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Izverg04
05-16-2005, 06:42 AM
This post is in part motivated by the poll started by 1800GAMBLER (http://forumserver.twoplustwo.com/showflat.php?Cat=&Number=2397002&page=0&view=colla psed&sb=5&o=14&fpart=1) over at the Mid- and High-Stakes Hold'em forum. He posed the following question to the degenerate gamblers that populate that forum:

[ QUOTE ]
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).

http://img214.echo.cx/img214/4731/advriskmanagement2cn.jpg (http://www.imageshack.us)

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?

adios
05-16-2005, 12:04 PM
Not sure I understand the concept of RTTMax. I think it means that it is such that your risk of ruin is at zero or close to it. Could you elaborate on it a little more than you have?

Bartholow
05-16-2005, 03:30 PM
Intuitively, eliminating ALL variance is much different than even greatly reducing variance, yes?

For instance, you can probably cut your variance by around 50% (maybe a bit more, maybe a bit less depending on game texture) by going to a limit half as large as the one you currently play, right? And you'd also be giving up around 50% of your earn. But this would NOT be even close to worth it for most players.

I was one of those who said around 10% I'd give up, you may have convinced me that I'd give up 30% or so. 50% still seems like I'm giving up too much EV.

alThor
05-16-2005, 03:45 PM
A brief comment:

How one evaluates a gamble depends on its stakes relative to one's wealth. Advantage blackjack players have talked about, e.g., the Kelly criterion for a long time. The underlying idea when using such things, though, is the understanding that you are trying to make your bankroll grow.

That sounds like a simple concept, but it is usually misunderstood. For instance, suppose you are playing 15-30 or 30-60 on PP, and you have no intention to ever move up to higher games. In that case, you are playing more for a wage than you are to accumulate a bigger bankroll to actually play with. Such a person might rationally be willing to give up less EV to reduce Var, because they are already playing with much less risk than they can handle. (I doubt most of the poll respondents fit this category, but it is valid nonetheless.)

Compare that to someone with no money (or in debt), who is playing a profitable multi table 3-6 game, but hopes to make it to 5-10 or 10-20 sometime soon. That person indeed should be willing to give up EV for Var. The usual approximation blackjack players use is that an optimal bettor should be willing to give up half of their EV in order to have zero variance. (This assumes they have already optimally chosen the level they're playing at!)

Maybe that wasn't so brief. That is all.

alThor

Siegmund
05-16-2005, 03:53 PM
Halving your betting limit, all other thing being equal, will result in 1/2 your former win rate and 1/4 your former variance, and therefore halve your Var/EV ratio that the OP is in love with.

It's a well-known fact that the 300BB recommended bankroll is considerably more conservative (for a player with a good win rate) than a Kelly-type maximum-RoR strategy.

I don't find the result of the original poll surprising at all: I think the players who responded they are unwilling to give up any of their current EV to reduce variance are simply saying "I have deliberately chosen to play 20/40 instead of 30/60 or 50/100 because even though I believe I would beat the bigger game I couldn't stand the swings at that limit.. now go away and quit trying to get me to move back down to 10/20."

Izverg04
05-16-2005, 04:36 PM
[ QUOTE ]
The usual approximation blackjack players use is that an optimal bettor should be willing to give up half of their EV in order to have zero variance. (This assumes they have already optimally chosen the level they're playing at!)


[/ QUOTE ]

Very nice, if you look at the graph, that's exactly the result that I get! In blackjack, unlike poker, if you increase your winrate, you increase Var/EV at exactly the same rate.

Now look at the point Winrate=2*Winrate0. If you increase or decrease EV at that point, while increasing Var/EV by the same rate, you are smack outside of grey area. EV=2EV0 is the optimal earning point on each curve. It is the point where

dRTT/dWr=RTT/Wr

The situation above is not the case in poker, where you increase Var/EV at a higher rate than winrate as you move up in limits or increase number of tables due to the increased difficulty of winning. Thus a poker player should sit lower on this curve than a blackjack player. The guy in the example clearly should move down in limits.

It follows that the answer to 1800GAMBLER's question has to be below 50% for a poker player that selected optimal stakes -- he should be willing to give up e.g. 30% of his winrate for a flat salary.

I feel like I am making important insights here: BruceZ, pzhon, etc -- any comments?

Izverg04
05-16-2005, 05:10 PM
[ QUOTE ]
Not sure I understand the concept of RTTMax. I think it means that it is such that your risk of ruin is at zero or close to it. Could you elaborate on it a little more than you have?

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Close to zero? Quite the opposite. RTTmax is such that you won't take on more risk to your bankroll no matter what the earning potential is.

pzhon
05-16-2005, 11:51 PM
[ QUOTE ]

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.

[/ QUOTE ]
This is ambiguous. You say the player would not play at the higher standard deviations, but the alternative was not clear. Was it to quit playing altogether, or to move down to an equally tough game with lower stakes? You ran into problems because you interpreted it both ways.

Here is a concrete situation from a recent NL thread (http://forumserver.twoplustwo.com/showthreaded.php?Cat=&amp;Number=2346073&amp;page=&amp;view=&amp;s b=5&amp;o=): Suppose you have AA in the big blind and far too much of your bankroll on the table. It's folded to the small blind, who has really neat reflective shades, and you see he has KK. He has you covered and pushes. The blinds are negligible. How much of your bankroll do you want to have on the table? How much of your bankroll needs to be on the table before you chicken out and fold as an 82:18 favorite? <font color="white">For simplicity, I'll ignore the possible ties.</font>

By the Kelly Criterion, you want 64% of your bankroll to be on the table. Any less, and you would regret the lost opportunity. Any more, and you'll regret that your opponent has you covered. However, the point at which you would fold is much higher. When a bit more than 95% of your bankroll is on the table, you are indifferent to calling and folding according to the Kelly Criterion.

Is the EV 200, SD 2000 situation supposed to be analogous to having 64% of your bankroll on the table, or 95%? You came up with a contradiction by assuming one time that it was the 95% meaning, and the other time that it was the 64% meaning.

[ QUOTE ]
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.

[/ QUOTE ]
No, it doesn't.

The Kelly Criterion says to maximize the expected logarithm of your bankroll. A fractional Kelly system corresponds to maximizing a different function of your bankroll (something like -1/bankroll^c). No time dependence is necessary. You can maintain a consistent utility function over all gambles, instantaneous and protracted.

There are some interesting complications to consider if you view time as a resource, and are thinking of trying to clear a casino bonus. Betting smaller amounts uses more time but decreases variance. However, that is not relevant here.

Here is something that might help explain the low percentages people say they would give up to eliminate variance: In poker, because your win rate is lower in higher games, you would be overbankrolled according to your utility function if the games were scalable. You optimize when the marginal gains of moving up equal the marginal costs of moving up. The marginal costs of moving up include the decreased (proportional) profits due to tougher games. Without this factor, the marginal gains from moving up would exceed the marginal costs of moving up. If the players polled are overbankrolled, then they don't care as much about controlling variance.

Izverg04
05-17-2005, 04:53 AM
[ QUOTE ]
[ QUOTE ]

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.

[/ QUOTE ]
This is ambiguous. You say the player would not play at the higher standard deviations, but the alternative was not clear. Was it to quit playing altogether, or to move down to an equally tough game with lower stakes? You ran into problems because you interpreted it both ways.

[/ QUOTE ]

I meant that if the player had a choice only between playing poker above his risk tolerance (EV/hr=$200; STD/hr&gt;$2000) and not playing, he would not play poker. That's how I define RTT -- this is the value of Var/EV at which a +EV game becomes unattractive to you. Note that this a real-life measure of your risk tolerance which effectively does incorporate the value of your time.

I guess, your post made me reinterpret what I was trying to say: RTT depends on the hourly wage, because your time is of value to you. For example, let's say you are making $10k/month playing poker with a STD=$10k. Let's say this is equivalent to you with making a flat $7k/month. This means that if offered a gaussian coin toss at the end of the month with EV=$3k and Std=$10k you would take it. So this is what defines your risk tolerance -- this is what I called RTTmax in the original post and interpreted as the risk tolerance when time is not involved. Makes more sense now?

I think this has been useful for me to understand because from experience (basically, by calibrating psychological pain after losses) I know what my RTT is at my current hourly winrate (it's quite a bit lower than in the example above). I have no empirical handle on my RTTmax but this is what I need to know if I want to be rational when choosing between alternative bets with different payoff structure and different hourly wage. To do that I need to know my risk tolerance if time was not a factor.