17. Social Choice and Welfare

Arrow's impossibility theorem and social welfare functions — the trade-offs of preference aggregation and inequality aversion.

• Arrow's Theorem — This scene stress-tests rules that merge many rankings into one. Switch profiles and rules and watch what breaks. Majority rule can spin in a circle. A beats B, and B beats C. Then C turns around and beats A. The Borda count wobbles for another reason. Delete a hopeless alternative and the ranking flips. No rule keeps all four fairness conditions. For example, three 🧑 friends run for class president. The class splits into three groups with clashing favorites. Head to head, A beats B and B beats C. Yet C beats A. Majority rule alone cannot pick a president.
• Social Welfare Functions — This scene merges two people's happiness into one score. It picks the best among all possible happiness pairs. One slider changes how the merging works. Count only the sum, and the bigger total wins. Hate inequality more, and the score changes. At the extreme, only the worse-off person counts. The best point slides toward the equal diagonal. The formula below fixes the merging rule. For example, two 🧑 friends split 15 🍬 candies. One friend gets 9 and the other gets 6. Judged by the sum alone, this split scores best. Care more about fairness, and they split 7.5 each. The worse-off friend's happiness sets the score.