buddha: Optimal Betting versus TC betting

[buddha]

As I read MJ's posting and the responses he received on the other site, it's evident that there's quite a bit of misunderstanding about optimal betting even among a successful group like MIT. It got me to questioning whether I truly understand the basics of how/why optimal betting is superior to betting in direct correlation to the TC (linear betting). Here are some questions I have:

What makes OB different from Kelly betting? I thought Kelly was the most effective way to maximize a bankroll (albeit with a certain amount of flux). As I understand, OB maximizes the logarithmic growth of a bankroll. For a statistical layman, what does this difference mean?

I assume there's no inherent conflict in OB and Kelly betting since CVCX allows for a Kelly-fraction option. Is the difference between OB and betting-in-direct-correlation-to-the-TC simply due to the fact that the TC is not directly related to the true advantage?

Does OB result in bets directly correlated to the true advantage? If so, it seems that OB is simply a more precise way of Kelly-betting.

I re-read BJA 9 and 10 but it doesn't seem to clarify my confusion. It seems to me that a tool like CVCX that started as a way to compare different rules has morphed primarily into a tool used to identify the appropriate bet at each TC. Any clarification would be appreciated.

[Don Schlesinger]

> As I read MJ's posting and the responses he received
> on the other site, it's evident that there's quite a
> bit of misunderstanding about optimal betting even
> among a successful group like MIT.

No, they understand it perfectly well. They chose not to use it for several different reasons, camouflage being one of the biggest.

> It got me to
> questioning whether I truly understand the basics of
> how/why optimal betting is superior to betting in
> direct correlation to the TC (linear betting). Here
> are some questions I have:

> What makes OB different from Kelly betting?

Different concepts. The optimal betting algorithm permits us to specify minimum and maximum bets. Kelly assumes, in its purest state, that we bet zero when we have no edge and that we have no upper bound to our wagers. Naturally, in real-world play, constraints usually need to be applied to the wagers we make, both for the min and the max.

> I thought
> Kelly was the most effective way to maximize a
> bankroll (albeit with a certain amount of flux).

Not the bankroll; rather its logarithmic growth.

> As I
> understand, OB maximizes the logarithmic growth of a
> bankroll. For a statistical layman, what does this
> difference mean?

No difference, if no constraints. But, if one chooses, for example, to bet a fractional amount of Kelly, one still needs to decide how the bets will be placed. And, if there are constraints, you will simply get smaller bets, yet optimally sized and placed, than if you were playing full Kelly, with or without constraints.

> I assume there's no inherent conflict in OB and Kelly
> betting since CVCX allows for a Kelly-fraction option.

Correct. See above.

> Is the difference between OB and
> betting-in-direct-correlation-to-the-TC simply due to
> the fact that the TC is not directly related to the
> true advantage?

No, that's not it.

> Does OB result in bets directly correlated to the true
> advantage? If so, it seems that OB is simply a more
> precise way of Kelly-betting.

All optimal bets other than the min and the max are equal to the bankroll times the edge divided by the variance. Same as Kelly. The trick comes when we need to specify a spread and, therefore, a max bet. The OB algorithm chooses the unit size and the max bet for the given bankroll.

> I re-read BJA 9 and 10 but it doesn't seem to clarify
> my confusion.

It doesn't present the math (messy!) behind the OB algorithm, which is contained in papers by several people whom I mention in the book.

> It seems to me that a tool like CVCX
> that started as a way to compare different rules has
> morphed primarily into a tool used to identify the
> appropriate bet at each TC. Any clarification would be
> appreciated.

Well, CVCX will, in fact, calculate OBs, but to say that this is its primary use is to vastly understate the versatility and range of its capabilities.

Don

[MJ]

What puzzles me is why would you want to use an optimal bet schedule if a linear bet schedule provides a higher SCORE with the SAME risk given a fixed BR??? Of course the minimum bets were the same. Perhaps I didn't give the optimal bet schedule enough of a spread? Whatever spread the custom bet schedule had out by a TC of +6 is the spread I used for the optimal schedule.

I believe you stated this might happen in the other thread. Well it did. I would post the graphic here but can't quite figure it out.

Another interesting point that was recently called to my attention: It turns out MIT wagers 1 unit per 0.5% advantage, advantage being measured as the RATIO of EV to Var (EV/Var). As you perfectly know, EV per true is not linear. However, the ratio of EV/Var per true is pretty darn close to linear (between true of 2-10)!!! Just graph the ratio of EV/Var on the y-axis and TC on the x-axis. So, (TC-1)/2 is a close approximation of advantage (measured as EV/Var).

Optimal bet = Kelly x BR x (EV/Var).

It follows that since our advantage IS linear, then so should our bet schedule.

Stay tuned for another simulation study......

MJ

[Don Schlesinger]

> What puzzles me is why would you want to use an
> optimal bet schedule if a linear bet schedule provides
> a higher SCORE with the SAME risk given a fixed BR???

It doesn't!! What would make you come to that conclusion? Why are we dragging this on?? "Optimal" means "optimal." It means you can't have a higher SCORE using any other bet scheme, once you fix the bankroll, the spread, and the ROR.

> Of course the minimum bets were the same. Perhaps I
> didn't give the optimal bet schedule enough of a
> spread? Whatever spread the custom bet schedule had
> out by a TC of +6 is the spread I used for the optimal
> schedule.

Obviously, you have to use the same spread as the other group used. You can't cap your bets when they don't cap theirs.

> I believe you stated this might happen in the other
> thread. Well it did. I would post the graphic here but
> can't quite figure it out.

> Another interesting point that was recently called to
> my attention: It turns out MIT wagers 1 unit per 0.5%
> advantage, advantage being measured as the RATIO of EV
> to Var (EV/Var). As you perfectly know, EV per true is
> not linear. However, the ratio of EV/Var per true is
> pretty darn close to linear (between true of 2-10)!!!
> Just graph the ratio of EV/Var on the y-axis and TC on
> the x-axis. So, (TC-1)/2 is a close approximation of
> advantage (measured as EV/Var).

> Optimal bet = Kelly x BR x (EV/Var).

I know all of this. I just don't understand what the problem is.

> It follows that since our advantage IS linear, then so
> should our bet schedule.

> Stay tuned for another simulation study......

You're not understanding that when you have an imposed minimum bet and an imposed max bet, and, therefore, a self-imposed maximum bet spread, this changes how you size your unit and when the max bet is placed.

Don

[MJ]

> Obviously, you have to use the same spread as the
> other group used. You can't cap your bets when they
> don't cap theirs.

I thought you wrote below whatever spread MIT has out by a TC of +5 is the spread to use for the optimal schedule.

> I know all of this. I just don't understand what the
> problem is.

I am trying to figure out how to set up an apples to apples comparison between an optimal and linear bet schedule using CVCX. That is the problem.

Now, if this is not possible, please be clear and say so; if this is possible, please tell me how to do so.

MJ

[Don Schlesinger]

> I thought you wrote below whatever spread MIT has out
> by a TC of +5 is the spread to use for the optimal
> schedule.

I said that MIT is fooling themselves thinking that they have a larger spread, because, effectively, they don't. That said, on the rare occasions that they bet, say, 36 units and you never do, that will, of course, make a little difference.

> I am trying to figure out how to set up an apples to
> apples comparison between an optimal and linear bet
> schedule using CVCX. That is the problem.

You can't.

> Now, if this is not possible, please be clear and say
> so; if this is possible, please tell me how to do so.

It isn't possible.

Don

[Magician]

I don't mean to fan the flames but I don't think the following comments have been made here yet (I haven't read any discussions on other sites) and I thought they might help.

> I thought you wrote below whatever spread MIT has out
> by a TC of +5 is the spread to use for the optimal
> schedule.

Technically their spread is 1-212, because their minimum bet is 1/4 "unit" and their maximum possible bet is 53 "units" at TC +52. But the frequency at which the higher bets are placed is so much smaller that comparing to an optimal 1-212 spread wouldn't make much sense.

> Now, if this is not possible, please be clear and say
> so; if this is possible, please tell me how to do so.

What you could do is show that, for a given bankroll & ROR and a sufficient spread (where sufficient is somewhat less than 1-212), optimal betting wins more money than linear betting. The optimal betting schedule will, of course, bet different amounts with different frequencies. But if it didn't it wouldn't be the optimal betting schedule.

[Don Schlesinger]

> What you could do is show that, for a given bankroll
> & ROR and a sufficient spread (where sufficient is
> somewhat less than 1-212), optimal betting wins more
> money than linear betting. The optimal betting
> schedule will, of course, bet different amounts with
> different frequencies. But if it didn't it wouldn't be
> the optimal betting schedule.

That's the whole point. And, I don't understand why we're beating a dead horse. We all should know the meaning of the word "optimal." If, for the same conditions, something else surpasses it, then it wasn't optimal, was it? :-)

If, on the other hand, something else surpasses it, then that something else has different assumptions and conditions than those used for the "optimal" scenario. I just can't think of any other way to express this.

Don

[Norm Wattenberger]

awhile back. The BJI guys advise that you leave the table on a low count. If you do that, their ramp is very close to optimal.