Model Answer
0 min readIntroduction
Kenneth Arrow’s impossibility theorem, presented in his 1951 book *Social Choice and Individual Values*, demonstrates the inherent difficulties in aggregating individual preferences into a collective social welfare function. This theorem highlights that no voting system can simultaneously satisfy a set of seemingly reasonable conditions. These conditions, if met, would ensure a fair and consistent method for making collective decisions. Understanding these conditions is crucial for evaluating the limitations of democratic processes and the challenges of social choice theory.
Arrow’s Five Reasonable Conditions
Arrow identified five conditions that a rational social welfare function should ideally satisfy. These are:
1. Unrestricted Domain
The social welfare function must be able to accept any possible ranking of individual preferences. This means that there should be no restrictions on what individuals can prefer; all possible orderings of alternatives are permissible inputs to the function. Essentially, it assumes individuals have well-defined and complete preferences.
2. Pareto Optimality (or Unanimity)
If every individual prefers alternative A to alternative B, then the social welfare function must also rank A higher than B. This condition ensures that if there is unanimous agreement on a preference, the social choice reflects that agreement. It’s a basic requirement for any reasonable social choice mechanism.
3. Independence of Irrelevant Alternatives (IIA)
The social ranking of alternatives A and B should depend only on the individual rankings of A and B, and not on the rankings of other alternatives (C, D, etc.). In other words, if we remove or add an irrelevant alternative, the relative ranking of A and B should remain unchanged. This condition prevents strategic manipulation of the voting system by introducing decoy options.
4. Non-Dictatorship
There should be no single individual whose preferences always determine the social ranking, regardless of the preferences of others. The social welfare function cannot simply impose the preferences of one person onto the entire society. This ensures that the process is not solely controlled by a single entity.
5. Non-Imposition (Universal Domain)
The social welfare function must not impose a particular social ordering on the society. It should be able to generate any possible social ordering consistent with the individual preferences. This condition ensures that the social welfare function is not biased towards a specific outcome or set of values. It allows for a wide range of possible social preferences.
Arrow’s theorem proves that it is mathematically impossible to simultaneously satisfy all five of these conditions. This has profound implications for the design of voting systems and the theory of social choice.
Conclusion
Arrow’s five conditions represent a set of intuitive and desirable properties for a social welfare function. However, his impossibility theorem demonstrates that these conditions are mutually incompatible. This highlights the inherent challenges in translating individual preferences into collective decisions and underscores the limitations of any voting system. The theorem doesn’t invalidate democratic processes, but rather provides a framework for understanding their inherent complexities and potential pitfalls.
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.