Effect of Bet Size on Aggregate DI

From Stanford Wong's BJ21

Posted by MathProf on 20 Aug 1998, 3:23 pm

In a discussion on the Misc. Board I had mentioned the idea of an aggregate DI, which was involved multiplying the DI by the square root of number of hands. The question of adjusting it for bet size also came. I thought about it and came up with a formula that corrects for this. Because we were getting off the main point of that thread, I thought it made more sense to start a thread here.

This formula is based upon the Certainty Equivalent described by Red Taylor. The idea is to compare games based upon the average CE that they produce. If you are a pure Kelly player, this CE corresponds to logarithmic rate of Bankroll growth.

First I need to introduce a new a parameter r. This is the ratio of your betting unit to your optimal unit. (This optimal unit is based on the size of Bank Roll and your Risk Aversion.) Then we may obtain an adjusted DI via

Adj DI  = DI  * SquareRoot( r * [2-r] ) 

We multiple this number by our square root of hands to form the aggregate DI that I discussed.

For convenience, let me call this new factor sqrt [r(2-r) ] your Bet Size Factor BSF. The maximal value of BSF is 1 and is obtained when r=1, indicating that you are at your optimal betting level. Some examples will clarify the use of this BSF.

Suppose there is a game with a $10 minimum and I believe I can bet $100 without any real problem. Let us say further that with the 10-1 spread that I will use, this game has a DI of 10.0. However I determine that to play the game with that spread, my Bankroll calls for an optimal unit size of $7. My ratio r is $10/7.5= 1.33. This means that I am over-betting by 33%. My BSF is sqrt (1.33*(.67) ] = .94. I multiply by DI by this and obtain 9.4 as my adjusted DI.

Suppose my wealthy friend Walt determines that for his bankroll the optimal unit is $25. But he feels that $100 is the maximum he can bet here, and so he uses the same $10-$100 spread that I do. He is under-betting; the ratio r is.4 and his BSF is sqrt[ 0. 4 * 1.6 ] =.8. So his adjusted DI is 8.0.

Now there is a bit of flaw in this. What I am doing is taking the original betting schedule (used to produce the raw DI) and simply shift the levels up or down proportionality. The player could get a better performance by changing the betting schedule based upoon their Bankroll. Walt, for example, would do better by ramping up the schedule and putting out the max $100 bets at a lower True Count. I, on the other hand, would do better to delay the maximum until much higher (and practically never attained) true count. Such rescheduling would lower the DI, but raise the BSF; the net effect would be a much higher adjusted DI.

There is some wisdom in ML's suggestion to look at DI^2. Not only does this remove the need for these ugly square roots, but it gives a more meaningful number. It will be proportional to our "risk-adjusted earnings" per hand per $ of bankroll. Let us look at Walt's DI of 8.0 and compare it with the Optimal DI off 10.0. That may not look like a big difference, but when we square them we get 64 and 100. This means that Walt must play 100 hands to accomplish what an optimal bettor could do in 64. Walt must play more that 1.5 hours to get what the optimal player could get in 1 hour.