Let’s say you save exactly one pill and it works on anyone. Also assume 100% pregnancy rate, so if you are paired with someone and neither have the pill, then it’s an automatic pregnancy. Our goal is to minimize number of pregnancies.

• “Max promiscuity”: Say we have a complete bipartite matching. if it’s given to a male, then no pregnancies have been prevented since every other male can impregnate every female. If it’s given to a female, then it reduces the number of pregnancies by 1 since none of the males can impregnate her.
• “Traditional”: Say we have a bijective matching (i.e. each male is paired with exactly one female, and vice versa). Then the pill can be given to anyone and it will always reduce the number of pregnancies by 1.
• “The Harem”: Say we have a matching where males have more than one pairing but females have at most one pairing each. In this case, giving the pill to the male with the largest number of pairings will reduce pregnancies by however many pairings they have.
• “Reverse Harem”: Same scenario as above but flip male and female. Giving the pill to any female will have the same effect of reducing pregnancies by 1. Giving it to a male will have no effect.
• “The Cliques”: The population is split into disjoint graphs, but each of these disjoint graphs are complete (bijective) bipartite graphs. In this case, if the pill is given to a male, then it will only have an effect if that male only has a single pairing, thereby reducing pregnancies by exactly 1. Otherwise, there will be no reduction in pregnancies. If given to a female, then it will always reduce pregnancies by exactly 1.

As far as I’m aware, the real world operates most like a mixture of “Traditional” and “Cliques”. At least, in places where birth control is an option. But in the real world, we have more than one pill.

If we have enough for either all males or all females, then the effect is the same regardless of who gets the pill. It will always lead to 100% pregnancy reduction.

Let’s say we have enough pills for all but one male, or all but one female.

• “Max promiscuity”: If the pill is given to the males, then we still have one male that can impregnate everyone, so there will be no reduction in pregnancies. If given to females, then you will end up with exactly one pregnancy.
• “Traditional”: As before, there’s no difference. Any decision will lead to reducing pregnancies to exactly 1.
• “The Harem”: giving to all the males except the one with the smallest number of pairings will reduce pregnancies to however many pairings that one male has (more than 1). If given to females, then it will reduce it to exactly 1.
• “Reverse Harem”: Giving it to the males will reduce pregnancies to exactly 1 since they’re only in 1 pairing. Giving it to females will also reduce it to exactly 1.
• “The Cliques”: if given to the males, then it will only make a difference if there exists a clique with exactly one male. It will reduce pregnancies by the largest number of females in a clique with a single male. If given to females, then it always reduces pregnancies to exactly 1.

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So with the goal of minimizing pregnancies, it either makes no difference or is optimal to have the pill on women (unless you’re in a harem). This is highly reductive though. We have many other considerations when deciding who should get access to birth control.