In poker, we often discuss the expected value (EV) of strategies, without considering the risk or variance involved. However, in the wider world of finance, there's a prevalent concept known as "Risk-Adjusted Return". This idea, while fundamental in finance, seems almost entirely overlooked in poker.

The simplest version of a Risk-Adjusted Return can be calculated with the Sharpe Ratio = (Expected Value) / (Standard Deviation)



The best investments are those that optimize your Risk-Adjusted Return. Let's set aside emotional factors for reducing risk and focus solely on the financial rationale for maximizing this ratio.

Consider, for example, two distinct games:

• Game 1: A guaranteed win of $50.
• Game 2: A coin flip where a win means $1000, but a loss costs you $900.

Both games have an EV of $50, yet Game 1 achieves this EV without any associated risk.

Now imagine that it takes a month for this game to complete. It's important to note that in reality, investments take time to pay off. Choosing Game 2 would require allocating a significant part of your bankroll, which could otherwise be invested elsewhere. Opting for Game 1 frees up your bankroll for other investment opportunities. So the fundamental idea is that maximizing Risk-Adjusted return improves your ability to make compound returns.

Here is an example of how you might calculate Sharpe Ratios in Poker

--

Now, the question: Is this concept applicable in poker?

• If two poker strategies offer the same EV, but one has lower variance, should the lower variance strategy be the preferred choice?
• Should you opt for a slightly lower EV strategy if it offers a better Risk-Adjusted Return?
• If so, what strategies could you employ to lower your variance? Shortstacking for example?

And the really hard question:

I believe this is connected with Kelly criteria. High shape ratio means you can put bigger part of your roll in.

You do this in poker too by using proper bankroll management. If you are winning NL50 online player with BR of 2k, you can find live 2/5 game that is higer ev but requires too big of an investment (shape ratio is too low).



And yes you should pick lower variance strategy, because it has lower BR requirements, making it easier to move up stakes. Problem is lower variance stra in poker are often lower ev or they might be easier to exploit.

I would say if 2 strategies truly had the same EV and you had the objectivity to know rather than go with what feels most comfortable to you, yes, lower variance would be better because you could play higher stakes with a smaller BR.
In reality, your criteria is so tainted by emotions that if you go for the lower variance strategy it's out of fear basically always and is actually lower EV

The most obvious example is bankroll management: if our only goal was to maximize pure EV, we would put all our money on the table every time.
Of course, this would also maximize our risk of ruin.

So instead, most of us use bankroll management in order to focus on maximizing the "median bankroll" and "IRL utility" instead.
As already mentioned, Kelly Criterion tells us we should prefer lower variance, allowing us to move up in stakes faster, thus increasing our long-term median bankroll
compared to the higher variance strategy.

As a human, though, I think it is often better to learn to handle a "high variance playstyle", because our human opponents tend to do worse when they are pushed outside their comfort zone.
Either they will make the mistake of being too risk-averse: overfolding, underbluffing, etc.
Or they will go to the other extreme and tilt.
Think of someone like Stefan who makes plenty of theoretical mistakes, yet crushes most human opponents because he pushes them outside their comfort-zone in a deliberate manner.

Some things we could do to lower variance:

* Run it twice. Saulo Costa actually argued against running it twice in a recent video, because he wants to maximize his chances of playing with deep stacks, which increases the winrate of the more skilled player.

I also heard someone argue against running it twice because it makes it less likely your opponent will go on tilt.
If you can handle a suckout without problem, but your opponent goes on monkey tilt if he loses a 60/40, then it would benefit us to run it once in order to have a 40% chance of
our opponent going on monkey tilt. If we run it twice, there's only a 16% chance of our opponent going on tilt.

* Make ICM deals on final tables in MTT's. Even if you have a skill edge, it may be worth to sacrifice a little EV in order to guarantee a big score, allowing you to move up in stakes or realize IRL-utility, like buying a house.

* Swap % with other players. In mathematics of poker, Bill mentions that it can beneficial to swap % with other players in high variance games, even if those players have slightly lower expected winrate than yourself, similar to how buying insurance is EV-, yet can be worth it from a risk-adjusted perspective.

Or if I invest in the stock market and believe that one particular stock is the very best, I would still diversify to other stocks in order to reduce volatility and risk of ruin and increase risk-adjusted return.

* Staking. One thing I believe is common among high stakes players is to be staked when playing at the highest stakes. Otherwise, the problem is that the very highest stakes rarely run,
so it is difficult to get enough hands in at the highest stakes to "even things out".

Let's say you play NL5k as your daily game and a massive whale sits down at NL40k. You have a huge edge, but the variance of a shot like this is also huge, so you get someone to
stake you for 50%, allowing you to still realize some of that edge without risking too much of your bankroll or risking having a single cooler ruin your entire month.

* All-in insurance. Let's say I somehow find myself with a large % of my bankroll on a table with a massive whale on it.
He shoves all-in pre with A6 and I have KK. If I win I will be able to move up in stakes, if I lose I will have to move down.
In this case it is worth giving up a couple of % EV in order to guarantee being able to move up in stakes instead of being forced to move down.

Similar to how a starving beggar on the street should prefer a guaranteed $70 over a 75% chance of winning $100.

1) Yes, given same EV, the lower variance strategy is the preferred one, and as part of a broad gameplaying strategy, is the most profitable, since it will allow you to use more aggressive BRM (less risk of ruin).

2) If it is close, yes. Again, the slightly lower EV might actually end up being more profitable for you on a broader gameplaying longterm point of view since it will allow you to use more aggressive BRM.

3) Need to think more.

Bonus question) I would try to use toygames, like the AA-QQ vs KK, 22233r scenario. You either have the equilibrium, or a MES if anyone deviates, no matter how little. No middle ground. Interestingly, the MES is lower variance Anyway, given there is no 2nd equilibrium in this case, and for your question to be true would require it to be true in any scenario, including this nuts/air vs bluffcatcher toygame, my answer is no.

vs a GTO Bot we could just check and fold all our EV0 mixed decisions.

If we are playing the polarized river toy game:

Villain bets $100 into $100 with 2/3 nuts and 1/3 air.

EV of calling is $0, with a standard deviation of $150(?)
EV of folding is $0, with a standard deviation of $0.

Same if we are the aggressor:
EV of bluffing is $0, with a standard deviation of $100(?)
EV of checking is $0, with a standard deviation of $0.

Of course, versus adaptable opponents we become very exploitable, but versus a GTO Bot we could significantly lower our variance without lowering our EV.

Solver has clairvoyance over what you’re doing, and tombos asked about a solver

E.g. The risk reward of always bluffing if I know you always fold is still pretty good.

*I am assuming we are talking about the last hidden question btw

In my example the solver does not adapt. It simply plays GTO.

(Didn't see the hidden question before)