# LW4EA: Value of Information: Four Examples

Jun 14 20222 min read 0

# 5

Written by LW user Vaniver.

This is part of LessWrong for EA, a LessWrong repost & low-commitment discussion group (inspired by this comment). Each week I will revive a highly upvoted, EA-relevant post from the LessWrong Archives, more or less at random

Excerpt from the post:

Value of Information (VoI) is a concept from decision analysis: how much answering a question allows a decision-maker to improve its decision. Like opportunity cost, it's easy to define but often hard to internalize; and so instead of belaboring the definition let's look at some examples.

## Gambling with Biased Coins

Normal coins are approximately fair.1 Suppose you and your friend want to gamble, and fair coins are boring, so he takes out a quarter and some gum and sticks the gum to the face of the quarter near the edge. He then offers to pay you \$24 if the coin lands gum down, so long as you pay him \$12 to play the game. Should you take that bet?

First, let's assume risk neutrality for the amount of money you're wagering. Your expected profit is \$24p-12, where p is the probability the coin lands gum down. This is a good deal if p>.5, but a bad deal if p<.5.  So... what's p? More importantly, how much should you pay to figure out p?

A Bayesian reasoner looking at this problem first tries to put a prior on p. An easy choice is a uniform distribution between 0 and 1, but there are a lot of reasons to be uncomfortable with that distribution. It might be that the gum will be more likely to be on the bottom- but it also might be more likely to be on the top. The gum might not skew the results very much- or it might skew them massively. You could choose a different prior, but you'd have trouble justifying it because you don't have any solid evidence to update on yet.2

If you had a uniform prior and no additional evidence, then the deal as offered is neutral. But before you choose to accept or reject, your friend offers you another deal- he'll flip the coin once and let you see the result before you choose to take the \$12 deal, but you can't win anything on this first flip. How much should you pay to see one flip? (Full Post on LW)

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