Warm Up There are four agents (1, 2, 3, 4)

Transcript:Warm Up There are four agents (1, 2, 3, 4):
Warm Up There are four agents (1, 2, 3, 4) and four objects (A, B, C, D). Preferences are as follows: How many Pareto efficient allocations can you find? Warm Up Solution There are five: ABCD ABDC ACDB CBDA CBAD Engineering the Allocation of Public Resources Lecture 2: Truthfulness and the Core Recall: Three Important Goals Efficiency Truthfulness Fairness Last Class Today Next Week Plan for the Day Pareto Efficiency Review: definition and homework Algorithms to find Pareto efficient allocations Introduction to Incentives (Truthfulness) Allocation with pre-existing ownership Application: Board Game “Math Trades” Key concepts: individually rational, the core Homework Review: Question 1 Is this allocation Pareto Efficient? Pair Efficient Definition of Pareto Efficiency An allocation is Pareto efficient if it is impossible to make somebody better off without making somebody else worse off. Pareto Efficient Rank Efficient This allocation cannot be improved by two-person trades, but is NOT Pareto efficient. (All agree that ACDB is better.) D E Which of the following are correct restatements of the definition of Pareto efficiency? You should check an answer only if BOTH of the following are true: (i) If the statement holds, then the allocation must be Pareto efficient. (ii) If the allocation is Pareto efficient, then the statement must hold. It is impossible to (strictly) improve the outcome for anybody. It is impossible to (strictly) improve the outcome for everybody. It is impossible to (strictly) improve the outcome for anybody without (strictly) worsening the outcome for somebody else. Is impossible to improve the sum of ranks. It is impossible to find a pair of people who want to swap items. An allocation could be Pareto efficient, even though (a) does not hold. Even if (b) holds, the allocation might not be Pareto efficient. (c) is equivalent to Pareto efficiency An allocation could be Pareto efficient, even though (d) does not hold. Even if (e) holds, the allocation might not be Pareto efficient. Possible Venn Diagrams Y X Y X X Y Y X X implies Y Y implies X Disjoint (X implies not Y) Overlapping Relationships From Homework Question 1 Y e b Pareto Efficient d a Equivalent Definitions of Pareto Efficiency A feasible allocation is Pareto Efficient if no other feasible allocation is at least as good for every agent, and strictly better for some agent. A feasible allocation is Pareto Efficient if every feasible allocation that is strictly better