Arrow impossibility theorem
The Arrow impossibility theorem, also known as Arrow's paradox, is a result in social choice theory that reveals certain limitations in trying to collectively make decisions based on individual preferences. It was formulated by economist Kenneth Arrow in 1950.
To understand the Arrow impossibility theorem, let's first establish some key concepts:
1. Social choice: It refers to the process of making a collective decision based on individual preferences. The goal is to aggregate individual preferences into a single societal choice.
2. Voting system: It is a method or rule used to translate individual preferences into a group decision. For example, majority rule, ranked voting, or approval voting.
3. Social welfare function: It is a mathematical representation of a voting system, which maps individual preferences to a societal preference. It takes as inputs the individual preferences and produces a collective preference.
Now, the Arrow impossibility theorem states that no voting system can satisfy all of the following three criteria simultaneously:
1. Pareto efficiency: If every individual prefers option A to option B, the collective decision should also rank A above B.
2. Independence of irrelevant alternatives: The relative ranking between two options should only depend on their individual rankings, and not on the presence or absence of other alternatives.
3. No dictator: No single individual should be able to determine the collective preference regardless of what all others prefer.
In simpler terms, the Arrow impossibility theorem essentially says that no voting system can be perfectly fair, unanimous, and unimposed by a single person's preferences simultaneously.
The proof of Arrow's theorem involves mathematical formalization and may be complex for a general explanation. However, it relies on constructing different scenarios and showing that in each case, the voting system violates one or more of the three criteria mentioned above.
It's important to note that the theorem doesn't necessarily imply that voting is pointless or that all voting systems are flawed. It highlights the inherent difficulties of aggregating individual preferences into a coherent and fair collective decision.
Overall, the Arrow impossibility theorem demonstrates the challenges involved in designing a perfect voting system that satisfies all desirable properties simultaneously.