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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