Rawls, Edgeworth, Shapley, Nash: Theories of “justice as fairness,” and to compare it...
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JOURNAL OF ECONOMIC THEORY 24, 1-39 (1981)
Rawls, Edgeworth, Shapley, Nash: Theories of Distributive Justice Re-examined
MENAHEM E. YAARI**'
Department of Economics, Hebrew University, Jerusalem, Israel
Received October 4, 1979; revised January 23, 1980
While preparing for a seminar on “Economics Justice,” I was rather perplexed to find the existing theory of distributive justice in what looked like a state of considerable disarray. Here was an enormous body of writing, dating back almost as far as writing itself, that was as amorphous as putty and almost as yielding when you tried to sink your teeth in it. One could hardly find two writers agreeing on what a third writer had said, to say nothing of the existence of an agreed body of assertions-no matter how small or ’ sketetal-that everyone could adopt as a common point of reference. And it was on this kind of ground that economists were merrily running along, producing such things as “Benthamite” or “Rawlsian” taxation schemes.
Having mentioned John Rawls, I should hasten to say that, in my opinion, one of the most valuable contributions of Rawls’ book [ 191 is indeed its attempt to answer the foregoing criticism. For Rawls had undertaken to state precisely what questions a theory of distributive justice is meant to answer and to lay down a general methodology that would make the answers to these questions deducible from certain ethical premises. Rawls then
* This paper is an outgrowth of a seminar that took place at the Hebrew University of Jerusalem in 1977/1978. I had the pleasure of sharing the responsibility for that seminar with Avishai Margalit, and it is here that I wish to record my debt to him, incurred through many conversations both in and out of the classroom. Thanks are due also to Maya Bar-Hillel, to Gideon Schwartz, and to Amartya Sen for their comments.
‘In this paper, I have made an attempt to provide a coherent account of the issues that make up the theory of distributive justice, to the best of my understanding. The paper is an economist’s way of reading the theory of distributive justice and it focuses on issues which economists tend to raise when discussing problems of distribution. Yet, I should like to emphasize that the paper is intended also for non-economists, and this fact has had an obvious influence upon the style of presentation. To a large extent, this is an essay in the history of thought, so the reader should be alerted not to look for the types of results which usually appear in papers being published in this Journal.
Copyright C 1981 by Academic Press. Inc. All rights of reproduction in any form reserved.
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proceeded to use this methodology to construct his own theory, called “justice as fairness,” and to compare it with other theories of distributive justice, most notably with utilitarianism. Rawls makes a very determined effort to define utilitarianism (as a theory of distributive justice) in terms of the framework that he himself had laid down, so that the differences between the two theories might come into focus. But here I find that Rawls meets with but limited success, and his discussion of utilitarianism, in many instances, is vague and imprecise. Witness the many comments that have appeared since 1971 on the relationship between Rawls and utilitarianism (Arrow , Barry [5, Chap. lo], Gordon , Harsanyi , Lyons 1151, Sen , among others) no two of which seem to agree on what the issues are, let alone on how they might be resolved. Seeing that this was the case, and with a seminar soon to get underway, I had no choice but to sit down and undertake the exercise of writing precisely what it was, to the best of my understanding, that utilitarianism as a theory of distributive justice did say, and then to do the same with what I understand to be Rawls’ position, so as to be able to carry out a rigorous comparison between the two. I started by taking advice from Rawls himself, who says [ 19, p. 221 that “in order to bring out the underlying differences in the simplest way. . . the kind of utilitarianism I shall describe is the strict classical doctrine which receives perhaps its clearest and most accessible formulation in Sidgwick.” Now, I did not find Sidgwick  all that clear, and at certain points I had the feeling that he was being deliberately vague. So, I decided to call in F. Y. Edgeworth as a kind of reinforcement. Edgeworth’s  articulate and methodical account of utilitarian distribution theory is more orthodox and audacious than Sidgwick’s and yet, at the same time, Edgeworth clearly sees himself as building upon the foundations that Sidgwick had laid. Therefore, any contrast that one might draw between Rawls and Sidgwick would no doubt be accentuated and brought more sharply into focus with Edgeworth in the picture.
My version of the confrontation between Sidgwick and Edgeworth on the one hand and Rawls on the other ended very differently from the confron- tation that Rawls himself had set up in his book. Using a standard minimax theorem, I was able to argue that, under very general assumptions, the Sidgwick-Edgeworth theory and the Rawlsian theory are equivalent. So, if one still wants to hold the view that Rawls and classical utilitarianism are fundamentally different from each other, then one would have to show either that I have misstated the Sidgwick-Edgeworth position or that I have misstated Rawls’ position, or else that the general assumptions under which the two positions are equivalent are fundamentally controversial and that the difference between the two theories rests on the failure of these assumptions.
The Sidgwick-Edgeworth brand of utilitarianism has long been passe, mainly because the pseycho-physical foundations of utility have been
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debunked and exposed as useless pseychologism. This immediately brings up a very important question: Can a theory of distributive justice that is founded upon utilitarian principles be consistent with modern utility theory? One would have thought that finding the answer to this question would by now be a matter of looking up the appropriate page in some standard text on social philosophy, social choice, or welfare economics. And a very promising source to look up is Chapter 9 of Sen’s fine book [2 11. But neither this source nor any other source that I have been able to consult gives a definitive answer to the question posed above. The feeling one gets is that, with so many “impossibility theorems” floating around, the answer is probably that a theory of distributive justice which would be both utilitarian and utility-theoretic is impossible. My own position on this question is less pessimistic, and I derive my mild sense of optimism from a framework suggested by Shapley  which appears to me to provide a powerful tool for constructing utilitarian theories of social interaction. (To make my position clear, I should emphasize that the reference here is to the general methodology of Shapley’s paper, and not to the specific conept-the so- called Shapley value-which he uses as a solution for situations of social choice with conflicting interests.)
It makes no sense to try and carry out a direct comparison between Rawls [ 191 and Shapley  because the two theories rest on different foundations. However, it does make sense so try and see what happens when Shapley’s methodology is applied to Rawls’ ethics. Such an experiment, one might hope, would lead to a theory of distributive justice that would be true to Rawls’ principles and yet be free of conflict with modern utility theory. Intrigued to see what would emerge, I performed the experiment, only to find myself re-discovering Nash’s [ 171 theory of bargaining.
These tales and others will be told in the sequel.
I propose to study distributive justice within the general framework of social choice theory. (See, for example, Sen .) To capture the essentials of a simple but non-trivial distribution problem, let me start with two building blocks: First, a non-empty set X and, second, a list of n real-valued functions u1 , u2 ,..., u,, defined on X. The set X is the universe of discourse and its elements are entities among which society could conceribably be called upon to exercise a choice. For the sake of concreteness, I shall refer to the elements of X as distribution schemes, with the interpretation of this term depending on the specific context of any given discussion. Thus, the elements of X could be given such divergent interpretations as alternative constitutions, alternative institutional structures, alternative tax structures,
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alternative ways of dividing up the net national product, alternative ways of cutting a cake at a bithday party, and so forth. The set X of all conceivable distribution schemes will be referred to as the choice space. As for the functions fi,, u2 ,..., I(,, they will be referred to as the utilities of the individuals who make up the society under consideration. In other words, “society” consists of n individuals who are identified by the utilities u, , u2 ,..., u, all of which are functions assigning real numbers to elements of the choice space X. Utilities are assumed to convery information about the individuals’ preferences. The formula
ui(x) > ui( Y)
has the obvious interpretation that individual i prefers the scheme x over the scheme y, in the sense that i would rather see