Economic Impossibility Theorems
“Science is only possible because some things are impossible.”
— John Barrow (1998, p. vii)
According to Karl Popper (1959, pp. 48, 247), a scientific law is a set of “prohibitions.” It prohibits or restricts a particular set of otherwise possible outcomes. From this perspective, scientific discovery identifies new limits, opening unchartered opportunities for innovation and creativity.
In the 20th century, scientific progress reached new heights. Werner Heisenberg’s Uncertainty Principle, Kurt Gödel’s Incompleteness Theorem, and Alan Turing’s Undecidability Theorem, among other discoveries, launched a new era whereby science and reason could discover their own limits. If there is a single property characterizing modern science, it is the recognition of its internal boundaries: boundaries of what we can know and of what we can do. These boundaries, as Barrow (1998, p. 252) points out, might be more valuable than the content of our knowledge:
“Our knowledge about the Universe has an edge. Ultimately, we may even find that the fractal edge of our knowledge of the Universe defines its character more precisely than its contents; that what cannot be known is more revealing than what can.”
Impossibility and incompleteness are deeply interlinked as if they were two sides of the same coin. This is most obvious in Gödel’s incompleteness theorem. The theorem shows that a formal axiomatic system rich in arithmetic cannot be both complete and consistent. That is, it cannot provide proof for every true sentence of the system without falling into contradiction. The price of consistency, therefore, is incompleteness. On the other hand, an impossibility result implies that some of the axioms or postulates need to be relaxed or modified to preserve consistency. This might potentially lead to the indeterminacy of some of the propositions of the system.
Impossibility and incompleteness appear to be two sides of the same coin
There has been a number of impossibility theorems in economics, but they are rarely studied or analyzed in one context. We try here to highlight three major theorems (or families of theorems) to learn the common lessons:
- The Impossibility of a Collective Choice Rule.
- The Impossibility of a Neoclassical Market Demand.
- The Impossibility of a Complete Contract.
This article will discuss the first theorem. The other two theorems will be addressed in two subsequent articles respectively.
The Impossibility of a Collective Choice Rule
Probably the most famous impossibility theorem in economics, and the first outside mathematics, is that of Kenneth Arrow (1921–2017). He developed this theorem while still a graduate student at Columbia University during 1948–1949. He published his results in the book Social Choice and Individual Values in 1951, then updated in 1963. Through this work, Arrow established a whole new field in economics and political science (Lützen, 2018).
The motivation for the study of social choice, as Arrow (1972, p. 128) explains, is that there are a great many situations in which the replacement of the market by collective decision-making is necessary or at least desirable, particularly in cases of externalities, increasing returns, and market failure. Technically, however, the theorem applies to both collective choice mechanisms such as voting and the market mechanism, as we shall see below.
The genesis of Arrow’s Theorem shares many similarities with earlier impossibility theorems in mathematics
The genesis of Arrow’s Impossibility Theorem shares many similarities with earlier impossibility theorems in mathematics, particularly those of Gödel, Tarski, and Turing. These theorems were initially motivated by set theory paradoxes like Russell’s. These paradoxes were thought initially to be marginal or peripheral, and they can be handled away sooner or later. However, these paradoxes turned out to be the tip of a vast iceberg that transformed mathematics and logic in the twentieth century (Kline, 1980).
Arrow majored in mathematics in his undergraduate study at the City College in New York. He was interested in mathematical logic, and in 1941, he took a course from the great mathematician Alfred Tarski. Tarski was considered one of the greatest logicians of the twentieth century, often regarded as second only to Gödel. He is famous for his 1933 “impossibility” theorem regarding truth: The truth of a language cannot be defined within that language. Arrow learned from Tarski, among other things, the axiomatic method of ordered sets which he applied extensively in his impossibility theorem.
As a graduate student, Arrow was investigating whether a formal procedure can construct a collective rational ordering from a set of individual rational orderings. He independently discovered the voting paradox that has been known for centuries. Suppose we have three individuals with the following preferences between three alternatives, x, y, and z:
- x, y, z
- y, z, x
- z, x, y
For these preferences, the majority rule will result in a cyclical ordering: the majority prefers x to y, y to z, and z to x. In other words, the majority rule may result in an “irrational” outcome.
Arrow wanted to find out if there were other possible rules for aggregating the preference orderings of individuals into a collective rational ordering. Arrow (1972) calls such a rule a “constitution,” by which he means a rule which associates with each possible set of individual preference orderings a collective or social choice rule.
Requirements for the Collective Rule
Arrow specified a set of reasonable, non-trivial requirements for the constitution or the collective choice rule (Arrow, 1963, 1972):
- Collective Rationality: For any set of complete and transitive individual rankings of social alternatives, the collective choice rule shall produce a complete and transitive ranking of the social alternatives.
- Pareto Principle: If each individual prefers x to y, then the collective rule will result in the preference of x to y.
- Independence of irrelevant alternatives: The collective ranking of x vs. y is independent of the ranking of other alternatives.
- Non-dictatorship: No single individual can impose his or her preference ordering on the collective ordering.
Arrow proved the stunning result that no formal rule satisfies all these four requirements. The centuries-old voting paradox is no more a curiosity. In Arrow’s (1963, p. 59) words, “There is no method of voting which will remove the paradox of voting, neither plurality voting nor any scheme of proportional representation, no matter how complicated.”
It might be relevant to point out that, in 1947, Kurt Gödel, in preparation for obtaining his American citizenship, told his fellow economist Oskar Morgenstern that he discovered a logical flaw in the US Constitution (Dawson, 1997, pp. 179–180). To the extent that the US Constitution represents a social choice rule, then Gödel was probably not too far from the discovery of Arrow. Incidentally, there is no evidence that the two men met despite being contemporaries with some common interests.
The Market as a Collective Choice Rule
Arrow was clear that the collective rule applies not only to voting mechanisms but to the market mechanism as well. For example, in the early pages of the book, he writes (1963, p. 2): “The methods of voting and the market, on the other hand, are methods of amalgamating the tastes of many individuals in the making of social choices.”
He then points out:
“In the following discussion of the consistency of various value judgments as to the mode of social choice, the distinction between voting and the market mechanism will be disregarded, both being regarded as special cases of the more general category of collective social choice.”
He cites several writers who considered the equivalence of the two mechanisms, including Frank Knight (p. 6): “Knight has also stressed the analogy between voting and the market in that both involve a collective choice among a limited range of alternatives.”
He reiterates this point after proving the impossibility theorem (p. 59): “Similarly, the market mechanism does not create a rational social choice.” As we shall see later in a subsequent article in this series, this can be linked to the theorems of Sonnenchein, Mantel, and Debreu (SMD) on the indeterminacy of the aggregate excess demand function.
The Undecidability of a Collective Choice Rule
Following Arrow (1963, p. 2), it should be emphasized that the impossibility theorem is concerned only with formal rules of collective choice. The theorem says that there is no formal procedure that fulfills the above four requirements simultaneously.
Arrow’s impossibility theorem could be framed within the theorem of Alan Turing in 1936 on the impossibility (in modern terms) of a computer program that can decide a priori whether a program, given an arbitrary input, would halt after execution. This is called “The Halting Problem.” The Halting Problem is undecidable: There is no computer program or algorithm that can answer “yes” or “no” to the question, “Would a program G halt upon executing an arbitrary input?”
The result was generalized by Henry G. Rice in his dissertation in 1951 to any “non-trivial” property and not only halting. A property is said to be “non-trivial” if:
- It applies to some programs but not others. A property that applies to all programs is a trivial one.
- It is semantic rather than syntactic. That is, it is related to the program’s behavior in terms of input-output rather than to its code or syntax.
(Harel, 2000, p. 54).
Thus, no program can decide upfront whether a given program has a non–trivial property, e.g., whether the program is a virus or malware (Chaitin et al., 2011, p. 13).
Since Arrow made it clear that his theorem is concerned only with a formal procedure that maps individual orderings to collective orderings, it is legitimate to consider a collective rule as an algorithm or a computer program that takes as input an arbitrary list of individual rational orderings and produces as output a collective rational ordering with a certain set of non-trivial properties.
The set of requirements that Arrow imposed on the collective choice rule was a set of non-trivial properties. Dictatorship, it should be noted, is a trivial property because it applies the same ordering regardless of the individual orderings. Non-dictatorship, therefore, is a non-trivial property.
Now, we may ask: Can we find an algorithmic collective rule that fulfills a set of non-trivial properties? By Rice’s theorem, the answer would be no. More accurately, there is no algorithm to verify the existence of such an algorithm. No mechanical procedure can tell us a priori if a collective rule satisfies a set of non-trivial properties, including those stated by Arrow.
The application of Rice’s theorems to collective choice generalizes the results of Arrow’s theorem
To elaborate, suppose someone claims that a particular algorithm, G, serves as a collective rule that satisfies a set of non-trivial properties P. How to verify this claim? Since there are potentially unlimited combinations of individual rational orderings, we will not be able to examine all these combinations. So, we need another algorithm, Gʹ, to verify that G fulfills the set P. But how to know if the algorithm Gʹ really works? We need another one, Gʹʹ, to verify that Gʹ verifies that G fulfills P, ad infinitum. So, no algorithm will verify the claim that G fulfills P.
Arrow’s theorem, on the other hand, shows the non-existence of a collective rule for a specific set of non-trivial properties. So it is more specific and direct.
The application of Rice’s theorems to the collective choice problem can be viewed as a generalization of Arrow’s theorem. This generalization reinforces Arrow’s perspective that his conclusions apply broadly across collective choice mechanisms, including the market mechanism, not only voting.
Conventions and Social Norms
Is there a way out of this impossibility? Recall that the impossibility theorem is concerned with a formal procedure or a mechanical process to decide the collective choice. However, this does not exclude non-mechanical or non-algorithmic processes. The most obvious of such processes is social norms.
Social norms are informal rules that govern the behavior and interactions of members of a community or a society. Arrow (1963, 1970) identified conventions and social norms as a method for collective choice whereby all members share the same “tastes” or preference orderings regarding social alternatives. Recent research shows why social norms are important in the economy and where they fit in economic analysis.
Social norms are essential in the presence of multiple equilibria, asymmetric information, and fundamental uncertainty. They also serve as an alternative to legal rules, as they may internalize negative externalities at a relatively low cost. Norms are critical in coordinating activities and converging expectations (Bicchieri et al., 2018).
In the presence of undecidability, norms are critical for filling in the inherent uncertainty. “Inertia” resolves the dilemma of indeterminacy and brings consistency and predictability to the decision process (Bewley, 1986).
Conclusion
Arrow’s impossibility theorem was probably the first impossibility theorem outside pure mathematics. It successfully redirected economics and political science toward a new conceptualization of collective choice.
While the theorem emphasizes the impossibility of a formal procedure to construct a collective rational ordering, it indirectly points to the role of non-algorithmic processes like social norms in smooth and stable decision-making in the economy. The significance of such non-algorithmic processes highlights the incompleteness of the axiomatic approach to modeling the collective choice problem. This reinforces the point made earlier that, in many cases, impossibility and incompleteness appear to be two sides of the same coin.
References
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