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Natural Science Forum / Physics / Relativity / February 2005S. Petersburg paradox
I would like to discuss about the S. Petersburg Paradox. This is an attempt to crossposting,(I have never done it). Do you agree on the folloing presentation of it ?? A coin is tossed untill head comes (N=number of tosses)the player is paid 1 if head turns out at 1° toss; 2 if it turns out at 2 toss, 4 if it turns out at 3° toss..... and so on. The Expectation value of the game is: E= 2^(1-1)*1/(2^1) + 2^(2-1)*1/(2^2) + 2^(4-1)*1/(2^4)+....... E=1 / 2 + 1 / 2 + 1 / 2 + 1 / 2 +......................= infinite Since the Expectation value of the game is infinite, what is the even bet a bettor should be willing to pay in order to play the game????? Since the even bet should be = the Expectation value = infinite the bettor should be willing to pay a infinite amount of money; but it happen that any reasonable person would pay not more than a very little amoumt of money.Ence the paradox. My thesis is that the S. Peterburg paradox in the above formulation is not a paradox but it contain inconsinstent Math premises. Let me know your opinions. best regards beda pietanza > I would like to discuss about the S. Petersburg Paradox. > [quoted text clipped - 29 lines] > > beda pietanza > > I would like to discuss about the S. Petersburg Paradox. > > [quoted text clipped - 18 lines] > > Since the even bet should be = the Expectation value = infinite > > the bettor should be willing to pay a infinite amount of money; That's assuming that people want to maximise their expected winnings.But what people are actually trying to maximize is 'utility' or 'happiness' - which isn't proportional to how much money you've got. > > but it happen that any reasonable person would pay not more > > than a very little amoumt of money.Ence the paradox. [quoted text clipped - 7 lines] > > > > beda pietanza Forwarded wrote: > beda pietanza wrote: > > I would like to discuss about the S. Petersburg Paradox. [quoted text clipped - 5 lines] > > A coin is tossed untill head comes (N=number of tosses)the > > player is paid 1 if head turns out at 1° toss; 2 if it turns out at > > 2 toss, 4 if it turns out at 3° toss..... and so on. > > [quoted text clipped - 10 lines] > > Since the even bet should be = the Expectation value = infinite > > the bettor should be willing to pay a infinite amount of money; >aft627@aol.com wrote: >That's assuming that people want to maximise their expected >winnings.But what people are actually trying to maximize is 'utility' >or 'happiness' - which isn't proportional to how much money you've got. beda reply: I desagree, the SPP doesn't tell you how much you are going to win after N tosses; that depend on the unit of the money used to pay off: you can use a unit having a infinitesimal value and the amount you win can be of great utility. The weirdness of the SPP lay as I said on its math inconsistency. Keep in touch. best regards beda pietanza > > but it happen that any reasonable person would pay not more > > than a very little amoumt of money.Ence the paradox. [quoted text clipped - 7 lines] > > > > beda pietanza The SPP incorrectly assumes that a dollar received at some distant point in the future is worth the same as the dollar bet today. This is patently unrealistic. If you introduce a discount rate, so that a dollar received in the future is worth less than a dollar received today, the "paradox" disappears. Whittle shows this clearly in his his book _Probability via Expectation_ > Forwarded wrote: > > beda pietanza wrote: [quoted text clipped - 50 lines] > > > > > > beda pietanza >Forwarded wrote: >> beda pietanza wrote: [quoted text clipped - 34 lines] >The weirdness of the SPP lay as I said on its math inconsistency. >Keep in touch. Look at it from the casino's point of view. In fact, give it a play and see how much the casino wins from you: http://www.mathematik.com/Petersburg/Petersburg.html If you are pulling money out of your wallet to play this game, you probably don't have enough money to win. It's analagous to the "doubling sytems" people keep bringing to Vegas to break the casinos. If one doubles bets long enough eventually one must win as a mathematical certainty. But it's no mathematical certainly that one has enough money to double bets long enough. With a finite, limited number of plays the average result of the St.P. P. is a good sized loss, and in the real world that's all the most that real people can play. >best regards > [quoted text clipped - 11 lines] >> > >> > beda pietanza >> I would like to discuss about the S. Petersburg Paradox. >> [quoted text clipped - 23 lines] >> My thesis is that the S. Peterburg paradox in the above formulation >> is not a paradox but it contain inconsinstent Math premises. I don't think so. The "paradox" is simply that people aren't computers. Suppose you offer the same bet, but the maximum payout is a zillion dollars. Now offer it again, but the max is a jillion dollars, which is a kadillion orders of magnitude more than a zillion. You see the problem? The numbers that dominate the math are irrelevant to the person. Not only is he not going to live long enough to see the coin flip that pays a jillion in any case, so he will not pay for that part of the expectation, but really, he is no better off with a jillion dollars than a zillion. -- Roy L > >> I would like to discuss about the S. Petersburg Paradox. > >> [quoted text clipped - 35 lines] > > -- Roy L You are following the official explanation of the so called paradox, this explanation is related to the marginal utility concept; I don't this is the key to understand the SPP, the incongruence is that while the bettor is asked to pay a infinite amount of money this very infinite payment is not to increase the possibility to win but to ensure the possibility of loosing the infinite bet paid. The game warrants a maximum win if no bet is paid as the bet paid increases the bettor looses more and more, with a infinite bet paid there is a sure infinite lost. The configuration of the wager is fallacious. What is asked to be paid doesn't correspond to the real value and mechanism of the game. regards beda pietanza >> >> I would like to discuss about the S. Petersburg Paradox. >> >> [quoted text clipped - 36 lines] >You are following the official explanation of the so called paradox, >this explanation is related to the marginal utility concept; It's more related to an accurate calculation of expectation. >I don't this is the key to understand the SPP, the incongruence is >that while the bettor is asked to pay a infinite amount of money >this very infinite payment is not to increase the possibility to >win but to ensure the possibility of loosing the infinite >bet paid. Economics can't handle infinite quantities. -- Roy L This is not a paradox AFAICT. It's simply that the casino proof works only asymptotically, I think it's in the introductory chapters of a common textbook for randomized algorithms. So, don't sit at a roulette table thinking of this. Cheers, -- Eray > This is not a paradox AFAICT. > [quoted text clipped - 6 lines] > -- > Eray Can you explain it better: what is the casino proof ??? regards beda pietanza > > This is not a paradox AFAICT. > > [quoted text clipped - 8 lines] > > > Can you explain it better: what is the casino proof ??? Well, I suppose you want to increase the amount invested in a binary bet, like in the red/black game on the roulette table. What happens if you double the bet every time you _lose_? You would definitely win the game at some unknown distant horizon. The unfortunate thing is that such a strategy is worthless in practice, because you can double your bet only so few times. Given that you don't have unbounded wealth, and that there is non-zero probability of losing, the odds aren't good that you are going to "actually" win. (Even if you have more wealth than the house!) An interesting problem would be of course to study "unfair" roulette tables on which multiple games are going on at once. In some ways, this is reminiscient to certain underinvestigated stock market prediction problems which depend on huge computational power to accomplish (I think). There are interesting possibilities for a computer player when it goes online, I believe. (But I'd rather not go into details) The relation to randomized algorithms is not too clear to me either, a friend had mentioned it, I don't think I read that textbook, but in this case, it seems plain expected cost analysis, it doesn't even have much to do with algorithms, but surely similar ideas are used frequently in the analysis of randomized / amortized algorithms... This is the simplest mathematical idea that Sum{k=0 to n}2^k + 1 = 2^{k+1}, no you wouldn't even win a lot, even if you won, so of course doubling the money probably isn't a good idea if you want to win a huge amount of money (which kind of runs contrary to the idea of a casino :) On newsgroups, I'd heard that "counting cards" was illegal in some US states, so figure... The conclusion, if you like gambling with dice: make sure you absolutely know how to trick the dice. :) Cheers, -- Eray PS: I'm not a gambler, as you can tell. I can't take that many risks. > What happens if you double the bet every time you _lose_? You would > definitely win the game at some unknown distant horizon. Only with probability 1, not with certainty. - Tim >> I would like to discuss about the S. Petersburg Paradox. >> [quoted text clipped - 23 lines] >> My thesis is that the S. Peterburg paradox in the above formulation >> is not a paradox but it contain inconsinstent Math premises. If this is what you want to discuss then spell out what you think are the inconsistencies. > > Since the Expectation value of the game is infinite, what is the > > even bet a bettor should be willing to pay in order to play the > > game????? Here's something off the top of my head: The probability of winning 2^n units is (1/2)^n, so on average you need to play 1/((1/2)^n) = 2^n times to win 2^n units. However to play that many times, you pay x*2^n units in entrance fees, where x is the entrance fee for one throw.. So for every amount you win from a draw, you expect to have lost x*2^n units. So to come out on top, you'd pay x<1 units to enter. There's probably some silly mistake there, but I'm sure someone will point it out soon enough. -Sunny > > > Since the Expectation value of the game is infinite, what is the > > > even bet a bettor should be willing to pay in order to play the [quoted text clipped - 12 lines] > > -Sunny I am not sure I got your reasoning, I only point out to you that the SPP is meant to be played only once: the Expectation value of the game is said to be infinite and it is expected that a bettor should be willing to pay a infinite amount of money in order to play the game just once !!! This an absurd consequence of using loosely the infinite in the game that makes the game incoherent. I hope I helped you Best regards Beda pietanza >>>>Since the Expectation value of the game is infinite, what is the >>>>even bet a bettor should be willing to pay in order to play the [quoted text clipped - 21 lines] > willing to pay a infinite amount of money in order to play the game > just once !!! I think the confusion here is using "expected value" too loosely as a decision device in a situation where only one probabilistic check is made. Consider the following simpler game: You flip a coin. On the 0.0001% chance it lands on its side, I give you $100000000000000000. On the 99.9999% chance it lands heads or tails, you give me $100. Now, you have no doubt that the expected value is positive (huge in fact, but I don't feel like doing simple arithmetic since I'm a lazy person), but are you really willing to play the game just once and pay me, say, $1000000 to do it? I don't think so, though the expected value is much higher than that! The St. Petersburg "Paradox" is not a paradox for the precise reason as above. It has a high expectation value, infinite, actually. But you are talking about a situation where we just play "once" and playing into the psychological factors involved. There is no paradox though, for if we keep playing, forever and ever, the player will indeed come out ahead in the long run on average. -Yan Z. > >>>>Since the Expectation value of the game is infinite, what is the > >>>>even bet a bettor should be willing to pay in order to play the [quoted text clipped - 42 lines] > > -Yan Z. I don't think the SPP has really a infinite expectation value if you pay a infinite bet in order to play it once: the really payoff scheme for a infinite paid bet is : 1° outcome payoff 1- infinite net win = -infinite 2° outcome payoff 2-infinite net win = -infinite 3° outcome payoff 4-infinite net win = -infinite and so on.... if you pay a infinite amount of money you will loose it for sure and if you play N numbers of games you surely will loose N time the infinite bets you have paid. I post a intresting post in which I explain that the expectation value 1/2+1/2+1/2+.....=infinite doesn't belong properly to the SPP but to a recursive game. I wonder why I don't get any reply. It ay not appear in the thread so I add it below regards beda pietanza I will claim here that the calculation of the Expectation value of this game doesn't belong to it. For the Expectation value calculated as E= 1 / 2 + 1 / 2 +1 / 2 + 1 / 2 +......= infinite There is a iterative game that perfectly correspond to that formula: Each possibility for which a 1 / 2 is (paid) calculated, the relative attempt has to be effectively carried out independently of all the others. 1° toss, if head, 1 unit of payoff is paid; the game doesn't end The game starts again to attempt for the chance that in 2 tosses if the second one shows a head, in this case 2 unit of payoff is paid; the game doesn't end; The game starts again to attempt that in 3 tosses the 3° one shows a head if this happens 4 unit of payoff is paid and so on. Of course if the head turns out before the expected toss the attempt is aborted and the game starts again with next attempt and so on. You can see that with all the infinite attempts there are, effectively a infinite possibility of winning is there, also with multiple winnings, and this is the game that really correspond to the SPP scheme of Expectation value, in this case paying a infinite amount of money is justified by the real possibility that the game warrants a real chance of infinite gain and not as the SPP that warrants only a infinite sure lost (remember the SPP finishes as soon as once the head shows up). So what do you think ???? Best regards beda pietanza >>>>>>Since the Expectation value of the game is infinite, what is the >>>>>>even bet a bettor should be willing to pay in order to play the [quoted text clipped - 53 lines] > and if you play N numbers of games you surely will loose N time > the infinite bets you have paid. Nonsense. It's not possible to bet an infinite amount. > >>>>>>Since the Expectation value of the game is infinite, what is the > >>>>>>even bet a bettor should be willing to pay in order to play the [quoted text clipped - 55 lines] > > Nonsense. It's not possible to bet an infinite amount. You don't have to you only calculate it. In any case substituting infinite with a very large amount the destiny of the bettor doesn't changes. Please read my post dated 14.02.05 best regards beda pietanza >>>>>>>>Since the Expectation value of the game is infinite, what is the >>>>>>>>even bet a bettor should be willing to pay in order to play the [quoted text clipped - 62 lines] > In any case substituting infinite with a very large amount the destiny > of the bettor doesn't changes. The amount of any bet is necessarily finite, and for any and all bets the gambler's Expected gain is infinite. Correct if I´m wrong, but the expected premium of the game isn´t infinite, as i saw in most posts: E(P)=1*1/2+2*1/2^2+...+n*1/2^n = sum(t/2^t), t=1,..,n lim sum(t/2^t)= 2 with t=1,..,n and n->infinite, says my calculator Therefore, one should bet less than 2 monetary units in order to expect some gain with this game Nuno T wrote: > Correct if I´m wrong, but the expected premium of the game isn´t > infinite, as i saw in most posts: > E(P)=1*1/2+2*1/2^2+...+n*1/2^n = sum(t/2^t), t=1,..,n > lim sum(t/2^t)= 2 with t=1,..,n and n->infinite, says my calculator > Therefore, one should bet less than 2 monetary units in order to > expect some gain with this game. It's more like: sum(2^(t-1)/2^t) = sum(1/2) > Correct if I'm wrong, but the expected premium of the game isn't > infinite, as i saw in most posts: > > E(P)=1*1/2+2*1/2^2+...+n*1/2^n = sum(t/2^t), t=1,..,n You've got the payoffs wrong. In the paradox you don't get n, you get 2^n. - Tim > > Correct if I'm wrong, but the expected premium of the game isn't > > infinite, as i saw in most posts: [quoted text clipped - 3 lines] > You've got the payoffs wrong. In the paradox you don't get n, you get > 2^n. the payoff is = 2^(n-1) for a expected value of 1/2+1/2+1/2..=infinite if you use a payoff = 2^n the expected value is 1+1+1+....=infinite if you use a payoff = 2^(n+1) the expected value is 2+2+2+ .=infinite The same infinite expected values give a hint on the math inconsistency of the S.P.P.: if a bettor pays a infinite sum or a very large one he is bound to loose the most part if it. So the wager has to be better defined in order to permit a bettor to participate with full knowledge of the game, the infinite has to be excluded from the formulation of the conditions of the game. Once defined the wager for a maximum number of tosses N, in case the payoff is = 2^(n-1)then the even bet is N/2, still for very large N no bettor would participate to the game: paying for changes to win (high value of N.th tosses) that are practically zero is not rational. A bettor mostly prefer a known risk and a known bet: i.e putting a bet that the n.th toss will be Head: he pays 1/2 and in case he wins he get a known amount of money of 2^(n-1). Asking someone to pay 1/2 (for a single bet) multiplied for all the infinite possible N tosses is senseless. best regards beda pietanza > - Tim >> > E(P)=1*1/2+2*1/2^2+...+n*1/2^n = sum(t/2^t), t=1,..,n >> [quoted text clipped - 4 lines] > if you use a payoff = 2^n the expected value is 1+1+1+....=infinite > if you use a payoff = 2^(n+1) the expected value is 2+2+2+ .=infinite None of these are Nuno's payoff = n, which does yield a finite expected value. - Tim > > > Correct if I'm wrong, but the expected premium of the game isn't > > > infinite, as i saw in most posts: [quoted text clipped - 20 lines] > bettor would participate to the game: paying for changes to win (high > value of N.th tosses) that are practically zero is not rational. I agree with what you have said. I think a (slightly) clearer way of understanding this supposed paradox is to limit the payout instead of the number of tosses. (It's almost the same thing.) If the maximum payout is 2^20 = 1M units, which is paid for all n > 20, what is the value of the game? If the maximum payout is 2^30 = 1G units, what is the value of the game? Or, to turn the problem around, if I am willing to pay X units to play, how large should the maximum payout be to make the game fair? It then becomes clear that larger bets correspond to payouts that have no bearing on reality. Scott  SignatureScott Hemphill hemphill@alumni.caltech.edu "This isn't flying. This is falling, with style." -- Buzz Lightyear > > > > Correct if I'm wrong, but the expected premium of the game isn't > > > > infinite, as i saw in most posts: [quoted text clipped - 35 lines] > Scott Hemphill hemphill@alumni.caltech.edu > "This isn't flying. This is falling, with style." -- Buzz Lightyear I run into this "paradox" some years ago and I kept it in my mind because "I knew" it was fictitious, in the wait of finding out all the wrong implications. Now, first of all, there is a pseudo explanation of the unwillingness for the bettor to pay a large amount of money for playing the game in the marginal utility of winning very large amount of money: this marginal utility of money is a extraneous argument (true in another contest) that doesn't explain the "paradox". The false "paradox" and the extraneous argument are a pretentious construction that objectively build a "mental entrapment": once the construction has been well arranged it lives a proper live in the collective cultural baggage, completely independent of the truth worthiness of it: individuals keep on going in using it as an acquired truth just in the sake of collective acceptance. This is a drama for the individual intelligences to put up with these sort of collective bias, every one of us undergoes through this as a proof of social maturity. The above is just a personal reflection. Back to the math inconsistency of the S.P.P.: Your suggestions are sound and interesting, let us try other possible reflections: 1) we could arrange the play having the bettor paying each single bet related to single N.th toss one by one: paying 1/2 for the 1° toss get the outcome whatever it is ; try (by starting all over again) the 2° toss by paying 1/2 for it ; then try the 3° toss and so on surely the bettor would quit as soon as the game reaches a high N tosses and the chances that there would be a positive result become practically impossible. Likely the bettor would ask to play only the bet that ensures him (by paying just 1 / 2 units) a winning amount necessary to make the desired trip to Himalaya and for which doesn't have the money. 2) What would be the chances that head never turns out in a infinite N tosses ??? the answer is zero !!!! : these are the very chances that a bettor would get of having a positive return after having paid a infinite (or a very large amount of money) in order to play the game. 3) the play could be arranged having the bettor not pay anything and alternatively use the money won at the first trial of the game to pay for next play and this forever: it is a sort of the game of life: we all go out in the morning without knowing if we would run into a smile of a beautiful woman or get rubbed of our pocket, the tossing goes on and on with its positive and negative outcome till the sad final certain loosing conclusion. 4) The "paradoxical" aspect of the game is that if instead a coin we use a dice and the payoff is made as soon as the number 1 turns out (1 chance out of 6) the payoffs of the game explodes to higher values; For a "dice" with a large number of facets the chances that at high number of tosses correspond a concrete positive results are more certain. To close this free dissertation, I can say that no one has the "power" to stop this S.P.P. from being a fake as a paradox and prevent the future use of it: we can only knowingly nod and carry on. best regards beda pietanza > I am not sure I got your reasoning, I only point out to you that the > SPP is meant to be played only once: the Expectation value of the game [quoted text clipped - 10 lines] > > Beda pietanza Yes, I was thinking about a game which you could play more than once. After playing with the link given before (http://www.mathematik.com/Petersburg/Petersburg.html) I was attempting to explain why you don't seem to come out on top, even though the expected value of the game is infinite. Still, personally I would pay less than 1 unit to play the game, since then I would be garanteed to win more than the entrance fee;P -Sunny > > I am not sure I got your reasoning, I only point out to you that the > > SPP is meant to be played only once: the Expectation value of the game [quoted text clipped - 20 lines] > > -Sunny The amount you should pay in order to play must be calculated as the even bet = the Expectation value or it should be the result of a bargain between the bank and the player. regards beda pietanza > The amount you should pay in order to play must be calculated as the > even bet = the Expectation value or it should be the result of a bargain [quoted text clipped - 3 lines] > > beda pietanza Why? > > The amount you should pay in order to play must be calculated as the > > even bet = the Expectation value or it should be the result of a [quoted text clipped - 6 lines] > > Why? If the decision is left to the bettor he would pay zero and win for sure. regards beda pietanza > > The amount you should pay in order to play must be calculated as the > > even bet = the Expectation value or it should be the result of a [quoted text clipped - 6 lines] > > Why? well, you have two possibility: paying the infinite Expectation value or to pay a sum after a bargain with the bank. Otherwise if the bettor was to decide by himself he would pay zero and have the absolute certainty to win. best regards beda pietanza As I said in a previous post, if utility is the same as money, then any finite amount is worth betting providing you play until you win. (You cannot, as one poster suggested, do intermediate calculations with infinity. Infinity expresses the idea of a limit: it is not a number. Limits should be taken at the end of a calculation, not at the beginning.) The rest of the paradox is psychological: psychologically it is known that utility is better measured as the logarithm of money, which then gives a reasonable account of the attitudes that people hold towards the game. However, another interesting psychological factor to consider is the probabilty that you make a profit. With any reasonable definition of utility, this will be the case if your winnings are larger than your bet. To simplify things, suppose that your bet is a power of two: B = 2^{n}. Then the probablity that you make a profit is the same as the probability that you win on turn m = n + 2 or higher (with winnings on turn m given by 2^{m - 1}). This is easy to calculate. The probabilty that you make a profit is 1/(2B). If you bet B = 1, you have probability 1/2 of making a profit, because if you win on the first turn, you break even. However, your probability to make a profit goes down quite fast as the bet goes up, and this is another way to see the psychological barrier involved in betting a large amount. This too can be interpreted in terms of a particular utility function. Ian. >> I am not sure I got your reasoning, I only point out to you that the >> SPP is meant to be played only once: the Expectation value of the game [quoted text clipped - 20 lines] > > -Sunny > The rest of the paradox is psychological: psychologically it is > known that utility is better measured as the logarithm of money In that case, pay 2^2^n instead of 2^n. Then the expectation in utility is infinite. - Tim > I would like to discuss about the S. Petersburg Paradox. > [quoted text clipped - 29 lines] > > beda pietanza I will claim here that the calculation of the Expectation value of this game doesn't belong to it. For the Expectation value calculated as E= 1 / 2 + 1 / 2 +1 / 2 + 1 / 2 +......= infinite There is a iterative game that perfectly correspond to that formula: Each possibility for which a 1 / 2 is (paid) calculated, the relative attempt has to be effectively carried out independently of all the others. 1° toss, if head, 1 unit of payoff is paid; the game doesn't end The game starts again to attempt for the chance that in 2 tosses if the second one shows a head, in this case 2 unit of payoff is paid; the game doesn't end; The game starts again to attempt that in 3 tosses the 3° one shows a head if this happens 4 unit of payoff is paid and so on. Of course if the head turns out before the expected toss the attempt is aborted and the game starts again with next attempt and so on. You can see that with all the infinite attempts there are, effectively a infinite possibility of winning is there, also with multiple winnings, and this is the game that really correspond to the SPP scheme of Expectation value, in this case paying a infinite amount of money is justified by the real possibility that the game warrants a real chance of infinite gain and not as the SPP that warrants only a infinite sure lost (remember the SPP finishes as soon as once the head shows up). So what do you think ???? Best regards beda pietanza > I would like to discuss about the S. Petersburg Paradox. > > This is an attempt to crossposting,(I have never done it). It would make more sense just in sci.math. This has nothing to do with relativity. > My thesis is that the S. Peterburg paradox in the above formulation > is not a paradox but it contain inconsinstent Math premises. The game contains no inconsistent mathematical premises, and the conclusion that the expected winnings are infinite is also consistent. There is a subtle flaw in the assertion that therefore the game would be fair if an infinite amount was paid to play. It is neither fair nor unfair. It is undefined. - Tim |
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