Arrow's Theorem and the Limits of Social Welfare

Introduction

Hello and welcome back. In our previous lessons, we explored how different social welfare functions (SWFs), specifically utilitarian and prioritarian ones, could be used to evaluate policy outcomes. We treated these SWFs as given mathematical tools for aggregating individual well-being into a social judgment.

Today, we take a step back to question a fundamental assumption of that entire project. Is it actually possible to construct a social welfare function that is both rational and respects basic democratic principles? This brings us to one of the most profound and unsettling results in 20th-century political and economic theory.

Our learning outcome for this lesson is to analyze the implications of Arrow's Impossibility Theorem for the philosophical project of aggregating preferences into a social welfare function. We will dissect Kenneth Arrow's famous theorem, understanding its conditions, its logic, and its far-reaching consequences for any theory of justice that seeks to derive a "will of the people" from the preferences of individuals. This exploration will directly challenge the welfarist foundations we've been examining and set the stage for theories of justice that reject preference aggregation altogether.

1. The Problem: From Individual to Social Preferences

The core challenge is translating a collection of individual preference rankings into a single, coherent social preference ranking. A simple majority vote seems like a natural way to do this, but it quickly runs into trouble. This issue, known as the Condorcet Paradox or the paradox of voting, was identified as early as the 18th century.

Let's watch a short video that illustrates this paradox and sets the stage for Arrow's more general investigation.

The video demonstrates that with three voters and three options (Biden, Jorgensen, Trump), majority rule can produce a cycle: Trump is preferred to Biden, Biden is preferred to Jorgensen, but Jorgensen is preferred to Trump. This is an intransitive social preference. Such a cycle is problematic for two key reasons:

1. Conceptual incoherence: It doesn't yield a "best" choice. Any option can be beaten by another.
2. Practical instability: The winner depends entirely on the order in which the pairwise votes are taken, making the process vulnerable to agenda-setting manipulation.

Kenneth Arrow's genius was to show that this isn't just a quirk of majority rule. He proved that any system for aggregating preferences that meets a few seemingly reasonable conditions will inevitably lead to an unacceptable outcome.

To understand his theorem, we first need to formalize the problem.

As the article clarifies, an SWF is a function $f$ that takes a preference profile $⟨R_1,\dots ,R_n⟩$ (a list of individual preference orderings) as input and produces a single social preference relation $R$ as output. Arrow's goal was to find an $f$ that was both rational and fair for any possible input profile.

2. The Conditions for a "Fair and Rational" SWF

Arrow proposed a set of conditions that any desirable SWF should satisfy. On their own, they each seem entirely reasonable.

1. Unrestricted Domain (U): The SWF must be able to produce a social ordering for any logically possible combination of individual preference orderings. A democratic system shouldn't break down just because people have unusual or conflicting preferences.

2. Social Ordering (SO): The output of the SWF must always be a complete and transitive ranking of the alternatives. It must not produce the kind of cycles we saw in the Condorcet paradox.

3. Weak Pareto Principle (P): If every single individual prefers alternative $x$ to alternative $y$, then the social ordering must also rank $x$ over $y$. This is a minimal condition of responsiveness to unanimous preferences.

4. Non-Dictatorship (D): The SWF cannot simply be the preference ordering of a single individual, ignoring everyone else. The social outcome should not be determined by a dictator.

5. Independence of Irrelevant Alternatives (IIA): The social ranking of any two alternatives, $x$ and $y$, should depend only on the individual preferences between $x$ and $y$. Preferences for a third, "irrelevant" alternative like $z$ should not affect the social ranking of $x$ versus $y$.

Of these, IIA is the most complex and controversial, but it has a powerful intuition. It's a consistency requirement: the social verdict on $x$ vs. $y$ should be the same whether the third candidate is Nader, Lincoln, or doesn't exist. This prevents strategic manipulation where, for instance, you could reverse the social ranking of Coke vs. Pepsi by changing your opinion on Sprite.

The SEP article provides a deep dive into the rationale and criticisms for each of these conditions. A thorough understanding of them is key to grasping the theorem's philosophical weight.

3. The Impossibility Theorem and its Proof

Having laid out these five reasonable conditions, Arrow proved a startling result:

Arrow's Impossibility Theorem: For any society with at least three alternatives and at least two individuals, no social welfare function can simultaneously satisfy Unrestricted Domain, Social Ordering, the Pareto Principle, Non-Dictatorship, and Independence of Irrelevant Alternatives.

In short, it is impossible to devise a preference aggregation system that meets all these criteria of fairness and rationality. If you satisfy U, SO, P, and IIA, you will find that your system is a dictatorship.

Given your mathematical background, you might appreciate seeing the logic of the proof. We won't go through a fully formal derivation, but the following article by Amartya Sen (who won his Nobel in part for work in this area) provides a beautifully elegant and intuitive proof.

Let's summarize Sen's proof sketch, as it's quite powerful:

1. Starting Point: The Pareto Principle implies that the set of all voters is a "decisive set" (if everyone in the set prefers $x$ to $y$, then society prefers $x$ to $y$).
2. Lemma 1: Spread of Decisiveness. Sen shows that if a set of voters $G$ is decisive over just one pair of alternatives, the axioms of U and IIA force it to be decisive over all pairs of alternatives. Decisiveness is an all-or-nothing property.
3. Lemma 2: Contraction of Decisive Sets. Sen then shows that if any decisive set $G$ with more than one person is split into two parts ($G_1$ and $G_2$), then one of those two smaller parts must also be a decisive set.
4. The Conclusion: You start with the set of all voters, which is decisive. By Lemma 2, you can always find a smaller, proper subset that is also decisive. You can apply Lemma 2 again to that smaller set, and again, and again. Since the number of voters is finite, this process must eventually terminate at a decisive set containing just one person. But a single-person decisive set is, by definition, a dictator. This violates the Non-Dictatorship condition.

Therefore, the five conditions are mutually inconsistent.

Test your understanding!

In Sen's proof of the "Contraction of Decisive Sets" lemma, he partitions a decisive set G into G1 and G2. He then constructs a scenario of preferences and considers two possibilities for the social ranking of alternatives $x$ and $z$. Why must one of the subsets, G1 or G2, end up being decisive, regardless of how the social ranking of $x$ and $z$ turns out?

Show answer

Sen's argument proceeds by cases. In the constructed preference profile, G is decisive for $x$ over $y$, so we know society prefers $x$ to $y$.

• Case 1: Society ranks $z⪰x$ (z is preferred or indifferent to x). By transitivity, since $z⪰x$ and $x\succ y$, it must be that society prefers $z\succ y$. But in the constructed profile, only the members of G2 were specified to prefer $z$ to $y$. By IIA, the social ranking of $z$ vs. $y$ can only depend on individual rankings of $z$ vs. $y$. Since G2's unanimous preference dictates the outcome, G2 is shown to be decisive over $\{z,y\}$. By the "Spread of Decisiveness" lemma, G2 is globally decisive.

• Case 2: Society does not rank $z⪰x$, which means society must rank $x\succ z$. In the constructed profile, only the members of G1 were specified to prefer $x$ to $z$. By the same logic as above, this means G1 is decisive over $\{x,z\}$, and therefore globally decisive.

Since these two cases are exhaustive, it must be that either G1 or G2 is decisive. The trap is sprung either way.

4. Implications for Justice and Social Welfare

What does this impossibility result mean for the project of utilitarianism, prioritarianism, and welfare economics in general? It seems devastating. However, the implications are more nuanced.

Arrow's theorem is about the impossibility of aggregating ordinal preferences (simple rankings). The SWFs we studied in the previous lessons, which form the basis of modern welfare economics, rely on a much richer informational base: cardinal and interpersonally comparable utility.

1. Cardinal Information: They assume we know the intensity of preferences (e.g., you prefer $x$ to $y$ by a lot, while I only prefer $y$ to $x$ by a little).
2. Interpersonal Comparability: They assume we can meaningly compare utility across people (e.g., the benefit of one dollar is greater for a poor person than for a rich person).

Arrow's framework explicitly forbids this kind of information. His IIA condition, in concert with Unrestricted Domain, effectively strips out any way to use information about preference intensity. The theorem, therefore, doesn't directly refute utilitarianism or prioritarianism. Instead, it serves as a powerful, formal justification for why they must use this richer, non-ordinal information. It shows that if you confine yourself to the impoverished world of pure preference rankings, the entire project of building a rational and fair SWF is doomed.

To escape Arrow's trap, you must relax one of the conditions. For welfare economics, the escape route is to reject the informational constraints imposed by Arrow's setup, which is effectively a rejection of his specific formulation of IIA in favor of a framework that allows for cardinal utility.

As Sen puts it in his reflection on the theorem:

"the combination of unrestricted domain, independence and the Pareto principle ... produce both the spread of decisiveness and the contractability of decisive sets. ... This is what makes the permitted social welfare functions be confined to the class of voting rules. So, in this sense, it is wrong to think of the Arrow result as merely extending the Condorcet paradox to all voting rules. It first establishes that the permitted social welfare functions must be voting rules (that is the big intermediate result), and then generalizes the Condorcet paradox."

The theorem proves that any procedure satisfying his axioms must ignore so much information that it acts like a simple voting rule, and all such rules are flawed. The philosophical project of aggregating well-being (as in utilitarianism) rather than just votes gets around this by using the very information Arrow's axioms exclude.

Conclusion

In this lesson, we have explored one of the cornerstones of modern political and economic theory.

Key Takeaways:

• Arrow's Impossibility Theorem states that no social welfare function can satisfy a set of simple, desirable conditions (Unrestricted Domain, Social Ordering, Pareto Principle, IIA, and Non-Dictatorship) when choosing among three or more alternatives.
• The result is not just a peculiarity of majority rule but a fundamental problem with any procedure that attempts to aggregate individual preference rankings into a collective decision.
• The proof demonstrates that the axioms, taken together, force any potential aggregation rule to become a dictatorship, where one person's preferences determine the social outcome.
• The primary implication for our study of justice is that the philosophical project of building a social welfare function (like a utilitarian or prioritarian one) cannot succeed if it is limited to ordinal, non-comparable preference information.
• Arrow's theorem thus provides a profound justification for why welfare economics must make assumptions about cardinal, interpersonally comparable utility. It is the price of entry for escaping the impossibility result and constructing a coherent theory of social welfare.

Preview of the next lesson:

This module has focused on consequentialist theories of justice, which aim to produce the best outcomes by aggregating some measure of individual good. We have seen the nuances of utilitarianism and prioritarianism, and now the deep structural challenges to aggregation itself posed by Arrow's theorem.

In the next module, we will pivot to a radically different conception of justice. We will begin our study of Libertarianism and Entitlement Theory with Robert Nozick. For Nozick, justice has nothing to do with patterns, outcomes, or aggregating preferences. Instead, it is about respecting historical processes and individual rights. This provides a powerful critique of the entire welfarist framework we have been exploring.