From the bjmath Workshop

Using RA-Fractions to Optimize DI or RoR.

Posted by MathProf on Thursday, 29 October 1998, at 6:07 p.m.

In the past, I have spoken of "critical fractions" in connection with RA playing strategy. For each risk-increasing play (like double or split) there is a critical bet size. Above that size, the risk-averse decision should be taken, and the player should give up EV to reduce Var.

Now suppose a player is confronted with a specific game, with specific rules dealt from a specific pack. This game has positive EV, and the player wishes to optimize her Desirability Index for this Game. There is an optimal playing strategy which does. This strategy will not be the convention one; it will be a Risk Averse Strategy which can be described using these critical fractions.

First we compute our optimal bet, f=k* Ev/Var (I personally prefer, Ev/E(X^2), but I'll talk about Variance since that is more widely recognized. The constant is 1 for Kelly players). We then assume that we bet "f", and determine the optimal RA strategy. That is, plays with critical fraction below f are avoided. This gives us a better strategy for the game, in the sense that it improves the DI over the conventional strategy.

We may have to iterate this process. With our new strategy, we have a new Ev and Var, and a different (usually higher) optimal bet. At this higher level, there may be additional plays to avoid. This allows us to refine our strategy again. When we are done, we will obtained the strategy that gives us the best DI.

Now this process does not depend on whether or not the player is a half-Kelly player (k=.5) or third-Kelly, or whatever. For example, a half-Kelly player will her bet size f cut in half. But her critical fractions will also be cut in half. So she will end up with the same strategy changes as the Kelly player. Thus the utility constant falls out in the wash.

So the key point here is that proper RA strategy leads to a game with the best Desirability Index. I don't think this is news to many of the correspondents on the board. So now I proceed to a different issue Risk of Ruin.

Suppose we have an under-capitalized player, who is given a chance to play a "great game" with high table limits. The player wishes to play, plans to follow a fixed-unit betting schedule, and wishes to minimize his Risk of Ruin. Clearly the person should opt for a more conservative playing strategy. What should that strategy be?

Here's the trick. Compute the optimal bet with this game. Then double it. That is, compute the "double Kelly" bet. Assume that the player is compelled to bet at this level. Determine the optimal RA strategy at such a betting level. That strategy will be the strategy which minimizes Risk of Ruin! The strategy which minimizes Risk of Ruin is the one which is RA-optimal at the double Kelly level.


We will have to use an iterative process as above. We delete those plays whose critical fraction exceeds double Kelly. This raises f and with it, 2*f. So we do it again. But when we are one, we minimize it.

I have a case study which illustrates this point: The optimal Basic Strategy when Black Jack pay 2 to 1! If there is interest in the topic, I will post it over the weekend.

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