Strong Duality for a Multiple-Good Monopolist

We characterize optimal mechanisms for the multiple-good monopoly problem and provide a framework to find them. We show that a mechanism is optimal if and only if a measure $\mu$ derived from the buyer's type distribution satisfies certain stochastic dominance conditions. This measure expresses the marginal change in the seller's revenue under marginal changes in the rent paid to subsets of buyer types. As a corollary, we characterize the optimality of grand-bundling mechanisms, strengthening several results in the literature, where only sufficient optimality conditions have been derived. As an application, we show that the optimal mechanism for $n$ independent uniform items each supported on $[c,c+1]$ is a grand-bundling mechanism, as long as $c$ is sufficiently large, extending Pavlov's result for $2$ items [Pavlov'11]. At the same time, our characterization also implies that, for all $c$ and for all sufficiently large $n$, the optimal mechanism for $n$ independent uniform items supported on $[c,c+1]$ is not a grand bundling mechanism

Minimum product set sizes in nonabelian groups of order pq

Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s , respectively. In this paper, we completely determine μG(r,s)μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1⩽r,s⩽pq1⩽r,s⩽pq such that μG(r,s)>μZ/pqZ(r,s)μG(r,s)>μ[subscript Z over pqZ(r,s)].National Science Foundation (U.S.) (Grant DMS-0447070-001)United States. National Security Agency (Grant H98230-06-1-0013

Mechanism Design via Optimal Transport

Optimal mechanisms have been provided in quite general multi-item settings [Cai et al. 2012b, as long as each bidder's type distribution is given explicitly by listing every type in the support along with its associated probability. In the implicit setting, e.g. when the bidders have additive valuations with independent and/or continuous values for the items, these results do not apply, and it was recently shown that exact revenue optimization is intractable, even when there is only one bidder [Daskalakis et al. 2013]. Even for item distributions with special structure, optimal mechanisms have been surprisingly rare [Manelli and Vincent 2006] and the problem is challenging even in the two-item case [Hart and Nisan 2012]. In this paper, we provide a framework for designing optimal mechanisms using optimal transport theory and duality theory. We instantiate our framework to obtain conditions under which only pricing the grand bundle is optimal in multi-item settings (complementing the work of [Manelli and Vincent 2006]), as well as to characterize optimal two-item mechanisms. We use our results to derive closed-form descriptions of the optimal mechanism in several two-item settings, exhibiting also a setting where a continuum of lotteries is necessary for revenue optimization but a closed-form representation of the mechanism can still be found efficiently using our framework.Alfred P. Sloan Foundation (Fellowship)Microsoft Research (Faculty Fellowship)National Science Foundation (U.S.) (CAREER Award CCF-0953960)National Science Foundation (U.S.) (Award CCF-1101491)Hertz Foundation (Daniel Stroock Fellowship

The Complexity of Optimal Mechanism Design

Myerson's seminal work provides a computationally efficient revenue-optimal auction for selling one item to multiple bidders [18]. Generalizing this work to selling multiple items at once has been a central question in economics and algorithmic game theory, but its complexity has remained poorly understood. We answer this question by showing that a revenue-optimal auction in multi-item settings cannot be found and implemented computationally efficiently, unless zpp ⊇ P[superscript #P]. This is true even for a single additive bidder whose values for the items are independently distributed on two rational numbers with rational probabilities. Our result is very general: we show that it is hard to compute any encoding of an optimal auction of any format (direct or indirect, truthful or non-truthful) that can be implemented in expected polynomial time. In particular, under well-believed complexity-theoretic assumptions, revenue-optimization in very simple multi-item settings can only be tractably approximated. We note that our hardness result applies to randomized mechanisms in a very simple setting, and is not an artifact of introducing combinatorial structure to the problem by allowing correlation among item values, introducing combinatorial valuations, or requiring the mechanism to be deterministic (whose structure is readily combinatorial). Our proof is enabled by a flow-interpretation of the solutions of an exponential-size linear program for revenue maximization with an additional supermodularity constraint.Alfred P. Sloan Foundation (Fellowship)Microsoft Research (Faculty Fellowship)National Science Foundation (U.S.) (CAREER Award CCF-0953960)National Science Foundation (U.S.) (Award CCF-1101491)Hertz Foundation (Daniel Stroock Fellowship

Educating and training a workforce for nutrition in a post-2015 world.

Nearly all countries in the world today are burdened with malnutrition, manifesting as undernutrition, micronutrient deficiencies, and/or overweight and obesity. Despite some progress, efforts to alleviate malnutrition are hampered by a shortage in number, skills, and geographic coverage, of a workforce for nutrition. Here, we report the findings of the Castel Gandolfo workshop, a convening of experts from diverse fields in March 2014 to consider how to develop the capacity of a global cadre of nutrition professionals for the post-2015 development era. Workshop participants identified several requirements for developing a workforce for nutrition, including an ability to work as part of a multisectoral team; communication, advocacy, and leadership skills to engage decision makers; and a set of technical skills to address future challenges for nutrition. Other opportunities were highlighted that could immediately contribute to capacity development, including the creation of a consortium to link global North and South universities, online training modules for middle managers, and practical, hands-on experiences for frontline nutrition workers. Institutional and organizational support is needed to enable workshop recommendations on education and training to be effectively implemented and sustained. The findings from the Castel Gandolfo workshop can contribute to the delivery of successful nutrition-relevant actions in the face of mounting external pressures and informing and attaining the forthcoming Sustainable Development Goals

Educating and Training a Workforce for Nutrition in a Post-2015 World

Nearly all countries in the world today are burdened with malnutrition, manifesting as undernutrition, micronutrient deficiencies, and/or overweight and obesity. Despite some progress, efforts to alleviate malnutrition are hampered by a shortage in number, skills, and geographic coverage, of a workforce for nutrition. Here, we report the findings of the Castel Gandolfo workshop, a convening of experts from diverse fields in March 2014 to consider how to develop the capacity of a global cadre of nutrition professionals for the post-2015 development era. Workshop participants identified several requirements for developing a workforce for nutrition, including an ability to work as part of a multisectoral team; communication, advocacy, and leadership skills to engage decision makers; and a set of technical skills to address future challenges for nutrition. Other opportunities were highlighted that could immediately contribute to capacity development, including the creation of a consortium to link global North and South universities, online training modules for middle managers, and practical, hands-on experiences for frontline nutrition workers. Institutional and organizational support is needed to enable workshop recommendations on education and training to be effectively implemented and sustained. The findings from the Castel Gandolfo workshop can contribute to the delivery of successful nutrition-relevant actions in the face of mounting external pressures and informing and attaining the forthcoming Sustainable Development Goals. Adv Nutr 2015;6:639–47

Quantum Correlated Equilibria in Classical Complete Information

Since the work of Aumann [1], the concept of correlated equilibrium (CE) has played an important role in the study of games. Correlated equilibria always exist, and unlike Nash equilibria, which are believed to be computational intractable (see [5]), a correlated equilibrium can be computed efficiently in a broad class of succinctly-representable games [19]. In a correlated equilibrium,