A 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.
“Are you the one?” is free money
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Re: “Are you the one?” is free money
#52If 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 expectation, it is mathematically necessary to use questions that invariably do produce exactly one bit of information, never more or less. If your questions do not have this property, they will in expectation produce less than the maximum amount of information, and your expected number of steps to the solution will increase, which contradicts your stated goal of minimizing that quantity.
You get the minimum number of steps needed in expectation by always using questions with maximum entropy. Yes. But those questions never have any variation in the amount of entropy they produce; a maximum entropy strategy can never take more - or fewer - steps than average.¹
¹ Unless the number of bits required to solve the problem is not an integer. Identifying one of three options requires 1.585 bits; in practice, this means that you'll get it after one question 1/3 of the time and after two questions the other 2/3 of the time. But identifying one of 128 options requires 7 bits, you'll always get it after 7 questions, and you'll never be able to get it after 6 questions. (Assuming you're using a strategy where the expected number of questions needed is 7.)
Re: “Are you the one?” is free money
#53I would love to know how the author went about constructing those lovely charts. Some library, or done by hand?
Re: “Are you the one?” is free money
#54As 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…
And sometimes they just don't do better as a plot point, staying together an extra week after finding out they are not the one because of the intensity of their love (they met 4 days before)
Re: “Are you the one?” is free money
#55Isn't 5 impossible in the 6-6 matching?
Re: “Are you the one?” is free money
#56When 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 and needs to find one of it to succeed). This raises the possible combination quite a bit.
Re: “Are you the one?” is free money
#57After 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.
Mastermind! What a great game. Played it for hours.
https://www.mcsweeneys.net/articles/the-mastermind-box-cover...
Re: “Are you the one?” is free money
#58I 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…
> No-one is going to watch a full season of a handful of math nerds take a few minutes to progress a worksheet towards a solution each week coupled with whatever they do to pass the time otherwise.
Um, what about those of us who watch Blood on the Clocktower streams?
Re: “Are you the one?” is free money
#59Earlier 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
Charlie Brooker did a good bit (in 2007!) about picking the right candidates for a mock reality TV show: https://youtube.com/watch?v=NGkJxju3uKo
Re: “Are you the one?” is free money
#60If 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…
You're correct but the complexity is in the things you're ignoring for simplicity.
This game is constructed such that the questions you can ask are not arbitrary, so you cannot choose them to always produce one bit of entropy (you need to frame your questions as ten matchups in parallel, using all the contestants exactly once) and the number of bits you need may indeed not be an integer.
Because you can't choose your questions to partition the state space arbitrarily, that affects not just the question you ask today, but also previous days: you want to leave yourself with a partitionable space tomorrow no matter what answers you get today.
In the Guess Who analogy, it's against the rules or at least the spirit to ask "does your character have a name which is alphabetically before Grace?". That would allow a strategy which always divides the state space exactly in two.