Abstract
Despite the importance of pioneering work by such precursors as Jean-Charles de Borda and Marquis de Condorcet in the 18th century, it was Kenneth Arrow and his general impossibility theorem that elevated the scientific status of social choice theory into an unprecedented plateau. This paper tries to highlight several unique features of his research program of social choice theory vis-à-vis the classical contributions of Borda and Condorcet, on the one hand, and the “new” welfare economics à la Bergson and Samuelson, on the other hand, as well as to identify several channels through which his impossibility impasse could be circumvented. It is concluded with several personal reminiscences of Kenneth Arrow based on the author’s own experiences with him over four decades.
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Notes
I asked Arrow over dinner at the San Antonio Meeting of the Public Choice Society held in 2001 to give me the first-best, the second-best and the third-best work of his own. The answer was immediate. His ranking was social choice theory and general impossibility theorem in the first place, followed by the theory of contingent commodities and the role of securities in the theory of risk-bearing, and thirdly the existence of an equilibrium in a competitive economy.
As a matter of fact, Arrow’s (1950) first exposition of his general impossibility theorem was for the minimal society in this sense.
If we allow individuals as well as society to express indifference relations, the total number of possible preference orderings on X becomes 13. This does not alter anything essential, and the crucial point we are making in what follows can be made, mutatis mutandis.
For example, \( \alpha \) is a linear preference ordering that ranks x first, y second, and z third.
The Avogadro constant is defined by \( N_{A} \) = 6.022 140 857(74) \( \times \)\( 10^{23} \) mol−1 in the International System of Units.
More generally, if there are m social alternatives and n individuals in the society, where 3 \( \le \)m\( < \) + \( \infty \) and 2 \( \le \)n\( < \) + \( \infty , \) the total number of the Arrow aggregation rules or processes amounts to \( (m!)^{{(m!)^{n} }} \), which reduces to \( 6^{36} \) if m = 3 and n = 2, but becomes an astronomically large number for reasonably large values of m and n. The unreality of enumerating all possible rules and checking their eligibility one after another is obvious.
I owe thanks to Eric Maskin who brought me to this observation.
The Condorcet paradox requires at least three voters, so that it is not relevant in the context of social decision-making in the minimal society.
Arrow (1983a, p. 23).
I am grateful to Amartya Sen who emphasized the importance of this distinction between the informational disciplines within which Condorcet and Arrow pursued their respective exercises.
The contrast between these two scenarios of normative economics is strongly reminiscent of Amartya Sen’s contrast between transcendental institutionalism and the comparative assessment approach, which Sen identified in the context of the ideas of justice [Sen (2009)]. Those who are interested in this contrast are referred to Suzumura (2018a, b) for more details in concrete contexts.
(B–P) stands for “Bentham and Pigou” for short, and individual utility functions are cardinally measurable and interpersonally unit-comparable.
Recollect that Arrow (1983b, p. 18) himself observed that “[s]urprisingly enough, there is only one mention of summing utilities [in Pigou’s The Economics of Welfare] and that is very incidental. Although the whole work is devoted to optimizing, there is no explicit formulation of a maximand. For the most part, the criterion is increase in the national income (“national dividend” in Pigou’s language). But he is at pains to point out national income is itself an imperfect approximation, though I am not clear what it was supposed to approximate.”
As a matter of fact, there is the second school of “new” welfare economics, which is based on the piecemeal welfare criteria of hypothetical compensation principles. It was Nicholas Kaldor (1939) and John Hicks (1940) who kicked off this school of “new” welfare economics. It is to be highlighted that there are two sharp distinctions between the social welfare function school led by Bergson and Samuelson, and the hypothetical compensation principles school led by Kaldor and Hicks. The first distinction is that the social welfare function school is a resurgence of the first scenario of welfare economics, viz. the constrained maximization paradigm, whereas the compensation principles school is a lineal descendent of the piecemeal welfare improvement paradigm that originates in Pigou. The second distinction is that the social welfare function school does not ask the origin and holder of the social value to be optimized, whereas the compensation principles school attempts to construct the piecemeal welfare criteria from within the economic system under scrutiny. See Suzumura (1999, 2018a, b) for more detailed contrast between these two schools of “new” welfare economics.
The Bergson–Samuelson social welfare function h is said to be individualistic if and only if it depends on the social state x through the mediation of the profile of individual utility functions u = (u1, …, ui, …, un).
Recollect that Arrow’s concept of collective rationality is such that the social choice function is rationalizable through the optimization of an underlying social value ordering over the set of feasible social states.
Serge-Christophe Kolm (1996, p. 439) made the following observation to a similar effect: “The requirement of a social ordering is indeed problematic at first sight: Why would we want to know the 193th best alternative? Only the first best is required for the choice.”
The simplest possible instance of the voting paradox consists of three candidates x, y, and z, and three voters 1, 2, and 3, where these voters have the following diametrically heterogeneous preference orderings on the set {x, y, z} of candidates:
$$ x \succ_{1} y \succ_{1 } z, y \succ_{2} z \succ_{2 } x, z \succ_{3} x \succ_{3 } y $$(VP)If we apply the simple majority decision rule to this profile \( \varvec{\succ} \) = (\( \succ_{1} , \succ_{2} , \succ_{3} \)), we obtain the cyclic social preference \( x \succ y \succ z \succ x \). It is clear that this cycle is caused by the extreme heterogeneity of voters’ preferences (VP).
This definition of non-consequentialism presupposes that the value of an opportunity set is entirely captured by its size measured by the number of alternatives within the set, which is admittedly restrictive. It requires elaborations in one way or the other in the future.
These four axioms for the extended social welfare function are counterparts to the four axioms that are required of the Arrow social welfare function. It should be obvious how these axioms for the extended social welfare function can be defined by modifying the original Arrow axioms for his social welfare function.
See, among others, Sen (1977) who provided an axiomatic characterization of the Rawlsian principle of leximin justice within this extended framework of social choice theory.
An attempt was made in Suzumura (1980) to explore this possibility.
For more details of this extended social choice framework in the specific context of game-form rights, those who are interested are cordially referred to Prasanta Pattanaik and Kotaro Suzumura (1996).
Recollect that violations of transitive indifference are quite likely to surface in practical choice situations. Indeed, Duncan Luce’s (1956) well-known coffee-sugar example provides a plausible argument against the transitivity of indifference relation: the inability of a decision-maker to perceive “small” differences in alternatives is bound to lead to intransitive overall preferences. Thus, full transitivity is often too strong to impose in the context of individual as well as social choice.
Sen’s Pareto extension rule fE is a preference aggregation rule that aggregates each profile R = (R1, R2, …, Rn) of individual preference orderings into a social weak preference relation Re such that, for each pair of social alternatives x, y\( \in \)X, (x, y) \( \in \)Re if and only if (y, x) \( \notin \)P(Rp), where Rp = \( \mathop \cap \nolimits_{i \in N } \)Ri. It is easy to verify that Re, thus defined, is quasi-transitive, and fE satisfies the axioms of unrestricted domain, Pareto principle, independence of irrelevant alternatives, and non-dictatorship.
To be precise, a preference relation R on X is Suzumura-consistent if and only if there exists no S-cycle of any finite order, where a t-tuple of alternatives (x1, x2, …, xt) is an S-cycle of order t (3 \( \le \)t\( < \)\( \infty ) \) if and only if (x1, x2) \( \in \)P(R), (xh, xh+1) \( \in \)R (h = 2, …, t\( - \) 1) and (xt, x1) \( \in \)R. See Suzumura (1983, 2009), Bossert and Suzumura (2010, Chapter 2), and Sen (2018) for several crucial properties of the concept of Suzumura-consistency.
A crucial property of Suzumura-consistency is Suzumura’s ordering extension theorem which asserts that a binary relation R on the universal set X of alternatives has an ordering extension R* such that R\( \subseteq \)R* and P(R) \( \subseteq \)P(R*) if and only if R satisfies Suzumura-consistency. In many contexts of individual and social choice, Suzumura’s ordering extension theorem has proved to be of crucial importance.
This quote is from Wesley Mitchell (1937), which was cited by Arrow in Social Choice and Individual Values, p. 11.
This Festschrift was subsequently published by Basu et al. (1995). Arrow’s contribution is Chapter 1 of this Festschrift, which is entitled “A Note on Freedom and Flexibility.”
This lecture was subsequently published in American Economic Review: Papers and Proceedings as Arrow (1994).
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Acknowledgements
I delivered a speech at the Academic Tribute to Kenneth Arrow held at the Frances C. Arrillaga Alumni Center of Stanford University on October 9, 2017, which was organized by Matthew Jackson, Alvin Roth, and John Shoven. This paper partly capitalizes on my Stanford speech in memory of Kenneth Arrow. I am most grateful to Eric Maskin, Amartya Sen, Walter Bossert, and Marc Fleurbaey for their helpful comments that enabled me to improve it. Needless to say, I am solely responsible for any defect and opaqueness that may remain. Financial support from a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan for the Project on Normative Economics with Extended Informational Bases, and the Reexamination of Its Doctrinal History (Grant number 16H 03599) is gratefully acknowledged.
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Paper prepared for the Special Issue of Social Choice and Welfare in Memory of Kenneth J. Arrow. Joint editors Marc Fleurbaey, Eric Maskin, Amartya Sen, and Kotaro Suzumura.
Kotaro Suzumura: Professor Emeritus of Hitotsubashi University, Professor Emeritus and Honorary Fellow of Waseda University, and Member of the Japan Academy.
