Model Answer
0 min readIntroduction
Arrow’s Impossibility Theorem, published in Kenneth Arrow’s *Social Choice and Individual Values* (1951), is a cornerstone of social choice theory. It demonstrates that when voters have three or more alternatives, no voting system can convert individual preferences into a collective social preference while simultaneously satisfying certain seemingly reasonable criteria. This theorem has profound implications for democratic decision-making. However, the theorem’s stringent assumptions have been challenged, notably by James Buchanan. Buchanan argued that the theorem’s insistence on a perfectly consistent social choice function is unrealistic and unduly restrictive. Amartya Sen later offered a modification to the theorem, relaxing the consistency requirement to address Buchanan’s critique.
Arrow’s Impossibility Theorem
Arrow’s theorem rests on four key axioms:
- Unrestricted Domain: The social welfare function must be able to accommodate any possible set of individual preferences.
- Non-Dictatorship: No single individual’s preferences should dictate the social preference.
- Pareto Efficiency: If every individual prefers alternative A to alternative B, then the social preference must also prefer A to B.
- Independence of Irrelevant Alternatives (IIA): The social ranking of alternatives A and B should depend only on individual preferences between A and B, and not on preferences for other alternatives.
Arrow proved that if there are three or more alternatives, any social choice rule satisfying these axioms will inevitably lead to a paradox – a situation where rational preferences can result in irrational collective outcomes. This implies that a perfectly fair and consistent democratic decision-making process is theoretically impossible.
Buchanan’s Criticism of Arrow’s Theorem
James Buchanan, a prominent figure in public choice theory, criticized Arrow’s theorem not by disputing its mathematical validity, but by questioning the relevance of its assumptions. Buchanan argued that the requirement of rational consistency of preferences is too strong and unrealistic. He posited that individuals often exhibit intransitive preferences in real life, and a social choice function that demands perfect consistency is therefore impractical and unnecessarily restrictive.
Buchanan’s critique centers on the idea that the theorem’s focus on a logically perfect social choice function overlooks the inherent complexities and imperfections of human decision-making. He believed that the theorem’s pessimistic conclusion stemmed from an overly idealized model of rational behavior. He suggested that a more realistic model should allow for the possibility of temporary inconsistencies in individual preferences, which would, in turn, relax the constraints on the social choice function.
Furthermore, Buchanan pointed out that the insistence on a universal domain – the ability to handle any possible set of individual preferences – is also problematic. He argued that in many real-world situations, certain preference profiles are simply not credible or likely to occur.
Sen’s Modification of Arrow’s Theorem
Amartya Sen addressed Buchanan’s criticism by modifying Arrow’s theorem. Sen demonstrated that if the requirement of overall consistency of social choice is relaxed, the impossibility result no longer holds. Specifically, Sen showed that if we allow for the possibility of limited inconsistency – meaning that the social choice function need not always produce a complete and transitive ranking of all alternatives – then a social choice rule can be constructed that satisfies the other three axioms (non-dictatorship, Pareto efficiency, and IIA).
Sen’s modification involves focusing on binary choice functions. Instead of requiring a complete ranking of all alternatives, Sen’s approach focuses on making pairwise comparisons. A social choice function is considered acceptable if it can consistently choose between any two alternatives when presented with a binary choice. This approach allows for the possibility that the social preference may not be fully defined for all possible combinations of alternatives, but it ensures that the social choice function is rational and consistent in the limited context of pairwise comparisons.
This modification doesn’t eliminate all the difficulties of social choice, but it provides a more realistic and practical framework for understanding how collective decisions can be made. It acknowledges that perfect consistency is unattainable but argues that a reasonable degree of consistency is sufficient for meaningful social choice.
| Feature | Arrow’s Theorem | Sen’s Modification |
|---|---|---|
| Consistency Requirement | Full, overall consistency of social choice | Limited consistency; focuses on binary choice |
| Impossibility Result | No social choice function can satisfy all axioms | Social choice function can be constructed under relaxed consistency |
| Realism | Less realistic due to stringent assumptions | More realistic by acknowledging limitations of rationality |
Conclusion
In conclusion, Buchanan’s critique of Arrow’s theorem highlighted the unrealistic nature of its assumptions, particularly the demand for perfect consistency in social choice. Sen’s modification, by relaxing this consistency requirement and focusing on binary choice functions, successfully addressed Buchanan’s concerns. While Arrow’s theorem remains a fundamental result in social choice theory, Sen’s work demonstrates that the impossibility result is not inevitable when we adopt a more nuanced and realistic understanding of human preferences and decision-making. This modification offers a more pragmatic approach to designing democratic institutions and processes.
Answer Length
This is a comprehensive model answer for learning purposes and may exceed the word limit. In the exam, always adhere to the prescribed word count.