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“Are you the one?” is free money

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Re: “Are you the one?” is free money

#31

Post by Yossarrian22 »

travisjungroth wrote: post #13

This brings up an area that’s been on the edge of my curiosity for years: how do you combine the knowledge of the experts (contestants) with logic to do better either than either strategy individually? It’s mostly about how to elicit the information from the contestants. Once you have the probabilities from them, it seems relatively straightforward.

I think you could do some form of Bayesian analysis with the prior being how likely each contestant thought that their available partners were "The One" for each other. Then the truth booth would be used to update your priors.

Re: “Are you the one?” is free money

#32

Post by bryanhogan »

pants2 wrote: post #26
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…

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.

Re: “Are you the one?” is free money

#33

Post by hatthew »

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.

The article states "Suppose we have calculated the expected information gained by potential truth booths like below: Expected information: 1.60 bits ..." This is impossible because of the general fact in information theory that (p(true) * bits_if_true) + (p(false) * bits_if_false) maximum information gained...", then 1.6 bits would be valid. They said "expected information" though, so 1.6 bits is invalid.

Re: “Are you the one?” is free money

#34

Post by adastra22 »

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.

It is definitional, which I mean in the strictest mathematical sense: the information content of a result is directly derived from how “unexpected” it is.

A result which conveys 2 bits of information should occur with 25% expected probability. Because that’s what we mean by “two bits of information.”

Re: “Are you the one?” is free money

#35

Post by MarkusQ »

kevindamm wrote: post #17
MarkusQ wrote: post #10

Earlier quoted context omitted.

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.

It's not a yes/no per contestent, it's per edge between contestants. There are n(n-1)/2 of these. A true answer for a potential match is actually a state update for all of the (n-1) edges connecting either contestant, that's 2(n-2) edges that can be updated to be false. Some of these may already be known from previous rounds' matchups but that's still more than a single binary.

But you don't receive more than a single binary value; you get a yes or no.

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.

Re: “Are you the one?” is free money

#36

Post by twoodfin »

MarkusQ wrote: post #35
kevindamm wrote: post #17

Earlier quoted context omitted.

It's not a yes/no per contestent, it's per edge between contestants. There are n(n-1)/2 of these. A true answer for a potential match is actually a state update for all of the (n-1) edges connecting either contestant, that's 2(n-2) edges that can be updated to be false. Some of these may already be known from previous rounds' matchups but that's still more than a single binary.

But you don't receive more than a single binary value; you get a yes or no. 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.

If I’m trying to guess a 9-letter English word, and test whether the first letter is “x”, there are only the same two answers: Yes/No.

But “Yes” obviously gives me much more than one bit of the information I need to know the answer.

Re: “Are you the one?” is free money

#37

Post by bronco21016 »

codebje wrote: post #20
jncfhnb wrote: post #15

I saw an episode of this and felt the contestants didn’t seem that interested in winning the money. Just romance. I was curious how suboptimally they tended to play.

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…

We need a ManningCast version of the show. For those unaware, ManningCast is a show following an NFL game with special guests and nontraditional commentary and analysis. Think of it kind of like having the Mannings in the living room while watching an NFL game.

In my hypothetical version of "Are you the one?", the math nerds would be giving commentary and explaining the math behind how they'll solve "Are you the one?" while also hilariously explaining how foolish the contestants' theories are.

Re: “Are you the one?” is free money

#38

Post by Dylan16807 »

twoodfin wrote: post #36
MarkusQ wrote: post #35

Earlier quoted context omitted.

But you don't receive more than a single binary value; you get a yes or no. 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.

If I’m trying to guess a 9-letter English word, and test whether the first letter is “x”, there are only the same two answers: Yes/No. But “Yes” obviously gives me much more than one bit of the information I need to know the answer.

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

Re: “Are you the one?” is free money

#39

Post by twoodfin »

Dylan16807 wrote: post #38
twoodfin wrote: post #36

Earlier quoted context omitted.

If I’m trying to guess a 9-letter English word, and test whether the first letter is “x”, there are only the same two answers: Yes/No. But “Yes” obviously gives me much more than one bit of the information I need to know the answer.

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.

Re: “Are you the one?” is free money

#40

Post by daturkel »

As a math guy who loves reality tv, I was also drawn to the show and wrote a blog post [0] about how to programmatically calculate the probabilities as the show progresses. It was a lot of fun optimizing it to be performant. You can `pip install ayto` to use it to follow along with the show or try out scenarios.

The linked post is a very thorough treatment of AYTO and a great read. I really like the "guess who" bit on how to maximize the value of guesses. It's a shame the participants aren't allowed to have pen and paper—it makes optimization a lot trickier! I'm impressed they do as well as they do.

[0]: https://danturkel.com/2023/01/25/math-code-are-you-the-one.h...

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