Solving "Are You The One?"
During the initial pandemic, my wife and I began watching MTV’s Are You The One. After watching the first season I was curious about the math behind it, is solving in 10 weeks impressive? Does it truly require connections to be made? Or is it closer to a mathematical inevitability?
Quick overview of the rules of the game (for season 1):
A group of men and women are secretly paired into couples by producers, via a matchmaking algorithm (10 couples).
The contestants try to identify all of their "perfect matches."
Each week, the contestants can propose one pairing and will receive confirmation whether or not the pairing is a perfect match (through the Truth Booth).
Each week, the contestants propose 10 couples and are told the number of couples that are correct, indicated by the number of lights which appear (they are not told which couples are correct)
They have ten weeks to guess all 10 couples correctly in order to win
The first step was solving the game for each week, which is fun in itself because you get to create a bit of dramatic irony as you gain more information than the contestants have. So I wrote a python script (https://github.com/shaunmac611/are_you_the_one) that takes in the pairings for each weeks and how many lights (or correct matches) there are with those pairings. I also added Truth Booth options. Given this input, the code will generate all possible scenarios and then identify which pairings are more likely based on how many of those scenarios include them as a pair.
Once we have this we can create a synthetic game, get 10 people in group 1 and 10 people in group 2, create a “secret” pairing amongst them and have them randomly guess pairings each night. We restrict their guesses to “possible” guesses, but not optimized beyond that. Their truth booths are more optimized and target pairings that would eliminate most options - whether they are right or wrong. We can run this a few hundred times to get an “expected” weeks this would take to solve
Are You The One possible solutions. Each grey line is a simulated game. The blue and red are the ‘luckiest’ game and ‘unluckiest’ game and the dotted black is the ‘median’ game.
We can see needing the full 10 weeks would require some unluckiness, and actually losing after 10 weeks is very uncommon
Percentage of simulations which arrive at one possible solution per night the single solution is guaranteed
Most games would likely end around night 8 or 9, with night 10 and beyond being possible but infrequent. If the weekly pairings were more optimized 10 weeks may even be the worst case scenario.
Obviously, the contestants don’t have perfect information. They will make mistakes and propose pairings that aren’t possible. While ten weeks seems to be “easy” algorithmically, it seems to be the prefect amount for the show. Currently, only one season of the show has lost the game. I’d imagine if you want to make a show about finding your perfect match, you’d want them to succeed most of the time so decreasing the number of weeks would result in more sad endings and increasing the number of weeks may make the game too easy (and cost more to produce). Ten weeks seems to be the sweet spot for this particular game.