相关论文: Product-Mix Auctions and Tropical Geometry

It is well known that the cup-product pairing on the complementary integral cohomology groups (modulo torsion) of a compact oriented manifold is unimodular. We prove a similar result for the $\ell$-adic cohomology groups of smooth algebraic…

The optimal pricing problem is a fundamental problem that arises in combinatorial auctions. Suppose that there is one seller who has indivisible items and multiple buyers who want to purchase a combination of the items. The seller wants to…

We investigate the realizability of balanced functions on tropical curves, establishing new sufficient criteria for superabundant functions on genus two curves, analogous to the well-spacedness condition in genus one. We find that…

We study the equilibria of uniform price auctions where many asymmetric bidders have flat demands up to their respective quantity constraints. We present an iterative procedure that systematically finds an equilibrium outcome as well as an…

This is a survey on tropical polytopes from the combinatorial point of view and with a focus on algorithms. Tropical convexity is interesting because it relates a number of combinatorial concepts including ordinary convexity, monomial…

This paper unifies two foundational constructs from economics and algorithmic game theory, the Arctic Auction and the linear Fisher market, to address the efficient allocation of differentiated goods in complex markets. Our main…

We show an auction-based algorithm to compute market equilibrium prices in a production model, where consumers purchase items under separable nonlinear utility concave functions which satisfy W.G.S(Weak Gross Substitutes); producers produce…

We consider optimization problems that are formulated and solved in the framework of tropical mathematics. The problems consist in minimizing or maximizing functionals defined on vectors of finite-dimensional semimodules over idempotent…

Tropical mathematics is used to establish a correspondence between certain microscopic and macroscopic objects in statistical models. Tropical algebra gives a common framework for macrosystems (subsets) and their elementary constituents…

We establish new results of first-order necessary conditions of optimality for finite-dimensional problems with inequality constraints and for problems with equality and inequality constraints, in the form of John's theorem and in the form…

We examine a multidimensional optimisation problem in the tropical mathematics setting. The problem involves the minimisation of a nonlinear function defined on a finite-dimensional semimodule over an idempotent semifield subject to linear…

Using tropical geometry one can translate problems in enumerative geometry to combinatorial problems. Thus tropical geometry is a powerful tool in enumerative geometry over the complex and real numbers. Results from $\mathbb{A}^1$-homotopy…

Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…

We introduce tropical dual numbers as an extension of tropical semiring. By this innovation, one can work with honest ideals, instead of congruences, and recover the Euclidean topology on affine tropical spaces similar to Zariski's approach…

We prove that the classic problem of finding a competitive equilibrium in an exchange economy with indivisible goods, money, and unit-demand agents is PPAD-complete. In this "housing market", agents have preferences over the house and…

Elements of the tropical vertex group, introduced by Kontsevich and Soibelman, are formal families of symplectomorphisms of the 2-dimensional algebraic torus. We prove ordered product factorizations in the tropical vertex group are…

We prove a tropical mirror symmetry theorem for descendant Gromov-Witten invariants of the elliptic curve, generalizing the tropical mirror symmetry theorem for Hurwitz numbers of the elliptic curve, Theorem 2.20 in [B\"ohm J., Bringmann…

This paper studies Markov perfect equilibria in a repeated duopoly model where sellers choose algorithms. An algorithm is a mapping from the competitor's price to own price. Once set, algorithms respond quickly. Customers arrive randomly…

Elements of the tropical vertex group are formal families of symplectomorphisms of the 2-dimensional algebraic torus. Commutators in the group are related to Euler characteristics of the moduli spaces of quiver representations and the…

We introduce a new technique to bound the fluctuations exhibited by a physical system, based on the Euclidean geometry of the space of observables. Through a simple unifying argument, we derive a sweeping generalization of so-called…