Articles Tagged ‘fungibility’ - Less Wrong Discussion

Main
- Posts
- Comments
Discussion
- Posts
- Comments

You're looking at Less Wrong's discussion board. This includes all posts, including those that haven't been promoted to the front page yet. For more information, see About Less Wrong.

Proof of fungibility theorem

3 Nisan 12 January 2013 09:26AM

Appendix to: A fungibility theorem

Suppose that $P$ is a set and we have functions $v_1, \dots, v_n : P \to \mathbb{R}$ . Recall that for $p, q \in P$ , we say that $p$ is a Pareto improvement over $q$ if for all $i$ , we have $v_i(p) \geq v_i(q)$ . And we say that it is a strong Pareto improvement if in addition there is some $i$ for which $v_i(p) > v_i(q)$ . We call $p$ a Pareto optimum if there is no strong Pareto improvement over it.

Theorem. Let $P$ be a set and suppose $v_i: P \to \mathbb{R}$ for $i = 1, \dots, n$ are functions satisfying the following property: For any $p, q \in P$ and any $\alpha \in [0, 1]$ , there exists an $r \in P$ such that for all $i$ , we have $v_i(r) = \alpha v_i(p) + (1 - \alpha) v_i(q)$ .

Then if an element $p$ of $P$ is a Pareto optimum, then there exist nonnegative constants $c_1, \dots, c_n$ such that the function $\sum c_i v_i$ achieves a maximum at $p$ .