Trust and Financial Trades: Lessons from an Investment Game Where Reciprocators Can Hide Behind Probabilities

In this paper we show that if a very small, exogenously given probability of terminating the exchange is introduced in an elementary investment game, reciprocators play more often the defection strategy. Everything happens as if they "hide behind probabilities" in order to break the trust relationship. Investors do no not seem able to internalize the reciprocators' change in behavior. This could explain why trades involving an exogenous risk of value destruction, such as financial transactions, provide an unfavorable environment for trust-buildingExperimental Economics; Financial Transactions; Investment Game; Objective Risk; Trust

Sparse Automatic Differentiation for Large-Scale Computations Using Abstract Elementary Algebra

Most numerical solvers and libraries nowadays are implemented to use mathematical models created with language-specific built-in data types (e.g. real in Fortran or double in C) and their respective elementary algebra implementations. However, built-in elementary algebra typically has limited functionality and often restricts flexibility of mathematical models and analysis types that can be applied to those models. To overcome this limitation, a number of domain-specific languages with more feature-rich built-in data types have been proposed. In this paper, we argue that if numerical libraries and solvers are designed to use abstract elementary algebra rather than language-specific built-in algebra, modern mainstream languages can be as effective as any domain-specific language. We illustrate our ideas using the example of sparse Jacobian matrix computation. We implement an automatic differentiation method that takes advantage of sparse system structures and is straightforward to parallelize in MPI setting. Furthermore, we show that the computational cost scales linearly with the size of the system.Comment: Submitted to ACM Transactions on Mathematical Softwar

Symmetric Functions in Noncommuting Variables

Consider the algebra Q> of formal power series in countably many noncommuting variables over the rationals. The subalgebra Pi(x_1,x_2,...) of symmetric functions in noncommuting variables consists of all elements invariant under permutation of the variables and of bounded degree. We develop a theory of such functions analogous to the ordinary theory of symmetric functions. In particular, we define analogs of the monomial, power sum, elementary, complete homogeneous, and Schur symmetric functions as will as investigating their properties.Comment: 16 pages, Latex, see related papers at http://www.math.msu.edu/~sagan, to appear in Transactions of the American Mathematical Societ

Quadratic Maps and Bockstein Closed Group Extensions

We study central extensions E of elementary abelian 2-groups by elementary abelian 2-groups. Associated to such an extension is a quadratic map which determines the extension uniquely. The components of the map determine a quadratic ideal in a polynomial algebra and we say that the extension is Bockstein closed if this ideal is invariant under the Bockstein operator. We find a direct condition on the quadratic map that characterizes when the extension is Bockstein closed. Using this we show for example that quadratic maps induced from the fundamental quadratic map on gl_n, Q(A)=A+A^2 yield Bockstein closed extensions. We also show that this condition is equivalent to a certain liftability of the extension and under certain conditions, to the fact that Q corresponds to a 2-power map of a restricted 2-Lie algebra. For the class of 2-groups coming from these type of extensions, under a 2-power exact condition we also compute the mod 2 cohomology ring of the corresponding groups - this includes the upper triangular congruence subgroups among other examples.Comment: To appear, Transactions of the A.M.

Strong Converse Theorems for Classes of Multimessage Multicast Networks: A R\'enyi Divergence Approach

This paper establishes that the strong converse holds for some classes of discrete memoryless multimessage multicast networks (DM-MMNs) whose corresponding cut-set bounds are tight, i.e., coincide with the set of achievable rate tuples. The strong converse for these classes of DM-MMNs implies that all sequences of codes with rate tuples belonging to the exterior of the cut-set bound have average error probabilities that necessarily tend to one (and are not simply bounded away from zero). Examples in the classes of DM-MMNs include wireless erasure networks, DM-MMNs consisting of independent discrete memoryless channels (DMCs) as well as single-destination DM-MMNs consisting of independent DMCs with destination feedback. Our elementary proof technique leverages properties of the R\'enyi divergence.Comment: Submitted to IEEE Transactions on Information Theory, Jul 18, 2014. Revised on Jul 31, 201