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Effect of Overhead on Risk of Ruin
Originally posted on Green Chip by MathProf
Many players are familiar with Kelly betting and/or classic Risk of Ruin analysis and use these to determine bankrolls and bet sizes. Now both of these systems are based on the assumption that all winnings are reinvested in the Bankroll. If this assumption is not true and Winnings are siphoned (to cover expenses, for example) then the RoR rises dramatically.
Consider a hypothetical Counter who plays on Weekends. (S)he estimates that the expected earnings are $30 an hour and has an estimated Standard Deviation of $300 an hour. The gambler typically plays 20 hours each; travel expenses come to $300 or $15 an hour. So our player is happy that his/her anticipated earnings will easily cover expenses. Applying the classical gambler's ruin formulas (from Chapter 10 of Blackjack Attack), the gambler estimates that the probability of losing a $3000 bankroll is 13.53%, well under our gambler's comfort zone.
However, a serious error has been made. If the earnings are being diverted to cover the expenses, then the RoR is much higher. To determine how much, we need to make some assumptions. Let us assume that these travel expenses are funded from the Black jack earnings and, on when weekends when they are insufficient, the bankroll is drawn down further. Let us also assume that there is no uncertainty as to what these expenses are (SD is 0). Then they essentially lower the counter’s net expected earnings from $30 to $15 an hour. If we now do the RoR calculations with mean =$15, SD =$300 and bank=$3000, then we obtain a RoR of 36.8%!
Now realizing this, what should our Counter do? Decrease the Bet Size? Unfortunately, if (s)he scales back his/her bets (reducing mean and SD) he will actually increase the RoR. That is because the Fixed Expenses will eat away at the bankroll.
Under the assumptions made above, that there fixed expenses F which must be met and lower the counter’s expected earnings, we can go back to the RoR equations and set up a function which gives us the RoR for various unit sizes. Then we may find the minimum of this function, using Calculus or other mathematical technique. We obtain a very simple result: The optimal betting scale is that for which hourly earnings equal to twice the Fixed Costs. Further increasing our unit size will increase RoR as in the classic case. However if we cut back below this optimal level, our RoR will go up because we will not be earning enough to offset the fixed expenses. If we bet at the optimal level, we may plug this unit size back into our function, and obtain the following equations for the corresponding bank and RoR:
|Formulas for Optimal Bank (given Overhead)|
Bank = -2 * ln(r) * F * [ s/m ] 2
These equations show how to compute the RoR for any given Bank. Alternatively, given a bank, how to compute the RoR. Again these results are the best that can be obtained under these circumstances. Please note the critical role played by the desirability index.
Also note that you do not need to use $ per hour, you may use $ per Hand or $ per Weekend, etc. as long as you are consistent.
Income: What if we want to produce net income from our play? If we put ourselves on "salary" and pay a certain income regardless of wins or losses, then we may fold that into our Fixed Expenses. We can also treat "dividends" to our investors the same way. Of course, these are not really Fixed Expenses, in that they could cut be back if the bankroll takes a beating.