# Poker is Surprisingly Generous

I realised something when I was walking my children to daycare: the worst odds
an opponent can give you in poker is 2:1. That means you will get *at least*
twice your money back if you win a hand. From another perspective:

*If you think you have more than a 50 % chance of winning the hand, you should
not back down, ever.*

That struck me as oddly generous, especially compared to my previous experience
of generally losing money in poker, and the statistic that 60 % of players lose
money^{1} Here, have a neat picture!.

In this article I’m going to talk about no limit Texas hold’em, which is probably one of the more popular forms of poker and has been for a while. A lot of it may apply also to other variants.

# The casino wins every hand

There are probably several things going on, and I’m not good enough at poker to
understand it all. I know one, which is fairly simple: in online poker, people
lose a lot of money to the casino. Every hand, the casino takes a small cut of
the pile of money on the table. As players keep playing, and the money goes back
and forth between them, the one consistently winning a small cut each time is
the casino. Even if the poker table was replaced with a coin toss, and two
players always wagered 2:1 on it, then in the long run *both* players would go
broke and the casino would hold all their money.

But that accounts only for less than half of the money flowing out of the system. The rest goes to players with superior skill. Superior skill at what? Let’s find out.

# Low-stakes poker is risk free

First, for any of this to make sense, we have to remove risk management from the situation. This means gambling only small amounts of money. Here’s one way to do that:

Let’s say we can afford a $100/month hobby^{2} It’s not the most expensive hobby
I have considered in the past few years, to be frank.. Then we set a deposit
limit of $25 per week^{3} In my jurisdiction, the online casino must offer this
as a configuration option even, meaning we don’t even have to keep track of it
ourselves. and figure out how often we want to play. If we want to play, say,
twice a week, we find a cash table^{4} Not tournaments! Tournaments are all
about risk management. with stakes that result in a full 100 big blind buy-in
corresponding to about an eighth of the money we have in our online poker
wallet.

Note that this means we can reinvest winnings. If we start with $0, deposit $100, get lucky and profit $50 after a couple of sessions, then our wallet contains $150 and the next time we buy in for about $20, not $12.

Though this is not growth optimal^{5} We are deliberately restricting the size
of the wagers to something that lowers both risk and reward, it allows us to
sit down at a table as often as we like, bet according to expected value, and
roll winnings on to higher stakes. And we don’t have to care about risk
management for every hand we’re dealt. We just place the best bet, whatever
amount that turns out to be.

# Poker is surprisingly simple

When we play risk-free poker, it becomes a surprisingly simple game, conceptually. In this article, I hope that experienced poker players get furious halfway through. I will take their complex game and lovingly strip it of all its interesting details, to look at the conceptuals in isolation. But don’t worry, we’ll put it back together again just as it was before.

Basically, any time you are forced to decide on an action in poker, it’s because the rest of the table has said something along the lines of

I bet you won’t win this hand, and I’m willing to offer you 4:1 odds on it.

Your only job is to either say,

- “Yup, you’re right” and fold (give up); or
- “I think you have judged the situation fairly” and call (accept their bet); or
- “Oh yeah? That sounds like a good deal to me” and raise (offer the same bet at even more cocky odds).

That’s it. That’s all of poker. We are repeatedly being offered bets, and, based on what cards we have, we evaluate whether these bets are in our favour. If so, we take them. If not, we don’t.

Now we get back to that realisation that opened this article: the worst odds
someone can offer us^{6} This happens if someone bets a comparatively large
amount of money into a comparatively small pot. is 2:1. In other words, if we
are more than 50 % sure we’ll win, there is no bet another player can make that
becomes a bad deal. With two players, 50 % is the *base rate*!

And this is a critical insight. Since we play without risk management, the
*size* of the bet in terms of dollars does not matter at all. We can afford to
look only at the odds. If we can convince ourselves of this, we have an upper
hand over people who are otherwise naturally risk-averse. If our hand has an
80 % chance of winning, we should accept bets until our poker money runs out.
They are always a good deal.

What this lens also lets us do is play poker with our forecasting hats on.

# Poker is satisfyingly complex

To get why that is interesting, we first need to put poker together again. Let’s back up a little and acknowledge the furious experienced players in the crowd. They will be yelling things like “implied odds!” and “game theory!” and they’re right.

The problems with our simplified poker model are threefold:

- The pot odds (money on the table vs. money we’re being asked to bet) is not the actual odds of the bet once the hands are played out fully. There will (maybe) be further betting rounds, and how we act now determines what those will look like. We should actually evaluate not just the bet in front of us, but the total odds of the entire series of bets – this is the idea known as implied odds, and it is much harder to figure out. This is one place experienced players have an edge.
- Determining the probability that our hand wins is not trivial. Experienced players know a lot of rules of thumb, and have memorised probability tables, and are good at mental maths, so they can figure this probability out more accurately than we can. This is another way in which they have an edge.
- Even if we could figure out our probability of winning very accurately,
here’s a curveball: our choice of action affects how our opponents act, and
thus the way we bet
*changes our probability of winning*! This is the domain of game theory, and, you guessed it, experienced players know it well.^{7}Bluffing is one of my favourite game theoretic ideas, by the way. Bluffing consists of deliberately making bets that are bad for us, and making sure other people see that we do that, so that they are forced to pay more for our profitable bets. This is what makes bluffing different from lying or deception:*we want to be caught bluffing*. (Strictly speaking, I guess we don’t*want*to be caught bluffing, but we’re fine with it. Either the bluff is not caught, in which case we gain as if it was plain deception, or the bluff is caught, in which case it increases the expected value of all legitimate bets we make. Proper bluffing is a win-win strategy: there is no bad outcome.)

This sounds sensible – what gives us the right to ignore that?

# Maybe being almost right is good enough

In forecasting, it is better to be almost right consistently, than to be very right most of the time and very wrong sometimes.

This is a hypothesis of mine, so take it with a grain of salt, but maybe the same goes for low-stakes online poker? Maybe if we can consistently guess the odds and probabilities within, say, a third, and consistently act rationally in response to those, we are ahead of the noisy crowd that is often more right than us, but also sometimes very wrong?

If that is true, then someone making very rough probability judgements without knowing much game theory and – indeed – not knowing much about poker at all – would in the long run win money. Small amounts, to be sure, but still above zero.

# Poker has notoriously high variance

I often try things out to see if they work before I share them with the public,
but here’s the problem: it takes a lot of time to figure out if you’re not
useless at poker. Cash poker success is often measured in *win rate*, which is
the average profit per hand, but it’s scaled and measured relative to the table
stakes – it’s called big blinds per 100 hands, and is notated bb/100. A player
that is not useless will have a win rate above zero bb/100. A good player may
have a win rate above, say, 4 bb/100.

Here’s the problem: with my limited time, I have been able to play 500 hands since getting to this realisation. This is something like six hours of poker, so it’s a significant time investment for me, even though it’s spread out over several weeks.

Even after these six hours, A 90 % credible interval around my win rate spans from -115 to +163 bb/100. In other words, six hours later I’m still incredibly far away from ruling out the null hypothesis. I don’t think I’ll ever get to the point where the credible interval does not straddle zero, because if my math is right, it would take hundreds of thousands of hands.

I might keep going though because it’s actually somewhat enjoyable to play, and
then we’ll see if I issue an update on this. I really *want* my hypothesis to be
true, though! It would make a lot of sense. If any of my readers are experienced
poker players, I’d love to hear your thoughts.