Open or not - do I have the best hand?

The title might be promising a bit too much. If this post could really tell you if you have the best hand or not, I would charge you for reading it. What it will give you, is a thought experiment I did lately and found useful for thinking about whether to open or not from a given position.

To estimate if it is likely that I have the best hand in a given position, let's do this thought experiment:
- Let's rank all opening hand combos from 0 to 1 linearly
- Let's say you are in the position with N players left to act behind you

So, how strong a hand do you need for it to be likely the best? If you are in the SB in an unopened pot, it is pretty obvious that the chance of you having the best hand is 50% before you look at your cards. And if you look at your cards and see a hand of strength 0.75 (let's call this S), it means you have a 75% chance of holding the strongest hand.

A bit more formally, we might define the function P(S, N) such that it gives the likelihood of your hand strength value being higher than all the ones behind you. You might already have guessed that given random distributions (that I hope your poker room or home game has!) that this probability will be (we'll treat the individual hands as independent and ignore the effect of folded cards, which is safe enough anyway - a word on so called bunching will follow in the last article of this series):

P(S,N) = S^N

You'll find range builder software that will give you answers to exactly what a particular range will translate to in terms of cards. I will not claim to have the final answer, but a "15% range" (meaning the bottom being S=0,85 the way we defined it) might look like this:

The bottom of this range, which might for instance be the hand QJo or so, is only 27% likely to be the best if opened from UTG (under-the-gun, the first position) 9-handed. Opening on the button with only the two blinds left to act, it is about 72% to be best, following our simple model.

This is very simplified, since playability and not only raw hand strength decides which hands to open and not. In any case it is pretty stunning how position affects the likelihood you actually have the best hand, right? Of course your opponent knows this as well, which might make the UTG bluff more effective than the button bluff... Trying to approach that mathematically will be for another article at a much later time, though - let's just say that it is a clear mathematical bound to how many hands you can open for a profit. Let me know if you think my approach here was too simplistic (or too complicated, for that matter).

Anyway, the calculations here are not intended to really give you formula to give you a correct range of hands to play for each position. They are however intended to show you that it is clearly correct to play a significantly different range from the first and last position. Missing how large this difference is supposed to be, is one of the major weak points of beginner players. After the flop UTG will also often play out of position (act before anyone but the blinds), which should also strengthen the requirement to open UTG or other early positions.

And then you might be thinking: If I follow this "recipe", everyone will know exactly what I am playing from every position? Yes and no, and we'll try to touch on this in a later article about balance and board coverage. The short answer is that we should make sure to include some less obvious hands and that beyond that it does not really matter, in fact if you play 100% mathematically sound you could inform your opponent about your entire strategy and still the best they could would be to break even against you.

Pot odds

Pot odds is simply a term for the price you are getting laid by a bet. You can compare this to your assumed equity in the pot to make a judgement to continue or not. So

Pot odds = Bet amount / Total pot after the bet is called

If our equity is higher than the pot odds, we can call (assuming no further action possible, or at least that we realize our equity exactly). In the previous example, our equity was around 17%. If we assume our opponent is betting all-in on the turn here (and that our J high never beats his bluffs), we can calculate the maximum bet we can call directly from the above formula when the bet is X times the pot before the bet:

17% >= Bet / Total pot

or

17% >= (X*Pot before) / (Pot before*(2X + 1)

which can be written

17% >= X/(2X + 1)
=> X <= 0.25

In other words, with this equity, we can call the bet given the assumptions above if the bet is less than 25% of the pot.

The expression X / (2X + 1) can be used directly to compute a few pot odds that can be useful to know by heart. To call a bet of half pot, you need to be good 25% of the time, pot sized leads to 33% and twice the pot to 40%. As the bet gets larger and larger, the needed equity will approach 50% as most of the pot to win will be in the bet and less already in the pot.

You'll sometimes find pot odds expressed in ratios, like 3 : 1, but personally I find it easier to think about total pot and percentages directly, rather than going via ratios such as the one mentioned for a half pot sized bet.

Basic poker probabilities

Poker probabilities can be done naively even at the tables, just counting the cards and doing the percentages in your head. I still find this kind of fun and sort of effective, even if I will show you some useful shorthand methods below.

What I mean by naively is simply to divide the cards you believe will make you win the pot by the total cards you have not seen. Look at this example turn situation, where you are last to act with a bet before you:

Hero: J♠T♠
Board: K♦Q♥2♠3♣

If you get an ace or a nine here, you are winning (or splitting) the pot. There are four aces and four nines not accounted for, and 46 unknown cards. Thus, you have 8/46 or a bit over 17% chance of making your hand. 

A shorthand way of computing this is to count the cards, or outs, you have to make your hand. Each out accounts for a roughly 2% chance of making the hand. In this case the "outs shorthand" would indicate 16%, which is probably close enough in most cases. If you are guaranteed to see two cards, for instance when your opponent is all-in on the flop, you can use 4% (which is simply 2% twice, or a rough shorthand for 1/47+1/46, which is about 4,3%).

The term usually used in poker for this "chance of winning" is pot equity. With that in place, we are ready to look at two other basic concepts, namely pot odds and later implied odds.

7 gems of poker magic (or at least math)

I've always enjoyed the game of poker. And I've always loved math. But it is not until recently[1] I have begun to combine the two much. Doing this, I did not really find a simple tutorial suited for me (what I found was either way too basic, way too advanced or did not read really well). I also use the word tutorial for what I was looking for on purpose, even when I am trying to explain math concepts, I hope to do so in a very practical way.

For some of you this set of articles is still going to be too basic, but I also have an ambition to come back with some more advanced pieces later (for instance covering stuff I am discovering when playing around with poker bots and lately reading the old, but still good, Mathematics of Poker book), given I reach that level and also given that I find the time.

Anyway, on to what my so called 7 gems of poker math will cover:

- Basic poker probabilities
- Pot odds
- Open or not - do I have the best hand from this position?
- Implied odds and EV estimation
- Counting combos
- Basic and badass understanding of blockers
- Balance and board coverage

All the content will be for NLH (No Limit Texas Hold'em). I will not explain core poker rules or basic probabilities in the articles, but I will go through the example step by step with a certain degree of hand-holding (no pun intended). I hope I got the balance right and that some of you will enjoy it. For sure it will be a useful exercise for me to structure my thoughts enough to write out the pieces.

Hope this proves useful for someone, please shout out if you have comments or other feedback.

[1] - Well, this set of articles have been in a rough draft state for about five years, so it has been a little while now (and in fact other books have inspired my game more than the good old MoP mentioned above, but I decided to keep it and rather write a separate post on books later). Inspired by making day 2 of the WSOP Main Event at GG yesterday (after double-satelliting in, first USD50 into a USD500 satellite and then from the latter into the actual event), I decided that it would be better to get it out rather than waiting to find time to make it "so, so much better". So here it is!

The cereal entrepreneur

I follow the blog of Seth Godin and admire this thoughts on marketing etc. almost daily. I particularly liked the piece today called The cereal entrepreneur.

This section is worth remembering, I think: 

Every dollar not spent is a dollar you don’t need to raise. Eat cereal, not sushi.

Kind of relevant to us, as we just raised money for our product company (article in Norwegian). We'll continue to eat cereals and work hard. 

Hats off to Microsoft

This is a blog post I kind over never thought I'd write, as I thought of MS being stuck in the old world with Office products, Sharepoint and struggling even with the latest Windows versions. As an example I was testing Azure two years ago and it was very clunky and not really a serious contender to Amazon's AWS feature-wise.

Fast forward to today. Nadella's MS seems to be very real about being on all kinds of devices and Azure is elegant and efficient - I'd say very close to being on-par with AWS (which I also love, by the way), even for us that is basically mainly a Linux-based shop on the server side. And locally we see a quite new humbleness in the organization of the giant and a willingness to work with and help startups.

So hats off to Microsoft. Well done. Be sure to keep on improving.

Full disclosure: My firm has received some free usage of Azure via MS's Bizspark Plus program and this is the way I got to use Azure more again, but the sense of radical improvement is very real.

Startup valuation hype cycle

There is a lot of discussion in the startup community these days about if we are on the top of a hype or close to seeing a bubble burst. From my point of view, running a bootstrapped startup, it is not feeling like VCs are throwing seed money at you. At least not in Norway. But valuations for the "Unicorns", which are truly not common to see just like their fairytale counterparts - even if we forget it when we use them as samples and idols, are surely high.

Anyway, I wanted to share a super fun tongue-in-cheek "press release" from Basecamp. The entire thing reads well and if you are like me, it will give you many giggles, but this is my favorite part:

In order to increase the value of the company, Basecamp has decided to stop generating revenue. “When it comes to valuation, making money is a real obstacle. Our profitability has been a real drag on our valuation,” said Mr. Fried. “Once you have profits, it’s impossible to just make stuff up. That’s why we’re switching to a ‘freeconomics’ model. We’ll give away everything for free and let the market speculate about how much money we could make if we wanted to make money. That way, the sky’s the limit!”

Sounds like a brilliant plan!