Multilabeled versions of Sperner's and Fan's lemmas and applications

We propose a general technique related to the polytopal Sperner lemma for proving old and new multilabeled versions of Sperner's lemma. A notable application of this technique yields a cake-cutting theorem where the number of players and the number of pieces can be independently chosen. We also prove multilabeled versions of Fan's lemma, a combinatorial analogue of the Borsuk-Ulam theorem, and exhibit applications to fair division and graph coloring.Comment: 21 pages, 2 figure

Two-Person Fair Division of Indivisible Items: Compatible and Incompatible Properties

Suppose two players wish to divide a finite set of indivisible items, over which each distributes a specified number of points. Assuming the utility of a player’s bundle is the sum of the points it assigns to the items it contains, we analyze what divisions are fair. We show that if there is an envy-free (EF) allocation of the items, two other desirable properties—Pareto-optimality (PO) and maximinality (MM)—can also be satisfied, rendering these three properties compatible, but other properties—balance (BL), maximum Nash welfare (MNW), maximum total welfare (MTW), and lexicographic optimality (LO)—may fail. If there is no EF division, as is likely, it is always possible to satisfy EFx, a weaker form of EF, but an EFx allocation may not be PO, BL, MNW, MTW, or LO. Moreover, if one player considers an item worthless (i.e., assigns zero points to it), an EFx division may be Pareto dominated by a nonEFx allocation that is MNW. Although these incompatibilities suggest that there is no “perfect” 2-person fair division of indivisible items, EFx and MNW divisions—if they give different allocations when there is no EF-PO-MM division—seem the most compelling alternatives, with EFx, we conjecture, satisfying the Rawlsian objective of helping the worse-off player and MNW, a modification of MTW, suggesting a more Benthamite view

Central Florida Future, Volume 42 No. 2, January 11, 2010

Report suggests mandatory insurance; Decade of UCF; 10 years of the Future; Fuel and cost key in fleet selection; LEAD Scholar chosen to be Olympic torchbearer.https://stars.library.ucf.edu/centralfloridafuture/3274/thumbnail.jp

Central High School, The Warrior, Memphis, 13:08, 1931

The Warrior , Vol. 13, No. 8, for March 31, 1931, published by the students of Central High School in Memphis, Tennessee. Established as the successor to Memphis High School, the school opened in its current location on Bellevue Boulevard (formerly Raleigh Avenue) in 1911.https://digitalcommons.memphis.edu/speccoll-pub-shelby/1097/thumbnail.jp

The BG News October 4, 1972

The BGSU campus student newspaper October 4, 1972. Volume 57 - Issue 14https://scholarworks.bgsu.edu/bg-news/3752/thumbnail.jp

Täydennysosa väitöskirjaan "Tietokoneavusteinen oppiminen perustuen karttuviin sanastoihin, käsiteverkostoihin ja Wikipedian linkitykseen"

A supplement to Lauri Lahti’s doctoral dissertation in 2015 "Computer-Assisted Learning Based on Cumulative Vocabularies, Conceptual Networks and Wikipedia Linkage" so that this supplement was referenced to by the original publication.Täydennysosa väitöskirjaan "Tietokoneavusteinen oppiminen perustuen karttuviin sanastoihin, käsiteverkostoihin ja Wikipedian linkitykseen"Not reviewe

Investigation of Game-Theoretic Mechanisms for the Valuation of Energy Resources

Electricity systems are facing the pressure to change in response to the effects of new technology, particularly the proliferation of renewable technologies (such as solar PV systems and wind generation) leading to the retirement of traditional generation technologies that provide stabilising inertia. These changes create an imperative to consider potential future market structures to facilitate the participation of distributed energy resources (DERs; such as EVs and batteries) in grid operation. However, this gives rise to general questions surrounding the ethics of market structures and how they could be fairly applied in future electricity systems. Particularly the most basic question "how should electricity be valued and traded" is fundamentally a moral question without any easy answer. We give a survey of philosophical attitudes around such a question, before presenting a series of ways that these intuitions have been cast into mathematics, including: the Vickrey-Clarke-Groves mechanism, Locational Marginal Pricing, the Shapley Value, and Nash bargaining solution concepts. We compared these different methods, and attempted a new synthesis that brought together the best features of each of them; called the 'Generalised Neyman and Kohlberg Value' or the GNK-value for short. The GNK value was developed as a novel bargaining solution concept for many player non-cooperative transferable utility generalised games, and thus it was intrinsically flexible in its application to various aspects of powersystems. We demonstrated the features of the GNK-value against the other mathematical solutions in the context of trading the immediate consumption/generation of power on small sized networks under linear-DC approximation, before extending the computation to larger networks. The GNK value proved to be difficult to compute for large networks but was shown to be approximable for larger networks with a series of sampling techniques and a proxy method. The GNK value was ethically compared to other mechanisms with the unfortunate discovery that it allowed for participants to be left worse-off for participating, violating the ethical notion of 'euvoluntary exchange' and 'individual rationality'; but was offered as an interesting innovation in the space of transferable utility generalised games notwithstanding. For sampling the GNK value, there was a range of new and different techniques developed for stratified random sampling which iteratively minimise newly derived concentration inequalities on the error of the sampling. These techniques were developed to assist in the computation of the GNK value to larger networks, and they were evaluated in the context of sampling synthetic data, and in computation of the Shapley Value of cooperative game theory. These new sampling techniques were demonstrated to be comparable to the more orthodox Neyman sampling method despite not having access to stratum variances