Someone For Everyone?
One of MTV’s infamous “reality” shows, Are You the One?, challenges 10 single males and 10 single females to find their “perfect match” before the end of 10 weeks. Dating experts claimed to find perfect matches in this group of 20 people, and if everyone can find his/her perfect match, all contestants get to split one million dollars. During each episode, contestants socialize, date, fight (it wouldn’t be a reality show without any drama!), and try to determine which one of the other contestants is his/her optimal partner. At the end of each episode, everyone is partnered up and told whether or not his/her partner this episode is, indeed, his/her perfect match.
As contestants mingle and date, they are evaluating each other, trying to determine which people may be potential matches. This is analogous to our class discussion of a perfect matching in a graph consisted of buyers and sellers. In graph theory, a perfect matching assigns buyers to sellers, taking into account the buyers’ valuations of the item being sold and calculating a market-clearing price where every buyer and seller is matched (assuming we have the same number of buyers and sellers). On a simplified level, the contestants of Are You the One? are executing this algorithm every episode. When the contestants interact and socialize, they are essentially determining their values. Regardless of who we consider to be buyers or sellers in this metaphor, the values and prices in the bi-partite graph correspond to levels of compatibility that the contestants feel with each other. For example, much like a buyer may have a high value for one seller’s item but a low value for another seller’s item, a contestant on this show could feel more compatible with one person over another. In this way, each iteration of the algorithm executed in determining market-clearing prices is essentially an episode of Are You the One? At the end of each episode, the feedback the couples receive lets them know if they were right about their suspected partner. If they are incorrect, they try again next episode, updating their valuations of the other contestants, thus executing the algorithm again. In the next episode, people now have more information and are able to reconsider and reevaluate their potential partners, and the cycle repeats until the season ends. Granted, there are many factors that influence the perceptions contestants have of each other – it would be extremely difficult to reduce feelings and confidence in compatibility to numbers. However, the general idea remains the same: the right prices in a buyer – seller network results in a perfect match much like the right evaluations of fellow contestants in Are You the One? reveal perfect matches.
Unlike the algorithm used to determine market-clearing prices, which is executed until the matches are made, Are You the One? contestants only have 10 iterations to determine the perfect matches. This constraint complicates the mathematics behind this problem and has led to many discussions on the probability of the contestants succeeding. A blog concerning mathematics in the real world explains some of the combinatorics relevant to the show and has some commentary on the best strategies the couples could use to increase their chances of winning. Although this aspect of the show moves further from topics relevant to our class, it is evident that the concept of market-clearing prices and a perfect matching is a simple model for the perfectly matched couples in MTV’s latest hit.