Very recently one of my floormates introduced me to Bumble, a Tinder-based dating app where girls must make the first move after a successful match is made. If a girl doesn’t contact a match within 24 hours, the match is deleted forever. It was refreshing to finally have a scenario where guys are the ones that sit down, wait for girls, and pick their favorite. I had believed this system to be more favorable to the traditional system where guys were mostly expected to make the first move. Boy was I wrong.

The Stable Marriage Problem is a well-studied problem concerned with finding the most stable individual matches between two groups of people. By a “stable” match — let’s call them Alice and Bob — the problem means that there does not exist a third partner — say Candace — in which Bob prefers Candace over Alice and Candace prefers Bob over her current partner. The problem is solved by the Gale–Shapley algorithm, a procedure in which a list of guys and girls are matched with each other based on their preferences so as to guarantee that the situation presented above never occurs. The Gale–Shapley algorithm works like so:

Two groups of n guys and girls form a ranked preference list of the other group. Every morning each man proposes to the most preferred woman on his list who has not yet rejected him. Each woman then collects all the proposals she received in the morning; to the man she likes best, she responds “maybe, come back tomorrow” (she now has him on a string), and to the others, she says “never”. After receiving the responses from the girls, each rejected man crosses off the woman who rejected him from his list. The procedure above repeats until each woman has a man on a string; on this day, each woman marries the man she has on a string. Credit: Adapted from lecture notes of UC Berkeley’s CS 70 course.

The guys ask the girls they like out, and the girls get to choose which of her suitors they prefer. If a girl rejects a guy, he moves on to a different — presumably less desirable — girl, asks her out, and waits for her response. Does this sound familiar to you? The Gale–Shapley algorithm is, in a loose sense, what Western society uses to find a suitable partner in the real world.

The interesting part of this problem comes from when we ask the question: Under this model, who benefits the most? The guys who must ask and face rejection? Or the girls who first wait and then make a choice? Let’s find out.

The algorithm guarantees that it will produce a list of stable matches. However, it does not say anything about who gets the optimal matches. An optimal match is when someone gets matched with the most-desirable match for him or her that does not create an unstable match. The algorithm can create multiple lists of stable matches, but who gets their optimal choice depends on who does the asking. Think about it. Guys move their way down their lists — they ask girls out according to their preference and stop at the first girl that says yes. Girls, however, must work their way up — they wait for a list of suitors and dump their current matches when better ones arrive. In essence, guys can only worsen their situation, while girls can only hope to improve it. When the music stops, guys are guaranteed to be with their best-possible match while girls must settle with whomever they are currently matched with.

Guys have it better in the dating world simply because they are, in general, expected to be more proactive to find themselves a good match — an option open to both guys and girls! So, next time you are rejected by the guy or girl you are crushing on keep in mind that you are truly the one who benefits the most.

Note: For those interested in the actual mathematical proof, I invite you to read the lecture notes from UC Berkeley’s CS 70 course.