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According to the fundamental theorem of poker, we should play any time we think we have the best hand. By my logic then, if there are 4 players at a table, I should play the top 1/4 starting hands, if there are 9 players at a table, I should play the top 1/9 hands, etc.

I know obviously that it is essential to take into consideration position, previous action, how tight the table is, etc etc. But, as a general rule of thumb, just as a launching off point, does my logic hold?

  • On average, you will have what will become the best hand at showdown 1/x of the time. However, not all of those hands will be 1/x top starting hands. – user1934 Nov 4 '16 at 1:44
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Yes, your logic is fine, in fact your Open Raise for positions stats must show that if you are playing balanced. At UTG you should be more tight because you are against 5 players, about 17%, but on SB vs BB you should play 50% of your hands more or less.

Obviously you have to adapt on your opponents, on early position is difficult because so many players are involved... if ALL the table are tight or lose you can adapt, but if there are a mix, players usually just go GTO playing a balanced number of hands. The positions where you can adapt is on Button or SB where less players are about to play.

Another consideration is, you talk about play your top 1/9 but... what does it mean? KTo is better than 65s? well the answer is depend on the stacks. If you are shortstack you would be more interested on play a big cards like KTo, but if you are deep you should play cards that can connect stronger hands.

  • That is very helpful, thank you. In particular the point you raise in your last paragraph, I had not considered. Thanks for sharing. – randouser Nov 5 '16 at 1:06
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As I think you realize and as others have pointed out, this is a sort of impractical approach to looking at starting hands and there's lots of other things to consider, but I suppose it could be worthwhile as an exercise. After all, what does it even mean to "play" a hand?--do you mean open raise? open limp? complete the small blind after everyone else has limped? etc etc.

Let's set some conditions to make things easier. Assume that you are first to act at a table of n players (you're one of the n) and that starting hands are easily rankable from best to worst--with what hands can you say that you're better than 50:50 to have the best hand at the table? Note that the knowledge of other players entering the pot would affect all of this since they are presumably playing the top X% of hands from their position (whatever they have set to X) and everything dissolves into some game theory type stuff.

Start with the case of 3 players. It would be natural to think that if you have a hand at the 33.3rd percentile of hand rankings, you're a favorite to have the best hand. But that's not exactly the case. Since you're at the 33rd percentile, the chances that the other two players are both below that are 0.67*0.67 -> 44.9%, so there's actually a 55.1% chance that either or both have a better hand. The breakeven point is actually 1 - (.5)^(1/2) -> 29.3rd percentile. In general, the formula would be 1 - (.5)^(1/(n-1)). The graph of this will be a little above a graph of 1/n at the points of interest, but like 1/n it decreases and approaches 0 as n gets higher. It turns out 1/n is actually not that bad as a rough approximation though.

As alluded to earlier, though, this doesn't have any precise application except to show that yes, in general you should have a tighter range with more players involved and a wider range with fewer players, and that the difference between adding a 4th player to a 3-person table should have more of an adjustment effect to your range than adding a 9th person to an 8-person table.

  • That is very helpful, though it'll take me a little time to absorb your answer. I really do love the game theory side of it all. :) – randouser Nov 5 '16 at 1:09
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1/x is a number that is only good for basically loose play and opening (UTG)

10 handed loose if you are opening on the Button you should do so a full 1/2 the hands

Learn 10 handed Open Raise Chart

Then 9 handed just treat UTG as UTG + 1 ... (on most charts they are the same anyway)
You can do this all the way down to heads up and UTG is Button

It is just not realistic to memorize charts for different table sizes

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