Herbert Scarf: a Distinguished American Economist

Herbert Eli Scarf (born on July 25, 1930 in Philadelphia, PA) is a distinguished American economist and Sterling Professor (Emeritus as of 2010) of Economics at Yale University. He is a member of the American Academy of Arts and Sciences, the National Academy of Sciences and the American Philosophical Society. He served as the president of the Econometric Society in 1983. He received both the Frederick Lanchester Award in 1973 and the John von Neumann Medal in 1983 from the Operations Research Society of America and was elected as a Distinguished Fellow of the American Economic Association in 1991.

A Practical Competitive Market Model for Indivisible Commo

A general and practical competitive market model for trading indivisible goods is introduced. There are a group of buyers and a group of sellers, and several indivisible goods. Each buyer is initially endowed with a sufficient amount of money and each seller is endowed with several units of each indivisible good. Each buyer has reservation values over bundles of indivisible goods above which he will not buy and each seller has reservation values over bundles of his own indivisible goods below which he will not sell. Buyers and sellers' preferences depend on the bundle of indivisible goods and the quantity of money they consume. All preferences are assumed to be quasi-linear in money and money is treated as a perfectly divisible good. It is shown in an extremely simple manner that the market has a Walrasian equilibrium if and only if an associated linear program problem has an optimal solution with its value equal to the potential market value. In addition, it is shown that the equilibrium prices of the goods and the profits of the agents are the optimal solutions of the linear program problem.Market, indivisibility, Walrasian equilibrium, linear program, potential market value

On Revealed Preference and Indivisibilities

We consider a market model in which all commodities are inherently indivisible and thus are traded in integer quantities. We ask whether a finite set of price-quantity observations satisfying the Generalized Axiom of Revealed Preference (GARP) is consistent with utility maximization. Although familiar conditions such as non-satiation become meaningless in the current discrete model, by refining the standard notion of demand set we show that Afriat's celebrated theorem still holds true. Exploring network structure and a new and easy-to-use variant of GARP, we propose an elementary, simple, intuitive, combinatorial, and constructive proof for the result.Afriat's theorem, GARP, indivisibilities, revealed preference.

A Double-Track Auction for Substitutes and Complements

We propose a new t^atonnement process called a double-track auction for efficiently allocating multiple heterogeneous indivisible items in two distinct sets S1 and S2 to many buyers who view items in the same set as substitutes but items across the two sets as complements. The auctioneer initially announces sufficiently low prices for items in one set, say S1, but sufficiently high prices for items in the other set S2. In each round, the buyers respond by reporting their demands at the current prices and the auctioneer adjusts prices upwards for items in S1 but downwards for items in S2 based on buyers' reported demands until the market is clear. Unlike any existing auction, this auction is a blend of a multi-item ascending auction and a multi-item descending auction. We prove that the auction finds an efficient allocation and its market-clearing prices in finitely many rounds. Based on the auction we also establish a dynamic, efficient and strategy-proof mechanism.Market design, dynamic auction, t^atonnement process, gross substitutes and complements, Walrasian equilibrium, incentives.

An Optimal Fair Job Assignment Problem

We study the problem of how to allocate a set of indivisible objects like jobs or houses and an amount of money among a group of people as fairly and as efficiently as possible. A particular constraint for such an allocation is that every person should be assigned with the same number of objects in his or her bundle. The preferences of people depend on the bundle of objects and the quantity of money they take. We propose a solution to this problem, called a perfectly fair allocation. It is shown that every perfectly fair allocation is efficient and envy-free, income-fair and furthermore gives every person a maximal satisfaction. Then we establish a necessary and sufficient condition for the existence of a perfectly fair allocation. It is shown that there exists a perfectly fair allocation if and only if an associated linear program problem has a solution. As a result, we also provide a finite method of computing a perfectly fair allocation.Perfectly fair allocation, equity, efficiency, indivisibility, multi-person decision, discrete optimization

Decentralized Market Processes to Stable Job Matchings with Competitive Salaries

We analyze a decentralized trading process in a basic labor market where heterogeneous firms and workers meet directly and randomly, and negotiate salaries with each other over time. Firms and workers may not have a complete picture of the entire market and can thus behave myopically in the process. Our main result establishes that, starting from an arbitrary initial market state, there exists a finite sequence of successive myopic (firm-worker) pair improvements, or bilateral trades, leading to a stable matching between firms and workers with a scheme of competitive salary offers. An important implication of this result is that a general random process where every possible bilateral trade is chosen with a positive probability converges with probability one to a competitive equilibrium of the market.Decentralized market, job matching, random path, competitive salary, stability

An Ascending Multi-Item Auction with Financially Constrained Bidders

A number of heterogeneous items are to be sold to a group of potential bidders. Every bidder knows his own values over the items and his own budget privately. Due to budget constraint, bidders may not be able to pay up to their values. In such a market, a Walrasian equilibrium typically fails to exist and furthermore no existing allocation mechanism can tackle this case. We propose the notion of an `equilibrium under allotment' to such markets and develop an ascending auction mechanism that always finds such an equilibrium assignment and corresponding price system in finitely many rounds. The auction can be viewed as an appropriate and proper generalization of the ascending auction of Demange, Gale and Sotomayor from settings without financial constraints to settings with financial constraints. We examine various properties of the auction and its outcome.Ascending auction, Financial constraint, Equilibrium under allotment.

On Fair Allocations and Indivisibilities

This paper studies the problem of how to distribute a set of indivisible objects with an amount M of money among a number of agents in a fair way. We allow any number of agents and objects. Objects can be desirable or undesirable and the amount of money can be negative as well. In case M is negative, it can be regarded as costs to be shared by the agents. The objects with the money will be completely distributed among the agents in a way that each agent gets a bundle with at most one object if there are more agents than objects, and gets a bundle with at least one object if objects are no less than agents. We prove via an advanced fixed point argument that under rather mild and intuitive conditions the set of envy-free and efficient allocations is nonempty. Furthermore we demonstrate that if the total amount of money varies in an interval [X,Y], then there exists a connected set of fair allocations whose end points are allocations with sums of money equal to X and Y, respectively. Welfare properties are also analyzed when the total amount of money is modeled as a continuous variable. Our proof is based on a substantial generalization of the classic lemma of Knaster, Kuratowski and Mazurkewicz (KKM) in combinatorial topology.Indivisibility, fairness, Pareto optimality, resource allocation, multiperson decision, KKM lemma

On a spontaneous decentralized market process

We examine a spontaneous decentralized market process widely observed in real life labor markets. This is a natural random decentralized dynamic competitive process. We show that this process converges globally and almost surely to a competitive equilibrium. This result is surprisingly general by assuming only the existence of an equilibrium. Our findings have also meaningful policy implications

Random Decentralized Market Processes for Stable Job Matchings with Competitive Salaries

We analyze a decentralized process in a basic labor market where finitely many heterogeneous firms and workers meet directly and randomly in pursuit of higher payoffs over time and agents may behave myopically. We introduce a general random decentralized market process that almost surely converges in finite time to a competitive equilibrium of the market. A key proposition en route to this result exhibits a finite sequence of successive bilateral trades from an arbitrary initial market state to a stable matching between firms and workers with a scheme of competitive salary offers