stable marriage problem

i can’t sleep because someone decided to say “stable marriage problem” and then my mind suddenly became so invested in this game theory and i stopped studying and now i can’t sleep. so here it is: there are a group of males and a group of females. same number of people in each. each female has a preference of males in a list. and each male has a preference of females in a list. during round one, all the males propose to their first choice. if a female has more than one male proposing, she says “yes (it’s more like a maybe)” to male that’s highest on her list and rejects the rest. if a female has just one guy, she says “yes (maybe)”.

round 2 begins and the males that don’t have a yes female go on and ask their second choice, even if the female said yes to some other male. if a female that’s already in a “yes” relationship gets asked by a male that is higher on her list than the guy she yes-ed to, she can dump her first guy (crycry, suicide) and move higher up the list. this keeps repeating and eventually, everyone is paired up. apparently, by this method, all the marriages are stable and there are no relationships that have people that would like to be with another person and have the other person want to be with them. so everything is stable.

but the results are kind of weird. it turns out that the coupling is optimal for males and pessimal (what an odd word, but wikipedia uses it) for females. this means males get their higher choices and females are usually not getting their highest choice. yet, females are the ones that are rejecting and moving higher. so it seems odd. but the popular girls that have a lot of guys can be choosy and get choices near the top. so that sucks for the girls that don’t get proposed to very much, because they are stuck with their lame few choices. so i guess those females can be more optimal if they are asking. (but that is against societal norms!)

but this assumes that everyone is in a room and knows all their options. which isn’t likely, and people come in and out of life, so this cross-section of time is not good. and there’s not always equal numbers of males and females (like in china!) and blahblabhabh. lists of preference change. variables everywhere.

anyway, i thought about the if there was an extra male. it would just end up in a loop, i think. always one leftover guy, just going down his list. and i guess… nevermind. just kidding. he would just be left out. poor guy. there would be some sorting around, but eventually no one would want that guy. well, extra girl is easy, because she’s just the girl that no guy wants. the fat ugly one. poor her.

also, it’s weird that if all the guys have a different girl as their first choice, then in one round, everyone is in a stable marriage. that is odd. doesn’t even consider the females. but as long as everything is stable. stable state. lowest energy state. boring. no excitation. oh, that’s dirty.

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