CE - Certainty Equivalent, what exactly is it's use?

[DSchles]

"Any actual occurrences of these payoffs mean the bankroll is growing faster than has been allowed for (assuming the one payoff that has been analysed occurs with its expected frequency).

"Hence my assertion that he can only ever underestimate the optimal fraction."

And again, I tell you that this isn't necessarily true. The other payoffs can have positive e.v. and contribute to bankroll growth, but also can have variances that are so large as to swamp the e.v., in which case they would decrease f*. Simple example: you get a 1,000 to 1 payoff for a 990 to 1 shot. Small edge, huge variance.

Don

 

[London Colin]

And again, I tell you that this isn't necessarily true. The other payoffs can have positive e.v. and contribute to bankroll growth, but also can have variances that are so large as to swamp the e.v., in which case they would decrease f*. Simple example: you get a 1,000 to 1 payoff for a 990 to 1 shot. Small edge, huge variance.

hmm Below is what I initially wrote in response. I'm now so befuddled that I am starting to doubt my own logic. But this is hopefully a clear description of what that logic is. So, if there is a flaw in it, it can hopefully be pointed out with surgical precision...


But he is simply pretending those other payoffs don't exist. As I said previously, it's as if he has decided to tip the dealer with them whenever they occur.

That's the notional game he has analysed. As a practical matter, I am sure sagefr0g has no intention of tipping the dealer in this way, so there will be additional income that he has not accounted for in his analysis.

The variance of the additional income may swamp the (calculated) e.v., but the additional income will vary above/below its actual (unknown) expected value, whereas we have assigned it an expected value of zero. So however much adverse variance there may be, we are still going to be doing better (or no worse) than expected.

Suppose you were playing blackjack and the casino introduced a promotion saying that you could randomly be awarded a 1,000 to 1 bonus payment on your blackjack bet, and that you had a 990 to 1 chance of being a winner. There's no way that can ever mean you should decease the size of your blackjack bet.

Analysis of the new situation, incorporating the possibility of the bonus, may show that you should bet more, but never less. In the meantime, you can just carry on playing blackjack exactly as before, and take whatever bonuses happen to come your way.

 

[DSchles]

You keep writing as if e.v.-maximizing play is all you're concerned with, variance be damned. Why would you do that? Suppose I can get a 5% edge playing a game with, say, x variance. or, I get get a 6% edge with, say, 2x variance. You want to argue that, because your edge has increased, there's no way you should be betting less in situation two than in situation one? That's absolutely wrong!

Would you make a double down of 10 vs. 10 at TC = +4.5 for, say $10,000? You'd be very foolish to do so, because the added e.v. doesn't justify the added variance. Again, your e.v. is increasing if you double, but it's the wrong play. You should NOT be increasing your bet in this situation.

Finally, you write: "The variance of the additional income may swamp the (calculated) e.v., but the additional income will vary above/below its actual (unknown) expected value, whereas we have assigned it an expected value of zero. So however much adverse variance there may be, we are still going to be doing better (or no worse) than expected."

And again, that is absolutely wrong. Overbet your bankroll by more than double Kelly, and you are mathematically certain to be ruined, all the while betting with an advantage. Moral of all of the above: expectation isn't everything.

Don

 

[DSchles]

One more example, to make sure you appreciate the concept. Suppose you have a $12,000 bank with which to apply your horse-racing system. You bet favorites exclusively, but you're so good at picking them that they win 50% of the time. And the average payoff for a $2 wager is $5 (3 to 2 odds), such that your edge is 25%. The correct Kelly f* for each wager is (0.25/1.5) x $12,000 = $2,000.

Now, suppose, instead, that I find a different system with a bigger edge, but it is based on picking longshots. Those winners hit with only 10% frequency, but when they do, they average a whopping payoff of $26 (12 to 1 odds), for a 30% edge. Should I be betting more on each selection, because of my increased e.v.? Absolutely not! Correct Kelly wager is now (0.30/12) x $12,000 = $300, MUCH LESS than before, despite the higher (30% > 25%) advantage.

So, do you understand why expectation isn't everything, and that, in our discussion, if you add a wager that increases edge but increases variance by even more, your optimal f decreases?

Don

 

[London Colin]

You keep writing as if e.v.-maximizing play is all you're concerned with, variance be damned. Why would you do that?

Yikes. That's a damning indictment of my writing skills, because that's not what I've been trying to suggest at all!

This whole discussion has been about the calculation of f*, the optimal bet, taking into account both e.v. and variance.

Finally, you write: "The variance of the additional income may swamp the (calculated) e.v., but the additional income will vary above/below its actual (unknown) expected value, whereas we have assigned it an expected value of zero. So however much adverse variance there may be, we are still going to be doing better (or no worse) than expected."

And again, that is absolutely wrong. Overbet your bankroll by more than double Kelly, and you are mathematically certain to be ruined, all the while betting with an advantage. Moral of all of the above: expectation isn't everything.

The Kelly bet has already been determined for a subset of the game (the subset in which we only take cognizance of one of the payoffs, and, indeed, discard any winnings that come from the other payoffs).

If we keep making the Kelly bet for the subset game, but now keep those extra winnings, then surely risk can only go down?

I guess my working assumption up to now, the thing I need help to get to the truth or falsity of (seemingly, the falsity!), has been -

If you have a multi-payoff game (where you make one bet and there are multiple possible winning outcomes, each paying different odds), and you calculate f* for a subset of the payoffs in that game, then f* for the game as a whole can never be less than this value.

 

[London Colin]

So, do you understand why expectation isn't everything,

That much, at least, I've always understood.

and that, in our discussion, if you add a wager that increases edge but increases variance by even more, your optimal f decreases?

What do you mean by 'add a wager'? All along, I've understood us to be talking about a single wager, with multiple payoffs associated with it.

Just to be clear, do you mean add a payoff? Assuming you do, then, in truth, my understanding is still somewhat hazy, but I think I may be slowly homing in on the flaw(s) in my previous reasoning.

 

[London Colin]

OK. After a certain amount of head-scratching, I think I've dispelled most of my confusion. Would the following be an accurate statement? -

If you analyse just a subset of the payoffs in a game, in the manner that Sagefr0g is doing, and come up with a positive e.v. and an associated f*, then you may be over betting. Not to the extent that you will be on the road to ruin, since you are correctly sizing your bets based on the information you have, but the expected rate of growth of your bankroll will not have been maximised. Taking account of all the payoffs in the game will be necessary for that, and may result in a smaller f*.

 

[DSchles]

OK. After a certain amount of head-scratching, I think I've dispelled most of my confusion. Would the following be an accurate statement? -

If you analyse just a subset of the payoffs in a game, in the manner that Sagefr0g is doing, and come up with a positive e.v. and an associated f*, then you may be over betting. Not to the extent that you will be on the road to ruin, since you are correctly sizing your bets based on the information you have, but the expected rate of growth of your bankroll will not have been maximised. Taking account of all the payoffs in the game will be necessary for that, and may result in a smaller f*.

Short answer: yes.

Don

 

[sagefr0g]

just want to say, that I believe I have indeed solved the real problem of which was the motive behind all of my convoluted questions and misdirected (because of my own misunderstanding) attempts to use factors such as f*, CE and the like. I didn't even divulge the real problem that I was trying to solve. but anyway, it's (I believe) solved. if I had of divulged the real problem, I believe either of those who tried to help would have solved it in a heartbeat.
also, none the less, me thinks I've learned a lot as a result of the discussion (examples: London's revelation that I probably have the info needed to solve for variance, Don's revelation that variance can be a bad, bad, bad thing, sorta thing). it's much appreciated, the help that was given.