tag:blogger.com,1999:blog-3891434218564545511.post4029399466043741214..comments2024-03-27T20:37:09.185-05:00Comments on Alexander Pruss's Blog: Expected utility maximizationAlexander R Prusshttp://www.blogger.com/profile/05989277655934827117noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-3891434218564545511.post-68832742221238721382022-11-08T09:48:15.838-06:002022-11-08T09:48:15.838-06:00Ian:
If Causal Finitism is false, you could tweak...Ian:<br /><br />If Causal Finitism is false, you could tweak the situation to make sure you can't just win all the bets. For instance, you could run the story in a supertask, make the payoffs come after the end of the supertask, and if you "won" all the bets (or even infinitely many of the bets), you lose all the benefits. This does not affect the statistical independence of all the events, because winning all the bets has zero probability, and changing things on a zero-probability set doesn't affect independence. But it does affect "intuitive" independence.Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-37861217085671238422022-10-28T16:01:47.076-05:002022-10-28T16:01:47.076-05:00Well, here is one way to look at headache setup…
...Well, here is one way to look at headache setup…<br /> <br />If you make only a finite number of bets, you and your friend together are certain to suffer an infinite number of headache days. If you accept all the bets, it’s possible that you might win them all. Then you and your friend will suffer no headaches. (I’m assuming that the headache-free days you win for your friend are taken sequentially without gaps.) Of course, this outcome has probability zero. But isn’t the <i>possibility</i> of no headaches, even at probability zero, to be preferred to the <i>certainty</i> of an infinite number? :-). And note, this sort of ‘reasoning’ applies even if the expected value of each bet is negative. :-)<br /><br />Hmm… This line of thought gives zero value to any merely finite change. But if you take the actual infinity seriously, and you compare by counting headache days, I’m not even sure that that is wrong. Maybe it’s better to stick with the original version with positive and negative payoffs.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-59381108278334523592022-10-26T10:39:41.978-05:002022-10-26T10:39:41.978-05:00Unbounded bets, while difficult to handle in decis...Unbounded bets, while difficult to handle in decision theory, don't seem metaphysically problematic. Suppose you have a friend you love as yourself but who is currently "scheduled" to have a headache for eternity, while you are currently "scheduled" to live forever without headache. Each time you win x units, your friend gets x days off from headache. Each time you lose x units, you get x days of headache. Specify that you don't get used to the headaches. (If you think it's metaphysically impossible not to be getting used to headaches, suppose your and your friend's memory of the previous day's headache or lack thereof is wiped each day.)Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-83933246510108893322022-10-26T03:34:26.332-05:002022-10-26T03:34:26.332-05:00What worries me about this is the actual infinity ...What worries me about this is the actual infinity of people and the unbounded bets. Suppose there are N people. It’s clear that for large N, most people are likely to lose small amounts. This is balanced by the very small probability that the last few people win very large amounts. To some people’s intuition (including mine) this does not seem like a good outcome. But I think that an <i>argument</i> for this has to be based on this finite case.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-30199966676618018042022-10-25T10:49:41.970-05:002022-10-25T10:49:41.970-05:00It's also interesting to think about whether o...It's also interesting to think about whether one's intuitions would be different in a reverse case. On day n, if you take the gamble, you are sure to get 1/2 unit and have a 1/2^n chance of losing 2^n. By expected utilities, you should refuse. But if you always accept, then almost surely you lose only a finite amount and gain an infinite amount. Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-82698856043705580572022-10-24T14:28:42.794-05:002022-10-24T14:28:42.794-05:00Ian:
I am trying to argue against the thesis that...Ian:<br /><br />I am trying to argue against the thesis that you should take the bets with positive expectation. Gambler's Ruin doesn't affect that because the bets have zero expectation. <br /><br />I also have a weak intuition that cases where what goes on in each wager is independent of what goes on in the others are more compelling. Triple-or-nothing doesn't have this independence: once ruined, you get nothing. Gambler's Ruin has a changing fortune. <br /><br />Another thing that makes my case particularly compelling to me is the interpersonal version, where each person faces a single wager, all the wagers completely independent, and yet if everyone maximizes their expected utility, with probability one, the result is disastrous--infinitely many paying and finitely many winning. It's like a tragedy of the commons, but with no interaction between the agents' decisions, no weird undefined probabilities, just everyone doing ordinary expected value maximization. Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-11066440471101772332022-10-24T01:55:41.633-05:002022-10-24T01:55:41.633-05:00None of that suggests that there is anything wrong...None of that suggests that there is anything wrong with Zhao and the others in a narrow formal sense – given their conditions, their formal results hold. Of course, the philosophical implications and practical relevance are a different matter. The same applies to your examples.<br /><br /><br />Against Zhao and the others, one could say this: it’s great that, given their conditions, choosing to accept every time ‘eventually’ becomes favoured, but is ‘eventually’ likely to be in your lifetime? It depends on the specifics.<br /><br />Against your examples, one could say that they require you and the other party to have unlimited money and unlimited time to gamble with it. What matters is the likely conditions when time or money run out.<br /><br />I’m doubtful about the applicability of standard decision theory for one-0ff high stakes choices, but I’m not sure that any of these cases are decisive.<br /><br />Our intuitions about sequences of fair (favourable, unfavourable) bets are reflected in the martingale (super-martingale, sub-martingale) optional stopping theorems. But these theorems have conditions which can be violated in quite ordinary setups. A simple example: repeated triple-or-nothing on fair coin flips. Each bet has positive expectation, but if you accept them all, you will lose you initial stake with probability 1.<br /><br />You don’t need exponential growth to get this sort of thing. A simple example is textbook Gambler’s Ruin. A fair coin is flipped. You win $1 on Heads, lose $1 on Tails. You play repeatedly against the house until either you go broke, or the house does. You start with $M, the house with $N. Your chance of ruin is M/(M+N), of ruining the house is N/(M+N). Your expected final fortune is of course the $M you started with.<br /><br />But what if the house has unlimited money? Then you will be ruined, and suffer a loss of $M, with probability 1.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-86186607865889092352022-10-20T10:01:20.776-05:002022-10-20T10:01:20.776-05:00There are two interrelated questions: the conditio...There are two interrelated questions: the conditions under which we have almost sure convergence to the mean and the conditions under which we have the particular kind of divergence that my argument uses--where almost surely if we accept the gambles, we lose an infinite amount and gain only a finite amount. <br /><br />The convergence question concerns necessary and sufficient conditions for the Strong Law of Large Numbers. Kolmogorov showed that the variance condition is sufficient (assuming throughout that all the random variables have finite expectations). But it is not necessary for convergence (indeed Prokhorov showed that no condition solely on variances can be a necessary and sufficient condition for convergence). Necessary and sufficient conditions were given by Nagaev ( https://epubs.siam.org/doi/abs/10.1137/1117072 ), but research on refinement continues ( https://www.jstor.org/stable/2160636 ). In any case, the failure to meet the condition Pekoz gives does not imply lack of convergence.<br /><br />Further, lack of convergence is not by itself enough to clearly show that it isn't rational to engage in expected utility maximization. After all, lack of convergence is compatible with the hypothesis that your total winnings will be infinite and your total losings will be infinite, in which case it's unclear if it's rational to play or not.<br /><br />However, it is much simpler to characterize cases that look like my example if we assume independence (by the way, my original example does not assume independence, because the Borel-Cantelli Lemma, unlike its converse which I am about to use, does not need independence). <br /><br />I can run my argument with any sequence Y_1,Y_2,... of random variables each of which has positive expected value, but where the sum P(Y_n > -epsilon) is finite for some positive epsilon. In that case, expected utility maximization says to accept each gamble, but if you follow that advice, you will (almost surely) get a result at least as bad as -epsilon in all but finitely many cases. If the random variables are independent, then by the converse Borel-Cantelli Lemma, the condition that the sum of those probabilities is finite is necessary for the claim that almost surely you will get a result at least as bad as -epsilon in all but finitely many cases. Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-16332889767046894072022-10-19T18:15:23.477-05:002022-10-19T18:15:23.477-05:00As I said, I can’t access Ross. But this paper [Er...As I said, I can’t access Ross. But this paper [Erol A. Peköz: Samuelson's Fallacy of Large Numbers and Optional Stopping. Journal of Risk and Insurance, March 2002], which builds on it, states a similar result. <br /><br />Peköz requires the condition than Σ((Nth variance)/N^2) is finite. I’m guessing the Ross’s condition may be similar. With even linear growth of prizes and reciprocal linear probabilities, the sum doesn’t converge. So it won’t converge with N (log N)^2 growth (and reciprocal probabilities) either. <br /><br /><br />For what little it’s worth, I share your doubts about EU for rare high-stakes gambles. My intent is to account for the apparent discrepancy between Zhao’s remarks and yours.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-74668973449306508882022-10-19T07:44:02.059-05:002022-10-19T07:44:02.059-05:00I guess I've tended to think that it's pre...I guess I've tended to think that it's precisely in cases of rare large stakes gambles that it makes sense to depart from expected utility. For small repeatable gambles, of course we have central limit theorem or law of large numbers considerations.<br />By the way, we don't need anything as radical as exponential growth. Barely more than linear growth is enough. My point goes through with n (log n)^2 as the nth prize with probability the reciprocal of that. Alexander R Prusshttps://www.blogger.com/profile/05989277655934827117noreply@blogger.comtag:blogger.com,1999:blog-3891434218564545511.post-49577490878643363892022-10-19T04:05:24.675-05:002022-10-19T04:05:24.675-05:00“… and suggests a limitation of the argument here....“… and suggests a limitation of the argument here.”<br /><br />From a formal point of view, I think the issue is that the sequence of gambles is not ‘well-behaved’ in the sense of Zhao’s footnote 14. I can’t be sure, because Zhao does not spell this out, but refers to <i>Stephen Ross “Adding Risks: Samuelson’s Fallacy of LargeNumbers Revisited.” Journal of Financial and Quantitative Analysis, 34:323–339, 1999 </i>, which is gated. (SUPPORT OPEN ACCESS!) Zhao says <i>The requirement is meant to rule out improbable cases like those where one decision has stakes that swamp all others, … .</i> In this case, the last gamble is always about the size of all the previous ones together.IanShttps://www.blogger.com/profile/00111583711680190175noreply@blogger.com