In the following paper proposing a solution to Conway's Angel Problem, I have a issue with the first claim.

The author says that if the angel can jump on uneaten squares for any finite amount of time, then they can do so indefinitely:


Claim 2.1. If the Devil can catch the Angel of power p, then there exists a positive integer N such that the Devil can entrap the Angel in B(N).

Proof. If for every positive integer n the Angel has a strategy to make n moves without jumping on eaten squares, then a suitable limit of these strategies gives a winning strategy for the Angel to move forever without jumping on eaten squares. Hence if the Devil can catch the Angel, then there exists an n such that the Devil can catch the Angel in at most n steps. Thus, the Devil can entrap the Angel of power p in B(np).

But if I can keep going for any finite amount of time, it doesn't mean that I can keep going indefinitely, right? Here's an example of what I mean:


I have a group of soldiers and $1$ unit of rations, and I'm trying to keep the soldiers alive as long as possible. The soldiers can survive on any nonzero amount of rations per day. But, if I give them less rations than they got the previous day, then they'll mutiny and kill me.

What fraction of the rations should I give the soldiers each day?

The Issue

Obviously I can keep the soldiers alive for any finite number of days $n$, by giving them $1/n$ units of rations per day. But this doesn't mean I can keep them alive indefinitely. They're always going to eventually die or mutiny.

Back to the paper

Is there some difference between the paper's claim and my example that I'm not seeing? Is there some foundational result in game theory that I'm oblivious to?

  • 2
    $\begingroup$ Those unfamiliar may also see Wikipedia's "Angel problem" entry. $\endgroup$
    – Blue
    Jan 28, 2019 at 5:00


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