Ex Post Implementation

We analyze the problem of fully implementing a social choice set in ex post equilibrium. Weidentify an ex post monotonicity condition that is necessary and -- in economic environments -- sufficient for full implementation in ex post equilibrium. We also identify an ex post monotonicityno veto condition that is sufficient. Ex post monotonicity is satisfied in all single crossing environments with strict ex post incentive constraints. In many economically significant environments, ex post implementation can be achieved in the direct mechanism. We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity (necessary and almost sufficient for complete information implementation). The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. Ex post implementation in the direct mechanism is also possible in this case. We describe an example where the Pareto correspondence fails ex post monotonicity but Maskin monotonicity is satisfied.Ex post equilibrium, Implementation, Single crossing, Interdependent values

Robust Implementation: The Case of Direct Mechanisms"

A social choice function is robustly implementable if there is a mechanism under which the process of iteratively eliminating strictly dominated messages leads to outcomes that agree with the social choice at every type profile. In an interdependent value environment with single crossing preferences, we identify a strict contraction property on the preferences which together with strict ex post incentive compatibility is sufficient to guarantee robust implementation in the direct mechanism. Strict EPIC and the contraction property are also necessary for robust implementation in any mechanism. The contraction property essentially requires that the interdependence is not too large. In a linear signal model, the contraction property is equivalent to an interdependence matrix having an eigenvalue less than one.Mechanism design, Implementation, Robustness, Common knowledge, Interim equilibrium, Iterative deletion, Direct mechanism

Robust Implementation in Direct Mechanisms

A social choice function is robustly implementable if there is a mechanism under which the process of iteratively eliminating strictly dominated messages leads to outcomes that agree with the social choice function for all beliefs at every type profile. In an interdependent value envi­ronment with single crossing preferences, we identify a contraction property on the preferences which together with strict ex post incentive compatibility is sufficient to guarantee robust imple­mentation in the direct mechanism. Strict ex post incentive compatibility and the contraction property are also necessary for robust implementation in any mechanism, including indirect ones. The contraction property requires that the interdependence is not too large. In a linear signal model, the contraction property is equivalent to an interdependence matrix having all eigenvalues smaller than one.Mechanism design, Implementation, Robustness, Common knowledge, Interim equilibrium, Iterative deletion, Direct mechanism

Probing many-body localization in a disordered quantum magnet

Quantum states cohere and interfere. Quantum systems composed of many atoms arranged imperfectly rarely display these properties. Here we demonstrate an exception in a disordered quantum magnet that divides itself into nearly isolated subsystems. We probe these coherent clusters of spins by driving the system beyond its linear response regime at a single frequency and measuring the resulting "hole" in the overall linear spectral response. The Fano shape of the hole encodes the incoherent lifetime as well as coherent mixing of the localized excitations. For the disordered Ising magnet, $\mathrm{LiHo_{0.045}Y_{0.955}F_4}$, the quality factor $Q$ for spectral holes can be as high as 100,000. We tune the dynamics of the quantum degrees of freedom by sweeping the Fano mixing parameter $q$ through zero via the amplitude of the ac pump as well as a static external transverse field. The zero-crossing of $q$ is associated with a dissipationless response at the drive frequency, implying that the off-diagonal matrix element for the two-level system also undergoes a zero-crossing. The identification of localized two-level systems in a dense and disordered dipolar-coupled spin system represents a solid state implementation of many-body localization, pushing the search forward for qubits emerging from strongly-interacting, disordered, many-body systems.Comment: 22 pages, 6 figure

Contingent Commodities and Implementation

In this note we consider the problem whether contingent commodity allocations can be used when the states are not directly contractible. In such a setting a contingent commodity allocation takes the form of a social choice function, and the question is whether this function is implementable (in the sense of full implementation). Using only very mild assumptions on the rule for selecting contingent commodity allocations, we derive a strong negative result which also proves to be robust with respect to different solution concepts employed for implementation. These findings have interesting implications for the interpretation of Arrow-Debreu economies.Contingent Commodities, Implementation, Single-Crossing