Dave Computes Poker Play Frequencies

A super-important problem in poker is determining whether your poker hand is likely to be the best at the table, without knowing what your opponents hold.  In this post, I'll assume that you know absolutely nothing about your opponents' hands, but you know the exact value of your own hand.

For example, suppose you're one of 3 players in a Hold'em poker game.  You expect to have the best hand about 33% of the time.  Suppose you hold K7o preflop and you're first to act (on the button).  Using the spreadsheet I made available in an earlier post, you see that only 32% of all possible hands are more likely to win than your K7o.  I'll therefore refer to K7o as a 32% hand.  Since this is less than 33%, you might assume you have a better-than-even chance of holding the best hand.  Are you correct?  No.  Let's see why.

There is a 68% chance (100% - 32%) that your K7o will beat any single opponent.  That means there's a 0.68 * 0.68 = 46% chance that your hand is better than both of your opponents', and a 54% chance that it isn't.  How strong must your hand be to have a 50% chance of beating both?

Let p be the probability that a single opponent's hand will beat yours.  (In the case of K7o, p = 0.32.)  Then the probability of beating a single opponent is (1 - p).  The probability of beating both opponents is (1 - p)².  Now set this equal to 0.5 and solve for p.

(1 - p)² = 0.5
p = 1 - (0.5)^1/2 = 29.29%

What does that mean for our poker play?  It means we can't raise a K7o for value in this situation, because there's a 54% chance we're just giving that money away to an opponent.  Actually, the situation is not so straightforward.  If I'm on the button, I'll be last to act on all remaining rounds, giving me an edge that probably brings my chances of winning the hand closer to 50%.  (There are a lot of other subtleties to consider here, too.  If both opponents will call me down no matter what, then I have enough pot equity with my better-than-33% hand to raise for value.  However, in a 3-way hand, my K7o goes down in value and is now beaten by 36% of possible hands.)

So, it is reasonable to raise a top-29.29% hand for value in a 3-player game.  Solving
(1 - p)³ = 0.5, we find that it's reasonable to raise a top-20.63% for value in a 4-player game.  Note that this reasoning holds for any poker game.  But you can only apply it when you know the exact value of your hand, as you might in Hold'em preflop play.

Suppose we're on the button in a 4-player Hold'em game, and the first player folds.  Now we're effectively in a 3-player game, so we can raise a 29.29% hand for value.  This explains much of why your position changes the number of hands you can play.  (By the way, my math has convinced me that you should nearly always open with a raise in a game of hold'em.  I rarely open by calling.  When someone does limp ahead of me, I'm more likely to limp in with hands I might have opened with a raise.)

In general, in an n-player game, we can find this threshold by solving (1 - p)^{n - 1} = 0.5 for p.

p = 1 - 0.5^{1/(n - 1)}

Here's what we find.

Number Of Players	Threshold
2	50.00%
3	29.29%
4	20.63%
5	15.91%

Now, suppose you're in a 5-player game of Hold'em.  The player under the gun opens with a raise (which we'll assume is not particularly monstrous).  The next player folds.  You're on the button.  You believe that raiser must have a top-15.91% hand.  You have ATo--a 9.5% hand with one opponent.  Do you reraise?  If we ignore the fact that you have position on the raiser, you should fold.  Why?  Because your opponent holds a 15.91% hand or better--anything from A8o to AA.  More than half of those hands are stronger than your ATo.  To raise, your hand must be in the top 15.91% / 2 = 7.96%.  So, you could raise a KQs (7.84%) or AJo (7.54%), but should fold ATo.

Now suppose you're the original raiser and the player on the button reraises you with what you assume to be a top-7.96% hand.  Using the same logic (and still ignoring position), you should reraise them back (potentially capping or putting yourself all-in at this point) with a top-3.98% hand (7.96% / 2) like 77 or AKs, but should probably call otherwise.

Number Of Players	Raise	Reraise	Cap
2	50.00% (J5s)	25.00% (QTo)	12.50% (A7s)
3	29.29% (J9s)	14.64% (QJs)	7.32% (AQo)
4	20.63% (A3s)	10.31% (A8s)	5.16% (AJs)
5	15.91% (A8o)	7.96% (KQs)	3.98% (77)
6	12.94% (A9o)	6.47% (ATs)	3.24% (88)
7	10.91% (KTs)	5.46% (AKo)	2.73% (99)
8	9.43% (66)	4.71% (AJs)	2.36% (TT)
9	8.30% (A9s)	4.15% (77)	2.07% (JJ)
10	7.41% (AQo)	3.71% (AKs)	1.85% (JJ)

These playing frequencies turn out to be fairly close to those suggested by successful poker players.

Let's use this table to work through one more example.  If you're the first to act among 6 players, then you can raise your top-12.94% hands (ignoring position again), which would be A9o or better in a tight game.  If you're reraised, you can expect your opponent to hold a top-6.47% hand (ATs or better), and you can therefore raise them back with a top 3.24% hand (88 or better).  Note again that we're not taking position into account or, more importantly, the playing styles of your opponents.

Incidentally, if you look at preflop Hold'em play in this way, you're not stealing the blinds--you're raising for value.

All of this begs the question:  How can I determine the strength of my hole cards without constantly turning to a spreadsheet?  I'll address this in my next post.