Advanced Risk Management (long)

[Izverg04]

This post is in part motivated by the poll started by 1800GAMBLER over at the Mid- and High-Stakes Hold'em forum. He posed the following question to the degenerate gamblers that populate that forum:

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If you could be paid per hour and you knew your win rate with 100% confidence, what percentage of your win rate would you give up so you didn't have to ride the variance wave?

i.e. after 500k hands you win at $200/hour but in the meantime you have to put up with $10k downstreaks etc or you could give up say $20/hour and just be paid per hour.

[/ QUOTE ]
The results of that poll are shocking. At the moment 53% of the respondents say that they would give up at most 10% of their EV to completely eliminate variance. 20% respond that they would not give up a cent. To me that says that either the respondents are joking, or in need of GA counseling, or simply have not given a second's thought about mature risk management.

Regular visitors to the casino forums over at bonuswhores.com know that I am obsessed about risk management, finding the right price for reducing variance, and selecting between alternative gambling propositions with different risk/reward structure. I rarely post on 2+2, but I think I could get better feedback on this subject from this forum than over at bw.com.

To start from basics, we know that the main risk-management concept in gambling is the long-term risk or ruin, which is given by

RoR=exp(-2BR/[Var/EV]),

where BR is the amount of money at risk, while Var(iance) and EV describe the bet that you are making.

The parameter of merit in gambling is Var/EV. A gambler can afford to make bets only if Var/EV is less than his critical threshold, which I coined as RTT (risk tolerance threshold) in a recent post on bw.com.

As an example, let's say a poker player has a bankroll of $100k, and so he is content with EV of $200/hr with STD of $2000/hr. However, if the same winrate was accompanied with STD of $3000/hr, the swings would be too big for the player's bankroll, and he would not play. His RTT is somewhere between $20,000 and $45,000.


To answer 1800GAMBLER's question, an important insight has to be made: RTT has to depend on the other parameter of merit in gabling, the hourly wage. It is rational to put your bankroll at a larger risk if you make more per hour. Let me explain why this has to be the case.

Let's say the poker player above is playing right at the threshold of his risk tolerance, that is his RTT=$20,000. In one month he plays 100 hours, making EV=$20k with STD=$20k. He would trade his earnings for a salary EV0<$20k with no variance. It makes no difference to him which payoff structure to choose, in his mind they are equivalent. Since EV and variance are additive, then the difference between the two payoff structures, meaning a one-time bet that returns an amount normally distributed around dEV=(EV-EV0) with standard deviation STD=$20k is also borderline-attractive to him.

However if we use the same RTT=$20,000 to evaluate the one-time bet then the player will require dEV=EV=$20k to justify taking on the variance, and we get EV0=0! If we follow this logic, playing right at the RTT is equivalent to getting a zero salary, but we know this is not true. The poker player will choose playing at his risk threshold over a null salary.

The only way to resolve the situation is to assume that when time is not involved, that is when your hourly wage is infinite (imagine getting paid as a result of a very quick simulation of your play), your RTT will be larger than if you have to put in the hours. You will accept EV<$20k to take on STD=$20k if the payoff is instantaneous.

So we need another parameter of merit: RTTmax -- your risk tolerance threshold for an infinite hourly wage.

If you know that your RTT(winrate=$200/hr)=$20,000 and RTTmax(winrate->inf)=$40,000 (for example) then you can quickly figure out that you should give up 50% of EV to eliminate variance. In general, you will get an earning equivalency graph that will look like this (blue line corresponds to the values in the example).



It shouldn't matter to you where you are on each line, although you'd really like to be on the green line rather than on blue or magenta. If your RTT at winrate=$200/hr is $20,000, and your RTTmax=$40,000, then the grey area represents the phase space of your earnings that should be acceptable to you.

The functional form of the lines is simply RTT(EV)=RTTmax*(1-EV0/EV).

The percentage of your EV that you should give up to enjoy a variance-free salary is simply (Var/EV)/RTTmax.

What do you all think?