You aren’t analyzing this game correctly. At the beginning of the game, you’re deciding between possible strategies for playing the game, and you should be evaluating the expected value of each of these strategies.
The strategy where you keep going until you lose has expected value −1. There is also a sequence of strategies depending on a positive integer n where you quit at the latest after the nth bet, and their expected values form an arithmetic progression. In other words, there isn’t an optimal strategy for this game because there are infinitely many strategies and their expected values get arbitrarily high.
In addition, the sequence of strategies I described tends to the first strategy in the limit as n tends to infinity, in some sense, but their expected values don’t respect this limit, which is what leads to the apparent paradox that you noted. In more mathematical language, what you’re seeing here is a failure of the ability to exchange limits and integrals (where the integrals are expected values). Less mathematically, you can’t evaluate the expected value of a sequence of infinitely many decisions by adding up the expected value of each individual decision. In practice, you will never be able to make infinitely many decisions, so this doesn’t really matter.
This issue is closely related to the puzzle where the Devil gives you money and takes it away infinitely many times. I don’t remember what it’s called.
Indeed they don’t, but the point is that while stopping at N+1 always dominates stopping at N, this thinking leads one to keep continuing and lose. As such, the only winning move is to do exactly NOT this and decide some arbitrary prior point to stop at (or decide indeterministically such as by coin flip). Attempting to maximize expected utility is the only strategy that won’t work. This game, prisoners’ dilemma, and newcomblike problems are all cases where choosing in such a way that does better (than the alternative) in all cases can still do worse overall.
The point isn’t that the strategy that is supposed to maximize expected utility is a bad idea. The point is that you’re computing its expected utility incorrectly because you’re switching a limit and an integral that you can’t switch. This is a completely different issue from the prisoner’s dilemma; it is entirely an issue of infinities and has nothing to do with the practical issue of being a decision-maker with bounded resources making finitely many decisions.
It isn’t a matter of switching a limit and an integral, or any means of infinity really. You could just consider the 1 number you’re currently on, your options are to continue or stop. To come out of the game with any money, one must at some point say “forget maximizing expected utility, I’m not risking losing what I’ve acquired”. By stopping, you lose expected utility compared to continuing exactly 1 more time. My point being that it is not always the case that “you must maximize expected utility”, for in some cases it may be wrong or impossible to do so.
All you’ve shown is that maximizing expected utility infinitely many times does not maximize the expected utility you get at the end of the infinitely many decisions you’ve made. This is entirely a matter of switching a limit and an integral, and it is irrelevant to practical decision-making.
You aren’t analyzing this game correctly. At the beginning of the game, you’re deciding between possible strategies for playing the game, and you should be evaluating the expected value of each of these strategies.
The strategy where you keep going until you lose has expected value −1. There is also a sequence of strategies depending on a positive integer n where you quit at the latest after the nth bet, and their expected values form an arithmetic progression. In other words, there isn’t an optimal strategy for this game because there are infinitely many strategies and their expected values get arbitrarily high.
In addition, the sequence of strategies I described tends to the first strategy in the limit as n tends to infinity, in some sense, but their expected values don’t respect this limit, which is what leads to the apparent paradox that you noted. In more mathematical language, what you’re seeing here is a failure of the ability to exchange limits and integrals (where the integrals are expected values). Less mathematically, you can’t evaluate the expected value of a sequence of infinitely many decisions by adding up the expected value of each individual decision. In practice, you will never be able to make infinitely many decisions, so this doesn’t really matter.
This issue is closely related to the puzzle where the Devil gives you money and takes it away infinitely many times. I don’t remember what it’s called.
Indeed they don’t, but the point is that while stopping at N+1 always dominates stopping at N, this thinking leads one to keep continuing and lose. As such, the only winning move is to do exactly NOT this and decide some arbitrary prior point to stop at (or decide indeterministically such as by coin flip). Attempting to maximize expected utility is the only strategy that won’t work. This game, prisoners’ dilemma, and newcomblike problems are all cases where choosing in such a way that does better (than the alternative) in all cases can still do worse overall.
The point isn’t that the strategy that is supposed to maximize expected utility is a bad idea. The point is that you’re computing its expected utility incorrectly because you’re switching a limit and an integral that you can’t switch. This is a completely different issue from the prisoner’s dilemma; it is entirely an issue of infinities and has nothing to do with the practical issue of being a decision-maker with bounded resources making finitely many decisions.
It isn’t a matter of switching a limit and an integral, or any means of infinity really. You could just consider the 1 number you’re currently on, your options are to continue or stop. To come out of the game with any money, one must at some point say “forget maximizing expected utility, I’m not risking losing what I’ve acquired”. By stopping, you lose expected utility compared to continuing exactly 1 more time. My point being that it is not always the case that “you must maximize expected utility”, for in some cases it may be wrong or impossible to do so.
All you’ve shown is that maximizing expected utility infinitely many times does not maximize the expected utility you get at the end of the infinitely many decisions you’ve made. This is entirely a matter of switching a limit and an integral, and it is irrelevant to practical decision-making.