The blocking lemma and group incentive compatibility for matching with contracts

This paper considers a general class of two-sided many-to-one matching markets, so-called matching markets with contracts. We study the blocking lemma and group incentive compatibility for this class of matching markets. We first show that the blocking lemma for matching with contracts holds if hospitals’ choice functions satisfy substitutes and the law of aggregate demand. The blocking lemma for one-to-one matching (Gale and Sotomayor, 1985) and that for many-to-one matching (Martínez et al., 2010) are special cases of this result. Then, as an immediate consequence of the blocking lemma, we show that the doctor-optimal stable mechanism is group strategy-proof for doctors if hospitals’ choice functions satisfy substitutes and the law of aggregate demand. Hatfield and Kojima (2009) originally obtain this result by skillfully using the strategy-proofness of the doctor-optimal stable mechanism. In this paper we provide a different proof for the group incentive compatibility by applying the blocking lemma.     

Competitive equilibrium and singleton cores in generalized matching problems

We study competitive equilibria in generalized matching problems. We show that, if there is a competitive matching, then it is unique and the core is a singleton consisting of the competitive matching. That is, a singleton core is necessary for the existence of competitive equilibria. We also show that a competitive matching exists if and only if the matching produced by the top trading cycles algorithm is feasible, in which case it is the unique competitive matching. Hence, we can use the top trading cycles algorithm to test whether a competitive equilibrium exists and to construct a competitive equilibrium if one exists. Lastly, in the context of bilateral matching problems, we compare the condition for the existence of competitive matchings with existing sufficient conditions for the existence or uniqueness of stable matchings and show that it is weaker than most existing conditions for uniqueness.     

Random paths to pairwise stability in many-to-many matching problems: a study on market equilibration

This paper considers a decentralized process in many-to-many matching problems. We show that if agents on one side of the market have substitutable preferences and those on the other side have responsive preferences, then, from an arbitrary matching, there exists a finite path of matchings such that each matching on the path is formed by satisfying a blocking individual or a blocking pair for the previous matching, and the final matching is pairwise-stable. This implies that an associated stochastic process converges to a pairwise-stable matching in finite time with probability one, if each blocking individual or pair is satisfied with a positive probability at each period along the process.     

Instability of matchings in decentralized markets with various preference structures

In any two-sided matching market, a stable matching can be found by a central agency using the deferred acceptance procedure of Gale and Shapley. But if the market is decentralized and information is incomplete then stability of the ensuing matching is not to be expected. Despite the prevalence of such matching situations, and the importance of stability, little theory exists concerning instability. We discuss various measures of instability and analyze how they interact with the structure of the underlying preferences. Our main result is that even the outcome of decentralized matching with incomplete information can be expected to be “almost stable” under reasonable assumptions.     

Properties of the nonsymmetric Robinson–Schensted–Knuth algorithm

We introduce a generalization of the Robinson–Schensted–Knuth insertion algorithm for semi-standard augmented fillings whose basement is an arbitrary permutation σ∈S _n . If σ is the identity, then our insertion algorithm reduces to the insertion algorithm introduced by the second author (Sémin. Lothar. Comb. 57:B57e, 2006) for semi-standard augmented fillings and if σ is the reverse of the identity, then our insertion algorithm reduces to the original Robinson–Schensted–Knuth row insertion algorithm. We use our generalized insertion algorithm to obtain new decompositions of the Schur functions into nonsymmetric elements called generalized Demazure atoms (which become Demazure atoms when σ is the identity). Other applications include Pieri rules for multiplying a generalized Demazure atom by a complete homogeneous symmetric function or an elementary symmetric function, a generalization of Knuth’s correspondence between matrices of non-negative integers and pairs of tableaux, and a version of evacuation for composition tableaux whose basement is an arbitrary permutation σ.     

Decentralized job matching

This paper studies a multi-stage decentralized matching model where firms sequentially propose their (unique) positions to workers. At each stage workers sequentially decide which offer to accept (if any). A firm whose offer has been declined may make an offer to another worker in the next stage. The game stops when all firms either have been matched to a worker or have already made unsuccessful offers to any worker remaining in the market. We show that there is a unique subgame-perfect equilibrium outcome, the worker-optimal matching. Firms in this game have a weakly dominant strategy, which consists of making offers in the same order as given by their preferences. When workers play simultaneously any stable matching can be obtained as an equilibrium outcome, but an unstable matching can obtain in equilibrium.     

Analysis of stochastic matching markets

Suppose that the agents of a matching market contact each other randomly and form new pairs if is in their interest. Does such a process always converge to a stable matching if one exists? If so, how quickly? Are some stable matchings more likely to be obtained by this process than others? In this paper we are going to provide answers to these and similar questions, posed by economists and computer scientists. In the first part of the paper we give an alternative proof for the theorems by Diamantoudi et al. and Inarra et al., which imply that the corresponding stochastic processes are absorbing Markov chains. The second part of the paper proposes new techniques to analyse the behaviour of matching markets. We introduce the Stable Marriage and Stable Roommates Automaton and show how the probabilistic model checking tool PRISM may be used to predict the outcomes of stochastic interactions between myopic agents. In particular, we demonstrate how one can calculate the probabilities of reaching different matchings in a decentralised market and determine the expected convergence time of the stochastic process concerned. We illustrate the usage of this technique by studying some well-known marriage and roommates instances and randomly generated instances.     

Equilibrium strategies for multiple interdictors on a common network

In this work, we introduce multi-interdictor games, which model interactions among multiple interdictors with differing objectives operating on a common network. As a starting point, we focus on shortest path multi-interdictor (SPMI) games, where multiple interdictors try to increase the shortest path lengths of their own adversaries attempting to traverse a common network. We first establish results regarding the existence of equilibria for SPMI games under both discrete and continuous interdiction strategies. To compute such an equilibrium, we present a reformulation of the SPMI game, which leads to a generalized Nash equilibrium problem (GNEP) with non-shared constraints. While such a problem is computationally challenging in general, we show that under continuous interdiction actions, an SPMI game can be formulated as a linear complementarity problem and solved by Lemke’s algorithm. In addition, we present decentralized heuristic algorithms based on best response dynamics for games under both continuous and discrete interdiction strategies. Finally, we establish theoretical lower bounds on the worst-case efficiency loss of equilibria in SPMI games, with such loss caused by the lack of coordination among noncooperative interdictors, and use the decentralized algorithms to numerically study the average-case efficiency loss.     

The dynamics of stable matchings and half-matchings for the stable marriage and roommates problems

We study the dynamics of stable marriage and stable roommates markets. Our main tool is the incremental algorithm of Roth and Vande Vate and its generalization by Tan and Hsueh. Beyond proposing alternative proofs for known results, we also generalize some of them to the nonbipartite case. In particular, we show that the lastcomer gets his best stable partner in both incremental algorithms. Consequently, we confirm that it is better to arrive later than earlier to a stable roommates market. We also prove that when the equilibrium is restored after the arrival of a new agent, some agents will be better off under any stable solution for the new market than at any stable solution for the original market. We also propose a procedure to find these agents.     

基于R-vine-copula-CoVaR模型的金融市场风险溢出效应研究

为了挖掘国际金融市场与中国金融市场的风险溢出效应,本文首先通过ARJI-GARCH模型捕捉单个市场收益率的跳跃等典型事实特征,然后采用最大生成树(Maximum Spanning Tree,MST)算法优化的R-vine来刻画多维金融资产的复杂相依结构;最后构建R-vine-copula-CoVaR模型,测度了国际原油市场、国际黄金市场、美国股票市场与中国股票市场、外汇市场之间的风险溢出效应。实证结果表明:各市场之间均存在双向风险溢出效应,但溢出程度差别很大,国际黄金市场是风险溢出的最大爆发源,仅有中国外汇市场与中国股票市场、国际黄金市场间存在负向风险溢出;市场之间的双向风险溢出效应呈非对称性,国际原油市场与黄金市场的风险溢出效应远大于中国股票市场与外汇市场风险溢出效应;Rosenb-Latt检验表明基于R藤的CoVaR风险溢出测度更具有灵活性和有效性;后验测试结果表明R-vine-copula-CoVaR模型能有效地测度国际金融市场对中国金融市场风险溢出效应,而对中国金融市场风险溢出效应的CoVaR测度存在被高估的可能。     

Binary operations and lattice structure for a model of matching with contracts

We consider a restricted model of many-to-one matching with contracts and we order the set of stable allocations according both to the unanimous-for-doctors partial ordering and Blair’s partial ordering for hospitals. We define two binary operations to calculate the least upper bound and greatest lower bound for each pair of elements of this set in a simple way. By using these operations, we show that the set of stable allocations has dual lattice structures, thus reflecting an expected counterposition of interests between both sides of the market.     

A note on the efficiency and fairness of decentralized matching

This note shows by means of simulation experiments that although two-sided matching markets can always converge to a stable outcome via a decentralized process of matching and rematching, the most likely outcome need not be the median matching or a minimum-choice-count matching, whether or not the two coincide.     

3阶幂图的广义导出匹配可扩性

A graph G is induced matching extendable if every induced matching of G is included in a perfect matching of G. A graph G is generalized induced matching extendable if every induced matching of G is included in a maximum matching of G. A graph G is claw-free, if G dose not contain any induced subgraph isomorphic to K1,3. The k-th power of G, denoted by Gu, is the graph with vertex set V（G） in which two vertices are adjacent if and only if the distance between them is at most k in G. In this paper we show that, if the maximum matchings of G and G3 have the same cardinality, then G3 is generalized induced matching extendable. We also show that this result is best possible. As a result, we show that if G is a connected claw-flee graph, then G3 is generalized induced matching extendable.     

Stochastic Enumeration Method for Counting Trees

The problem of estimating the size of a backtrack tree is an important but hard problem in the computational sciences. An efficient solution of this problem can have a major impact on the hierarchy of complexity classes. The first randomized procedure, which repeatedly generates random paths through the tree, was introduced by Knuth. Unfortunately, as was noted by Knuth and a few other researchers, the estimator can introduce a large variance and become ineffective in the sense that it underestimates the cost of the tree. Recently, a new sequential algorithm called Stochastic Enumeration (SE) method was proposed by Rubinstein et al. The authors showed numerically that this simple algorithm can be very efficient for handling different counting problems, such as counting the number of satisfiability assignments and enumerating the number of perfect matchings in bipartite graphs. In this paper we introduce a rigorous analysis of SE and show that it results in significant variance reduction as compared to Knuth’s estimator. Moreover, we establish that for almost all random trees the SE algorithm is a fully polynomial time randomized approximation scheme (FPRAS) for the estimation of the overall tree size.     

在线多租赁选择问题的最优竞争策略

在线算法与竞争分析是研究信息不确定决策问题的一种新工具,应用该方法研究在线租赁问题是近年来国内外的一个研究热点。传统的在线租赁问题以经典的"雪橇租赁模型"为基础,考虑在线决策者可以选择购买或按单位时间租赁的方式来使用设备。然而现实租赁市场(比如汽车租赁,房屋租赁)往往提供多种租赁方式供在线决策者选择,除了按单位时间进行租赁,通常可以以一个较优惠的价格租赁多个单位时间。在这种现实背景下,本文建立了多种租赁形式下的在线租赁模型,给出了这种租赁模型下的确定性竞争策略,并证明该策略具有最优竞争比。     

Distributed Price Adjustment Based on Convex Analysis

In this paper, we suggest a distributed process of price adjustment toward a partial market equilibrium. As the main contribution, our algorithm of price adjustment is computationally efficient and decentralized. Its convergence properties are crucially based on convex analysis. The proposed price adjustment corresponds to a subgradient scheme for minimizing a special nonsmooth convex function. This function is the total excessive revenue of the market’s participants and its minimizers are equilibrium prices. As the main result, the algorithm of price adjustment is shown to converge to equilibrium prices. Additionally, the market clears on average during the price adjustment process, i.e., by historical averages of supply and demand. Moreover, a global rate of convergence is obtained. We endow our algorithm with decentralized prices by introducing the trade design with price initiative of producers. The latter suggests that producers settle and update their individual prices, and consumers buy at the lowest purchase price.     

Hydrodynamic limit of the Robinson–Schensted–Knuth algorithm

We investigate the evolution in time of the position of a fixed number in the insertion tableau when the Robinson–Schensted–Knuth algorithm is applied to a sequence of random numbers. When the length of the sequence tends to infinity, a typical trajectory after scaling converges uniformly in probability to some deterministic curve.     

What price stability? Social welfare in matching markets

In two-sided matching markets, stability can be costly. We define social welfare functions for matching markets and use them to formulate a definition of the price of stability. We then show that it is common to find a price tag attached to stability, and that the price of stability can be substantial. Therefore, when choosing a matching mechanism, a social planner would be well advised to weigh the price of stability against the value of stability, which varies from market to market.     

中国高考招生匹配市场中的算法设计及公平激励机制

用匹配博弈的方法,研究中国高考招生市场的算法设计及公平激励机制.基于高考招生程序,构建高考招生匹配算法,证明该算法的可行性.证明一个稳定匹配,可以由一个纳什均衡策略经高考招生算法生成,但反之不一定成立.证明一个稳定匹配一定是公平的,反之不一定成立.构建拒绝-接受算法,证明该算法是一个稳定的、策略防御的匹配机制,因而是一个公平的激励机制.     

On the convergence of the proximal algorithm for nonsmooth functions involving analytic features

We study the convergence of the proximal algorithm applied to nonsmooth functions that satisfy the ?jasiewicz inequality around their generalized critical points. Typical examples of functions complying with these conditions are continuous semialgebraic or subanalytic functions. Following ?jasiewicz’s original idea, we prove that any bounded sequence generated by the proximal algorithm converges to some generalized critical point. We also obtain convergence rate results which are related to the flatness of the function by means of ?jasiewicz exponents. Apart from the sharp and elliptic cases which yield finite or geometric convergence, the decay estimates that are derived are of the type O(k ^?s ), where s ∈ (0, + ∞) depends on the flatness of the function.