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Expected value paradox? 
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#1
Oct2812, 01:02 PM

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Not long ago I was surprised to learn that when trying to maximize the expected longterm growth rate of your money, it is sometimes necessary to bet on an outcome that has negative expected value (in addition to outcomes that have positive expectation). See http://www.physicsforums.com/showthread.php?t=637064. I think I get that now, although I still don't consider it as obvious as everyone else seemed to. I'm used to problems where you are trying to maximize the total expected value on one round of betting. Surely, I thought, you would never bet any amount of your money on an outcome with negative expectation if you were trying to maximize your total expected value.
Can you think of an exception to this? Can you think of a situation where there are multiple outcomes to bet on, and in order to maximize your expected value (not expected rate of return or longterm growth rate) you must bet some fraction of your money on an outcome with negative EV, along with the other positive EV bets you make? 


#2
Oct2812, 04:35 PM

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Expectation values (for a single round) are additive. You cannot increase them with a bet with negative expectation value.
Two fair dice is rolled. If they show "1,1", you get 35.9 times your bet. If they show anything else, you get 1.5 times your bet. The second one has a positive expectation value, the first one a negative one. You want to bet a lot of money on the second (80%+? Did not calculate it) and keep the rest. But then you risk losing much which throws you back by several rounds. Give ~2% of your money to the first one, and you lose at most 30%, while the money increase in the other case is just a little bit smaller. 


#3
Oct2812, 05:47 PM

P: 254

If the odds you are getting are fixed, then you definitely can't increase your expected value by betting on a negative EV outcome. But what if the amount you bet affects the odds ("moves the line"), as is the case in a bet pool? Then your expected value is no longer linear in the amounts you bet on each outcome. 


#4
Oct2812, 06:07 PM

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Expected value paradox?



#5
Oct2812, 07:11 PM

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#6
Oct2912, 06:20 AM

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If your bets are correlated, it is meaningless to talk about the expectation value of a single one. You can compare "expectation value with it" and "expectation value without"  the former one should be larger with an ideal strategy, so this bet (given all others) has a positive expectation value.



#7
Oct2912, 08:48 AM

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#8
Oct2912, 09:20 AM

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Correlations between the expected money and other bets can be relevant, e. g. "if you bet 1 € on option A and win, you get 2 €. However, if you bet an additional 1 € on option B and option A wins, you get 10 €". If A has a positive expectation value, you would want to bet 1€ on B as well as it increases the expectation value  even if B itself and alone would have a negative expectation value. 


#9
Oct2912, 12:48 PM

P: 254

Okay, you convinced me. When your bet affects the odds you get on the various outcomes, it is somewhat arbitrary to try to assign expected values to the individual outcomes. So there's not much of a paradox here, just an oddity. Thank you, and good work :)
I don't want to disappoint people who looked at this thread hoping for a puzzle though. If you want a challenge, try to figure out the optimal bets to make on mutually exclusive outcomes when you are the last person to place your bets in a bet pool. For example, say a group of people are betting on the outcome of a single die roll. Everyone has placed their bets except you, and the breakdown of the pool so far is $1110 on 1 $1333 on 2 $1754 on 3 $1792 on 4 $1961 on 5 $2050 on 6 Assume that the die is fair and that you have a large enough budget to make any bet necessary. How much do you bet on each outcome to maximize your expected value? 


#10
Nov712, 07:59 PM

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Are you talking about poker?
I think that's the classic case of betting on an expected negative outcome (at least by the odds) in order to maximize your long term gain. The obvious case being bluffing to beat someone out of the pot. The subtler case being caught bluffing in an affordable pot to tempt others to challenge you when you do have a strong hand. 


#11
Nov812, 06:34 PM

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