URGENT! Kelly Betting...

I am going to be hitting AC Tables for the weekend and need some advice.

I've followed Stanford Wong's derivation of Optimal Bet sizes for BlackJack.
Bet=Adv * 0.76 * BankRoll.
It seems to me that the derivation assumes head-on playing i.e. just a single player against the house.

Question(s): If I sit at third base and there are 6 other players, how is the optimal bet affected? For example, the count may be 2 before the deal, but it may be drastically different by the time I get my card(s). If it does affect the bets, how should I compensate for it?

Thanks.

It does not matter where you sit when you bet. The first baseman has just as much information on what the count is as the third baseman with regards to betting. However, the third baseman has more information on when to vary his playing strategy decisions(index numbers). The gain in expectation is negligeable in a shoe game though.

I'm fairly new at this, but from what I know, just bet according to the true count as you know it when you bet. That means, if the TC is 2 before the deal, bet your 2 units or whatever, and hope like heck the high cards come to you. Just as if you were at 1st base, you bet according to the composition of the remaining deck. Of course, the other cards played would affect your playing decisions if you use the indicies.

Cheers,

Running Count

Thanks for the quick responses.

I just finished practicing in prep for the trip tomorrow. I realize AC is not the best place to get advantage, but that's all I have access to at the moment.

Keep Counting!
The Engineer.

If there are three other players at the table with you then you expect about 13 or 14 cards to be dealt that round. The true count gives you an expectation of the composition of these cards but tells you nothing of the order. If all the players to your right get high cards then it indicates that the cards to be dealt that round were very rich in tens and you would likely have bet even higher, not lower, if you knew this quantity of high cards would be dealt this round. The precise order of the cards and who gets them is a matter of total luck.

That said, betting the full Kelly fraction is a very, very aggressive betting strategy. It is a higher level of risk than most people are comfortable with. If you bet half the Kelly fraction then your long term growth rate will be about 75% as much as full Kelly but your risk of ruin (instantaneous) drops from about 13.5% to less than 2%.

If you bet half the Kelly fraction then your long term growth rate will be about 75% as much as full Kelly but your risk of ruin (instantaneous) drops from about 13.5% to less than 2%.

I'm not sure what you mean by "long term" growth rate. What you say is true for the log of Kelly's G, which some call "exponential growth rate," and also for (G-1) (profit growth). I've heard these called "instantaneous" growth rates!

Unfortunately, G works a lot like compound interest in the long term -- a small change makes a big difference. For example, for a biased coin flip, with p = .51, we have f* = .02 and G = (1+f*)^p x (1-f*)^q = 1.000200... For half Kelly, f = .01, and G = 1.0001500... But look what happens after a years time (assume 100 hands per hour, 20 hours a week, 50 weeks a year, start with a unit blanroll):
Median Bankroll for full Kelly = 1.000200...^100,000 = 485,812,619 units (nice to dream!)
Median Bankroll for half Kelly = 1.000150...^100,000 = 3,270,379 units. The profit is less than 1% that of full Kelly.

If you never resize your bets, then betting half as much cuts your expected profits in half, just as you'd expect.

I agree full Kelly is far too reckless for most people. It's really strange to me so many people "choose" a 13+% risk of ruin, just because it happens to coincide with Kelly betting without resizing (which is a complete oxymoron!).

ETF

I agree with DD' on this because I once disagreed and he showed me where I was wrong :-). If one resizes upward or downward at approximately 25% levels, bankroll growth of flatbetting is about indistinguishable from instant resizing (flatbetting resolves about 5% quicker) so the proportional betting equation ((1+f)^W)*((1-f)^L applies for that situation over the long term for the median win. If you look at a 1% game for example and use the median wins and losses and look at the results for f=.01 and f=.005 growth will be about .75 for .005. It was that phenomonon to which DD' was referring.

If one resizes upward or downward at approximately 25% levels, bankroll growth of flatbetting is about indistinguishable from instant resizing (flatbetting resolves about 5% quicker) so the proportional betting equation ((1+f)^W)*((1-f)^L applies for that situation over the long term for the median win.

If you mean to say resizing at 25% levels is about indistinguishable from continuous resizing, I agree completely. Resizings correspond to compounding dates on your savings account (except savings accounts have zero variance, assuming the banking system remains standing). The key statistic is how many resizings are done in the long term, not so much how often.

But this only serves to broaden the gap between DD' and your humble servant, ETF. I was only objecting to the use of the words "long term." Long term, the difference between full and half Kelly is quite significant -- much more significant than 75% would tend to suggest -- however often we choose to reramp.

If you look at a 1% game for example and use the median wins and losses and look at the results for f=.01 and f=.005 growth will be about .75 for .005. It was that phenomonon to which DD' was referring.

Got it the first time! :-)

ETF

because I assume that at some point the bets will be resized. You are right, if one never resizes then betting smaller just means slower growth with less risk. But whether someone resizes when they increase the bank by 10%, resize instantaneously after each bet, or resize when they double or lose half, in all these scenarios half Kelly will give a median growth rate of .75 of full Kelly.

As soon as you decide that one day you will resize bets then the entire interval between resized bets essentially constitutes a single bet, with the resize target being analogous to the win payout and the probability of hitting the upsize target before the downsize target becomes the win probability. To bet optimally, the resize interval will be a fraction of your bank equal to the optimal f for the hypothetical bet. Betting half Kelly in this scenario will return a long term growth of bankroll of 75% the rate of full Kelly.

I can honestly say that I used this term because it is the one I thought would be most easily grasped. I couldn't think of another term to use here. Those who might nit pik already understand the principle. If you can find a more accurate way to present the idea to the general readership that is easily understood, great.

... "instantaneous growth rate." I believe Lefty used that term on another board we both access. Everyone seemed to know what he meant.

Let me just say that I read every one of your posts, and continue to feel you are the most helpful poster on the boards. If I seem to nitpick, it's because in the rare cases where we disagree, I think there may be genuine opportunity to Lift Up Counting Knowledge.

L:U:C:K
ETF

is because we are really talking about median growth rate. The full Kelly bettor has twice as big of an expectation. But over the long term, as results converge upon the median, the half Kelly bettor will grow his bank at 75% the rate of the 1K bettor. This is a rate that is converged upon.

I do use instantaneous to refer to the ROR, because that's all it really is. Nobody maintains the same bet sizes forever without ever removing any money from the bank, which is what the ROR equations are based upon.

the half Kelly bettor will grow his bank at 75% the rate of the 1K bettor

This is like saying a savings account with 1% interest "grows" at 75% the rate of an account with 2% interest. There's a clear implication that half Kelly is 75% as profitable as full Kelly. This doesn't just miss the mark, it misses it as much as a hundred fold over the course of a lifetime!

Sorry. Looks like we'll have to agree to disagree.

ETF

Not really explaining this to you, but for others who may be following who have not studied proportional betting.

Using the game p=.505, q=.495, bank=1000 units, optimal Kelly bet=.01. Let's look at the two banks after 500 bets:

Bn=Bi*[(f+1)^(p*n)] * [(f-1)^q*n]

Where Bn is bank afer n bets, Bi=initial bank, f=fraction of bank wagered on each trial, p=win probability, q=loss probability

Full Kelly, f=.01

Bn=1000 * 1.01^(.505 *500) * .99^(.495*500)
=1000*1.01^252.5 * .99^247.5
=1025.31555

half Kelly, f=.005
Bn=1000 * 1.005^252.5 * .995^247.5
=1018.93

Full Kelly grows the bank about 25 units, half Kelly grows the bank by about 19 units. Half Kelly grew at about 75% the rate of full Kelly. This will be the long term effect whether bets are resized instantaneously or at intervals. This is what I was talking about as long term growth rate. It is the growth per bet based on long term results. ETF uses instantaneous to refer to "per bet".

We look at G: G=(f+1^p) * (f-1^q)

Full Kelly: G=1.01^.505 * .99^.495= 1.000050002083429171768660362523
Half Kelly: G=1.005^.505 * .995^.495=1.0000375009635638676864203570062

You can see that the expected growth of bankroll per bet for half Kelly is 75% of the expected growth per bet for full Kelly.

What ETF is just pointing out is that with exponential growth rates
the smaller fraction will not have a net win of 75% as much. There will be a geometric difference over time. After 10k bets

Kelly = Bn*G^n=1000*1.000050002083429171768660362523^10000=1649
Half K= 1000 *1.0000375009635638676864203570062^10000=1455

After 10K bets full Kelly wins 649, half Kelly wins 455, only 70% as much. Given enough time, an exponential growth rate of only a small fraction more will have a geometrically bigger net win.

The instantaneous rate is based upon long term convergence. You cannot express a figure in terms of percentages of how much more profitable one fraction will be over another unless you know N.