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“Are you the one?” is free money

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Re: “Are you the one?” is free money

#41

Post by Dylan16807 »

twoodfin wrote: post #39
Dylan16807 wrote: post #38

Earlier quoted context omitted.

But that "yes" is so unlikely that your expected/average information is still 1 bit or less.

The claim was that one bit was the maximum amount of information you could gain, which is clearly false. Just to make this unambiguous: If you ask me to guess a number between one and one billion, and by fantastic luck I guess right, your “yes/no” answer obviously gives me more than one bit of information as to the right answer.

> The claim was that one bit was the maximum amount of information you could gain, which is clearly false.

That's not what I see.

https://news.ycombinator.com/item?id=46282007 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

https://news.ycombinator.com/item?id=46282343 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.

The specific comment you replied to had one sentence that didn't say "expected" or "average", but the surrounding sentences and comments give context. The part you objected to was also trying to talk about averages, which makes it not false.

Re: “Are you the one?” is free money

#42

Post by twoodfin »

Dylan16807 wrote: post #41
twoodfin wrote: post #39

Earlier quoted context omitted.

The claim was that one bit was the maximum amount of information you could gain, which is clearly false. Just to make this unambiguous: If you ask me to guess a number between one and one billion, and by fantastic luck I guess right, your “yes/no” answer obviously gives me more than one bit of information as to the right answer.

> The claim was that one bit was the maximum amount of information you could gain, which is clearly false. That's not what I see. https://news.ycombinator.com/item?id=46282007 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 https://news.ycombinator.com/i…

If both of these are equally likely, you gain one bit of information, the maximum possible amount. If you already have other information about the situation, you might gain _less_ than one bit on average (because it confirms something you already knew, but doesn't provide any new information), but you can't gain more.

Can’t gain more!

The core confusion is this idea that the answer to a yes/no question can’t provide more than one bit of information, no matter what the question or answer. This is false. The question itself can encode multiple bits of potential information and the answer simply verifies them.

Re: “Are you the one?” is free money

#43

Post by Dylan16807 »

twoodfin wrote: post #42
Dylan16807 wrote: post #41

Earlier quoted context omitted.

> The claim was that one bit was the maximum amount of information you could gain, which is clearly false. That's not what I see. https://news.ycombinator.com/item?id=46282007 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 https://news.ycombinator.com/i…

If both of these are equally likely, you gain one bit of information, the maximum possible amount. If you already have other information about the situation, you might gain _less_ than one bit on average (because it confirms something you already knew, but doesn't provide any new information), but you can't gain more. Can’t gain more! The core confusion is this idea that the answer to a yes/no question can’t provide…

> Can’t gain more!

"you might gain _less_ than one bit on average [...], but you can't gain more."

On. Average.

That's a true statement. Can't gain more than one bit on average.

Re: “Are you the one?” is free money

#44

Post by yaur »

nrhrjrjrjtntbt wrote: post #28
codebje wrote: post #20

Earlier quoted context omitted.

Everything is lined up for sub-optimal play. For a start, the setting is an emotive one. It's not just a numeric game with arbitrary tokens, it's about "the perfect romantic partner." It would take an unusually self-isolating human to not identify who they feel their perfect match should be and bias towards that, subconsciously or consciously. We (nearly) all seek connection. Then, it's reality TV. Contestants will b…

Need to find out their psycopath screening technique

Just Ask them to describe Shannon Entropy. If they start talking about information they are out, if they start talking about their crazy cousin they are in.

Re: “Are you the one?” is free money

#45

Post by twoodfin »

Dylan16807 wrote: post #43
twoodfin wrote: post #42

Earlier quoted context omitted.

If both of these are equally likely, you gain one bit of information, the maximum possible amount. If you already have other information about the situation, you might gain _less_ than one bit on average (because it confirms something you already knew, but doesn't provide any new information), but you can't gain more. Can’t gain more! The core confusion is this idea that the answer to a yes/no question can’t provide…

> Can’t gain more! "you might gain _less_ than one bit on average [...], but you can't gain more." On. Average. That's a true statement. Can't gain more than one bit on average.

I’m not arguing with that, it’s basic information theory.

One bit, however, is not “the maximum possible amount” you can gain from an oracular answer to a yes/no question. The OP covers exactly this point re: the “Guess Who?” game.

Re: “Are you the one?” is free money

#46

Post by Dylan16807 »

twoodfin wrote: post #45
Dylan16807 wrote: post #43

Earlier quoted context omitted.

> Can’t gain more! "you might gain _less_ than one bit on average [...], but you can't gain more." On. Average. That's a true statement. Can't gain more than one bit on average.

I’m not arguing with that, it’s basic information theory. One bit, however, is not “the maximum possible amount” you can gain from an oracular answer to a yes/no question. The OP covers exactly this point re: the “Guess Who?” game.

The start of this comment thread was a complaint that OP is showing more than one bit expected for certain yes/no answers. Not best case, expected.

That's why people are talking about the maximum expected value.

Re: “Are you the one?” is free money

#47

Post by thaumasiotes »

tobyjsullivan wrote: post #29
hatthew wrote: post #25

Earlier quoted context omitted.

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; the maximum possible is precisely 1.

The total probabilities add up to 1. But I’m not following how that relates to the average bits. Despite summing to 1, the exact values of P(true) and P(false) are dependent on the options which have previously been discounted. Then those variables get multiplied by the amount of information gained by either answer.

So, you have n options, you ask a question, and now you're down to m options.

The number of bits of information you gained is -log₂ (m/n).

If you ask a question which always eliminates half of the options, you will always gain -log₂ (1/2) = 1 bit of information.

If you go with the dumber approach of taking a moonshot, you can potentially gain more than that, but in expectation you'll gain less.

If your question is a 25-75 split, you have a 25% chance of gaining -log₂ (1/4) = 2 bits, and a 75% chance of gaining -log₂ (3/4) = 0.415 bits. On average, this strategy will gain you (0.25)(2) + (0.75)(0.415) = 0.8113 bits, which is less than 1 bit.

The farther away you get from 50-50, the more bits you can potentially gain in a single question, but - also - the lower the number of bits you expect to gain becomes. You can never do better than an expectation of 1 bit for a trial with 2 outcomes.

(All of this is in the article; see footnote 3 and its associated paragraph.)

The article explicitly calls out the expectational maximum of one bit:

>> You'll also notice that you're never presented with a question that gives you more than 1 expected information, which is backed up by the above graph never going higher than 1.

So it's strange that it then goes on to list an example of a hypothetical (undescribed, since the scenario is impossible) yes/no question with an expected gain of 1.6 bits.

Re: “Are you the one?” is free money

#48

Post by empathy_m »

bryanhogan wrote: post #32
pants2 wrote: post #26

Earlier quoted context omitted.

I'd watch a game show where you put a variety of math nerds on each team and watch them argue about the optimal strategy. Who's strategy will win? The quant analyst or the bioinformatician? Tune in next week!

Reminds me a bit of Devils Plan, or other similar reality game shows in Korea / Asia.

Watched a fantastic film about this on the plane a few years ago, "Liar Game - Reborn". There is some fairly sophisticated logic puzzling and scheming going on (see e.g. sample illustration https://imgur.com/a/0AOb67G from an interlude about 50min in where there are 3 groups of people who mutually distrust each other, each know a secret collection of 3-4 integers unique to their group, and want to deniably pass share integers with each other which are "not my team's". Another participant watches what happened and realizes in retrospect this is how the info was shared.)

A lot more upbeat than "Alice in Borderland".

Re: “Are you the one?” is free money

#50

Post by rahimnathwani »

After he described the rules, my immediate reaction was 'this is like mastermind'. Sure enough, further down the page:

  Other than that, in my research I came across a boardgame called Mastermind, which has been around since the 70s. This is a very similar premise - think of it as "Guess Who?" on hard mode.
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