tag:blogger.com,1999:blog-8471376121734156326.post5682765388395864507..comments2011-06-24T07:02:24.687-07:00Comments on WNIO: Two-Envelope Paradox IIOdatafanhttp://www.blogger.com/profile/04335737875089687401noreply@blogger.comBlogger5125tag: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.com