Incentive compatibility and pricing under moral hazard

We study a simple insurance economy with moral hazard, in which random contracts overcome the non-convexities generated by the incentive-compatibility constraints. The novelty is that we use linear programming and duality theory to study the relation between incentive compatibility and pricing. Using linear programming has the advantage that we can impose the incentive-compatibility constraints on the agents that are uninformed (the insurance firms). In contrast, most of the general equilibrium literature imposes them on the informed agents (the consumers). We derive the two welfare theorems, establish the existence of a competitive equilibrium, and characterize the equilibrium prices and allocations. Our competitive equilibrium has two key properties: (i) the equilibrium prices reflect all the relevant information, including the welfare costs arising from the incentive-compatibility constraints; (ii) the equilibrium allocations are the same as when the incentive-compatibility constraints are imposed on the consumers

PPP-Completeness with Connections to Cryptography

Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with profound connections to the complexity of the fundamental cryptographic primitives: collision-resistant hash functions and one-way permutations. In contrast to most of the other subclasses of TFNP, no complete problem is known for PPP. Our work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input, and thus we answer a longstanding open question from [Papadimitriou1994]. Specifically, we show that constrained-SIS (cSIS), a generalized version of the well-known Short Integer Solution problem (SIS) from lattice-based cryptography, is PPP-complete. In order to give intuition behind our reduction for constrained-SIS, we identify another PPP-complete problem with a circuit in the input but closely related to lattice problems. We call this problem BLICHFELDT and it is the computational problem associated with Blichfeldt's fundamental theorem in the theory of lattices. Building on the inherent connection of PPP with collision-resistant hash functions, we use our completeness result to construct the first natural hash function family that captures the hardness of all collision-resistant hash functions in a worst-case sense, i.e. it is natural and universal in the worst-case. The close resemblance of our hash function family with SIS, leads us to the first candidate collision-resistant hash function that is both natural and universal in an average-case sense. Finally, our results enrich our understanding of the connections between PPP, lattice problems and other concrete cryptographic assumptions, such as the discrete logarithm problem over general groups

An Energy-Minimization Finite-Element Approach for the Frank-Oseen Model of Nematic Liquid Crystals: Continuum and Discrete Analysis

This paper outlines an energy-minimization finite-element approach to the computational modeling of equilibrium configurations for nematic liquid crystals under free elastic effects. The method targets minimization of the system free energy based on the Frank-Oseen free-energy model. Solutions to the intermediate discretized free elastic linearizations are shown to exist generally and are unique under certain assumptions. This requires proving continuity, coercivity, and weak coercivity for the accompanying appropriate bilinear forms within a mixed finite-element framework. Error analysis demonstrates that the method constitutes a convergent scheme. Numerical experiments are performed for problems with a range of physical parameters as well as simple and patterned boundary conditions. The resulting algorithm accurately handles heterogeneous constant coefficients and effectively resolves configurations resulting from complicated boundary conditions relevant in ongoing research.Comment: 31 pages, 3 figures, 3 table

Stochastic Differential Games and Energy-Efficient Power Control

One of the contributions of this work is to formulate the problem of energy-efficient power control in multiple access channels (namely, channels which comprise several transmitters and one receiver) as a stochastic differential game. The players are the transmitters who adapt their power level to the quality of their time-varying link with the receiver, their battery level, and the strategy updates of the others. The proposed model not only allows one to take into account long-term strategic interactions but also long-term energy constraints. A simple sufficient condition for the existence of a Nash equilibrium in this game is provided and shown to be verified in a typical scenario. As the uniqueness and determination of equilibria are difficult issues in general, especially when the number of players goes large, we move to two special cases: the single player case which gives us some useful insights of practical interest and allows one to make connections with the case of large number of players. The latter case is treated with a mean-field game approach for which reasonable sufficient conditions for convergence and uniqueness are provided. Remarkably, this recent approach for large system analysis shows how scalability can be dealt with in large games and only relies on the individual state information assumption.Comment: The final publication is available at http://www.springerlink.com/openurl.asp?genre=article\&id=doi:10.1007/s13235-012-0068-