Effect of Trip Ruin on Trip EV (A longy but a goody)

From Stanford Wong's BJ21

Posted by MathProf on 25 Aug 1998, 4:54 am


Recently I have been looking at the problem of short-term (Trip) Gambler's Ruin, and specifically its impact on Trip Statistics. Let me give an example of the type of problem.

Greta plans to play 16 hours of BJ on a trip. Her hourly EV is 1.5 units with a SD of 20. (DI = 7.5) However she only brings 60 units of Bank with her, and she will stop playing when the cash runs out. What are Expected Earnings for the Trip? One might answer that it is 16 hours of play times 1.5 units equals 24 units. However that assumes that she will get to play 16 hours. With her low bank, her chances of tapping out are approximately 36%! The net result of this is that her EV for the trip is only 19.1 units. Her low bank has cost her about 20% of her EV potential.


Description of Data

I will post below some data which I have just computed. The column labeled "TI" represent what I call the Trip Index. That is the ratio of Trip EV to Trip SD. It is a kind of DI for the trip. You may compute it by taking DI/100 * sqrt(Hours), if you play 100 hands per hour. If you played a good game with DI of 10 for 100 hours, your TI would 1.0. If you played a mediocre game with DI of 5 for only 4 hours, your TI would be 0.1. I think that most "short-term" games would fall in between that range. So I have tabulated the results for values of TI between .1 and 1.0, but I have also included some bigger values, in case you want to sue this for longer-term planning.

I have listed the data for different size banks; which I sometimes call a Stop Loss. These are measured relative to the trip Standard Deviation. My algorithm (described below) requires a Stop Win. This is not a serious limitation; one can simply put in an unattainably high SW to avoid this. In the runs below, I used a SW of 6 trip SDs.

For each entry I have tallied the probability of Trip ruin and the probability of hitting the Stop Win (which is approximately 0 for these runs.) I also express the actual EV and TI which the player would encounter, if they were constrained by the Stop Loss and Stop Win. These are expressed as ratios to the unconstrained statistics. An example will clarify this.


Example Revisited

For Greta's trip described above, her trip EV is 1.5 units time 16 hours = 24u. Her trip SD is 20u times sqrt(16) = 80u. So the Trip Index is 24/80 =0.3. Her bank is 60 units. We divide this by the trip SD to get 0.75. So we scroll down to the data for TripBank =0.75 and look up the entry for 0.3. We see that it is 



Trip Bank = 0.75 Trip SDs    ( Stop Win  = 6 Trip SDs ) 

    TI    PrRuin    PrSW     EV Ratio     TI Ratio  

  0.10    0.420     0.000     0.762        0.852

  0.20    0.387     0.000     0.781        0.844

  0.30    0.355     0.000     0.798        0.837


This says the Probability of Trip Ruin is about 35.5% and there is no chance of hitting the Stop Win. Her trip EV is only 79.8% of her unconstrained EV. That is, her actual EV is .798*24u = 19.1 units. Also, her actual TI is 83.7 of her unconstrained .30. This works out to .20. You can use this to compute the actual Trip SD if you wish.

I will post the data as a response to this. Right now, it is in the form above. It may be useful to change the format and to have a sub-table for each TI level, showing the effects of different banks. I will also post some comments on the methodology.

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