We study the implications for a simple public good model with single- peaked preferences of a "replacement principle". We apply the principle to situations where the preferences of one of the agents may change. Then, the principle results in a condition used by Moulin in the contex of binary public decision: as a consequence of the replacement, all other agents should be affected in the same direction; they all gain or they all lose. We characterize the family of efficient solutions satisfying the property. It is a subfamily of the family of strategyproof solutions introduced by Moulin under the name of "generalized Condorcet-winner solutions".
Key-words: replacement principle, public decision, single-peaked preferences, generalized Condorcet solutions.
Population-monotonicity is a property of resource allocation rules which says that if the number of agents increases while the resources at their disposal remain fixed, so that the profile of welfare levels chosen for the initial group remains feasible only by ignoring the newcomers, then none of the agents initially present should gain. The implications of abstract versions of this requirement have been investigated in game-theoretical models such as bargaining problems and coalitional form games; the requirement, and a number of variants, have also been the object of several recent studies in the context of resource allocation, in classical models as well as in non-classical models, including economies with public goods, economies with indivisible goods, and economies with single-peaked preferences, both in the private good and in the public good cases. The purpose of this paper is to survey this literature.
Key-words: population-monotonicity, bargaining theory, coalitional form games, allocation problems.
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 with subsolutions of the no-envy solution. The set of equilibrium allocations of any one of them coincides with the set of envy-free allocations for the true preferences.
Key-words: fair allocation, indivisible goods, no-envy solution, manipulation games, Nash implementation.
The purpose of this essay is to survey the current state of the literature on cooperative models of bargaining.
Key-words: bargaining theory, cooperative models, axiomatic methods, Nash solution, Kalai-Smorodinsky solution, egalitarian solution.
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 continuum of such allocations, we search for selections from the no-envy solution. First, we establish the non-existence of population monotonic selections. Then we propose a variety of selections motivated by intuitive considerations of fairness.
Key-words: fair allocation, indivisible goods, population-monotonicity, selections.
We consider the problem of choosing a level of a public good when agents have continous and single-peaked preferences. We search for solutions satisfying efficiency and population-monotonicity (Thomson, 1983): the requirement that upon the arrival of additional agents, all agents initially present be made weakly worse-off, or equivalently, that upon the departure of some of the agents, all remaining agents be made weakly better-off. We characterize the class of solutions satisfying these requirements. It is a subclass of the class of generalized Condorcet-winner solutions (Moulin, 1980).
Key-words: population-monotonicity, public decision, single-peaked preferences, generalized Condorcet-winner solution.
The purpose of the paper is to present the current status of the literature devoted to the axiomatic analysis of concretely specified economic models. We introduce the main principles that have been used in this literature and the various models that have been recently analyzed in the axiomatic mode, briefly describe the main results, and sketch directions for further research.
Key-words: axiomatic method, resource allocation.
This paper is devoted to a presentation of the various notions of implementation. We discuss the required relationship between the set of equilibrium outcomes of the game form and the set of outcomes chosen by the correspondence that it is supposed to implement. Our points are the following: First, when the concept of a correspondence is understood in its full generality, as a mapping from a domain of preference profiles to some outcome space embodying desirable social norms, it is full implementation that is the correct objective. Second, the theoretical problem of characterizing the class of partially implementable correspondences is mathematically identical to the problem of characterizing the class of fully implementable correspondences. Third, we point out a useful connection between implementation theory and the theory of non-cooperative games. Here too, full implementation is the relevant notion. When full implementation is not possible, however, one may want to consider partial implementation; but then we suggest "maximal partial implementation" or "minimal overimplementation" as the correct second-best objective. These concepts are all well-defined whenever the conditions for implementation are closed under intersection and union, as is the case for the conditions that play a central role for Nash implementation. Finally, we also discuss a notion of approximate implementation.
Key-words: partial implementation, full implementation, overimplementation, approximate implementation.
The objective of this paper is to understand the implications of the "replacement principle" for the fair allocation of an infinitely divisible commodity among agents with single-peaked preferences. A general formulation of the principle is as follows: 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 affected in the same direction. We apply the principle to situations in which the preferences of one of the agents may change, under the name of replacement-domination. We show that there is no selection from the no-envy and Pareto solution satisfying replacement-domination. Then, we weaken the requirement by limiting its applications to situations in which the change in the preferences is not so disruptive that it turns the economy from one in which there is "too little" of the commodity to one in which there is "too much", or conversely. We show that there is only one selection from the no-envy and Pareto solution satisfying this property of one-sided replacement-domination and the standard condition of replication-invariance. It is a solution, known as the "uniform rule", that has played a central role in previous analyses of the problem.
Key-words: replacement-domination, single-peaked preferences, no-envy, uniform rule.
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, consumptions, permutations, points in simplices of dimensionalities equal to the number of goods or to the number of agents, and integers. We also propose a simple game implementing the Pareto solution and games implementing the intersections of the Pareto solution with each of these solutions.
Key-words: Nash implementation, no-envy, divide-and-permute.
We consider the problem of allocating indivisible goods and some amount of an infinitely divisible good among agents with equal rights on these resources. We 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 (weakly) gain, or they all (weakly) lose. This condition is a particular application of a general principle of solidarity discussed in Thomson (1990) under the name of "replacement principle". We look for selections from the no-envy solutions satisfying this property. We show that in the general case, there is no such selection. However, in the one-object case (a single prize), there is only one selection from the no-envy solution satisfying the property. It is the solution that always selects the worst envy-free allocation for the winner of the prize.
Key-words: No-envy, indivisible goods, replacement principle.
This essay is an introduction to the literature devoted to the axiomatic analysis of bankruptcy problems. We present the rules that are commonly used in practice or discussed in the theoretical literature (the proportional rule, the Talmudic solution, the constrained equal award solution), formulate a number of appealing properties that one may want rules to enjoy, first in a setting in which the number of claimants is fixed, then in a richer framework in which this number is allowed to vary, compare the rules on the basis of these properties, and search for rules satisfying the greatest number of the properties together. We also show how most of the important rules in the literature can be obtained by applying solution concepts developed in the theory of cooperative games (the theory of bargaining and the theory of coalition form games). We present several models of bankruptcy as a non-cooperative game. Finally, we consider "dual" situations where the amount to divide is more than sufficient to cover the claims - this is the problem of surplus sharing - and models where the feasible set is specified in utility space.
The problem of identifying well-behaved taxation rules is formally identical to that of identifying bankruptcy rules, and all of the results that we present here can be reinterpreted in that context.
Key-words: axiomatic analysis, bankutpcy rules, proportional taxation, Talmudic solution.
The objective of this work is to present a principle that 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 consistency and converse consistency. We survey the applications of the principles to a variety of problems in game theory, economics, and political science. They are bargaining problems, coalitional form games, strategic games; bankruptcy and taxation problems, quasi-linear cost allocation problems, public good decision problems, pricing problems; fair division in classical economies, in economies with indivisible goods, and in economies with single-peaked preferences; apportionment problems, and finally matching problems. We also present related issues and sketch directions for further research.
Key-words: consistency principle, converse consistency, bargaining, coalitional form games, strategic form games, bankruptcy, cost allocation, fair allocation, apportionment, matching.
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