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

Community Colleges: School Community Relationships

The community college functions of community service and continuing education persistently tie the colleges\u27 goals and objectives to their surrounding communities. The community colleges have an opportunity to invest in their own future by embracing and nurturing their relationship with the community. This fostering of an enhanced school- community connection occurs when the colleges involve themselves in the educational, cultural, recreational, and social services of the community. The economic and business links to the community must be strengthened where they already exist and new programs promoted with an eye toward mutually beneficial endeavors

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

A Quantitative Comparative Analysis of EdD Persistence Factors

Whether studying physical sciences, social sciences, engineering, mathematics, humanities, or education, approximately one in every two doctoral students fail to persist to degree completion (Bowen & Rudenstine, 1992; Lovitts, 2001; Tinto, 2012). A quantitative comparative study focused on two populations; students currently enrolled in the professional doctorate EdD program and former EdD students, including students who started but did not finish the program. Research-based variables, characterized as personal and program factors driving doctoral student attrition, were tested for significance. The participation criteria defined at least 80% of the program’s course content in totality was or is currently delivered online from a university offering the professional EdD degree, including affiliation with the Carnegie Project on the Education Doctorate (Allen & Seaman, 2015; Rockinson-Szapkiw et al., 2019). About half of the survey respondents attended an EdD program affiliated with the Carnegie Project on the Education Doctorate (CPED). In contrast, the other half attended an EdD program with no affiliation with CPED. The Community of Inquiry for Online Learning comprised four elements, teaching presence, social presence, cognitive presence, and emotional presence, and was the study’s theoretical framework. A total of [n = 725] individuals responded to surveys, which yielded a sample size of [n = 475] usable responses from former and current EdD students. The data from 30 former students, who did not persist, was analyzed for comparative purposes. Survey respondents represented a diverse population of age, gender, ethnicity, and marital status, attending public, private, and for-profit colleges and universities from geographic locations throughout the United States. The independent variable for all but the last of 16 hypothesis tests were current and former EdD students. The dependent variables were the personal and program factors. Five hypothesis tests included the effect of a moderating or second independent variable to reveal differences between the primary independent and dependent variables. The last hypothesis test compared time-to-degree between former students who attended an EdD program affiliated with the CPED and students who attended an EdD program with no affiliation with CPED. Within the 16 statements of hypothesis were 32 sub-hypotheses tests, of which the results indicated 19 were significant

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

The Regulation of Leptin, Leptin Receptor and Pro-opiomelanocortin Expression by N-3 PUFAs in Diet-Induced Obese Mice Is Not Related to the Methylation of Their Promoters

<p>Abstract</p> <p>Background</p> <p>The expression of leptin is increased in obesity and inhibited by n-3 polyunsaturated fatty acids (n-3 PUFAs), but the underlying molecular mechanisms have not been firmly established.</p> <p>Methods</p> <p>In this study, we investigated the effects of dietary n-3 PUFAs on the methylation of CpG islands in the promoter regions of the leptin, leptin-R and POMC genes, as well as the effects of n-3 PUFA status in early life on the modification of the promoters of these three genes. Male C57 BL/6J mice were fed a high-fat diet with one of four different fat types: sunflower oil (n-3 PUFA deficient), soy oil, fish oil, or a mixture of soy and fish oil (soy:fish oil = 1:1). Two low-fat diets with sunflower oil or soy oil served as controls. Female mice were fed two breeding diets, sunflower oil or a mixture of soy and fish oil (soy:fish oil = 1:1), during pregnancy and lactation to breed new pups.</p> <p>Results</p> <p>Compared to mice fed the control diets, the expression of leptin in fat tissue and leptin-R and POMC in the hypothalamus was higher in the diet-induced obesity (DIO) mice, and the n-3 PUFAs in the diets reversed these elevated expression levels. The mean methylation levels of CpG sites in the promoter regions of the leptin and POMC genes showed no difference between the DIO and the control diet groups nor between the n-3 PUFA-containing and -deficient diet groups. For the CpG sites in the promoter regions of leptin-R, no methylation was found in any of the DIO or control groups. Feeding mice with the n-3 PUFA diet during pregnancy and lactation did not affect CpG methylation in the leptin or POMC promoters.</p> <p>Conclusions</p> <p>Our findings indicate that promoter DNA methylation may not be related to the expression of leptin, leptin-R or its related hypothalamic satiety regulator POMC.</p