how much free money are we talking here? what are they awarded?
> If the group can correctly guess all the perfect matches, they win a cash prize of $1M.
Is that each, or divided between 20 people?
“Are you the one?” is free money
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Re: “Are you the one?” is free money
#62Earlier quoted context omitted.
Because when it's true, you also learn about any prior match ups involving those two people.
You also learn about other pairings now being impossible.
No, that doesn't make sense either. For a truth booth, you're taking all the possible pairing arrangements, and dividing them into two sets. After the answer, one of those two sets is false. There is no way that this can provide more than 1 bit of information.
The match-ups can however give more information, as it isn't giving a yes/no answer.
Re: “Are you the one?” is free money
#63Fun post. I'd be interested to know: How many consecutive Truth Booths (or: how many consecutive Match Ups) are needed to narrow down the 10! possibilities to a single one? Discussing "events" (ie, Truth Booth or Match Up) together muddles the analysis a bit. I agree with Medea above that a Truth Booth should give at most 1 bit of information.
If you can only check pairings one at a time I’m not sure it’s possible to do better than greedily solving one person at a time.
Agreed. There's an argument elsewhere about how a truth booth can possibly have an expected return of more than 1 bit of information, but in reality most of the time it's going to give you way less than that.
Re: “Are you the one?” is free money
#64As 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 o…
Let's be friends :')
Loved your post, really enjoyed getting into the meat of it. I wanted to position mine to a layman, kept asking myself "can I explain this to my Dad?"
I think where the post falls short is the absence of a silver bullet that contestants can use to win the game sooner.
Re: “Are you the one?” is free money
#65This was great, but it skipped over the most interesting bit - how you actually choose which matchups and truth booths. That is, an actual strategy that contestants could use that doesn't require a computer.
Thank you! This is consistent with feedback I got from the pudding, and is ultimately the reason they didn't go ahead with the post. I tried reverse-engineering the information-theory approach to try see what sort of decisions it made.
I noticed that for any match up score of X, the following match up would keep exactly X pairs in common. So if they scored 4/10 one week, they would change 6 couples before the next one. Employing that approach alone performed worse than the contestants did in real life, so didn't think it was worth mentioning!
Re: “Are you the one?” is free money
#66When my wife and I watched the show I wrote a solver on the side so we always had the current probabilities and impossible combinations on the side. I am thinking about making a website for it when the next season starts. Also: in Germany at least they have 10 x 10 candidates from the start, but sometimes they add a 11th or even 12th of one gender so that there are double matches (e.g. 1 woman has two man as match an…
Would love to see this!
Yes there's a gender fluid season and a season where someone had > 1 match, as well as people leaving part way through the season (apparently perfect matches are interchangeable...). All very interesting spins on the core problem to solve; would be really interested if anyone tries to tackle those seasons.
Re: “Are you the one?” is free money
#67If the goal is to find the perfect matching in some maximum number of turns or less, it's possible to do even better by using a full game tree that minimises the maximum height of the tree ( = number of turns required), instead of using information/entropy as done here. Basically, using the entropy produces a game tree that minimises the number of steps needed in expectation -- but that tree could be quite unbalanced…
> For concreteness, a game requiring 6 bits of information to identify the perfect matching will take 6 steps on average, and may sometimes require many more I'm not following your logic. Consider the setup we actually have: 1. You get to ask a series of yes/no questions. (Ignoring the matchups.) 2. Each question can produce, in expectation, up to one bit of information. 3. In order to achieve this maximum expectatio…
> Unless the number of bits required to solve the problem is not an integer.
That is one case where root-to-leaf path lengths can vary, though it's not obvious to me that it exhausts all such cases -- in particular, even if we have "ideal leaves" (numbering a power of 2, and each equally likely), it's not clear that there is always a question we can ask that divides a given node's leaves exactly in half.
Re: “Are you the one?” is free money
#68A thing to note - the contestants are not allowed to have even pen and paper, as mentioned in the other blogpost. So they need to do these computations in their heads.
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Re: “Are you the one?” is free money
#69They 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
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!
Re: “Are you the one?” is free money
#70They 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
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.