# Silly Malthusian probability question

As panache45 and others note, if it indeed is the case that there is a fixed finite number of people, each of whom kills precisely one person, with no one killed by more than one person, then it must indeed be the case that everyone is killed. If this is the stipulation, then this is what happens.

As to the question of who kills the last person in such a scenario, it must either be that someone manages to kill after they themselves are killed (e.g., by planting a time-bomb), or that several people manage to kill simultaneous with their own death (e.g., a Mexican standoff with no survivors). The latter might well be seen as a special case of the former, with slight jitter of timing (e.g., a bullet released from one’s gun might kill others after one’s own death).

If we outlaw the possibility of simultaneous or posthumous killings, then… it cannot actually be the case that everyone kills one other person (with a fixed finite number of people, with no one killed by more than one person). Them’s the breaks.

[We can discuss other scenarios, of course, but they won’t actually involve everyone killing one other person (many scenarios under discussion, for example, involve some people killing before they manage to kill anyone). Once we allow scenarios where not everyone kills one other person, there is lots of room for considering different such scenarios with different traits, so we should not expect any One True Answer among them.]

The entire question ignores the infants born that day.

At the current rate of 255 children born per minute, that is 367,200 newborns added to the population that day – and all of them are too weak & uncoordinated to kill another person that day (or even for many days, months, or years).

Per my response:

And regarding who kills the last person, it has not been stipulated that death must be immediate. A person may be fatally injured, yet still have plenty of time to kill another.

I wonder what the probability is, of all of one gender being killed in this way, thereby dooming the entire species anyway.

Unless there’s some strong gender-based pattern in the choice of victims or timing of killings, the probability of this is so close to zero that we might as well ignore it.

Let’s assume a large number, n of cannibal bugs. All bugs are hungry, but killing and eating eating one other bug is sufficient for some time. All bugs are pretty much identical so no group has any advantage over any other. The bugs are in a confined space.

At the end of the experiment, all remaining bugs are full so the killing stops. No bug will kill or eat more than one other bug.

• All bugs will kill and eat only one other bug
• There are no injured bugs - death is instantaneous.
• Bugs can eat bugs that have eaten, so all bugs will have killed and eaten at the end.

Does this have enough information to arrive at a formulae? If so, what is it?

This only makes sense if you assume that no one can be killed unless they have already made their kill. That doesn’t seem logical to me, and I don’t think that is the question the OP is asking. The OP seems to be asking the question, if everybody tries to kill one other person, how many kill without being killed? How many kill and get killed? And how many are killed before they had a chance to kill?

This is not meant to be an organized mass suicide.

I think you need to add some specifications about how bugs target/attack each other and who wins (if they immediately pair off and fight each other, with winner eating loser, that’s a different answer than, say if the weakest bug is immediately eaten by a random non-weakest bug, then the second-weakest is eaten by a random remaining hungry bug, etc). You’ll want to be clear about what outcomes are random in your chosen scenario, too.

Certainly, in any scenario at least half the bugs must have been eaten, but you can arrange ones that result in exactly half remaining at the end (pair off), only one remaining at the end (weakest is eaten by second-weakest, repeat), or something in between.

Sure, sure. And, as a piece of interesting math, Oukile’s model is intriguing; I hope to have more to say on it soon.

These are the rules that make sense to me and Oukile arrived at the correct conclusion.

E(K(n)) = sum(i from 0 to n) (-1)^i * (n-i+1)/(i!)

and

lim(n to infinity) E(K(n))/n = exp(-1)

I think this related to the secretary problem in some way, but the connection is not absolutely clear to me.

Oh, that’s nice! Why is this?

But panache45 didn’t say that each person succeeded in killing another, only that they “must”. One would assume they were excused from this imperative if they were themselves dead.

As you say, the only way for total casualties in such a scenario is if the method of killing allows someone to still kill after being killed themselves. Which is not guaranteed by the scenario, so 100% casualties is not a certain outcome.

n.m Ninja’d

I don’t think you were ninja’ed, Buck Godot. I thought that was a very nice observation, not explicitly made by anyone else yet.

For what it’s worth, Buck Godot’s observation was that, if everyone randomly picked someone to kill, and then simultaneously went to work killing (some people multiple-killed and all), then the expected number of survivors out of n total people is (1 - 1/n)^n * n, which is asymptotically n/e for large n. Not exactly the same as Oukile’s model, but perhaps related.

[Also:

Whoops, typo corrected in bold. (Anyway, of course the most interesting version of this problem isn’t the pedantic version where everyone dies all the time)]

It’s nice and wrong. I shifted indices.

Should be E(K(n)) = sum(i from 0 to n-2) (-1)^i * (n-i-1)/(i!).

As for why… I have a couple of pages of incoherent derivation of this. If I have time I’ll try to boil it down to something understandable and post later.

Do you happen to have E(k, i) for general k as well?

The possibility of one survivor, in the scheme of only one kill allowed per person, does not fit the time frame. You cannot have a reverse-geometric-progression, because that only works if each person can kill more than one other person, which is not the edict. To have one person left, either huge numbers of people must engage in mutual murder of each other, or all the killings would have to be in a straight, sequential line to the last person. For almost the entire population to die over the full 172,800 second course of a day, over forty thousand people would have to die every second. In a linear sequence, that seems very unlikely, perhaps impossible.

Indistinguishable,

Further fooling around with a recursion relation like that of Oukile seems to indicate that for large i and k, E(k,i) ~ exp(-i/(k+i)).

Sorry I was working directly with the fraction. In this formulation it is

E(k,i) ~ (k+i) * exp(-i/(k+i))