What is “Condorcet Paradox” known as a decision-making paradox?
Scenario
Three friends, Alice, Bob, and Carol are deciding where to go for dinner. They have three options: Pizza (P), Sushi (S), and Burgers (B). Each friend has a different preference ranking for these options:
- Alice’s preference: Sushi > Pizza > Burgers
- Bob’s preference: Pizza > Burgers > Sushi
- Carol’s preference: Burgers > Sushi > Pizza
They decide to use a pairwise voting system to determine the winner. In this system, each option is compared against each other, and the option that wins the most pairwise comparisons is the overall winner.
Pairwise Comparisons
1. Pizza vs Sushi:
- Alice prefers Sushi over Pizza.
- Bob prefers Pizza over Sushi.
- Carol prefers Sushi over Pizza.
Result: Sushi wins 2–1.
2. Pizza vs Burgers:
- Alice prefers Pizza over Burgers.
- Bob prefers Pizza over Burgers.
- Carol prefers Burgers over Pizza.
Result: Pizza wins 2–1.
3. Sushi vs Burgers:
- Alice prefers Sushi over Burgers.
- Bob prefers Burgers over Sushi.
- Carol prefers Burgers over Sushi.
- Result: Burgers win 2–1.
Analysis
In the pairwise comparisons:
- Sushi beats Pizza.
- Pizza beats Burgers.
- Burgers beat Sushi.
This creates a cycle where:
- Sushi > Pizza
- Pizza > Burgers
- Burgers > Sushi
This cycle means that there is no clear winner, as each option is both preferred and not preferred when compared to another option. This is known as the Condorcet Paradox. It demonstrates how collective preferences can be cyclic, even if individual preferences are not.
Mathematical Formulation
Let’s denote the options by P (Pizza), S (Sushi), and B (Burgers), and the preference of each friend as follows:
From the matrix, we see:
- Sushi (S) wins over Pizza (P) with a 2–1 majority.
- Pizza (P) wins over Burgers (B) with a 2–1 majority.
- Burgers (B) wins over Sushi (S) with a 2–1 majority.
Conclusion
The Condorcet Paradox reveals that even when individual preferences are consistent, the collective decision can be cyclic and inconsistent. This paradox can cause issues in voting systems and collective decision-making processes where the goal is to identify a clear and consistent winner.
Why Does This Matter?
Understanding such paradoxes is crucial for designing fair and efficient decision-making processes, especially in voting systems, committee decisions, and other group choices. It highlights the need for careful consideration of the methods used to aggregate individual preferences into a collective decision.
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