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The many-to-many stable matching problem (MM), defined in the context of a job market, asks for an assignment of workers to firms satisfying the quota of each agent and being stable, pairwise or setwise, with respect to given preference lists or relations. In this paper, we propose a time-optimal algorithm that identifies all stable worker--firm pairs and all stable assignments under pairwise stability, individual preferences, and the max-min criterion. We revisit the poset graph of rotations to obtain an optimal algorithm for enumerating all solutions to the MM and an improved algorithm finding the minimum-weight one. Furthermore, we establish the applicability of all aforementioned algorithms under more complex preference and stability criteria. In a constraint programming context, we introduce a constraint that models the MM and an encoding of the MM as a constraint satisfaction problem. Finally, we provide a series of computational results, including the case where side constraints are imposed.

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... There is a broad literature on algorithms for MM. In brief, Baïou and Balinski [6] provide an algorithm that finds an MM solution and identifies some non-stable pairs (i.e., pairs of agents appearing in no solution), while Eirinakis et al. [12] provide an algorithm for finding all such pairs. Bansal et al. [10] present algorithms for the egalitarian as well as the 'optimal' stable matching, while faster such algorithms are proposed in [12]. ...

... In brief, Baïou and Balinski [6] provide an algorithm that finds an MM solution and identifies some non-stable pairs (i.e., pairs of agents appearing in no solution), while Eirinakis et al. [12] provide an algorithm for finding all such pairs. Bansal et al. [10] present algorithms for the egalitarian as well as the 'optimal' stable matching, while faster such algorithms are proposed in [12]. ...

... In such cases, our result provides a minimal linear relaxation, which admits additional constraints per variant, and can be used efficiently in the framework of general solution methods (e.g., Branch & Cut). In contrast, the problemspecific combinatorial algorithms that also minimize a linear function over MM [12] become non-applicable in the presence of additional constraints. Hence, we consider a minimal linear description of stable b-matchings valuable in terms of solving several related optimization problems. ...

The realization of stable b-matchings as matroid kernels yields the existing linear description of the stable b-matching (MM) problem. We revisit that description to derive the dimension, the facets, and the minimum equation system of the stable b-matching polytope. The derived minimal description includes O(m) constraints, m being the number of pairs, thus being significantly smaller than the existing one and linear with respect to the problem size. The result carries over to the stable admissions (SA) problem, whose existing linear description relies on an exponential number of comb inequalities; i.e., we identify O(m) among these inequalities that define a minimal description of SA.

... In 2012 Eirinakis et al. [15] used the poset graph of rotations to enumerate all solutions of HR, and presented an improved version of the direct CP model of Manlove et al. [35]. Subsequently, Siala and O'Sullivan [47] used the rotation poset to model stable matchings as SAT formulation for all three types of problems: one-to-one, one-to-many, and many-tomany. ...

We introduce five constraint models for the 3-dimensional stable matching problem with cyclic preferences and study their relative performances under diverse configurations. While several constraint models have been proposed for variants of the two-dimensional stable matching problem, we are the first to present constraint models for a higher number of dimensions. We show for all five models how to capture two different stability notions, namely weak and strong stability. Additionally, we translate some well-known fairness notions (i.e. sex-equal, minimum regret, egalitarian) into 3-dimensional matchings, and present how to capture them in each model. Our tests cover dozens of problem sizes and four different instance generation methods. We explore two levels of commitment in our models: one where we have an individual variable for each agent (individual commitment), and another one where the determination of a variable involves pairing the three agents at once (group commitment). Our experiments show that the suitability of the commitment depends on the type of stability we are dealing with, and that the choice of the search heuristic can help improve performance. Our experiments not only brought light to the role that learning and restarts can play in solving this kind of problems, but also allowed us to discover that in some cases combining strong and weak stability leads to reduced runtimes for the latter.

... 6 Among other results, they use cycles to compute the full set of stable match-ings. Eirinakis et al. (2012) revise and improve the algorithm presented in Bansal et al. (2007). Moreover, they extend the algorithm for a model in which agents' preferences satisfy the "max-min criteria". ...

In a many-to-many matching model in which agents’ preferences satisfy substitutability and the law of aggregate demand, we present an algorithm to compute the full set of stable matchings. This algorithm relies on the idea of “cycles in preferences” and generalizes the algorithm presented in Roth and Sotomayor (1990) for the one-to-one model.

We provide a linear description of the unconstrained stable allocations problem by proving that the corresponding polytope is affinely congruent to the order polytope of a partially ordered set. The same holds for stable matchings hence simplifying the derivation of known polyhedral results. We also show that this congruence no longer holds for the constrained version of stable allocations. As side outcomes, we characterise the neighbouring vertices of the order polytope and the partially ordered set associated with stable allocations.

Sharing economy is significant for economic development, stable matching plays an essential role in sharing economy, but the large-scale sharing platform increases the difficulties of stable matching. We proposed a two-sided gaming model based on probabilistic linguistic term sets to address the problem. Firstly, in previous studies, the mutual assessment is used to obtain the preferences of individuals in large-scale matching, but the procedure is time-consuming. We use probabilistic linguistic term sets to present the preferences based on the historical data instead of time-consuming assessment. Then, to generate the satisfaction based on the preference, we regard the similarity between the expected preferences and actual preferences as the satisfaction. Considering the distribution features of probabilistic linguistic term sets, we design a shape-distance-based method to measure the similarity. After that, the previous studies aimed to maximize the total satisfaction in matching, but the individuals’ requirements are neglected, resulting in a weak matching result. We establish the two-sided gaming matching model from the perspectives of individuals based on the game theory. Meanwhile, we also study the competition from other platforms. Meanwhile, considering the importance of the high total satisfaction, we balance the total satisfaction and the personal requirements in the matching model. We also prove the solution of the matching model is the equilibrium solution. Finally, to verify the study, we use the experiment to illustrate the advantages of our study.

We introduce new CP models for the many-to-many stable matching problem. We use the notion of rotation to give a novel encoding that is linear in the input size of the problem. We give extra filtering rules to maintain arc consistency in quadratic time. Our experimental study on hard instances of sex-equal and balanced stable matching shows the efficiency of one of our propositions as compared with the state-of-the-art constraint programming approach.

An implicit linear description of the stable matching polytope is provided in terms of the blocker and antiblocker sets of constraints of the matroid-kernel polytope. The explicit identification of both these sets is based on a partition of the stable pairs in which each agent participates. Here, we expose the relation of such a partition to rotations. We provide a time-optimal algorithm for obtaining such a partition and establish some new related results; most importantly, that this partition is unique.

The many-to-many Stable Matching (MM) problem is defined on a set of workers and a set of firms and asks for an allocation of workers to firms that satisfies the firms' quotas and the preferences of workers for firms and vice-versa. This article proposes a time-optimal algorithm for solving the minimum-regret problem for the MM, i.e. the problem of finding the MM solution in which the agent that is least well off is matched with the best possible set of agents of the opposite set. The proposed algorithm utilizes the notion of rotations and extends an analogous result for the Stable Marriage problem.

Competitive adjustment processes in labor markets with perfect information but heterogeneous firms and workers are studied. Generalizing results of Shapley and Shubik [7], and of Crawford and Knoer [1], we show that equilibrium in such markets exists and is stable, in spite of workers' discrete choices among jobs, provided that all workers are gross substitutes from each firm's standpoint. We also generalize Gale and Shapley's [3] result that the equilibrium to which the adjustment process converges is biased in favor of agents on the side of the market that makes offers, beyond the class of economies to which it was extended by Crawford and Knoer [1]. Finally, we use our techniques to establish the existence of equilibrium in a wider class of markets, and some sensible comparative statics results about the effects of adding agents to the market are obtained. THE ARROW-DEBREU THEORY of general economic equilibrium has long been recognized as a powerful and elegant tool for the analysis of resource allocation in market economies. Not all markets fit equally well into the Arrow-Debreu framework, however. Consider, for example, the labor market-or the housing market, which provides an equally good example for most of our purposes. Essential features of the labor market are pervasive uncertainty about market opportunities on the part of participants, extensive heterogeneity, in the sense that job satisfaction and productivity generally differ (and are expected to differ) interactively and significantly across workers and jobs, and large set-up costs and returns to specialization that typically limit workers to one job. All of these features can be fitted formally into the Arrow-Debreu framework. State-contingent general equilibrium theory, for example, provides a starting point for studying the effects of uncertainty. But this analysis has been made richer and its explanatory power broadened by the examination of equilibrium with incomplete markets, search theory, and market signaling theory. The purpose of this paper is to attempt some improvements in another dimension: we study the outcome of competitive sorting processes in markets where complete heterogeneity prevails (or may prevail). To do this, we take as given the implications of set-up costs and returns to specialization by assuming that, while firms can hire any number of workers, workers can take at most one job. We also return to the simplification of perfect information. In the customary view of competitive markets, agents take market prices as given and respond noncooperatively to them. In this framework equilibrium cannot exist in general unless the goods traded in each market are truly homogeneous; heterogeneity therefore generally requires a very large number of markets. And since these markets are necessarily extremely thin-in many cases containing only a single agent on each side-the traditional stories supporting the plausibility of price-taking behavior are quite strained.

The Stable Marriage problem (SM) is an extensively-studied combinatorial prob-lem with many practical applications. In this paper we present a range of Constraint Programming (CP) models of an instance I of SM as an instance J of a Constraint Satisfaction Problem. We prove that, in a precise sense, establishing arc consistency in J is equivalent to the action of the established Extended Gale/Shapley algorithm for SM on I. As a consequence of this, the man-optimal and woman-optimal stable matchings can be derived immediately. Furthermore we show that, in both encod-ings, all solutions of I may be enumerated in a failure-free manner. The encodings presented are natural developments of those appearing in [8].

The stable admissions polytope– the convex hull of the stable assignments of the university admissions problem – is described by a set of linear inequalities.
It depends on a new characterization of stability and arguments that exploit and extend a graphical approach that has been
fruitful in the analysis of the stable marriage problem.

A global cardinality constraint (gcc) is specified in terms of a set of variables X = {x1,..., xp} which take their values in a subset of V = {v1,...,vd}. It constrains the number of times a value vi ∈ V is assigned to a variable in X to be in an interval [li, ci. Cardinality constraints have proved very useful in many real-life problems, such as scheduling, timetabling, or resource allocation. A gcc is more general than a constraint of difference, which requires each interval to be [0,1]. In this paper, we present an efficient way of implementing generalized arc consistency for a gcc. The algorithm we propose is based on a new theorem of flow theory. Its space complexity is O(|X| × |V|) and its time complexity is O(|X|2 × |V|). We also show how this algorithm can efficiently be combined with other filtering techniques.

An instance I of the Hospitals / Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a stable matching, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. Our study suggests that Constraint Programming is indeed an applicable technology for solving this problem, in terms of both theory and practice.

We present the first complete algorithm for the SMTI problem, the stable marriage problem with ties and incomplete lists. We do this in the form of a constraint programming encoding of the problem. With this we are able to carry out the first empirical study of the complete solution of SMTI instances. In the stable marriage problem (SM) (?) we have men and women. Each man ranks the women, giving himself a preference list. Similarly each woman ranks the men, giving herself a preference list. The problem is then to marry men and women such that they are stable i.e. such that there is no incentive for individuals to divorce and elope. This problem is polynomial time solvable. However, when preference lists contain ties and are incomplete (SMTI) the problem of determining if there is a stable matching of size is then NP-complete, as is the optimisa- tion problem of finding the largest or smallest stable matching (?, ?). In this paper we present constraint programming solutions for the SMTI decision and optimisation problems, a problem generator for random instances of SMTI, and an empirical study of this problem.

The Stable Marriage problem (SM) is an extensively-studied combinatorial problem with many practical applications. In this paper we present two encodings of an instance I of SM as an instance J of a Constraint Satisfaction Problem. We prove that, in a precise sense, establishing arc consistency in J is equivalent to the action of the established Extended Gale/Shapley algorithm for SM on I. As a consequence of this, the man-optimal and woman-optimal stable matchings can be derived immediately. Furthermore we show that, in both encodings, all solutions of I may be enumerated in a failure-free manner. Our results indicate the applicability of Constraint Programming to the domain of stable matching problems in general, many of which are NP-hard.

Competitive adjustment processes in labor markets with perfect information but heterogeneous firms and workers are studied. Generalizing results of Shapley and Shubik [7], and of Crawford and Knoer [1], we show that equilibrium in such markets exists and is stable, in spite of workers' discrete choices among jobs, provided that all workers are gross substitutes from each firm's standpoint. We also generalize Gale and Shapley's [3] result that the equilibrium to which the adjustment process converges is biased in favor of agents on the side of the market that makes offers, beyond the class of economies to which it was extended by Crawford and Knoer [1]. Finally, we use our techniques to establish the existence of equilibrium in a wider class of markets, and some sensible comparative statics results about the effects of adding agents to the market are obtained.

We develop a theory of stability in many-to-many matching markets. We give conditions under wich the setwise-stable set, a core-like concept, is nonempty and can be approached through an algorithm. The setwise-stable set coincides with the pairwisestable set, and with the predictions of a non-cooperative bargaining model. The set-wise stable set possesses the canonical conflict/coincidence of interest properties from manyto -one, and one-to-one models. The theory parallels the standard theory of stability for many-to-one, and one-to-one, models. We provide results for a number of core-like solutions, besides the setwise-stable set. JEL classification numbers: C78 Key words: Matching markets, Core, Lattice, Stability, Substitutability, Tarski's Fixed Point Theorem Markets # 1

Some Examples The Logic of Propositions The Logic of Discrete Variables The Logic of 0-1 Inequalities Cardinality Clauses Classical Boolean Methods Logic-Based Modeling Logic-Based Branch and Bound Constraint Generation Domain Reduction Constraint Programming Continuous Relaxations Decomposition Methods Branching Rules Relaxation Duality Inference Duality Search Strategies Logic-Based Benders Decomposition Nonserial Dynamic Programming Discrete Relaxations References Index.

We show that the set of stable many-to-many matchings under responsive preferences is a complete distributive lattice but does not possess some of the other nice properties – monotonicity and strategyproofness – associated with the set of stable one-to-one matchings.

This expository paper develops the principal known results (and some new ones) on the stable matchings of marriage games in the language of directed graphs. This both unifies and simplifies the presentation and renders it more symmetric. In addition, it yields a new algorithm and a new proof for the existence of stable matchings, new proofs for many known facts, and some new results (notably concerning players' strategies and the properties of the stable matching polytope).

We study the Stable Marriage problem (SM), which is a combinatorial problem that arises in many practical applications. We present two new models of an instance I of SM with n men and n women as an instance J of a Constraint Satisfaction Problem. We prove that establishing arc consistency in J yields the same structure as given by the established Extended Gale/Shapley algorithm for SM as applied to I. Consequently, a solution (stable matching) of I can be derived without search. Furthermore we show that, in both encodings, all stable matchings in I may be enumerated in a failure-free manner. Our first encoding is of O(n 3) complexity and is very natural, whilst our second model, of O(n 2) complexity (which is optimal), is a development of the Boolean encoding in [6], establishing a greater level of structure.

College Admissions and the Stability of Marriage

This paper generalizes the selection problem discussed by J. M. Rhys [Rhys, J. M. W. 1970. Shared fixed cost and network flows. Management Sci. 17 (3, November).], J. D. Murchland [Murchland, J. D. 1968. Rhys's combinatorial station selection problem. London Graduate School of Business Studies, Transport Network Theory Unit, Report LBS-TNT-68, June 10.], M. L. Balinski [Balinski, M. L. 1970. On a selection problem. Management Sci. 17 (3, November).] and P. Hansen [Hansen, P. 1974. Quelques approches de la programmation non lineaire en variables 0-1. Conference on Mathematical Programming, Bruxelles, May.]. Given a directed graph G, a closure of G is defined as a subset of nodes such that if a node belongs to the closure all its successors also belong to the set. If a real number is associated to each node of G a maximal closure is defined as a closure of maximal value.

The Stable Marriage Problem and its many variants have been widely studied in the literature (Gusfield and Irving, The Stable Marriage Problem: Structure and Algorithms, MIT Press, Cambridge, MA, 1989; Roth and Sotomayor, Two-sided matching: a study in game-theoretic modeling and analysis, Econometric Society Monographs, vol. 18, Cambridge University Press, Cambridge, 1990; Knuth, Stable Marriage and its Relation to Other Combinatorial Problems, CRM Proceedings and Lecture Notes, vol. 10, American Mathematical Society, Providence, RI, 1997), partly because of the inherent appeal of the problem, partly because of the elegance of the associated structures and algorithms, and partly because of important practical applications, such as the National Resident Matching Program (Roth, J. Political Economy 92(6) (1984) 991) and similar large-scale matching schemes. Here, we present the first comprehensive study of variants of the problem in which the preference lists of the participants are not necessarily complete and not necessarily totally ordered. We show that, under surprisingly restrictive assumptions, a number of these variants are hard, and hard to approximate. The key observation is that, in contrast to the case where preference lists are complete or strictly ordered (or both), a given problem instance may admit stable matchings of different sizes. In this setting, examples of problems that are hard are: finding a stable matching of maximum or minimum size, determining whether a given pair is stable—even if the indifference takes the form of ties on one side only, the ties are at the tails of lists, there is at most one tie per list, and each tie is of length 2; and finding, or approximating, both an ‘egalitarian’ and a ‘minimum regret’ stable matching. However, we give a 2-approximation algorithm for the problems of finding a stable matching of maximum or minimum size. We also discuss the significant implications of our results for practical matching schemes.

We characterize the bipartite stable b-matching polytope in terms of linear constraints. The stable b-matching polytope is the convex hull of the characteristic vectors of stable b-matchings, that is, of stable assignments of a two-sided multiple partner matching model. Our proof uses a generalization by M. Baïou and M. Balinski [Discrete Appl. Math. 101, No. 1–3, 1–12 (2000; Zbl 0946.90033)] of the comparability theorem of A. E. Roth and M. Sotomayor [Econometrica 57, No. 3, 559– 570 (1989; Zbl 0668.90004)] and follows a similar line as the proof of Rothblum’s characterization of the stable matching polytope.

The paper proposes an algorithm to compute the full set of many-to-many stable matchings when agents have substitutable preferences. The algorithm starts by calculating the two optimal stable matchings using the deferred-acceptance algorithm. Then, it computes each remaining stable matching as the firm-optimal stable matching corresponding to a new preference profile, which is obtained after modifying the preferences of a previously identified sequence of firms.

This paper presents algorithms for the assignment problem, the transportation problem, and the minimum-cost flow problem of operations research. The algorithms find a minimumcost solution, yet run in time close to the best-known bounds for the corresponding problems without costs. For example, the assignment problem (equivalently, minimum-cost matching in a bipartite graph) can be solved in O(v/’rn log(nN)) time, where n, m, and N denote the number of vertices, number of edges, and largest magnitude of a cost; costs are assumed to be integral. The algorithms work by scaling. As in the work of Goldberg and Tarjan, in each scaled problem an approximate optimum solution is found, rather than an exact optimum.

The stable marriage problems is a well-known problem of matching n men to n women to achieve a certain type of 'stability; ' the O(n**2) time Gale-Shapley algorithm for finding two particular, but extreme, stable marriages (out of a possibly exponential number of stable marriages) is also well known. In this paper we consider four problems concerned with finding information about the set of all stable marriages, and with finding stable marriages other than those obtained by the Gale-Shapley algorithm. In particular, we give an O(n**2) time algorithm which, for any problem instance of n men and n women, finds every man-woman pair that is contained in at least one stable marriage; we show that the same algorithm finds all the 'rotations' for the problem instance in O(n**2) time; we give an O(n**2 plus n vertical S vertical ) time and O(n**2) space bounded algorithm to enumerate all stable marriages, where S is the set of them; and we give an O(n**2) time algorithm to find the minimum regret stable marriage.

The major results known for the marriage and university admissions problems — the one-to-one and many-to-one stable matching problems — are shown to have equivalents in the general many-to-many setting. Some of these results depend upon a particular, natural definition of individual preferences over sets of mates: notably, characterizations of “optimal” stable assignments in terms of “efficiency”, “monotonicity”, and “strategy-proofness”.

The original work of Gale and Shapley on an assignment method using the stable marriage criterion has been extended to find all the stable marriage assignments. The algorithm derived for finding all the stable marriage assignments is proved to satisfy all the conditions of the problem. Algorithm 411 applies to this paper.

Let I be a stable matching instance with N stable matchings. For each man m, order his (not necessarily distinct) N partners from his most preferred to his least preferred. Denote the ith woman in his sorted list as pi(m). Let αi consist of the man-woman pairs where each man m is matched to pi(m). Teo and Sethuraman proved this surprising result: for i = 1 to N, not only is αi a matching, it is also stable. The αi's are called the generalized median stable matchings of I. Determining if these stable matchings can be computed efficiently is an open problem. In this paper, we present a new characterization of the generalized median stable matchings that provides interesting insights. It implies that the generalized median stable matchings in the middle - α(N+1)/2 when N is odd, αN/2 and αN/2+1 when N is even - are fair not only in a local sense but also in a global sense because they are also medians of the lattice of stable matchings. We then show that there are some families of SM instances for which computing an αi is easy but that the task is NP-hard in general. Finally, we also consider what it means to approximate a median stable matching and present results for this problem.

This paper considers the many-to-many version of the stable marriage problem where each man and woman has a strict preference ordering on the members of the opposite sex that he or she considers acceptable. Further, each man and woman wishes to be matched to as many acceptable partners as possible, up to his or her specified quota. In this setup, a polynomial time algorithm for finding a stable matching that minimizes the sum of partner ranks across all men and women is provided. It is argued that this sum can be used as an optimality criterion for minimizing total dissatisfaction if the preferences over partner-combinations satisfy a no-complementarities condition. The results in this paper extend those already known for the one-to-one version of the problem.

Constraint programming is a generalised framework designed to solve combinatorial problems. This framework is made up of a set of predefined independent components and generalised algorithms. This is a very versatile structure which allows for a variety of rich combinatorial problems to be represented and solved relatively easily. Stable matching problems consist of a set of participants wishing to be matched into pairs or groups in a stable manner. A matching is said to be stable if there is no pair or group of participants that would rather make a private arrangement to improve their situation and thus undermine the matching. There are many important "real life" applications of stable matching problems across the world. Some of which includes the Hospitals/Residents problem in which a set of graduating medical students, also known as residents, need to be assigned to hospital posts. Some authorities assign children to schools as a stable matching problem. Many other such problems are also tackled as stable matching problems. A number of classical stable matching problems have efficient specialised algorithmic solutions. Constraint programming solutions to stable matching problems have been investigated in the past. These solutions have been able to match the theoretically optimal time complexities of the algorithmic solutions. However, empirical evidence has shown that in reality these constraint solutions run significantly slower than the specialised algorithmic solutions. Furthermore, their memory requirements prohibit them from solving problems which the specialised algorithmic solutions can solve in a fraction of a second. My contribution investigates the possibility of modelling stable matching problems as specialised constraints. The motivation behind this approach was to find solutions to these problems which maintain the versatility of the constraint solutions, whilst significantly reducing the performance gap between constraint and specialised algorithmic solutions. To this end specialised constraint solutions have been developed for the stable marriage problem and the Hospitals/Residents problem. Empirical evidence has been presented which shows that these solutions can solve significantly larger problems than previously published constraint solutions. For these larger problem instances it was seen that the specialised constraint solutions came within a factor of four of the time required by algorithmic solutions. It has also been shown that, through further specialisation, these constraint solutions can be made to run significantly faster. However, these improvements came at the cost of versatility. As a demonstration of the versatility of these solutions it is shown that, by adding simple side constraints, richer problems can be easily modelled. These richer problems add additional criteria and/or an optimisation requirement to the original stable matching problems. Many of these problems have been proven to be NP-Hard and some have no known algorithmic solutions. Included with these models are results from empirical studies which show that these are indeed feasible solutions to the richer problems. Results from the studies also provide some insight into the structure of these problems, some of which have had little or no previous study.

A vast and often confusing economics literature relates competition to investment in innovation. Following Joseph Schumpeter, one view is that monopoly and large scale promote investment in research and development by allowing a firm to capture a larger fraction of its benefits and by providing a more stable platform for a firm to invest in R&D. Others argue that competition promotes innovation by increasing the cost to a firm that fails to innovate. This lecture surveys the literature at a level that is appropriate for an advanced undergraduate or graduate class and attempts to identify primary determinants of investment in R&D. Key issues are the extent of competition in product markets and in R&D, the degree of protection from imitators, and the dynamics of R&D competition. Competition in the product market using existing technologies increases the incentive to invest in R&D for inventions that are protected from imitators (e.g., by strong patent rights). Competition in R&D can speed the arrival of innovations. Without exclusive rights to an innovation, competition in the product market can reduce incentives to invest in R&D by reducing each innovator's payoff. There are many complications. Under some circumstances, a firm with market power has an incentive and ability to preempt rivals, and the dynamics of innovation competition can make it unprofitable for others to catch up to a firm that is ahead in an innovation race.

We propose a general definition of stability, setwise-stability, and show that it is a stronger requirement than pairwise-stability and core. We also show that the core and the set of pairwise-stable matchings may be non-empty and disjoint and thus setwise-stable matchings may not exist. For many labor markets the effects of competition can be characterized by requiring only pairwise-stability. For such markets we define substitutability and we prove the existence of pairwise-stable matchings. The restriction of our proof to the College Admission Model is simple and short and provides an alternative proof for the existence of stable matchings for this model.

This paper adopts a \revealed preference" approach to the question of what can be inferred about bias in a political system. We model an in nite horizon, dynamic economy and its political system from the point of view of an \outside observer." The observer sees a nite sequence of policy data, but does not observe the underlying distribution of political power that produced this data. Neither does he observe the preference pro le of the citizenry.The observer makes inferences about distribution of political power as if political power were derived from a wealth-weighted voting system with weights that can vary across states. The weights determine the nature and magnitude of the wealth bias. Positive weights on relative income in any period indicate an \elitist" bias in the political system whereas negative weights indicate a \populist" one. We ask: what class of weighted systems can rationalize the policy data as weighted majority outcomes each period? We show that without further knowledge, all forms of bias are possible: any policy data can be shown to be rationalized by any system of wealth-weighted voting. An additional single crossing restriction on preferences can, however, rule out certain weighting systems. We then augment policy data with polling data and show that the set of rationalizing wealth-weights are bounded above and below, thus ruling out extreme biases. In some cases, polls can provide information about the change in political inequality across time. Classification-JEL Codes: C73, D63, D72, D74, H11

A network is a graph with numeric parameters such as edge lengths, capacities, costs, etc. We present efficient algorithms for network problems that work by scaling the numeric parameters. Scaling takes advantage of efficient nonnumeric algorithms such as the Hopcroft-Karp matching algorithm. Let n, m and N denote the number of vertices, number of edges, and largest numeric parameter of the network, respectively; assume all numeric parameters are integers. A scaling algorithm for maximum weight matching on a bipartite graph runs in O(n3/4 m log N) time. This can improve the traditional Hungarian method which runs in O(n m log n) time. This result gives similar improvements for the following problems: single-source shortest paths for arbitrary edge lengths (Bellman's algorithm); maximum weight degree-constrained subgraph; minimum cost flow in a 0-1 network (Edmonds and Karp). Scaling also gives simple algorithms that match the best time bounds (when log N = O(log n)) for shortest paths on a directed graph with nonnegative lengths (Dijkstra's algorithm) and maximum value network flow (Sleator and Tarjan).

This paper presents an algorithm achieving hyperarc consistency for the stable admissions problem and discusses computational results.

We introduce a new approach to the maximum flow problem. This approach is based on assigning arc lengths based on the residual flow value and the residual are capacities. Our approach leads to an O(min(n <sup>2/3</sup>, m<sup>1/2</sup>)m log(n<sup>2</sup>/m) log U) time bound for a network with n vertices, m arcs, and integral arc capacities in the range [1,…,U]. This is a fundamental improvement over the previous time bounds. We also improve bounds for the Gomory-Hu tree problem, the parametric flow problem, and the approximate s-t cut problems

- S A Ilog

ILOG S.A., ILOG Solver Callable Library 6.1, 2005.