A Simply Exponential Upper Bound on the Maximum Number of Stable Matchings

Stable matching is a classical combinatorial problem that has been the subject of intense theoretical and empirical study since its introduction in 1962 in a seminal paper by Gale and Shapley. In this paper, we provide a new upper bound on $f(n)$, the maximum number of stable matchings that a stable matching instance with $n$ men and $n$ women can have. It has been a long-standing open problem to understand the asymptotic behavior of $f(n)$ as $n\to\infty$, first posed by Donald Knuth in the 1970s. Until now the best lower bound was approximately $2.28^n$, and the best upper bound was $2^{n\log n- O(n)}$. In this paper, we show that for all $n$, $f(n) \leq c^n$ for some universal constant $c$. This matches the lower bound up to the base of the exponent. Our proof is based on a reduction to counting the number of downsets of a family of posets that we call "mixing". The latter might be of independent interest

The egalitarian sharing rule in provision of public projects

In this note we consider a society that partitions itself into disjoint jurisdictions, each choosing a location of its public project and a taxation scheme to finance it. The set of public project is multi-dimensional, and their costs could vary from jurisdiction to jurisdiction. We impose two principles, egalitarianism, that requires the equalization of the total cost for all agents in the same jurisdiction, and efficiency, that implies the minimization of the aggregate total cost within jurisdiction. We show that these two principles always yield a core-stable partition but a Nash stable partition may fail to exist.Comment: 7 page

Past and Present Patterns of Neutral and Adaptive Genetic Diversity in Wild Mandrills (Mandrillus Sphinx)

Although primates have fascinated researchers and the public alike for generations, one species that has remained enigmatic is the mandrill (Mandrillus sphinx), a large Cercopithecine monkey endemic to Central Africa. Mandrills are currently in decline due to bushmeat hunting, urbanization, and habitat loss. Neutral and adaptive genetic diversity are important tools for understanding evolutionary history and future viability, since diversity influences a species’ ability to adapt to a changing environment. However, thus far, minimal genetic information has been available for wild mandrills. Because of the dense vegetation in their tropical forest habitat, studying wild mandrills has proven to be a challenge, and the majority of research on this species has been performed on semi-captive populations. Here, I present findings from three studies using the first genetic data to be generated from a wild mandrill population, primarily using non-invasive sampling of feces. First, we use demographic history modeling to test for evidence of population bottlenecks in mandrills and three other forest-associated species (blue duiker [Philantomba monticola], Peters’s duiker [Cephalophus callipygus], and western lowland gorillas [Gorilla gorilla gorilla]). Despite a severe loss of forest cover in central Africa approximately 2,500 years ago, our results suggest that none of the four species experienced major population declines. Second, we perform next-generation sequencing of the class II major histocompatibility complex (MHC), a gene family involved in adaptive immunity. We test for a difference in replicability between sequences generated from DNA extracted from feces and that extracted from higher quality tissue. We also present a new method of assigning MHC alleles to individuals using degraded samples, and we use that method to characterize the MHC genes in the focal population. Lastly, we test for sex-specific selection on the MHC in male and female mandrills. Male mandrills are thought to be more vulnerable to pathogens than females, which may result in stronger selective pressure on the MHC genes in males. Results from these three studies will contribute to our understanding of mandrill evolutionary history and conservation by providing insight on the role of demographic processes and selective pressures in shaping their past and present populations

The Egalitarian Sharing Rule in Provision of Public Projects

In this note we consider a society that partitions itself into disjoint jurisdictions, each choosing a location of its public project and a taxation scheme to finance it. The set of public project is multi-dimensional, and their costs could vary from jurisdiction to jurisdiction. We impose two principles, egalitarianism, that requires the equalization of the total cost for all agents in the same jurisdiction, and efficiency, that implies the minimization of the aggregate total cost within jurisdiction. We show that these two principles always yield a core-stable partition but a Nash stable partition may fail to exist.Jurisdictions, Stable partitions, Public projects, Egalitarianism

Heterogeneity gap in stable juridiction structures

Stability, Jurisdictions, Public Projects, Heterogeneity Gap