Expected value paradox?

techmologist

Not long ago I was surprised to learn that when trying to maximize the expected long-term 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?

mfb

Expectation values (for a single round) are additive. You cannot increase them with a bet with negative expectation value.

Here an example which might be clearer:

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.

techmologist

Thanks for the reply.

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.

That's a good example. I understand the reasoning behind it, but I still wouldn't consider it obvious for someone who hadn't spent some time thinking about it. In fact, I think someone with a little mathematical knowledge is even more likely to get that wrong. One of the first things you learn when you start learning probability is that smart gamblers look for bets with a postive EV and avoid those with a negative EV. Once you understand why that is, it is surprising to learn that it is not strictly true.

mfb

Then you have to consider all correlated bets at the same time and find the ideal distribution.

techmologist

Yes. But would you be surprised to find yourself betting some of your money on an outcome with a negative expectation? It would seem strange to me.

mfb

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.

techmologist

I see what you are saying, but I still think it is possible to talk about the expected value of each bet separately. Your total winnings is the sum of the winnings from each bet, so the expected value is the sum of the expected values of each bet. This is true even if the outcomes are correlated (even mutually exclusive, for an extreme case). This was also the case for the problem of maximizing longterm growth rate when the odds were fixed. The outcomes were mutually exclusive, but we still talked about some of them having positive EV and others having negative EV.

mfb

No, correlations between the outcomes do not matter.
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.

techmologist

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?

BobG

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.

techmologist

Actually, I hadn't even thought about situations where psychology comes into play. That is a little tricky because you have to assign a subjective probability for you opponent to fold. It might be worth it if the pot odds you're getting are good. So that would really be positive EV. The subtler case you mentioned really would be negative EV, at least for that hand. That is very similar to mfb's example. You bet on a small, negative EV bet now for the opportunity to make a large +EV bet later. mfb convinced me that it is difficult to disentangle the amount of EV that comes from each individual bet in cases like that. So your answer is a good answer to my question. I originally thought this was strange enough to qualify as "parodoxical" in the informal sense (think Parrondo's paradox). But it appears to be much more common than I thought. I am used to thinking that smart gamblers make only positive EV bets, but there are so many exceptions to this rule that maybe I shouldn't have been surprised when I stumbled on a particular one.