tag:blogger.com,1999:blog-8471376121734156326.comments2011-06-24T07:02:24.687-07:00WNIOOdatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comBlogger11125tag:blogger.com,1999:blog-8471376121734156326.post-84928575244987352172011-06-24T07:02:24.687-07:002011-06-24T07:02:24.687-07:00Regarding probability... I'll take whatever wi...Regarding probability... I'll take whatever wikipedia says or a textbook says. Did it seem that I have a strange assumption about the meaning of the word? I glanced at http://en.wikipedia.org/wiki/Probability_theory and saw that they immediately start talking about distributions. Wiki says: <br /><br />Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results ... be assigned a value of one.<br /><br />That is... the notion of probability assumes an integrable distribution or an convergent discrete assignment of weights to conditions (or events).Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-59644985091885890192011-06-24T06:57:48.906-07:002011-06-24T06:57:48.906-07:00I read point 5 as a conditional probability. Let ...I read point 5 as a conditional probability. Let x be the condition "the envelope is known to contain value X." Let y be the condition "the other envelope is worse." <br /><br />5. For all x, p(y|x) = 2/3 . <br /><br />I read 4 as a conditional probability, too: let x2000 be the condition "the envelope is known to contain 2000 gameshow points." Then <br /><br />4. p(y|x2000) = 2/3. <br /><br />But let x0 be the condition "the envelope contains X0," where X0 is the gameshow's minimal value. Then p(y|x0) = 0. Of course, the player may not know that X0 is the minimal value. <br /><br />Let xTOP be the condition "the envelope contains XTOP", where XTOP is the maximal value the gameshow ever considers offering. Then p(y|xTOP) = 1. Again, the player does not know this.<br /><br />6. I read this as p(y) = 2/3.<br /><br />You begin with an assumption such as "indifference" or "we don't know the conversion ratio between game-points and utility" and the conclusions are strange. One conclusion you got is that <br /><br />> the probability of switching down is exactly 1/3 ... and a "hard-wired" distribution. We began by saying that we know nothing about the distribution of prizes... so how could we end up concluding that we know something about the distribution? Only because our usual arguments are invalid when we don't have an integrable prior.<br /><br />> there is no minimal value, and arbitrarily small values are much more likely than any larger values. A distribution like that can't translate into real-world money or utility. <br /><br />> the distribution is a constant multiple of X^(-0.5) and therefore is not integrable. <br /><br />There are a lot of possible values between 0 and infinity... and to suppose that they are all equally likely leads to strange conclusions. Maybe that is part of what makes the paradox compelling -- that we strongly want to assume a uniform distribution on the natural numbers, or the positive real numbers. <br /><br />Maybe we should have the player make some assumption about c, and then revise it during the game.Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-9241879187382669412011-06-24T06:38:50.112-07:002011-06-24T06:38:50.112-07:00@ResCogitans: Though I'm rather new to bloggin...@ResCogitans: Though I'm rather new to blogging, I know that people want credit for their ideas. Thanks for pointing out the missing reference.Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-12522722828137577372011-06-24T06:36:52.229-07:002011-06-24T06:36:52.229-07:00@ResCogitans: Please find your blog referenced at ...@ResCogitans: Please find your blog referenced at the top of this post. <br />@others: argument 1-6 comes from ResCogitans' blog.Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-76425647385010486852011-06-24T05:45:26.459-07:002011-06-24T05:45:26.459-07:00it is polite to at least acknowledge referenced ma...it is polite to at least acknowledge referenced material you know!<br /><br />I'll grant you that point 5 is definitely suspect, but surely point 4 is the same as point 6?<br /><br />Of course there is a probability distribution for how high an envelope value can be, but the probability factors of 1/3 and 2/3 are hardwired in and so such a distribution has to be very specific - whereas in reality it never will be.<br /><br />With moral paradoxes I'll happily conclude that the premise of morality is wrong. And it is probability paradoxes like this that lead me towards concluding that 'probability' itself is a shaky concept at times.<br /><br />I defy you to come up with a definition of the word 'probability' such that i couldn't come up with a scenario of the form "the probability of X is p" which isn't covered by your definition!ResCogitanshttp://www.blogger.com/profile/16098462922178341583noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-55413691669517498052011-06-13T07:23:55.118-07:002011-06-13T07:23:55.118-07:00The encyclopaedia of philosophy article about the ...The encyclopaedia of philosophy article about the St.P paradox argues from a philosophical point of view against bounded utility, limiting games to finite jackpots (or to probability distributions and payoff functions for which the expected value integral converges) or other constructions which would give the game a finite value. <br /><br />This might lead one to consider whether to pay "any finite amount" or whether to try to pay an infinite amount. But apparently in the philosophical community, the argument is over whether the St. Petersburg lottery is worthwhile at a price of, say, 100 coin.<br /><br />I'm personally pulled to accept "any finite value" as a good price for the St.P game; I would then try to get my savings (and loans, etc.) back by making side bets beforehand, leveraging those unlikely possibilities. But then my own opinion seems rather silly to me, which is good; this is a "paradox."Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-78888866588888307762011-06-13T04:58:04.151-07:002011-06-13T04:58:04.151-07:00Perhaps variable-ROI can explain how much an indiv...Perhaps variable-ROI can explain how much an individual should gamble, as a function of that person's alternatives -- consumption, investment -- and investment goals. <br /><br />This <a href="http://irserver.ucd.ie/dspace/bitstream/10197/539/3/farrelll_article_pub_001.pdf" rel="nofollow"> academic article </a> mentions that variable ROI, if it causes increasing marginal utility, makes gambling rational. Maybe the argument was like this: increasing utility makes gambling rational. Bailey et al. (1980) argued that delayed consumption would achieve the same average utility as gambling. (That is, Bailey wrote that rather than gamble 1 coin with 49:51 odds of winning another coin, the consumer might first consume 0, earn another coin, and then consume 2 coins. The consumer's average utility would seem to be the gambler's average utility). The present article argues that the gambler has the advantage of obtaining the 2 coins now. Marginal consumption utility probably diminishes anyway, but marginal investment utility is what increases, and of course it is better to buy an investment now rather than later; the average of buying it now or not buying it is better than the average of not buying it now and then buying it later. The article goes on to discuss how we prefer one time to another; my basic mode for these would be: that we want to smooth out consumption over time, while we want all our investments to be made as soon as possible. <br /><br />Of course, this 14-page article from the American Economic Review contains winning strategies, proofs, and commentary at the cutting edge of economic thought, whereas we are presenting what must be an old argument: that increasing ROI makes gambling-then-investment rational.Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-38262365539292468122011-06-13T04:01:43.966-07:002011-06-13T04:01:43.966-07:00That site is: Utility of money to a poker playerThat site is: <a href="http://www.quantitativepoker.com/2011/01/utility-part-1-basics.html" rel="nofollow"> Utility of money to a poker player</a>Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-51951154239018633832011-06-13T00:55:01.924-07:002011-06-13T00:55:01.924-07:00At http://www.quantitativepoker.com/2011/01/utilit...At http://www.quantitativepoker.com/2011/01/utility-part-1-basics.html I asked whether "moves with negative expected value even below the expected value of paying the ante and always folding" (-EV moves) occur in poker. Since poker players are losing their money constantly due to ante, I supposed that the money can be considered as "invested with a -ROI" and that players who are winning have the luxury of paying a lower ante % and can wait and only play the best moves. Their money is effectively invested with a better ROI. So -EV moves should occur, according to the argument I made above. But -EV moves may make you vulnerable to exploitation. <br /><br />The author of that blog mentions that when poker is viewed from the outside, then a -EV move can occur -- if you see a good investment opportunity, you might put your cash on the roulette wheel, since going home without money and going home having played the good opportunity are, on average, better than going home with your cash.Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-27691570639556481782011-06-11T14:02:54.633-07:002011-06-11T14:02:54.633-07:00Thank you, I read that article. The author compar...Thank you, I read that article. The author compares the St. Petersburg game to a cash machine that can print arbitrarily large sums of money...<br /><br />"The payoff of any conceivable game is always finite.... The paradoxical result can be put this way: no matter what (finite) entry price X is charged, it can be shown that the expected payoff of the game is larger than that..."Odatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comtag:blogger.com,1999:blog-8471376121734156326.post-92030675473728805352011-06-10T20:12:40.554-07:002011-06-10T20:12:40.554-07:00This looks great; you might want to look a little ...This looks great; you might want to look a little in to Loss Aversion and the dopamine system of the brain; and here is another source if you need one: http://plato.stanford.edu/entries/paradox-stpetersburg/Joseph Lewishttp://www.blogger.com/profile/17619923769410887420noreply@blogger.com