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| In [[probability theory]], '''Proebsting's paradox''' is an argument that appears to show that the [[Kelly criterion]] can lead to ruin. Although it can be resolved mathematically, it raises some interesting issues about the practical application of Kelly, especially in investing. It was named and first discussed by [[Edward O. Thorp]] in 2008.<ref name="Wilmott II">E. O. Thorp, <cite>Understanding the Kelly Criterion: Part II</cite>, Wilmott Magazine, September 2008</ref>
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| == Statement of the paradox ==
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| If a bet is equally likely to win or lose, and pays b times the stake for a win, the Kelly bet is:
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| : <math> f^{*} = \frac{b - 1}{2b} \! </math>
| | It depends on the quality of the Wordpress theme but even if it's not a professional one you will be able to average 50-60$ EACH link. You can either install Word - Press yourself or use free services offered on the web today. The effect is to promote older posts by moving them back onto the front page and into the rss feed. s and intelligently including a substantial amount of key words in the title tags, image links, etc. provided by Word - Press Automatic Upgrade, so whenever you need to update the new version does not, it automatically creates no webmaster. <br><br>The Internet is a vast open market where businesses and consumers congregate. If you are a positive thinker businessman then today you have to put your business online. Our Daily Deal Software plugin brings the simplicity of setting up a Word - Press blog to the daily deal space. So, if you are looking for some option to build a giant e-commerce website, then e-shopping preferable CMS tools will be helpful for you. Word - Press makes it possible to successfully and manage your website. <br><br>Your Word - Press blog or site will also require a domain name which many hosting companies can also provide. Now if we talk about them one by one then -wordpress blog customization means customization of your blog such as installation of wordpress on your server by wordpress developer which will help you to acquire the SEO friendly blog application integrated with your site design as well as separate blog administration panel for starting up your own business blog,which demands a experienced wordpress designer. Possibly the most downloaded Word - Press plugin, the Google XML Sitemaps plugin but not only automatically creates a site map linking to everyone your pages and posts, it also notifies Google, Bing, Yahoo, and Ask. Our skilled expertise, skillfulness and excellence have been well known all across the world. Article Source: Stevens works in Internet and Network Marketing. <br><br>Google Maps Excellent navigation feature with Google Maps and latitude, for letting people who have access to your account Latitude know exactly where you are. Here is more in regards to [http://htxurl.info/backupplugin354764 backup plugin] check out the page. As an example, if you are promoting a product that cures hair-loss, you most likely would not wish to target your adverts to teens. This allows for keeping the content editing toolbar in place at all times no matter how far down the page is scrolled. The company gains commission from the customers' payment. Digital digital cameras now function gray-scale configurations which allow expert photographers to catch images only in black and white. <br><br>Website security has become a major concern among individuals all over the world. Being a Plugin Developer, it is important for you to know that development of Word - Press driven website should be done only when you enable debugging. It's not a secret that a lion share of activity on the internet is takes place on the Facebook. ) Remote Login: With the process of PSD to Wordpress conversion comes the advantage of flexibility. For your information, it is an open source web content management system. |
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| times wealth.<ref name="original Kelly article">J. L. Kelly, Jr, <cite>A New Interpretation of Information Rate</cite>, Bell System Technical Journal, 35, (1956), 917–926</ref> For example, if a 50/50 bet pays 2 to 1, Kelly says to bet 25% of wealth. If a 50/50 bet pays 5 to 1, Kelly says to bet 40% of wealth.
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| Now suppose a gambler is offered 2 to 1 payout and bets 25%. What should he do if the payout on new bets changes to 5 to 1? He should choose ''f''* to maximize:
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| : <math> 0.5 \ln(1.5 + 5f^{*}) + 0.5 \ln(0.75 - f^{*}) \! </math>
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| because if he wins he will have 1.5 (the 0.5 from winning the 25% bet at 2 to 1 odds) plus 5''f''*; and if he loses he must pay 0.25 from the first bet, and ''f''* from the second. Taking the derivative with respect to ''f''* and setting it to zero gives:
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| : <math> 5(0.75 - f^{*}) = 1.5 + 5f^{*} \! </math>
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| which can be rewritten:
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| : <math> 2.25 = 10f^{*} \!</math>
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| So ''f''* = 0.225.
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| The paradox is that the total bet, 0.25 + 0.225 = 0.475, is larger than the 0.4 Kelly bet if the 5 to 1 odds are offered from the beginning. It is counterintuitive that you bet more when some of the bet is at unfavorable odds. Todd Proebsting emailed [[Edward O. Thorp|Ed Thorp]] asking about this.
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| Ed Thorp realized the idea could be extended to give the Kelly bettor a nonzero probability of being ruined. He showed that if a gambler is offered 2 to 1 odds, then 4 to 1, then 8 to 1 and so on (2<sup>''n''</sup> to 1 for ''n'' = 1 to infinity) Kelly says to bet:
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| : <math> \frac{3^{n - 1}}{4^n} \! </math>
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| each time. The sum of all these bets is 1. So a Kelly gambler has a 50% chance of losing his entire wealth.
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| In general, if a bettor makes the Kelly bet on a 50/50 proposition with a payout of b<sub>1</sub>, and then is offered b<sub>2</sub>, he will bet a total of:
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| : <math> f^{*} = \frac{b_2 - 1}{2b_2} + \frac{b_1 - 1}{4}(\frac{1}{f_1} - \frac{1}{f_2}). \! </math>
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| The first term is what the bettor would bet if offered b<sub>2</sub> initially. The second term is positive if f<sub>2</sub> > f<sub>1</sub>, meaning that if the payout improves, the Kelly bettor will bet more than he would if just offered the second payout, while if the payout gets worse he will bet less than he would if offered only the second payout.
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| == Practical application ==
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| Many bets have the feature that payoffs and probabilities can change before the outcome is determined. In sports betting for example, the line may change several times before the event is held, and news may come out (such as an injury or weather forecast) that changes the probability of an outcome. In investing, a stock originally bought at $20 per share might be available now at $10 or $30 or any other price. Some sports bettors try to make income from anticipating line changes rather than predicting event outcomes. Some traders concentrate on possible short-term price movements of a security rather than its long-term fundamental prospects.<ref name="Handbook of Asset and Liability Management">S.A. Zenios and W.T. Ziemba, <cite>Handbook of Asset and Liability Management</cite>, North Holland (2006), ISBN 978-0-444-50875-1</ref>
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| A classic investing example is a trader who has exposure limits, say he is not allowed to have more than $1 million at risk in any one stock. That doesn't mean he cannot lose more than $1 million. If he buys $1 million of the stock at $20 and it goes to $10, he can buy another $500,000. If it then goes to $5, he can buy another $500,000. If it goes to zero (as stocks sometimes do), he can lose an infinite amount of money, despite never having more than $1 million at risk.<ref name="The Dhandho Investor">Mohnish Pabrai, <cite>The Dhandho Investor: The Low - Risk Value Method to High Returns</cite>, Wiley (2007), ISBN 978-0-470-04389-9</ref>
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| == Resolution ==
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| One easy way to dismiss the paradox is to note that Kelly assumes that odds do not change. A Kelly bettor who knows odds might change should factor this into a more complex Kelly bet. For example suppose a Kelly bettor is given a one-time opportunity to bet a 50/50 proposition at [[odds]] of 2 to 1. He knows there is a 50% chance that a second one-time opportunity will be offered at 5 to 1. Now he should maximize:
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| : <math> 0.25 \ln(1 + 2f_1) + 0.25 \ln(1 - f_1) + 0.25 \ln(1 + 2f_1 + 5f_2) + 0.25 \ln(1 - f_1 - f_2) \! </math>
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| with respect to both ''f''<sub>1</sub> and ''f''<sub>2</sub>. The answer turns out to be bet zero at 2 to 1, and wait for the chance of betting at 5 to 1, in which case you bet 40% of wealth. If the probability of being offered 5 to 1 odds is less than 50%, some amount between zero and 25% will be bet at 2 to 1. If the probability of being offered 5 to 1 odds is more than 50%, the Kelly bettor will actually make a negative bet at 2 to 1 odds (that is, bet on the 50/50 outcome with payout of 1/2 if he wins and paying 1 if he loses). In either case, his bet at 5 to 1 odds, if the opportunity is offered, is 40% minus 0.7 times his 2 to 1 bet. | |
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| This is not entirely satisfactory, however. If a Kelly bettor has incorrect beliefs about what future bets may be offered, he can make suboptimal choices, and even go broke. The Kelly criterion is supposed to do better than any essentially different strategy in the long run and have zero chance of ruin, as long as the bettor knows the probabilities and payouts.<ref name="original Kelly article">J. L. Kelly, Jr, <cite>A New Interpretation of Information Rate</cite>, Bell System Technical Journal, 35, (1956), 917–926</ref> The fact that it can be frustrated by unexpected new offers is puzzling. It is also puzzling that the Kelly bettor bets more at blended 2 to 1 and 5 to 1 odds than at 5 to 1 odds, and that it is improving odds that lead to the possibility of ruin.
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| More light on the issues was shed by an independent consideration of the problem by [[Aaron Brown (financial author)|Aaron Brown]], also communicated to [[Edward O. Thorp|Ed Thorp]] by email. In this formulation, the assumption is the bettor first sells back the initial bet, then makes a new bet at the second payout. In this case his total bet is:
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| : <math> f^{*} = \frac{b_2 - 1}{2b_2} - \frac{b_1 - 1}{4}\left(\frac{1}{f_1} - \frac{1}{f_2}\right)\frac{f_2-1}{f_2+1} \! </math>
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| which looks very similar to the fomula above for the Proebsting formulation, except that the sign is reversed on the second term and it is multiplied by an additional term.
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| For example, given the original example of a 2 to 1 payout followed by a 5 to 1 payout, in this formulation the bettor first bets 25% of wealth at 2 to 1. When the 5 to 1 payout is offered, the bettor can sell back the original bet for a loss of 0.125. His 2 to 1 bet pays 0.5 if he wins and costs 0.25 if he loses. At the new 5 to 1 payout, he could get a bet that pays 0.625 if he wins and costs 0.125 if he loses, this is 0.125 better than his original bet in both states. Therefore his original bet now has a value of -0.125. Given his new wealth level of 0.875, his 40% bet (the Kelly amount for the 5 to 1 payout) is 0.35.
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| The two formulations are equivalent. In the original formulation, the bettor has 0.25 bet at 2 to 1 and 0.225 bet at 5 to 1. If he wins, he gets 2.625 and if he loses he has 0.525. In the second formulation, the bettor has 0.875 and 0.35 bet at 5 to 1. If he wins, he gets 2.625 and if he loses he has 0.525. | |
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| The second formulation makes clear that the change in behavior results from the mark-to-market loss the investor experiences when the new payout is offered. This is a natural way to think in finance, less natural to a gambler. In this interpretation, the infinite series of doubling payouts does not ruin the Kelly bettor by enticing him to overbet, it extracts all his wealth through changes beyond his control.
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| == Cited references ==
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| <!--This section uses the Cite.php citation mechanism. If you would like more information on how to add references to this article, please see http://meta.wikimedia.org/wiki/Cite/Cite.php -->
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| <div class="references-small">
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| <references/>
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| </div>
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| [[Category:Games (probability)]]
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| [[Category:Information theory]]
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| [[Category:Statistical paradoxes]]
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| [[Category:Utility]]
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