David P. Kraines

MASSEY HIGHER PRODUCTS

In this paper, we shall investigate some properties of a class of higher order cohomology operations of several variables. These operations, the higher products, were defined by Massey as a generalization of his triple product. There is a correspondence between the higher products and iterated Whitehead products in homotopy groups [5], [11]. It has been noted that the differentials in certain spectral sequences involving the Ext and Tor functors are related to Massey higher products [6], [9]. In particular, the differentials in a spectral sequence relating the cohomology of a space with that of its space of loops are generalized higher products. In the first part we establish a number of properties of these higher products. These properties indicate the similarities between the higher products and the cup product. In the final section, the higher products are specialized to an operation of one variable. In certain cases, this operation may be evaluated in terms of primary Steenrod operations (Theorems 14, 19). These results are useful in the computation of the higher product structure of a space with coefficients in a field.

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.

Rotations in Space and Orthogonal Matrices

One application of elementary linear algebra that never fails to surprise my students is that a composition of rotations in space is itself a rotation. The more general result that any 3 by 3 orthogonal matrix of determinant one (a proper orthogonal transformation) is a rotation can be demonstrated most convincingly if one has a computer package which will do the messy arithmetic involved. In this lesson we find the axis and angle of rotation of such a transformation. It brings together such different concepts from linear algebra as orthogonal bases and matrices, the geometric interpretation of eigenvalues and eigenvectors (both real and complex), and the geometric meaning of changing bases. We presume that the students have seen the 2 by 2 matrix of a rotation in the plane and that they know a little about orthogonal matrices and eigenvalues and eigenvectors.

Classroom Computer Capsules: Binary Operations

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

Rational cohomology operations and Massey products

Let Q be the group of rational numbers. Then H*(Q, n; Q) is either an exterior algebra or polynomial algebra on a class u of dimension n. By the Kunneth formula, if PX>=K(Q, nj), that is if P is a rational generalized Eilenberg-MacLane space (GEM), then every class in Hk(P; Q) is a polynomial on the fundamental classes {ui}. Thus every rational primary cohomology operation on (xi, ** *, x) can be written d) {Xj I} = ZXjxj+ Eyrzr when XjE Q and yr, zrEI Q). The decomposable term is the twofold matrix Massey product