### More EV/Variance/Ruination Crap

Just got a very interesting comment on my post on the topic earlier this month, from someone named "David" (I play poker with several of those but they all go by their own nicknames so I wonder if I know this one).

I found it very interesting, so much so I start rambling off into a long comment. Long enough I just decided to promote it to its own post.

David said:

Your willingness to accept the consequences of statistical variance should depend on how deep your wallet is. If you are playing close to $0 EV in massive pots, then your risk of ruin is very high.

In this case, if you push, you risk $175 now to net $38 on average, giving

you about a 22% return (there's a number called the Sharp Ratio, which describes

your risk with respect to return and volatility, I have bastardized its use here to make a point). If you call, you risk $75 to net $19, giving you something like a 25% return. So it gives you a little more bankroll safety.

So, in the end, from a variance (and consequently, risk of

ruin) point of view, it is a wash, and is really up to your personal style

(which is fairly well documented).

My response:

Thanks for a very interesting comment. It made me think about things I haven't before.

I will need to read up on this Sharp ratio--I've heard of it in the most general of terms as applied to evaluating investment strategies. [after looking it up I see it's actually Sharpe and I see that as you said, you have indeed simplified it; because it's complicated beyond this discussion (and my present desire to comprehend it), I'll stick to the way I think you used it and drop the E.]

I am not fully following you however. What you have said at the beginning of the comment remains consistent with how I think about the problem in general--the willingness to take that risk has to be measured in terms of your bankroll.

Yet you go on to point to the similar adjusted returns of 22% and 25% and state that the implication is that it comes down in this case to a matter of style.

Don't you need to factor in bankroll and opportunity cost as well rather than just reward relative to amount risked? I mean as to bankroll, if this was your whole bankroll, risk of ruination would be extreme versus the push being for example 1/1000th of your bankroll where you'd really have to be pretty abysmally unlucky to go broke making this play in any iterated fashion.

As for opportunity cost, assuming risk of ruination of taking this risk relative to your bankroll is acceptably small (as it was for, though not quite 1/1000th to be sure), if I having nothing else to invest my bankroll in at that moment, isn't still a mistake not to take the higher EV?

To illustrate my attempted point, if you had $1000 and time for 100 coin flips and someone offered you the chance to win 11 cents against 10 cents per flip ("Sharp" 10%?) or 1.05 against 1.00 ("Sharp" 5%?) wouldn't you be silly to take the first bet over the second bet not withstanding the better ratio?

EV for the first choice is 1.00 with very low variance compared to $1000 and no chance of ruination, and EV of the second is $5.00 also with no chance of ruination, and a significantly higher variance relative to $1000 than the other bet (but still quite small relative to the bank roll).

More thought, the Sharp

__e__Ratio also factors in a rate of riskless return, I wonder if that can be worked in as a missing piece to quantify what I am trying to get at with opportunity cost. Oh, this is starting to feel like I am still at work. Oh wait a minute...

Anyway, I'm glad you got me thinking about this.

## 4 Comments:

My comment was off the cuff, so let me clarify. Actually, this comment is off the cuff too, so feel free to tell me what looks wrong.

You aren't playing $0 EV in this situation, it is just something that I force myself to think that whenever I start off analyzing a poker situation. But it is worth it to remember this, the EV numbers that you crunched are for your $75/$175 choice. You have already lost $45 to the pot with your preflop and flop action, so the EV of the way that you have played the hand overall, is -$25 or -$7.

So, the Sharpe ratio (whoops on the e), I screwed up my thinking on it and missed the point. I one-d out the standard deviation of your actions in my head, and just thought about the rate of return. That is bad, because the Sharpe ratio is supposed to help characterize the risk, not ignore it.

To really think about it in terms of Sharpe, you probably have make an estimate of S.D. ((x% of time win)(amount won))^2 - (average net)^2 yada yada yada, and then crunch that into the formula. I haven't done the numbers, but it probably isn't the risk-wash I thought it was.

I honestly don't know what at good Sharpe ratio for a poker player is. Maroon wrote one time, as an example, that a limit hold 'em player may average 1BB/100 hands, with a SD of 10BB/100, which would give you an atrocious Sharpe ratio, compared to investing in stocks.

As an aside, riskless rate of return is usually taken to be a long government bond rate, but in the poker game context, I'd say it is zero. You will play, or you won't.

I'm just a random internet David.

I am not sure whether your reply was meant to respond to my response and may have lost you on the relevance of standard deviations in this context (dealing with traders talking about this in the context of volatility versus variance, I don't doubt it's relevant, I just don't quite get it).

I do have to disagree actively on the not $0 EV point--the money in the pot is gone, sunk cost, and the analysis bears only to the decision tree from here on in.

To say that raising preflop is -EV without considering the potential outcomes after the flop does not make sense. By this reasoning, if preflop I had AA, raised 100 with 100 behind, and was called, my EV in the hand would be minus 100. This cannot be correct.

EV analysis for a decision at hand is relative to the future outcomes, but prior actions are only relevant to the extent those actions a) put money in the pot and b) are liable to influence the reactions of opponents at this point. Beyond that, the correct analysis of any decision tree must start from where that decision begins.

Also I am not completely convinced zero is a correct application of the riskless rate of return in poker as time itself has a non-trivial utilty cost.

If you assume zero, I suspect the "correct" decision will be biased to the least risky plus EV move since you would have infinite time to wait around and collect. In my coinflip example, I would certainly prefer the 0.11/0.10 flip over the 1.05/1.00 flip if I could chose the number of flips without cost.

In reality, the "riskless rate" is actually thus probably slightly negative, and varies depending on your outside life, representing the cost of time spent playing.

Gotta be quick, gotta get to work.

In your Aces example, you EV is -$100 if you raise preflop and fold. If you raise to $100 with a hundred behind, you hope that you have some nonzero probability of winning the pot after the flop. Moreover, you really hope that this equity wins you the $100 you sunk into the game already plus some.

My EV analysis is restricted to this situation. In reality, when you raise preflop, you are speculating that when you flop top pair or better, you will make more money than you lose in situations like this.

I think that the money loser for you here is the continuation bet, but that'll have to be saved for when I have a minute to crunch the numbers (how does your equity/EV change if you check-call or check-push?).

In the financial sense, volatility is the standard deviation of the performance of your portfolio. Or in poker, it's the SD of your performance.

This is pretty much stream-of-consciousness-David, so not a response to anything in particular.

U r such a nerd!! Just kidding. I downloaded that site so we are all set.

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