Karel: Calculating ROR given Kelly fraction $p$

[Karel]

The analysis of risk of ruin has always been a very interesting topic. In this post I give the formulas for calculating ROR when the player initially chooses the betting unit as some Kelly fraction given the game parameters and his bankroll.

It is a common practice that a player forms blackjack bankroll, and then chooses appropriate Kelly fraction to bet. After choosing the Kelly fraction the player calculates the betting unit given the betting spread and other constraints. Denote the Kelly fraction by $p$, where $p=2$ for half Kelly, etc.

Suppose that the player plays a fixed unit size with Kelly fraction $p$. Thus, full Kelly, also so called logarithmic utility, corresponds to $p=1$. Kelly fraction $p$ corresponds to the so-called power utility, also called constant relative risk aversion utility, with risk aversion parameter $p$.

If the player finds the unit bet size as the corresponding Kelly fraction $p$ from the initial bankroll, and fixes the betting unit afterwards, then the risk of ruin can be calculated as
$$ROR = exp{-2 p}.$$

Note that, perhaps surprisingly, the ROR does NOT depend on the edge and risk of the game, nor on the bankroll. The reason for that is that the desirability of the game and the size of the bankroll are already reflected in the bet size. The only thing that matters is player?s risk aversion reflected by the Kelly fraction $p$.

For example, suppose that there are otherwise two identical games, except one has an edge of 1% and the other has an edge of 2%. The optimal bet size for the second game will be double the first game. But the ROR will be the same! Intuitively, one can understand this as saying that ?time runs twice that fast? for the second game. To obtain the same results in terms of risk and return, one just needs to play twice as many games with HALF the bet size in the first case.

Another interesting question is the probability of ruin before reaching a certain goal, i.e., before doubling the bankroll, after which point the player may for example want to increase the unit size.

Denote $B >= 1$ to be the desired multiple of bankroll. The corresponding risk of ruin before reaching the goal $B$ then equals
$$ROR(B) = (\exp{-2 p} ? exp{-2 B * p}) / (1- exp{-2 B * p} ).$$

Note that if $B$ is large (if $B$ converges to infinity) we get exactly the simpler formula above.

Here are some numbers:
Suppose full Kelly bettor ($p=1$). If the goal is doubling the bankroll, the player runs ROR(2) = 11.92%. The ROR(3) for $B=3$ (tripling the bankroll) is 13.32%, while it is 13.53% for $B = infinity$. $B = infinity$ means never hitting zero.

For half Kelly and $B = 2$ we get ROR(2) = 1.8%, while the ROR is 1.83% for $B = infinity$. This number changes very little ? if the initial bet size is chosen as half Kelly, it is indeed extremely unlikely to ever hit zero again, once the player doubles the bankroll.

For those mathematically inclined I will make another post on these pages shortly, showing how to derive the ROR formulas above.

Best regards,

Karel

[Artguy]

Karel,

Is it inadvisable to size the "Kelly" bet up as the bankroll is increasing before reaching the goal of doubling it? I ask this because of remarks Don made in BJA to the effect that the ROR can be lowered in half by sizing the bet down when half the bankroll is lost. So If you can size down, why not size up as well?

Beginning with a BR of $5000. I have been sizing my bets down $5 for each loss of $1000. This provides a lot more potential for losses/opportunities before hitting the half way point...or lower. This seems to be effective so far but is not supported any math I have seen.

[Don Schlesinger]

"Pure" Kelly resizes constantly, but, of course, there are practical limitations to playing this way. There is no rule as to when to increase or decrease stakes. You might wait for doubling or halving, to increase or decrease, or you might do either sooner. Personally, I don't suggest it to the downside, and conservative people do like to wait to double banks before increasing stakes. It's up to you.

By he way, playing half-stakes when half the bank is lost, as a plan before you begin any play at all, more than cuts ROR in half (see BJA, p. 143).

Don

[Karel]

The answer to the question of optimal bet sizing is a bit more complicated. Let me first explain what the theoretically optimal strategy for experienced and pro or semi-pro players is.

For really experienced players, a formal definition of a ?bankroll? is actually redundant. There is really no theoretical need for defining a bankroll, other than perhaps psychological purposes, having a reference point, and similar. For example, what happens if a pro player loses the current bankroll (and it must ever happen just by pure chance)? Well, I guess that most pro players just form another bankroll and go on.

The reference point is, in fact, the player?s entire investment wealth. The definition of investment wealth is rather complicated, but let me just say that it is usually more than currently available cash or liquid assets, even though one should also subtract some minimum required consumption, etc.

Of course, nobody is suggesting that one should play full Kelly or half Kelly with one?s entire wealth. The proper risk aversion can be calculated as follows: Consider player?s entire wealth and the size of his formal bankroll as a proportion of the wealth. The proportion of bankroll times the Kelly fraction used for betting is the theoretical risk aversion.

For example, suppose that the pro player?s bankroll is 10% of his wealth, and the pro player plays half Kelly. Then the actual risk aversion is ? * 10% = 1/20, i.e., the risk aversion if 1/20 Kelly with the reference point being entire player?s wealth. Or, suppose that the bankroll is 1/15 of entire wealth but the player plays full Kelly. Then the actual risk aversion is 1/15 Kelly.

The theoretically optimal strategy

The theoretically optimal strategy is to define the unit bet size as an exact proportion, for example 1/15 Kelly, of the entire wealth. Thus, the unit size would change all the time. This is, of course, not practical and not even useful. For practical purposes it does not matter at all if the bet size is 10% more or less than theoretical optimum. Thus, it is correct to use the following strategy:

1) Form a formal ?bankroll? as, for example, 10% of wealth. Calculate full Kelly betting with respect to this artificial bankroll, which corresponds to risk aversion 1/10 Kelly with respect to entire wealth. (Or half Kelly with respect to the bankroll, which corresponds to risk aversion 1/20 Kelly.)
2) Do not change the unit bet size until you either double the bankroll or lose the bankroll. In case of an unfortunately loss (probability of 11.92%, see the first post), form a new bankroll and recalculate the unit bet size. Since the wealth is now 10% lower, the new optimal bet size should be also 10% smaller. In case of a win (doubling the bankroll), the player has now 10% higher wealth, and the new optimal bet size is 10% higher.

From analysis above it follows that, for pro and experienced players, it is not advisable to half the bet size after losing half the formal bankroll. It is too conservative. Especially, consider the case if you half the bet size and still lose entire bankroll. A pro player would perhaps form a new bankroll and start again with higher unit size. This cannot be possibly correct (higher bet after having lost more).

All the analysis above applies for experienced and pro players. The situation is somewhat different for beginners or less experienced players. Above all, beginners cannot be certain about their edges. Thus, it is advisable to form a maximum bankroll the player is willing to lose. In case of unfortunate bad luck the player may want to quit blackjack. This may be rational, since perhaps the player only thinks that he has an edge. For beginners it is certainly advisable to bet proportionally to the formal bankroll rather than entire investment wealth. In this case I would recommend halving the bet size after having lost half of the bankroll.

Regards,

Karel

[Don Schlesinger]

> 2) Do not change the unit bet size until you
> either double the bankroll or lose the
> bankroll. In case of an unfortunate loss
> (probability of 11.92%, see the first post)

I'm interested in this, Karel. If we play without regard to doubling, we know that the "infinite hours, no upper barrier" ROR for full-Kelly players who don't change the starting unit throughout play is 13.52%.

Your 11.92% number implies that, about 1.6% of the time, a player would actually double a bank, NOT change his stakes, and yet still slide all the way back to zero, tapping out. At the doubling point, this, in effect, would be the ROR for a 0.5 Kelly player, which we would then have to subtract from the 13.52% figure. Is that correct?

If so, I thought we got 0.5 Kelly ROR by simply squaring 1.0 ROR, which would be .1352^2 = 0.0183, which is close to, but not quite the same, as the above surmised 0.016.

What am I doing wrong?

Don

P.S. What's more, the 13.52% is a global figure that includes ruin even after going beyond doubling! So, there is even more (very small) probability to be subtracted, which would include the times one taps out after MORE than doubling the bank. Where is my error in thinking?

[Karel]

It is a little tricky calculation.

The probability of 11.92% says that about 1.6% of time the player doubles the bank and also busts (later). However, this is NOT the same question as: "What is the probability that, with a double bank, the player busts?" The latter probability is indeed 1.83%, which is 0.1352^2, which is the 1/2 Kelly ROR.

The first probability is smaller since it asks the odds of "busting with double bank AND doubling first". In fact, calculation of conditional probability gives us
$$P[A interestion B] = P[A|B] * P[B].$$

In our case, we should have
13.52%-11.92% = 18.3% * (1-11.92%),
which is indeed true. The (1-11.92%) is the probability of doubling the bankroll (with or without busting later).

Regards,

Karel

> I'm interested in this, Karel. If we play
> without regard to doubling, we know that the
> "infinite hours, no upper barrier"
> ROR for full-Kelly players who don't change
> the starting unit throughout play is 13.52%.

> Your 11.92% number implies that, about 1.6%
> of the time, a player would actually double
> a bank, NOT change his stakes, and yet still
> slide all the way back to zero, tapping out.
> At the doubling point, this, in effect,
> would be the ROR for a 0.5 Kelly player,
> which we would then have to subtract from
> the 13.52% figure. Is that correct?

> If so, I thought we got 0.5 Kelly ROR by
> simply squaring 1.0 ROR, which would be
> .1352^2 = 0.0183, which is close to, but not
> quite the same, as the above surmised 0.016.

> What am I doing wrong?

> Don

[Don Schlesinger]

[xxi]

> In our case, we should have
> 13.52%-11.92% = 18.3% * (1-11.92%),

You mean 1.83% x (1 - 11.92%), right? Your explanation is very clear and I want to be sure I'm following.

[Don Schlesinger]

[Karel]

> You mean 1.83% x (1 - 11.92%), right? Your
> explanation is very clear and I want to be
> sure I'm following.

Yes, I do mean what you wrote above, but I am kind of surprised! I have always used "*" to denote multiplication, in other than mathematical (LaTeX) texts. (In LaTeX one can use \cdot, which is a big dot.) My point is that "x" could easily get confused with the commonly used letter "x", or for example with the vector product in mathematics. I am not aware of any other meaning of the character "*".

Regards,

Karel

Karel

[Artguy]

Karel,

Thanks for such a detailed and informative response to my question re: kelly bet sizing. It was far more than I had bargained for. A playing bank, which is a proportion of one's wealth, is an important concept to my psychological well-being as an advantage player.

One more question, however; how do you play "half-Kelly" without destroying the concept that makes it work? (ie. is it 1/2 the dollars you would play? Or would you play only half the ramp? Or would you lower the incline of the ramp? Obviously, I am quite mystified.

[Karel]

It is psychologically useful to have a playing bankroll, although it is theoretically redundant.

> Thanks for such a detailed and informative
> response to my question re: kelly bet
> sizing. It was far more than I had bargained
> for. A playing bank, which is a proportion
> of one's wealth, is an important concept to
> my psychological well-being as an advantage
> player.

> One more question, however; how do you play
> "half-Kelly" without destroying
> the concept that makes it work? (ie. is it
> 1/2 the dollars you would play? Or would you
> play only half the ramp? Or would you lower
> the incline of the ramp? Obviously, I am
> quite mystified.

Suppose following: You form a BJ bankroll, which is, say, 10% of your "investment wealth" (which is pretty much "net worth"). With the bankroll you can choose to play half Kelly, calculating the half Kelly unit size with respect to the defined bankroll.

Then, it is almost precise, almost perfectly correct, to just keep the fixed unit size until you either
1) Double the bankroll. In this case, your wealth is 1.1 times the original wealth, since your bankroll is 0.2 (20%) instead of original 0.1 (10%). Now, you can form another bankroll, again 10% of your new wealth, which will be 1.1*10% = 0.11. Your new optimal unit size should now be 10% higher than it was originally.

As you can see, you increase your unit size by only 10% even though you DOUBLED your bankroll.

2) In the unfortunate and unlikely case you may lose the whole bankroll. Then, your wealth is 0.9 since you lost 0.1=10%. You can form another bankroll, again 10% of wealth, which is 10%*0.9 = 0.09. Your optimal bet size should be 0.09/0.1 times smaller than the original bet size, i.e., 10% smaller.

Again, you decrease your unit size only by 10% even though you may have lost the whole bankroll!

All the considerations above assume an experienced or pro player, who does not need to question or verify his playing abilities.

Regards,

Karel

[Double21]

> Karel could you please tell me what your ROR would be for a fixed bank playing to a 1.5% ROR when, after doubling it, you withdrew the winnings and reverted back to the original bank using an identical playing/betting strategy?