An Evaluation of Bunching

by Daniel Kimberg | Published: Sep 12, 2003

In late position at a full ninehanded hold'em table, the actions of the players ahead of you dictate what kinds of hands can be played. If five players call, your 10-9 suited may have added value, but stealing with K-4 may not be the best idea. With an informative raise to your right, A-10 can become a big dog, while a maniac's raise might make a reraise a solid play.

When everyone ahead of you has passed, you're basically playing shorthanded. The one qualification is that because you know the players ahead of you didn't like their hands, the distribution of the remaining cards is not completely random. This effect, sometimes described as "bunching," is hard to evaluate without knowing a bit about your opponents. If they are extremely tight, the folding tells you relatively little. If they are the types of players to play any ace, the folds mean the remaining deck is rich in aces.

I've always wondered how big an effect this really is. So, I decided to find out for myself, using a computer simulation. The basic idea behind the simulation is simple. Deal out a large number of random hands. Discard the ones when someone would have entered the pot ahead of you. For the remaining hands, see how your hand fares against the remaining players to act. Compare this to your hand's performance against truly random hands (as in a shorthanded table when you're under the gun). The difference between the two is due to the bunching factor.

Of course, there may be other ways to evaluate bunching, such as actually looking at the distributions of cards directly. This is just one approach, but one that I think will be informative. There are still a few critical details to go over before getting to the results. First, I decided to address the problem from the point of view of a player on the button at a ninehanded table. So, six players have folded, and our hero has only the blinds to worry about. This maximizes the bunching factor while still keeping us in late position. Second, rather than letting our starting cards vary randomly, I set them to particular hands of interest, and reran the whole simulation multiple times for each hand. Third, the estimated win percentages for each hand were averaged across 10,000 deals (not including the ones that had to be discarded).

A final difficult issue is how to decide whether or not any of the earlier-position players would have entered the pot. For this, I relied on Lou Krieger's starting-hand chart from More Hold'em Excellence. Just for fun, I also ran the simulations a second time, using a slight modification – assuming players will also play any ace from any position, but otherwise stick with Krieger's advice.

The table below presents the results for a variety of hands, assuming no information ("SH," as in shorthanded play), assuming a full table playing according to Krieger's chart ("K"), and assuming a full table playing the "Krieger plus aces" strategy ("K+").

We can make a few observations immediately. First, the overall impact of bunching is small. It would be perfectly reasonable to look at this table and decide never to worry about bunching again. Although, it's worth noting that it's comparable to the effect of holding suited cards. A-K offsuit in a bunching situation has slightly better equity than A-K suited shorthanded.

The biggest winners, the big aces (A-K and A-Q), gain about 3 percent in pot equity, probably because they dominate the bunched big cards. The biggest losers, losing in the neighborhood of 2 percent-3 percent equity, are the hands most vulnerable to aces, like the big pocket pairs and big unpaired cards like K-Q. Most of the remaining hands in my survey lost 1 percent-2 percent equity. Equity may not be the best way of looking at things, but the small size of this effect suggests that bunching is unlikely to affect your decision-making drastically.

One thing worth noting is buried in the last column, which shows the percentage of hands folded around to our player (average of K and K+). The percentage is actually very high for pocket aces, reflecting the fact that if you have two aces, the likelihood anyone else will have a playable hand goes down dramatically. This is simply because such a high proportion of playable hands have aces in them. Anyone who's ever whined about not getting any action with pocket aces can let out a big "I told you so."

What can we really learn from showdown percentages? In this case, I think they're most useful not because you actually expect to get all of your chips in before the flop (although you might in a tournament), but just as an indication of the overall impact of bunching. Certainly, it's rarely if ever going to be a decisive factor in how you play your hand preflop.

Of course, there are many more things we could try in this kind of simulation. We could try different positions and different numbers of players, and we could try all 169 possible hold'em starting hands. We could adjust how we simulate the other players' actions and we could certainly be more sophisticated in terms of how we evaluate the impact of bunching. But I think this set of numbers paints enough of a picture to get started, and perhaps suggests focusing on other issues in deciding how to act preflop. If you feel differently, feel free to e-mail me and suggest a follow-up.

For completeness, I should mention a few other things. First, win percentages were calculated counting two-way ties as half a win and three-way ties as one-third of a win. Although counting them as whole wins would make only a miniscule difference, this is nominally more accurate for equity calculations. Second, for each deal, the showdown percentages are exact. That is, the simulator dealt out all possible five-card boards to calculate the equity of the three hands. However, the eight preflop hands were dealt out randomly, with the hope that 10,000 deals would be enough for a representative sample.

Lastly, although I had to do some programming to get these numbers, I didn't do all the hard work myself. The programming for these simulations was made much easier by the "pokersource" project, a free software project for poker hand evaluation and iteration. The code was originally written by Cliff Matthews and is currently maintained by Brian Goetz and Michael Maurer. Pokersource isn't really software you can download and run. It's a resource that makes it easy for other people to write useful software (although it comes with some examples that are pretty useful). Although it's most directly useful to programmers, it's an incredibly valuable resource to the poker community, because it facilitates simulations like this (it took me only a few hours to write the code for these simulations) and the development of other useful poker-related free software. It's released under the GPL, a free software license, and can be downloaded from http://sourceforge.net/projects/pokersource. The code for my own simulations can be found on my web site at www.seriouspoker.com.