Seshadri constants, Diophantine approximation, and Roth's Theorem for arbitrary varieties

In this paper, we associate an invariant $\alpha_{x}(L)$ to an algebraic point $x$ on an algebraic variety $X$ with an ample line bundle $L$. The invariant $\alpha$ measures how well $x$ can be approximated by rational points on $X$, with respect to the height function associated to $L$. We show that this invariant is closely related to the Seshadri constant $\epsilon_{x}(L)$ measuring local positivity of $L$ at $x$, and in particular that Roth's theorem on $\mathbf{P}^1$ generalizes as an inequality between these two invariants valid for arbitrary projective varieties.Comment: 55 pages, published versio

E-Choice option for Elsevier's Science/Direct

If there's one thing that seems to be as reliable as the rising sun, it's that each year brings a new ScienceDirect pricing scheme from Elsevier. This might be seen in the positive context of flexibility and a willingness to adapt and/or learn. With E-Choice Elsevier has formally adopted the perspective of those institutions for which quality trumps quantity

The exchange-stable marriage problem

In this paper we consider instances of stable matching problems, namely the classical stable marriage (SM) and stable roommates (SR) problems and their variants. In such instances we consider a stability criterion that has recently been proposed, that of <i>exchange-stability</i>. In particular, we prove that ESM — the problem of deciding, given an SM instance, whether an exchange-stable matching exists — is NP-complete. This result is in marked contrast with Gale and Shapley's classical linear-time algorithm for finding a stable matching in an instance of SM. We also extend the result for ESM to the SR case. Finally, we study some variants of ESM under weaker forms of exchange-stability, presenting both polynomial-time solvability and NP-completeness results for the corresponding existence questions

Size versus stability in the marriage problem

Given an instance I of the classical Stable Marriage problem with Incomplete preference lists (smi), a maximum cardinality matching can be larger than a stable matching. In many large-scale applications of smi, we seek to match as many agents as possible. This motivates the problem of finding a maximum cardinality matching in I that admits the smallest number of blocking pairs (so is “as stable as possible”). We show that this problem is NP-hard and not approximable within n1−ε, for any ε>0, unless P=NP, where n is the number of men in I. Further, even if all preference lists are of length at most 3, we show that the problem remains NP-hard and not approximable within δ, for some δ>1. By contrast, we give a polynomial-time algorithm for the case where the preference lists of one sex are of length at most 2. We also extend these results to the cases where (i) preference lists may include ties, and (ii) we seek to minimize the number of agents involved in a blocking pair

Two algorithms for the student-project allocation problem

We study the Student-Project Allocation problem (SPA), a generalisation of the classical Hospitals / Residents problem (HR). An instance of SPA involves a set of students, projects and lecturers. Each project is offered by a unique lecturer, and both projects and lecturers have capacity constraints. Students have preferences over projects, whilst lecturers have preferences over students. We present two optimal linear-time algorithms for allocating students to projects, subject to the preference and capacity constraints. In particular, each algorithm finds a stable matching of students to projects. Here, the concept of stability generalises the stability definition in the HR context. The stable matching produced by the first algorithm is simultaneously best-possible for all students, whilst the one produced by the second algorithm is simultaneously best-possible for all lecturers. We also prove some structural results concerning the set of stable matchings in a given instance of SPA. The SPA problem model that we consider is very general and has applications to a range of different contexts besides student-project allocation

A Computational Algorithm for Optimal Collusion in Dynamic Cournot Oligopoly

Recent work in oligopoly theory has focused on the optimal degree of tacit collusion sustainable in a dynamic oligopoly. Such analysis for price-setting oligopoly has advanced substantially; however, analysis of Cournot oligopoly has not proceeded apace. this paper analyzes best and worst strongly symmetric equilibria for dynamic quantity-setting oligopoly with demand dynamics, such as business cycles, that are independent of previous quantity choices. Two primary results are obtained. First, the paper shows through construction that best and worst strongly symmetric equilibria exist, and that the worst equilibria may be constructed, as in Abreu (1983, 1986), so that they prescribe optimal equilibrium play from periods 2 onward. That is, they have a stick-and-carrot structure. Second, it provides a simple algorithm for the computation of the extremal equilibria.Center for Research on Economic and Social Theory, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100932/1/ECON379.pd

A Theory of Partnership Dynamics: Learning, Specific Investment, and Dissolution

This paper explores the benefits and drawbacks of potential partnership dissolution through an infinite-period, dynamic game-theoretic model of learning and endogenous dissolution. As partners learn about the quality of their partnership relative to their outside opportunities, the rents associated with the partnership change, effecting a related change in the strangth of incentives to provide effort. The paper develops an incentive-constrained dynamic programming algorithm for the computation of optimal symmetric equilibria of dynamic games with known worst punishments (such as dissolution here). The scheme is much simpler than the more general set-valued approach pioneered by Abreu, Pearce, and Stacchetti in that it only requires the computation of one value function at each iteration. The algorithm is then used to show that rather mild supermodularity conditions lead to effort levels in the optimal equilibria which rise in the expected quality of the partnership.Center for Research on Economic and Social Theory, Department of Economics, University of Michiganhttp://deepblue.lib.umich.edu/bitstream/2027.42/100933/1/ECON380.pd