Bunching in hold'em

In the 2+2 thread "Button play: Number of folds matter?", Mason Malmuth posted the following on 27 August 1997 :

The bunching factor is the idea that as people throw away their hands it tends to leave a remaining deck that is better in "good" cards. Years ago I did some programming on draw lowball and was able to show that the bunching factor did have a noticeable effect once many hands were passed. However, in hold 'em I have always felt that the bunching factor's effect would be insignificant. This is because you are only dealt two cards as opposed to five and many of the hands that you throw away will consist of a "good" card and a "bad" card.

Recently, Wayne Russel did some programming for us and he essentially verified our results. That is despite what you may occasionally read elsewhere, the bunching factor has essentially no effect in hold 'em. Put another way, if you are on the button in a full game and everyone passes, the distribution of hand strength that you will face from the blinds will be approximately the same as the distribution of hand strength you would face if you were on the button in a three handed game.

A few months later JP Massar came along and showed that according to his simulations, there was evidence of bunching in Hold'em ("Bunching in Texas Holdem: Simulation Results", 25 January 1999) :

Depending on your assumptions about how the other players play, there is a 'bunching' effect in Hold 'em, as I will show via simulation results below.

Whether or not the magnitude of this effect is enough to influence how you play is left to the reader, or for further analysis by experts...
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I confirmed this in a separate post in the same thread :

Nice work. I can also confirm the "bunching" effect for a sim I just completed, the assumptions being that the 7 players before the button play S&M Groups 1-5 only.

Probability of button having Group 1-5 hand = 0.181

Probability of button having Group 1-5 hand given all 7 players folded before him = 0.197

Like you, I leave the interpretation of these results to the readers.

So while the "bunching effect" was there, we were hesitant to conclude whether it was exploitable or not. Mason Malmuth has just written a very interesting article in the December 2005 issue of 2+2 Internet Magazine in which he shows how bunching can be effectively used in hold'em :

Last time we looked at why the idea of bunching, as it is normally used, is fairly worthless in hold 'em games. That is if you are at a full table, are in late position, and everyone has passed to you, it does not mean that the chances of running into a strong hand have gone up over what standard probability would dictate. Part of the reason for this is that hands that players normally fold are made up of big and little cards, while other hands which are frequently played are not necessarily made up of just large cards. Today let's look at a couple of different examples.

Suppose you are in late position and hold

3c 3d

There is a raise by a player in early position, and three other players call him. Should you call the two bets cold?

First notice that you almost always need to flop a set if you play to win the pot, and that it is 7½-to-1 to flop a set with a standard 52-card deck. Furthermore, just because you do catch that third trey, it doesn't mean that you have a guaranteed winner. We have all flopped sets and have gotten them cracked, and that's not any fun. So what this means is that your implied odds when you flop a set and win need to be higher than 7½-to-1. I think that 10-to-1 is probably about right.

Notice that in this spot you're likely to get immediate odds from the pot of approximately 5-to-1 since there is blind money in there as well as the other four active players, and one or more of the remaining players, including the blinds, may come. So this means you need to make on average an additional five double-sized bets those times you flop a set and have your hand hold up for this call to be correct. In many games that seems like a tough order to me, so the obvious conclusion is that the pair of treys should quickly hit the muck.

But not so fast. Let's think a little about bunching. Since the initial raiser is in early position, he should have a good hand which probably does not include a trey. Furthermore, by the same argument, none of the callers should hold a trey. Of course a trey could be out there in the discarded hands, but in this situation there are fewer of them than normal. So it seems to me that it is a little more likely to flop your set here than it would typically be. My educated guess is that instead of being 7½- to-1 to catch that third trey, a better estimate is more like 5-to-1.

This means that your implied odds don't need to be 10-to-1 to make this hand playable. Indeed 7-to-1 might be acceptable, and that should be easily achievable those times you make a set and it holds up. So I'm definitely playing any small pair here.
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