Terror in Poker and Finance Part II
 

Forgive me if I have erred. I am not financially educated, so perhaps I misunderstand, but it seems to me that a remark in "Poker and Finance Part II" by Dan Mezick is so profoundly incorrect, that any decent poker player should be deeply offended, and it certainly should not appear in an article associatied with the fine name of "2+2".

Apparently, "Sharpe ratio is basically the net reward divided by the volatility." Mr. Mezick then later goes on to conclude, "The player with the better Sharpe Ratio is the better player." It would appear that the thumpers of the weak-tight bible finally have their day! Nitty McSupertight is at home cheering, "Yes! I knew that I was the best player in my game, I almost almost always quit while I'm still ahead! Some guys are up way more than me over the last 5000 hours or so, but their results are so volatile... sometimes they lose 30 big bets in one session! I knew all this time I was the best player in the game; I earn way less than some guys, but I don't have to go through those big swings!"

OK, admittedly I have vented some of my frustration in that last tirade, but do you see my point? To clarify, suppose two of us have $10,000,000,000 bankrolls, and we're playing $1-$2 limit HE. Who is the better player? The one with the higher Sharpe Ratio, or the one with the higher EV?

There are clearly some important and points to be made, but claiming that "the player with the better Sharpe Ratio is the better player" is wildly incorrect, and only serves to champion the cause of the weak-tight masses.

Am I wrong?

Re: Terror in Poker and Finance Part II
 

You're absolutely wrong.

P. 754 of Investments by Bodie, Kane and Marcus.

"Sharpe's measure devides average portfolio excess returns over the sample period by the standard deviation of the returns over that period. It measures the reward to (total) volatility trade-off.

Playing super tight certainly lowers your standard deviation, but at the expense of playing marginal hands in profitable situations.

Playing super loose is the opposite, increasing the standard deviation and playing marginal hands in less than profitable situations.

The object is to get your maximize your return to standard deviation ratio, thus maximizing your Sharpe ratio. Depending on the situation, that could mean a super nitty style or a super loose one. Depends on how high the variance is in the game. So once again, the right approach to the game is situational.

I knew all that crap I learned in MBA school and all those books I refused to throw away would come in handy one day.

Re: Terror in Poker and Finance Part II
 

I'm sorry, I don't see it. You claim, "The object is to get your maximize your return to standard deviation ratio, thus maximizing your Sharpe ratio," but that is not correct. The object is to make the most money. As I said, I think there are valid points to be made, but it still seems to me that saying that the goal is to maximize your Sharpe Ratio is stone dead false.

Re: Terror in Poker and Finance Part II
 

With all due respect, I believe you lack a fundamental understanding of the Sharpe ratio and it's application. Further, I think you're trying to project something on the Sharpe ratio that isn't there. You're suggesting that the Sharpe ratio vindicates playing a particular style, which is patently false.

While the super nit will have low variance and a low expectation, somebody who plays correctly will likely have a slightly higher variance and a much higher expectation. A maniac may have the highest variance and no higher expectation than somebody who plays correctly.

Or, another way - increasing your standard deviation without a corresponding increase in your win rate is lunacy. Further, decreasing your standard deviation in a way that causes a greater decrease in your win rate is just as dumb.

The Sharpe ratios of many bond funds and many penny stocks are about the same. It's pretty easy to go busto with penny stocks (high standard deviation, high return) and tough to get rich with bond funds (low standard deviation and low returns). There are lots of spots in the middle that yeald better with a reasonable standard deviation, thus higher Sharpe ratios.

Re: Terror in Poker and Finance Part II
 

Sweet, what you're missing is that when you have a ratio of:

N/D

Decreasing D increases the value of the fraction, but so does increasing N. Since "n" in this value would be the expected "reward" (i.e. how much money you make overall), then EV is still very important according to this model and can still trump decreased variance much of the time.

Re: Terror in Poker and Finance Part II
 

Absolutely correct.

Assuming that two plays have the same EV, you would be insane to choose the play with the higher variance.

Assuming that two plays have the same variance, you would be insane to choose the play with the lower EV.

The world does not often work like that, and one is left to contemplate a variety of choices with corresponding EV and variance. This is why Sharpe's model came around in the first place.

Re: Terror in Poker and Finance Part II
 

[ QUOTE ]
Sweet, what you're missing is that when you have a ratio of:

N/D

Decreasing D increases the value of the fraction, but so does increasing N. Since "n" in this value would be the expected "reward" (i.e. how much money you make overall), then EV is still very important according to this model and can still trump decreased variance much of the time.

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Well, I didn't actually miss that. I didn't claim that the player with the higher Sharpe ratio cannot be the best player. I did observe that there are ways for your EV to decrease while your Sharpe index increases, which, from my position, means that claiming the player with the higher Sharpe ratio is "the better player" is digustingly false. I gave a $10,000,000,000 example that I thought made this pretty clear. No?

Re: Terror in Poker and Finance Part II
 

[ QUOTE ]
I believe you lack a fundamental understanding of the Sharpe ratio and it's application.......
While the super nit will have low variance and a low expectation, somebody who plays correctly will likely have a slightly higher variance and a much higher expectation.

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You may be right about my understanding. For example, I might've guessed that playing "correctly" would give a slightly higher expectation but a much higher variance. Usually the low-earn high-variance hands are exactly the ones that nits don't play, right? These are relative terms though, so maybe I don't understand the units in the numerator and denominator.

In any case, I think my $10,000,000,000 example clearly demonstates that the player with the higher Sharpe ratio need not be the best player.

Re: Terror in Poker and Finance Part II
 

You can do a lot to lowering your personal sharpe ratio by playing poorly.

Re: Terror in Poker and Finance Part II
 

[ QUOTE ]
To clarify, suppose two of us have $10,000,000,000 bankrolls, and we're playing $1-$2 limit HE. Who is the better player? The one with the higher Sharpe Ratio, or the one with the higher EV?

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In your question, you should add:
One of the players has a higher standard deviation, but also a higher EV such that the Sharpe's Ratio is lower.

So:

Player 1 EV > Player 2 EV
Player 1 Sharpes Ratio < Player 2 Sharpes Ratio
Player 1 SD > Player 2 SD

(Unless I missed something, I think the combination shown above is possible and not an impossible situation)

Who would you rather be if you had 10mil, and you are playing $1/$2?

I'd rather be Player 1.

Re: Terror in Poker and Finance Part II
 

I guess I'd rather be Player 1 also. As long as my bankroll was sufficiently large that risk of ruin was minimal. Otherwise, a slight increase in EV along with a relatively greater increase in SD implies that the risk outweighs the benefits (and I'd rather be Player 2 in that spot).

Re: Terror in Poker and Finance Part II
 

Is it possible? Yes.

Is it likely? Maybe.

Maybe player two is super talented with some huge holes and could smoke player one in EV and SD. Never consider the best of both worlds as an unattainable option.

Variance is a bitch, and some of us will do a lot to stay away from it if it doesn't mess with our EV much and it tends to help keep us off of tilt.

Re: Terror in Poker and Finance Part II
 

[ QUOTE ]
I'm sorry, I don't see it. You claim, "The object is to get your maximize your return to standard deviation ratio, thus maximizing your Sharpe ratio," but that is not correct. The object is to make the most money.

[/ QUOTE ]

Right. Don't let these guys fool you, Sweet. Poker differs from finance, because your time is inherently consumed playing poker, and EV can go down very quickly if you try to play higher stakes.

Where is the feedback in the formula? If the author had shown how the EV of the game adjusted (and hence the Sharpe ratio) as a result of making -EV decisions, that would have been some new material. There would still be lots of issues, but that would probably be sufficient to drive home the only real point - people playing on short bankrolls have extra incentive to avoid wacko tables, because variance reducing play at those tables does not really help.

Re: Terror in Poker and Finance Part II
 

I am very pleased that my second article sparks real debate about evaluating poker plays and actions on a risk-adjusted, Sharpe-ratio basis, rather than EV alone.

I like this thread a lot and I sincerely thank the OP, Sweet, for opening it up.

I admit that I may be 100%, completely dead wrong in my beliefs about the merits of Sharpe ratio relative to EV.

Will I ever have to? Probably not. We’ll see.

What I’m really looking for is for the best thinkers on 2+2 to enter this discussion.



Generally speaking, I am very surprised that so few readers are not deeply offended by, and rabidly opposed to, my assertion that EV alone is NOT the best basis for totally evaluating a given poker situation.

The second sentence of the essay challenges the the reader’s beliefs about EV right from the start:

“Poker plays are best evaluated on a risk-adjusted basis, rather than simply considering Expected Value alone.”

Don’t you find it kind of interesting that so few 2+2 readers are actually challenging this statement?

Here is another one: at the end of the article, I throw down this bald assertion:

“The player with the better Sharpe Ratio is the better player. He makes more money per unit of bankroll on a risk-adjusted basis. It can be said that he deploys capital better, by managing risk more effectively.”




Finance guys know for a fact that Sharpe ratio, Sortino ratio and other reward/risk ratios have EVERYTHING to do with the proper evaluation of poker games, starting poker hands and individual poker plays. That’s because every poker situation is, at the core, financial in nature.

Sharpe ratio and its derivatives are being used to the best players, now, to make plays that increase reward per unit of risk, or lower risk per unit of reward, or both.

Risk has many components, and some are hard to measure or even identify. For example, highly volatile returns tend to trigger emotions and ultimately, real “tilt” in some players. Self-aware players identify and manage that risk. In practical terms, this “tilt risk” adds an additional dimension of risk to the denominator of the reward-to-risk (Sharpe) ratio for certain kinds of plays. My point is that risk has many components, and that some of these risks may be difficult to measure. But, to correctly figure the risk-adjusted value of any poker action, you must sincerely **try** to quantify ALL the components of risk—and reward.

Reward also has many components, some of which are non-cash but convertible to cash by highly skilled players. For example, a given play can gain you some bluff-equity, even if you lose the hand. That is a very real component of reward that a skilled player always considers. Recognizing the potential gain in bluff-equity makes some otherwise marginal moves very playable. This is because the value of the potential bluff-equity gained increases the theoretical reward, relative to the theoretical risk. This is Sharpe applied to poker.

Bluff-equity is a component of “overall theoretical reward” the same way propensity-to-tilt is a component of “overall theoretical risk”. Both of these are examples of non-cash rewards and risks inherent in a given play. Both must be included when thinking about the risk-adjusted (Sharpe ratio) value of a given play. Thinking about poker theory in terms of Sharpe provides a framework for a more quantitative, non-discretionary approach to the game. Discussing poker in terms of Sharpe provides rigor and a framework for explaining the reasoning behind complex poker decision making.

I often hear the answer “it depends” to questions about many poker problems. Depends on what, exactly? In my view, it almost always depends on the risk-adjusted quality of the play after all of the rewards and all the risks, cash and non-cash, have been identified, considered and weighed. Sharpe ratio and its derivatives (such as Sortino) provide a framework for discussing complex poker problems quantitatively. Sharpe is just a starting point. We have a long way to go in terms of advancing the state of the art in poker theory.

Finance and especially the Modern Portfolio Theory (MPT) can help get us there.

I assert that, if you choose to turn a blind eye to Sharpe and continue to play by EV alone, you are sure to be taking the worst of it relative to opponents who truly understand how to apply Sharpe’s work to poker.

Re: Terror in Poker and Finance Part II
 

I don't fully understand a lot of this discussion as I have no background in finance, but here is my view.

Suppose a particular high variance play in my arsenal slightly increases my EV. Taking it out of my game would lead to a lower variance and a greater sharpe ratio. Taking only one move into account here it's not going to change either by much, but there are probably several such plays.

Let's do some simple cost/benefit analysis. The benefits seem to be:

1. Decreasing my variance by adjusting my play would allow me to play in higher stakes games at my given bankroll.
2. At a given bankroll I'm less likely to have a severe downswing leading to me going bust and/or having to move down.

However, the cost is that you lose EV at the current game you're in. Another cost is that you probably will learn more as a player by making these high variance plays.

Let's study them individually.

1. Ability to move up

The span of possible games is not continuous so 1 may not be a factor. For example, considering my bankroll it may be possible for me to move up from 5/10 to 5.5/11 by lowering my EV and decreasing my variance but such a game doesn't exist. Furthermore, given my skill level if I had an infinite bankroll 5/10 may be a higher EV game (obviously infinitely rolled EV is the only consideration) than the next actual available level. In fact, many players are playing in the game that gives them the highest expected earn.

2. This could be a factor. However, many players are overbankrolled for the highest EV game available to them. For these people, this isn't. Furthermore, if people are in a situation where losing money could force them to go down it is very likely that they will do as you suggest and give up +EV but high variance plays. There are books, IIRC one of them is Greenstein's AotR, that suggest playing conservatively if going on a downswing means you have to drop in stakes.

So in short, your theory only applies to people that are slightly to moderately underbankrolled for the highest EV game available to them. Note that this would exclude stakes where people are ridiculously underrolled for their best game.

Re: Terror in Poker and Finance Part II
 

JaredL,

Thanks for your highly detailed post. You describe and bring up financial points in conversational English, and you know PLENTY about these topics. One of the issues about these topics is the lack of sufficient terminology we have to discuss them. Precise terms helps with generating collaborative discussion and can advance any art or science, such as poker theory. We are hobbled by a lack of precise terminology. Finance helps because the terminology is there to speak with precision. This is one of the main points of my article series.

The main point you raise is the potential error of giving up +EV when the BR size is sufficient. This is true, especially (as you point out) when the range of opportunities is for sizing variance to the BR (via bet/stakes sizing) are not ideal. I must agree with you here.

Remember though, I am speaking in highly theoretical terms. My basic theoretical premises are sound; adjustments to theory must fit the situation in practice. You make a key point about adjusting that does not invalidate my poker theory thesis about Sharpe.

Secondly you raise many points about the complex interrelationships between Sharpe, EV, variance and RoR. My article series is ALL about these poker issues, spoken in financial terminology and using financial concepts.

Lastly, in terms of the potential error of giving up +EV assuming a BR that can enforce acceptable RoR, I believe that in general most players are underrolled and that most players take on massive quantities of "uncompensated volatility" when they play poker.

Thanks again for your posted reply. It is sure to attract more interest to 'Poker and Finance'.



Links:

Uncompensated Volatility
"One way to control risk drag would be to eliminate volatility entirely. But doing so provides a hollow victory because..."

Re: Terror in Poker and Finance Part II
 

[ QUOTE ]

The second sentence of the essay challenges the the reader’s beliefs about EV right from the start:

“Poker plays are best evaluated on a risk-adjusted basis, rather than simply considering Expected Value alone.”

[/ QUOTE ]

This statement is trivial and useless in practice at the same time. Dan, how much would you guess someone would lose, in terms of expected utility, if he played in a way that perfectly maximized expected value instead of expected utility? Would it be 1% of the winrate? 2%? Now how much do you think a typical expert player loses because he doesn't maximize his return (makes mistakes)?


[ QUOTE ]
“The player with the better Sharpe Ratio is the better player. He makes more money per unit of bankroll on a risk-adjusted basis. It can be said that he deploys capital better, by managing risk more effectively.”

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Could you give a numerical example of what you mean? Because using Sharpe ratio in the context of poker really doesn't make sense in any situation I can think of.


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Sharpe ratio and its derivatives are being used to the best players, now, to make plays that increase reward per unit of risk, or lower risk per unit of reward, or both.

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I think you are inventing a class of poker players that really doesn't exist.

[ QUOTE ]
[...] risk-adjusted (Sharpe ratio) value of a given play.

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Are you just using "Sharpe ratio" as shorthand for "risk-adjusting expected returns to obtain the expected utility of a gamble"? You are throwing Sharpe and other financial risk measures all over the place, and these are usually specific results to specific risk-adjustment problems that don't come up in the same form anywhere in poker.

Re: Terror in Poker and Finance Part II
 

By the way, here is a very simple way, based on Expected Utility theory, to appraise you risk. For small bets (small compared to bankroll), expected utility or the Certainty Equivalent of a play is:

CE=EV-Var/2R,

where EV and Var are expected value and StdDev^2 of a gamble, and R is the Kelly bankroll (equal to e.g. bankroll/4 for Kelly/4 bettors).

Actually, more formally, R is the curvature of your utility function R=-U'(x)/U''(x). I assumed here that R remains about the same whatever the result of the gamble is (thus the caveat about small bets).

No need for fancy ratios from quantitative finance. Find CE for the choices that you have and make the decision that maximizes CE. Most of the time you'll make the same choice as if you were maximizing EV, while taking occasional EV hits in marginal situations, where cost of variance is too high.

Re: Terror in Poker and Finance Part II
 

[ QUOTE ]

Generally speaking, I am very surprised that so few readers are not deeply offended by, and rabidly opposed to, my assertion that EV alone is NOT the best basis for totally evaluating a given poker situation.

[/ QUOTE ]
You are attacking a straw man. The objections people have are not that E$ is the only thing that matters. There are contexts in which a different combination of E$ and risk is more appropriate than the Sharpe ratio. Please don't assume that anyone who disagrees with you is saying that only E$ matters.

The OP pointed out a context in which E$ is much more important than the standard deviation. As your bankroll grows, maximizing E$ is relatively more important than minimizing variance for the purpose of maximizing expected utility. If you are underbankrolled, you may want to be more conservative than the Sharpe ratio would suggest. According to polls, 2+2ers tend to be overbankrolled, so the Sharpe ratio should tend to overemphasize managing risk while underemphasizing E$.

Rather than consider only EV/SD, you can consider (EV^n)/SD. For any n, you get a risk vs. reward ratio, and only n=1 corresponds to the Sharpe ratio. The most appropriate n to use depends on both your Kelly fraction and your bankroll.

There are other objections to the Sharpe ratio, such as that the shape of the distribution of results about the mean matters. To a slightly favorable coinflip, it is possible to add a purely favorable gamble which decreases the Sharpe ratio because it increases the variance much more than it increases the expected value, e.g., a 1/1,000,000 chance to win $10,000 adds a penny to EV, but adds about as much variance as a coin-flip for $100. This might ruin the Sharpe ratio of someone crushing low stakes games, but it would be nothing to fear. Less obviously, buying lottery tickets is not the same as selling lottery tickets.

[ QUOTE ]

“Poker plays are best evaluated on a risk-adjusted basis, rather than simply considering Expected Value alone.”

Don’t you find it kind of interesting that so few 2+2 readers are actually challenging this statement?


[/ QUOTE ]
Not really. Perhaps your expectations are off. You underestimated your audience.

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The player with the better Sharpe Ratio is the better player...

Finance guys know for a fact that Sharpe ratio, Sortino ratio and other reward/risk ratios ...


[/ QUOTE ]
Stop right there. If the Sharpe ratio is everything, why would there be other measures like the Sortino measure? The answer is that the Sharpe ratio isn't everything, which is intuitive for many poker players who don't know the finance terminology. The Sharpe ratio is inappropriate for many situations, such as the one the OP brought up. That strongly suggests that your assertion, that a better Sharpe ratio means you are a better player, is wrong.

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I assert that, if you choose to turn a blind eye to Sharpe and continue to play by EV alone, you are sure to be taking the worst of it relative to opponents who truly understand how to apply Sharpe’s work to poker.

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You are attacking a straw man. We aren't saying EV is everything. We're saying something more complicated that your assertion that the Sharpe ratio is everything.

It is possible to ignore the Sharpe ratio and not to be taking the worst of it in any serious sense. E$ is roughly zero sum, but variance is not. If I evaluate my risks in a different consistent fashion from the way you do, that doesn't mean I lose, and it doesn't mean you gain from playing me.