Social Welfare Function

Arrow’s Impossibility Theorem

Arrow’s Impossibility Theorem, named after economist Kenneth Arrow, is a fundamental result in social choice theory and welfare economics. It shows the inherent difficulties in aggregating individual preferences into a collective or social preference that satisfies certain fairness criteria.

Arrow introduced this theorem in his landmark work, Social Choice and Individual Values (1951), which laid the foundation for modern welfare economics and political economy. The theorem demonstrates that no voting or decision-making system can satisfy all of the desirable criteria for collective decision-making simultaneously.

Statement of the Theorem

Arrow’s Impossibility Theorem states that:

• No voting or social choice rule can convert individual preferences into a social preference that satisfies all of the following conditions simultaneously:
  1. Unrestricted Domain (Universal Admissibility)
  2. Non-Dictatorship
  3. Pareto Efficiency (or Pareto Optimality)
  4. Independence of Irrelevant Alternatives (IIA)
  5. Transitivity of Social Preferences (Collective Rationality)

If a system is to fulfill all these conditions, it is impossible to design a fair and consistent social decision-making rule that respects all individuals’ preferences.

The Five Key Conditions

1. Unrestricted Domain (Universal Admissibility):
   • This condition states that the social welfare function (SWF) should be able to process any set of individual preferences. No preference order is restricted or ruled out; all possible preferences should be admissible.
2. Non-Dictatorship:
   • This condition asserts that no single individual should have the power to dictate the social preference, meaning that no one’s preferences should count more than everyone else’s. The collective decision should not be based solely on the preference of one person, even if their preference is different from all others.
3. Pareto Efficiency (or Pareto Optimality):
   • Pareto Efficiency implies that if every individual prefers one option to another, then the social preference must also reflect that choice. This condition ensures that if everyone agrees on a preference, the social preference should align with it, as the choice would be Pareto improving (i.e., it would make no one worse off).
4. Independence of Irrelevant Alternatives (IIA):
   • This condition states that the social preference between any two alternatives should depend only on the individual preferences regarding those two alternatives, and not on other irrelevant alternatives. The presence or absence of a third option should not affect the social ranking of the first two options.
5. Transitivity of Social Preferences (Collective Rationality):
   • If a society prefers option A to B and B to C, then it must also prefer A to C for consistency and rationality. This ensures that the social preferences are transitive and do not create contradictions.

Implications of Arrow’s Impossibility Theorem

Arrow’s theorem demonstrates that no social choice function can satisfy all of these conditions. This creates an inherent tension between fairness, efficiency, and rationality in collective decision-making processes.

1. Unrealistic Assumptions:
   • The conditions laid out by Arrow are very stringent, and in practice, they may conflict with one another. For example, the non-dictatorship condition contradicts the Pareto efficiency condition in certain situations. As a result, these conditions often cannot be fulfilled simultaneously in real-world decision-making systems.
2. Limits of Democracy and Voting Systems:
   • Arrow’s theorem has significant implications for voting systems and democratic decision-making. It suggests that no voting system can produce outcomes that meet all of the desired fairness criteria. Thus, each voting system will inevitably involve trade-offs between fairness and efficiency.
3. Necessity of Compromises:
   • Due to the impossibility of satisfying all five conditions, social choice mechanisms must make compromises. For example, in real-world political systems, we often observe the trade-off between efficiency (like Pareto optimality) and fairness (like equal weight for all voters).

Real-World Applications of Arrow’s Impossibility Theorem

1. Voting Systems:
   • In electoral systems, such as plurality voting or ranked-choice voting, the impossibility theorem suggests that these systems are inherently flawed in their ability to capture collective preferences perfectly. No voting system is immune to issues like strategic voting or condorcet paradoxes (when there is no clear winner).
2. Public Policy:
   • In policymaking, Arrow’s theorem suggests that it’s difficult for policymakers to make decisions that perfectly reflect the preferences of society. This insight helps explain why governments often face challenges in satisfying all citizens, as their preferences often conflict.
3. Economic and Welfare Decisions:
   • When it comes to welfare economics, where the goal is to aggregate individual welfare into a social welfare function, the theorem highlights the difficulty of creating a fair and efficient system that accommodates everyone’s preferences. Economists often need to consider alternative methods to measure social welfare, like using utilitarian or Rawlsian frameworks, which balance efficiency with equity.

Criticism and Alternatives

1. Focus on Simpler Models:
   • Some economists argue that Arrow’s conditions are too strict and that a more realistic approach to decision-making should acknowledge that people’s preferences are not always transitive or independent.
2. Alternative Aggregation Methods:
   • After Arrow’s work, alternative social choice mechanisms, such as the Borda count, Condorcet method, and ranked-choice voting, have been developed. These methods try to mitigate some of the issues identified by the theorem, but they still face their own trade-offs.
3. Flexibility in the Real World:
   • Critics argue that real-world decision-making does not always strictly follow Arrow’s assumptions. Preferences in actual societies may be more fluid or context-dependent, making the strict requirements of the theorem less applicable.