Why asking her out is better than waiting: algorithms to live by

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In dating, as a guy, we have it pretty good. Dating norms usually means the man usually initiates or proposes for a date most of the time. There’s no biological time limit for men as there is for women — meaning more time to focus on other activities (like ‘trying to make it’) instead of having to tie down with a ball & chain.

Sure, there’s the cringe from rejections but when you look at it real closely — men have it pretty good. Here’s why:

Imagine there are three boys and three girls at a party. We call our six singles: Joey, Chandler, Ross, Phoebe, Monica, and Rachel. We rationally assume that each individual has their own preferences; an ordered list on who they would like to date.

Ross and Chandler have ranked Rachel as their first choice. This clash means that if everyone wants a date, there will need to be some sort of compromise.

Let’s assume it’s better to be with someone than leaving without a date. Also, a girl can’t revoke a proposal.

Playing out the scenario, each boy proposes to his first-choice girl. Rachel gets proposed by both Ross and Chandler, so she needs to pick between them. Ross is higher on her preferences, so Rachel and Ross hook-up. Rachel secretly hopes to hookup with Joey.

Chandler, still single, goes for the 2nd-choice girl: Monica. Since Monica has no other proposals, she hooks-up with Chandler but hopes to hook-up with Joey.

Phoebe, without an offer from Ross or Chandler hook-ups with Joey.

The dust is settled and the final couples are:

1. Ross - Rachel
2. Chandler — Monica
3. Joey — Phoebe

Our solution is optimal for the boys; in no other situations can the boys improve their preferences. Chandler is without his top choice — but she already shot him down. The boys have zero incentive to switch partners, even if the girls wanted to switch — Rachel might prefer Joey, but Joey’s first choice was Phoebe.

Here’s why it’s better for the boys: Ross ended up with his first, choice, Chandler second and Joey’s first. Rachel, Phoebe and Monica end up with their second, third and second choices.

If the girls approached the boys, the couples would be:

1. Rachel — Joey (1st choice)
2. Phoebe — Ross (1st choice)
3. Monica — Chandler (2nd choice)

This setup is known as the ‘stable marriage problem’ and the process to pick their partners is known as the Gale-Shapley algorithm.

Regardless of the number of boys or girls, there are four conclusions that will always be true:

1. Everyone will find a partner
2. Once all partners are determined, no man and woman in different couples could improve their happiness by leaving their current partner. (Phoebe may want Ross, but he’s happy with Rachel)
3. Once all partners are determined, every man will have the best partner available to him.
4. Once all partners are determined, every woman will end up with the least bad of all men who approach her.

The last two points demonstrate a surprising result: the group (ie. boys) who does the asking and risk continual rejection end up far better than the group who sits back and accept a suitor’s advances. Intuitively, this makes sense — if you take initiative, start at the top of the list and work your way down, you will end up with the best possible person who’ll have you. So aim high, aim frequently: the maths says so.

Notes on the Gale-Shapley algorithm:

Essentially the algorithm involves rounds:

• In the first round, a) each unengaged guy proposes to his first choice b) women says ‘maybe’ and ‘no’ to everyone else. Suitor is provisionally engaged.
• Subsequent round, a) each unengaged man proposes to the next most-preferred who he has not yet proposed. b) women says ‘maybe’ if she’s not engaged or she prefers this guy over her current partner.
• Subsequent rounds repeated until everyone is engaged.

Runtime complexity is O(n²).

Sources:

https://www.nobelprize.org/uploads/2018/06/popular-economicsciences2012.pdf

https://en.wikipedia.org/wiki/Stable_marriage_problem

https://www.cse.unsw.edu.au/~tw/prvwcomsoc10.pdf

https://www.cs.princeton.edu/~wayne/kleinberg-tardos/pdf/01StableMatching.pdf