On the degree and half degree principle for symmetric polynomials

In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte. It says that a symmetric real polynomial $F$ of degree $d$ in $n$ variables is positive on $\R^n$ (on $\R^{n}_{\geq 0}$) if and only if it is so on the subset of points with at most $\max\{\lfloor d/2\rfloor,2\}$ distinct components. We deduce Timofte's original statement as a corollary of a slightly more general statement on symmetric optimization problems. The idea we are using to prove this statement is to relate it to a linear optimization problem in the orbit space. The fact that for the case of the symmetric group $S_n$ this can be viewed as a question on normalized univariate real polynomials with only real roots allows us to conclude the theorems in a very elementary way. We hope that the methods presented here will make it possible to derive similar statements also in the case of other groups.Comment: (v2) revision based on suggestions by refere

Symmetric nonnegative forms and sums of squares

We study symmetric nonnegative forms and their relationship with symmetric sums of squares. For a fixed number of variables $n$ and degree $2d$, symmetric nonnegative forms and symmetric sums of squares form closed, convex cones in the vector space of $n$-variate symmetric forms of degree $2d$. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate the asymptotic behavior when the degree $2d$ is fixed and the number of variables $n$ grows. Here, we show that, in sharp contrast to the general case, the difference between symmetric nonnegative forms and sums of squares does not grow arbitrarily large for any fixed degree $2d$. We consider the case of symmetric quartic forms in more detail and give a complete characterization of quartic symmetric sums of squares. Furthermore, we show that in degree $4$ the cones of nonnegative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows. We conjecture that this is true in arbitrary degree $2d$.Comment: (v4) minor revision and small reorganizatio

A Note on Extrema of Linear Combinations of Elementary Symmetric Functions

This note provides a new approach to a result of Foregger and related earlier results by Keilson and Eberlein. Using quite different techniques, we prove a more general result from which the others follow easily. Finally, we argue that the proof given by Foregger is flawed.Comment: (v2) revision based on suggestions by refere

Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic subsets of $\mathrm{R}^k$, which are defined by symmetric polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing the generalized Euler-Poincar\'e characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in $\mathrm{D}$, are polynomially bounded in terms of $k$ and the number of polynomials in the input, assuming that the degrees of the input polynomials are bounded by a constant. This is in contrast to the best complexity of the known algorithms for the same problems in the non-symmetric situation, which are singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result ($\#\mathbf{P}$-hardness) coming from discrete complexity theory.Comment: 29 pages, 1 Figure. arXiv admin note: substantial text overlap with arXiv:1312.658

Group Identity and Discrimination in Small Markets: Asymmetry of In-Group Favors

We experimentally study the inuence of induced group identity on the determination of prices and beliefs in a small market game. We create group identity through a focal point coordination game. Subjects play a three-person bargaining game where one seller can sell an indivisible good to one of two competing buyers under four different treatments varying the buyer-seller constellation. We find evidence of in group favoritism on the buyer side. However we do not detect a lower ask prices for in-group sellers for in-group buyers, indicating that in-group favoritism is in favor of the more powerful market participant.Group identity, Experiments, Markets, Bargaining

Path-Based Program Repair

We propose a path-based approach to program repair for imperative programs. Our repair framework takes as input a faulty program, a logic specification that is refuted, and a hint where the fault may be located. An iterative abstraction refinement loop is then used to repair the program: in each iteration, the faulty program part is re-synthesized considering a symbolic counterexample, where the control-flow is kept concrete but the data-flow is symbolic. The appeal of the idea is two-fold: 1) the approach lazily considers candidate repairs and 2) the repairs are directly derived from the logic specification. In contrast to prior work, our approach is complete for programs with finitely many control-flow paths, i.e., the program is repaired if and only if it can be repaired at the specified fault location. Initial results for small programs indicate that the approach is useful for debugging programs in practice.Comment: In Proceedings FESCA 2015, arXiv:1503.0437

Ambiguity aversion as a reason to choose tournaments

We test the implications of ambiguity aversion in a principal-agent problem with multiple agents. When output distributions are uncertain, models of ambiguity aversion suggest that tournaments may become more attractive than independent wage contracts, in contrast to the case where output distributions are known. We do so by presenting agents with a choice between tournaments and independent contracts, which are designed in a way that under uncertainty about output distribution (that is, under ambiguity), ambiguity averse agents should typically prefer tournaments, while ambiguity neutral agents prefer independent contracts, independent of their degree of risk aversion. This is the case, because the tournament removes all ambiguity about the equilibrium wages. We compare the share of participants who choose the tournament under ambiguity with the share of participants choosing the tournament in a control treatment, where output distributions are know. As the theory predicts, we find indeed that under ambiguity the share of agents who choose the tournaments is higher than in the case of known output distributions.Ambiguity aversion, tournaments, Ellsberg urn, contract design