Mechanism Design with Strategic Mediators

We consider the problem of designing mechanisms that interact with strategic agents through strategic intermediaries (or mediators), and investigate the cost to society due to the mediators' strategic behavior. Selfish agents with private information are each associated with exactly one strategic mediator, and can interact with the mechanism exclusively through that mediator. Each mediator aims to optimize the combined utility of his agents, while the mechanism aims to optimize the combined utility of all agents. We focus on the problem of facility location on a metric induced by a publicly known tree. With non-strategic mediators, there is a dominant strategy mechanism that is optimal. We show that when both agents and mediators act strategically, there is no dominant strategy mechanism that achieves any approximation. We, thus, slightly relax the incentive constraints, and define the notion of a two-sided incentive compatible mechanism. We show that the $3$-competitive deterministic mechanism suggested by Procaccia and Tennenholtz (2013) and Dekel et al. (2010) for lines extends naturally to trees, and is still $3$-competitive as well as two-sided incentive compatible. This is essentially the best possible. We then show that by allowing randomization one can construct a $2$-competitive randomized mechanism that is two-sided incentive compatible, and this is also essentially tight. This result also closes a gap left in the work of Procaccia and Tennenholtz (2013) and Lu et al. (2009) for the simpler problem of designing strategy-proof mechanisms for weighted agents with no mediators on a line, while extending to the more general model of trees. We also investigate a further generalization of the above setting where there are multiple levels of mediators.Comment: 46 pages, 1 figure, an extended abstract of this work appeared in ITCS 201

The Jones polynomials of 3-bridge knots via Chebyshev knots and billiard table diagrams

This work presents formulas for the Kauffman bracket and Jones polynomials of 3-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of 2-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.Comment: 20 pages, 4 figures, 2 table

Dipole Symmetry Near Threshold

In celebrating Iachello's 60th birthday we underline many seminal contributions for the study of the degrees of freddom relevant for the structure of nuclei and other hadrons. A dipole degree of freedom, well described by the spectrum generating algebra U(4) and the Vibron Model, is a most natural concept in molecular physics. It has been suggested by Iachello with much debate, to be most important for understanding the low lying structure of nuclei and other hadrons. After its first observation in $^{18}O$ it was also shown to be relevant for the structure of heavy nuclei (e.g. $^{218}Ra$). Much like the Ar-benzene molecule, it is shown that molecular configurations are important near threshold as exhibited by states with a large halo and strong electric dipole transitions. The cluster-molecular Sum Rule derived by Alhassid, Gai and Bertsch (AGB) is shown to be a very useful model independent tool for examining such dipole molecular structure near thereshold. Accordingly, the dipole strength observed in the halo nuclei such as $^6He, ^{11}Li, ^{11}Be, ^{17}O$, as well as the N=82 isotones is concentrated around threshold and it exhausts a large fraction (close to 100%) of the AGB sum rule, but a small fraction (a few percent) of the TRK sum rule. This is suggested as an evidence for a new soft dipole Vibron like oscillations in nuclei.Comment: Presented at Iachello's Fest, Symmetry in Physics, Erice, March 23-30, 2003. Supported by USDOE Grant No. DE-FG02-94ER4087

On mutual information, likelihood-ratios and estimation error for the additive Gaussian channel

This paper considers the model of an arbitrary distributed signal x observed through an added independent white Gaussian noise w, y=x+w. New relations between the minimal mean square error of the non-causal estimator and the likelihood ratio between y and \omega are derived. This is followed by an extended version of a recently derived relation between the mutual information I(x;y) and the minimal mean square error. These results are applied to derive infinite dimensional versions of the Fisher information and the de Bruijn identity. The derivation of the results is based on the Malliavin calculus.Comment: 21 pages, to appear in the IEEE Transactions on Information Theor