August 25, 2020

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.

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