[Medea]

They have an example that calculates the expected information gained by truth booths and all of the top ones are giving more than one bit. How can this be? It is a yes/no question a max of 1 bit should be possible

[tobyjsullivan]

The author defines one “bit” as ruling out half the remaining options.

So a yes might rule out 75% of remaining options (for example) which provides 2 bits of information.

[hatthew]

We have to make a distinction between "expected information gain" vs "maximum information gain". An answer of "yes" generally gains >1 bit, but an answer of "no" generally gains <1 bit, and the average outcome ends up <1. It is impossible for a single yes/no question to have an expected gain of >1; the maximum possible is precisely 1.

[owenlacey]

Great spot! The max expected information is 1. I've updated this part of the post to only show examples that are < 1, thank you for raising!

[akoboldfrying]

Do you mean the diagram following the sentence "Suppose we have calculated the expected information gained by potential truth booths like below:"?

Yes, that looks like a mistake -- a truth booth only has 2 outcomes, so it can produce at most 1 bit of information.

Regarding the other mentions on the page of information levels exceeding 1 bit: Those are OK, since they allow match-ups, which for 6 people have 7 possible outcomes, thus can yield up to log2(7) ≈ 2.81 bits.

[latortuga]

Because when it's true, you also learn about any prior match ups involving those two people.

[MarkusQ]

That's not how information works. Learning more from one outcome than the other decreases the probability of that outcome occurring, so the expected information (which is the sum of the outcome probability times the outcome information for each of the two possible outcomes) is always less than or equal to one.

If all you can get is a "true" or "false" you expect, at most, one bit of information.

[stevage]

You also learn about other pairings now being impossible.