William Thomson

• University of Rochester

We consider classical problems of fair division and generalize the standard requirement on solution mappings known as “replication invariance”. Consider an economy and an allocation chosen by a solution for it. Allowing the replication parameter to vary from agent to agent, we correspondingly adjust the social endowment so that it be possible to as...

In the context of claims problems, we formulate an axiom of invariance of a rule with respect to its “partial implementation”: having applied the rule to a problem in its domain of definition, we imagine some claimants picking up their awards and we consider the problem of distributing what remains of the endowment among the other claimants: in thi...

An allocation rule is “withholding-proof" if no agent ever benefits from withholding some of the resources they own, their final bundle consisting of what the rule assigns to them together with whatever they withheld. It was known that on the “classical" domain of continuous, monotone, and convex preferences, no rule is efficient and withholding-pr...

In the context of classical exchange economies, we study four ways in which agents can strategically take advantage of allocation rules by affecting who participates and on what terms (Thomson in Soc Choice Welf 42:289–311, 2014). (1) An agent transfers their endowment to someone else and withdraws. The two of them may end up controlling resources...

We study the geometry of the Pareto set in Edgeworth box economies when both agents have continuous, strictly monotonic, strictly convex, smooth, and homothetic preferences. We show that this set is either the diagonal of the box or it is a “doubly visible” curve connecting the origins of the box: imagining the curve to be opaque, an observer stand...

In a “queueing problem”, a group of agents are waiting for a service. Each agent incurs a cost of waiting that is proportional to the time they wait. Monetary transfers can take place. We study the subsolutions of the no-envy solution that are anonymous, consistent, conversely consistent, and continuous. We show that there are infinitely many prope...

This is a survey of the literature of (approximately) the last 10 years on the axiomatic approach to conflict resolution formulated by Nash (Econometrica 28:155–162, 1950). It updates two previous surveys (Thomson, in: Aumann, Hart (eds) Handbook of game theory, North-Holland, 1994, in: Thomson (ed) Bargaining and the theory of cooperative games: J...

We consider the problem of choosing a location in an interval so as to best take into account the preferences of two agents who will use this location. One has single-peaked preferences and the other single-dipped preferences. The most preferred location for the agent with single-peaked preferences is known and it is the least preferred location fo...

This paper explores the robustness of a characterization of the Nash bargaining solution in a variable-population framework in which the population of players is bounded (Lensberg in J Econ Theory 45:330–341, 1988). A central axiom in this characterization is “consistency”, which says that the solution outcome of any problem should always be “confi...

The purpose of this note is to propose a two-way classification of the axioms of the theory of economic design, and to map out directions for future research that we perceive as particularly promising.

Cambridge Core - Statistics for Econometrics, Finance and Insurance - How to Divide When There Isn't Enough - by William Thomson

We consider the adjudication of conflicting claims over a resource. By mapping such a problem into a bargaining problem à la Nash, we avail ourselves of the solution concepts developed in bargaining theory. We focus on the solution to two-player bargaining problems known as the “equal area solution.” We study the properties of the induced rule to s...

The purpose of this paper is to provide a critical examination of some of the terminology that is common in economic design and to suggest ways in which it can be improved.

An allocation rule is “non-bossy” if whenever a change in an agent’s preferences does not bring about a change in his assignment, then it does not bring about a change in anybody’s assignment. We discuss the multiple interpretations that have been proposed for this property. We question their validity, arguing that in most cases, non-bossiness eith...

For the problem of adjudicating conflicting claims, we develop a new proof of an important result due to Young (1995), a characterization of the proportional rule, rather, we prove a dual characterization of this rule. Our axioms are composition down, which states the equivalence of two ways of dealing with possible decreases in the amount to divid...

Introduction The purpose of this chapter is to briefly introduce the theory of fair allocation in the context of concretely specified economic environments. There, requirements of fairness are expressed in terms of resources and opportunities understood in their physical sense. Thus, it contrasts with abstract social choice theory as this expressio...

We survey the literature on coalitional games, focusing on recent developments and on the axiomatic approach to the subject.

For the problem of adjudicating conflicting claims, we define a family of two-claimant rules that offer a compromise between the proportional and constrained equal awards rules. We identify the members of the family that satisfy particular properties. We generalize the rules to general populations by requiring “consistency”: The recommendation made...

A group of agents have claims on a resource but there is not enough of it to honor all of the claims. How should it be divided? A group of agents decide to undertake a public project that they can jointly afford. How much should each of them contribute? This essay is an update of Thomson (2003), a survey of the literature devoted to the study of su...

We identify previously unnoticed ways in which agents can strategically distort allocation rules, by affecting the set of “active” agents. (i) An agent withdraws with his endowment. (ii) He gives control of his endowment to someone else and withdraws. (iii) He invites someone in and let him use some of his endowment. (iv) He pre-delivers to some ot...

A group of agents have claims on a resource, but there is not enough of the resource to honor all of the claims. How should it be divided? We survey the recent game theoretic approaches that have been followed in solving such "claims problems".

An allocation rule is ‘consistent’ if the recommendation it makes for each problem ‘agrees’ with the recommendation it makes for each associated reduced problem, obtained by imagining some agents leaving with their assignments. Some authors have described the consistency principle as a ‘fairness principle’. Others have written that it is not about...

In the context of a simple model of public good provision, we study the requirement on an allocation rule that it be immune to manipulation by augmenting one’s endowment through borrowing from the outside world. We call it open-economy borrowing-proofness (Thomson, 2009). We ask whether the Lindahl rule satisfies the property. The answer is yes o...

In the division problem with single-peaked preferences, we analyze the consequences of manipulation for allocation rules. It is conducted by charac-terizing the equilibrium allocations of the direct revelation game of a given rule. For each rule in a wide class of rules, the uniform allocation (i) is the unique strong Nash equilibrium allocation an...

We consider the problem of fairly allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. We address the issue of (Nash)-implementability of solutions to this problem. As the lack of monotonicity of preferences often causes solutions to violate no veto power, the classic Maskin-type theorems cannot be in...

We formulate and study the requirement on an allocation rule that no agent should be able to benefit by augmenting his endowment through borrowing resources from the outside world (alternatively, by simply exaggerating it). We show that the Walrasian rule is not "borrowing-proof" even on standard domains. More seriously, no efficient selection from...

This essay surveys the literature on the axiomatic model of bargaining formulated by Nash ("The Bargaining Problem," Econometrica 28, 1950, 155-162).

This paper constructs a two-country (Home and Foreign) general equilibrium model of Schumpeterian growth without scale effects. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: the permanent effect on growth (PEG) specification, which allows policy effects on long-run growth; and...

We consider the problem of adjudicating conflicting claims in the context of a variable population. A property of rules is “lifted” if whenever a rule satisfies it in the two-claimant case, and the rule is bilaterally consistent, it satisfies it for any number of claimants. We identify a number of properties that are lifted, such as equal treatment...

We consider the problem of allocating some amount of an infinitely divisible and homogeneous resource among agents having claims on this resource that cannot be jointly honored. A “rule” associates with each such problem a feasible division. Our goal is to uncover the structure of the space of rules. For that purpose, we study “operators” on the sp...

We define two families of rules to adjudicate conflicting claims. The first family contains the constrained equal awards, constrained equal losses, Talmud, and minimal overlap rules. The second family, which also contains the constrained equal awards and constrained equal losses rules, is obtained from the first one by exchanging, for each problem,...

For resolving claims problems, we consider the rule that makes awards proportional to claims truncated at the endowment. Using a technique developed by Thomson [Thomson, W., On the existence of consistent rules to adjudicate conflicting claims: a geometric approach, mimeo, 2001.], we show that the two-claimant rule so defined has no consistent exte...

We consider the problem of dividing the cost of a facility when agents can be ordered in terms of the need they have for it, and accommodating an agent with a certain need allows accommodating all agents with lower needs at no extra cost. This problem is known as the “airport problem”, the facility being the runway. We review the literature devoted...

For the problem of adjudicating conflicting claims, we offer simple criteria to compare rules on the basis of the Lorenz order. These criteria pertain to three families of rules. The first family contains the constrained equal awards, constrained equal losses, Talmud, and minimal overlap rules (Thomson, 2007a). The second family, which also contain...

We review the theory of fairness as it pertains to concretely specified problems of resource allocations. We present punctual notions designed to evaluate how well individuals, or groups, are treated in relation to one another: no-envy, egalitarian-equivalence, individual and collective lower or upper bounds on welfare, notions of equal or equivale...

We survey the theory of equity in a variety of concretely specified resource allocation models: classical economies with private goods, economies with production, economies with indivisible goods, when monetary compensations are feasible and when they are not, economies with single-peaked preferences, and economies in which the dividend is a non-ho...

This paper constructs a two-country (Home and Foreign) general equilibrium model of Schumpeterian growth without scale effects. The scale effects property is removed by introducing two distinct specifications in the knowledge production function: the permanent effect on growth (PEG) specification, which allows policy effects on long-run growth; and...

We consider the problem of dividing a non-homogeneous one-dimensional continuum whose endpoints are topologically identified. Examples are the division of a birthday cake, the partition of a circular market, the assignment of sentry duty or medical call. We study the existence of rules satisfying requirements of efficiency, fairness (no-envy), and...

For the problem of adjudicating conflicting claims, a rule is consistent if the choice it makes for each problem is always in agreement with the choice it makes for each “reduced problem” obtained by imagining that some claimants leave with their awards and reassessing the situation from the viewpoint of the remaining claimants. We develop a genera...

We consider the problem of dividing a non-homogeneous one- dimensional continuum whose endpoints are topologically identi¯ed. Examples are the division of a birthday cake, the partition of a circular market, the assignment of sentry duty or medical call. We study the existence of rules satisfying various requirements of fairness (no-envy, egalitari...

We consider the problem of adjudicating conflicting claims, and characterize the family of rules satisfying four standard invariance requirements, homogeneity, two composition properties, and consistency. It takes as point of departure the characterization of the family of two-claimant rules satisfying the first three requirements, and describes th...

For the problem of adjudicating conflicting claims, we consider the requirement that each agent should receive at least 1/n his claim truncated at the amount to divide, where n is the number of claimants (Moreno-Ternero and Villar, 2004a). We identify two families of rules satisfying this bound. We then formulate the requirement that for each probl...

We construct “simple” games implementing in Nash equilibria several solutions to the problem of fair division. These solutions are the no-envy solution, which selects the allocations such that no agent would prefer someone else's bundle to his own, and several variants of this solution. Components of strategies can be interpreted as allocations, co...

We study the behavior of rules for the adjudication of conflicting claims when there is a large number of claimants with small claims. We model such situations by replicating some basic problem. We show that under replication, the random arrival rule [Math. Soc. Sci. 2 (1982) 345] behaves like the proportional rule, the rule that is the most often...

When a firm goes bankrupt, how should its liquidation value be divided among its creditors? This essay is an introduction to the literature devoted to the formal analysis of such problems. We present the rules that are commonly used in practice or discussed in theoretical work. We show how many can be obtained by applying solution concepts develope...

We investigate the existence of consistent rules for the resolution of conflicting claims that generalize the Talmud rule but do not necessarily satisfy equal treatment of equal. The first approach we follow starts from the description of the Talmud rule in the two-claimant case as “concede-and-divide”, and an axiomatic characterization for the rul...

We consider the problem of fair allocation in economies with indivisible objects that may or may not be desirable (for instance, activities that may or may not be pleasurable but have to be carried out unless there are not enough agents for that). We search for efficient solutions satisfying two additional properties. First, each agent should find...

We investigate the existence of rules for the adjudication of conflicting claims satisfying ‘group order preservation’: given two groups of claimants, suppose that the sum of the claims of the members of the first group is greater than or equal to the sum of the claims of the members of the second group. Then, similar inequalities should hold for t...

This is a study of the axiomatic method and its recent applications to game theory and resource allocation. It begins with a user's guide. This guide first describes the components of an axiomatic study, discusses the logical and conceptual independence of the axioms in a characterization, exposes mistakes that are often made in the formulation of...

This book is an invaluable guide for young economists working on their dissertations, preparing their first articles for submission to professional journals, getting ready for their first presentations at conferences and job seminars, or facing their first refereeing assignments. In clear, concise language--a model for what he advocates--William Th...

We offer a guide to the recent literature on the problem of fair allocation, and we present various concepts that have been developed in this literature and the various economic environments in the context of which they have been analyzed.

We offer a guide to the recent literature on the problem of fair allocation, and we present various concepts that have been developed in this literature and the various economic environments in the context of which they have been analyzed.

Le but de cet essai est de présenter la littérature consacrée à l'étude de la monotonie des règles de choix: si les options accessibles à un groupe d'agents se multiplient, chacun d'eux y gagne. Cette condition a été analysée dans les modèles abstraits de la théorie des jeux tels que les problèmes de marchandage et les jeux de coalition. Elle a aus...

The purpose is this essay is to review the literature devoted to understanding the implications of requiring of an allocation rule the property of "monotonicity", which says that if the options available to a group of agents improve, then none of them should lose. Abstract versions of this requirement have been investigated in game-theoretical mode...

The objective of this paper is to describe various applications of a requirement of solidarity pertaining to situations in which the preferences of some of the agents may change. It says that the welfares of all agents whose preferences are fixed should be affected in the same direction: they should all weakly gain, or they should all weakly lose....

The property of "monotonicity" is necessary, and in many contexts, sufficient, for a solution to be Nash implementable (Maskin 1977). In this paper, we follow Sen (1995) and evaluate the extent to which a solution may fail monotonicity by identifying the minimal way in which it has to be enlarged so as to satisfy the property. We establish a genera...

I formulate recommendations to young authors for writing economic theory well. I state general principles of good writing, and emphasize the role of the different components of a paper and the importance of the structure of one's work being clear. I explain how to choose notation and language, and how to present and illustrate proofs.

This paper explains how to represent economies with one private good, one public good, and two agents when the public good is produced from the private good by operating a linear technology, by means of the so-called "Kolm triangle." It also shows the usefulness of this representation in analyzing this class of economies. Copyright 1999 by Blackwel...

This essay is a didactic introduction to the literature on the "consistency principle" and its "converse". An allocation rule is consistent if for each problem in its domain of definition and each alternative that it chooses for it, then for the "reduced problem" obtained by imagining the departure of an arbitrary subgroup of the agents with their...

We consider the problem of allocating a list of indivisible goods and some amount of an infinitely divisible good among agents with equal rights on these resources, and investigate the implications of the following requirement on allocation rules: when the preferences of some of the agents change, all agents whose preferences are fixed should (weak...

We consider the problem of allocating a single infinitely divisible commodity to agents with single-peaked preferences, and establish two properties of the rule that has played the central role in the analysis of this problem, the uniform rule. Among the efficient allocations, it selects (1) the one at which the difference between the largest amoun...

Our objective is to investigate the implications of the “replacement principle” for the fair allocation of an infinitely divisible commodity among agents with single-peaked preferences. The principle says that when one of the components of the data entering the description of the problem to be solved changes, all of the relevant agents should be af...

In these notes we deal with the so called “bargaining problem” (Nash, 1950). Our approach is axiomatic. We search for solutions satisfying some desirable properties (axioms).

The model discussed in Part I is written under the assumption of a fixed number of agents. Here, the number of agents is allowed to vary. How solutions should respond to such changes are formulated as axioms, and additional characterizations of the main solutions are developed. A detailed account of these recent developments can be found in Thomson...

The objective of this work is to present a principle taht has recently played a fundamental role in axiomatic analysis, and a converse of this principle, which has also played an important role. They are now most commonly known under the names of consistence and converse consistency. We survey the applications of the principles to a variety of prob...

I found this paper to be a very innovative and interesting contribution to a topic that in my view has been the object of too little axiomatic analysis, namely intertemporal allocation.

We consider the problem of fairly allocating an infinitely divisible commodity among agents with single-peaked preferences. First, we examine the implications of the requirement that a change in the population affect all agents that are present before and after the change in the same direction. We show that this requirement is met by no selection f...

Fair division is as old as mathematics. According to the Roman historian Proclus, the litigious division of land after the yearly flood of the Nile triggered the invention of geometry by the Egyptians, and the necessities of trade and commerce that of arithmetic by the Phoenicians (see Guilbaud, 1952). The modern literature on fair allocation is ho...

We consider the problem of allocating a list of indivisible goods and some amount of an infinitely divisible good among agents with equal rights on these resources, and investigate the implications of the following requirement on allocation rules: when the preferences of some of the agents change, all agents whose preferences are fixed should (weak...

We consider the problem of fairly allocating an indivisible good to one of several agents equally entitled to it when monetary compensations to the others are possible. Our primary normative concept is no-envy. First, we show that there is no non-manipulable selection from the no-envy solution. Then we study the direct revelation games associated w...

We consider the problem of fair allocation in economies with indivisible goods. Our primary concept is that of an envy-free allocation, that is, an allocation such that no agent would prefer anyone else's bundle to his own. Since there typically is a large set (a continuum) of such allocations, the need arises to identify well-behaved selections fr...

This chapter discusses cooperative models of bargaining. The axiomatic theory of bargaining originated in a fundamental paper by J.F. Nash. Nash introduced an idealized representation of the bargaining problem and developed a methodology that gave the hope that the undeterminateness of the terms of bargaining could be resolved. The canonical bargai...

We consider the problem of fairly allocating an infinitely divisible commodity among a group of agents with single-peaked preferences. We search for solutions satisfying resource-monotonicity, the requirement that all agents be affected in the same direction when the amount to divide changes. Although there are resource-monotonic selections from th...

We consider the problem of fairly allocating an infinitely divisible commodity among agents with single-peaked preferences. We search for methods of performing this division, or solutions, that satisfy the following property of consistency: any recommendation made for any economy is in agreement with the recommendation made for any "reduced" econom...

We formulate and study three concepts of equity designed to capture certain notions of equal, or equivalent, opportunities. The central concept is that of a family of choice sets. Given such a family , a feasible allocation z is alternatively required to be such that (i) there is B such that each agent i maximizes his satisfaction in B at z i , (i...

The authors consider the problem of allocating a bundle of commodities among a group of agents who are collectively entitled to them. It is proved that, for an atomless economy with possibly satiated preferences, any solution that is efficient, equitable, and consistent must select allocations that are supported by equal-budget Walrasian equilibria...

We consider the problem of allocating a single indivisible good to one of several agents when monetary compensations to the losers are possible. We look for solutions to this problem satisfying the property of consistency, which says that the recommendation made for any problem is never contradicted by the recommendation made for any reduced proble...

Consider a heterogeneous but divisible commodity, bundles of which are represented by the (measurable) subsets of the good. One such commodity might be land. The mathematics literature has considered agents with utilities that are nonatomic measures over the commodity (and hence are additive). The existence of ‘α-fair’ allocations, in which each ag...

The paper reexamines the foundations of the axiomatic Nash bargaining theory. More specifically it questions the interpretation of the Nash bargaining solution and extends it to a family of non-expected utility preferences. A bargaining problem is presented as [X, D, greater-than-or-equal-to 1, greater-than-or-equal-to 2] where X is a set of feasib...

An allocation rule, or solution, is consistent if what it recommends for any economy is always ‘in agreement’ with what it recommends for associated ‘reduced economies’. I propose here to evaluate the extent to which a solution may fail to be consistent by identifying its minimal consistent extension, that is, the smallest consistent solution that...

We study the implications for a simple public good model with single-peaked preferences of a ‘replacement principle’. We apply the principle to situations when the preferences of one of the agents may change. Then the principle results in a condition used by Moulin in the context of binary public decision: as a consequence of the replacement, all o...

We enrich the traditional model of bargaining by adding to the disagreement point and the feasible set a point representing claims (or expectations) that agents may have when they come to the bargaining table. We assume this point to be outside of the feasible set: the claims are incompatible, We look for solutions to the class of problems so defin...

In this book, Professor Thomson and Professor Lensberg extrapolate upon the Nash (1950) treatment of the bargaining problem to consider the situation where the number of bargainers may vary. The authors formulate axioms to specify how solutions should respond to such changes, and provide new characterizations of all the major solutions as well as g...

The authors study fair allocation in economies with indivisible goods. They look for subsolutions of the no-envy solution satisfying the property of consistency which says that the desirability of an allocation for some economy is unaffected by the departure of some of the agents with their allotted bundles. The authors show that essentially there...

We consider economies with indivisible goods (houses, jobs) and one infinitely divisible good. An allocation is envy-free if no agent prefers someone else's consumption to his own. An allocation is egalitarian-equivalent if there is a reference bundle that every agent finds indifferent to his consumption. We show that in such economies there may be...

We consider bargaining problems with uncertain disagreement points. We search for solutions providing agents the incentive to reach agreements before the uncertainty is resolved. We identify the weighted Egalitarian solutions as being the only ones in the family of ‘directional solutions’ characterized in an earlier paper [Chun and Thomson (1987)]...