office address: **Box 90320, Durham, NC 27708-0320**Phone: **(919) 660-2849
### Research Areas and Keywords

##### Biological Modeling

##### Topology

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.

##### Education & Training

### Selected Grants

## Pages

## Pages

## Mathematics

## Academics

## Current Students

## Future Students

- Associate Professor Emeritus of Mathematics

Prisoner's dilemma, sequestering CO2

cohomology operations

Ph.D., University of California at Berkeley 1965

M.A., University of California at Berkeley 1963

B.A., Oberlin College 1961

VIGRE awarded by National Science Foundation (Co-Principal Investigator). 2000 to 2003

Kraines, D, and 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)* 1193 (January 1, 2015): 219-231.

Kraines, DP, and Kraines, VY. "Natural selection of memory-one strategies for the iterated prisoner's dilemma." *J Theor Biol* 203.4 (April 21, 2000): 335-355.
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Kraines, D, and Kraines, V. "Evolution of Learning among Pavlov Strategies in a Competitive Environment with Noise." *Journal of Conflict Resolution* 39.3 (September 1995): 439-466.
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Kraines, D, and Kraines, V. "Learning to cooperate with Pavlov an adaptive strategy for the iterated Prisoner's Dilemma with noise." *Theory and Decision* 35.2 (1993): 107-150.
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Kraines, D, and Kraines, V. "Pavlov and the prisoner's dilemma." *Theory and Decision* 26.1 (1989): 47-79.
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Geist, R, Kraines, D, and Fink, P. "NATURAL LANGUAGE COMPUTING IN A LINEAR ALGEBRA COURSE." (1982): 203-208.

Kraines, D. "The Kernel of the loop suspension map." *Illinois Journal of Mathematics* 21.1 (January 1, 1977): 91-108.

Kraines, D. "The A(p) cohomology of some k stage Postnikov systems." *Commentarii Mathematici Helvetici* 48.1 (1973): 56-71.
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Kraines, D, and Schochet, C. "Differentials in the Eilenberg-Moore spectral sequence." *Journal of Pure and Applied Algebra* 2.2 (1972): 131-148.
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Kraines, D. "On Excess in the Milnor Basis." *Bulletin of the London Mathematical Society* 3.3 (November 1971): 363-365.
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On a cool, cloudy Saturday in early November, 45 teams of high school students from Virginia, the Carolinas, and even three from China came to the Duke Campus to participate in the annual Duke Math Meet. After a light breakfast, each team was led to... read more »

Students and faculty enjoyed a break before exams to share food and conversation at the annual math party on April 24, 2017. Those who competed in the MCM/ICM competition received Duke Math shirts and certificates. Officers of DUMU and research... read more »

Vinit Ranjan, Kelly Zhang, and Junmo Ryang are a group of prospective math majors who all met in high school at the North Carolina School of Science and Mathematics, where they first competed in MCM, but on separate teams. This year, they came... read more »

Steven J. Miller, Associate Professor of mathematics at Williams College, delivered the first Duke University Math Union (DUMU) guest lecture of the 2016-2017 year to an audience including nearly 50 undergraduates on Wednesday, September 7, 2016..... read more »

The Duke University Math Union (DUMU) is devoted to organizing and providing resources to students interested in mathematics. Throughout the year, we invite guest lecturers, organize social events, and host a high school contest in mathematics.... read more »

120 Science Drive

117 Physics Building

Campus Box 90320

Durham, NC 27708-0320

phone: 919.660.2800

fax: 919.660.2821

dept@math.duke.edu