Commentary on Optimal Heads-up Preflop Holdem

by Nick Christenson

This article was previously published in the April 2010 issue of TwoPlusTwo Online Magazine.

Background

This article series has been about gleaning poker insight from academic research on the subject. So far, this research has been in the form of articles in prestigious journals or from highly regarded conferences, and most of these papers have been on the order of a dozen pages in length. This article is a little different. Here, I'll be looking at a Usenet news posting of a few hundred words from more than a decade ago.

It might not seem like much, but this posting has had significant repercussions in the world of analytical poker. I know for a fact that this note has been cited by at least a half dozen refereed papers, it has been referenced in at least three theses, and I know of two books that have cited it. Despite it's casual tone, it is a very well respected piece of work, and has had more influence on poker research than most journal articles.

On May 5, 1999, Paul Hankin posted a message titled "Optimal heads-up preflop holdem" to the rec.gambling.poker Usenet newsgroup. It detailed some work that that English mathematician Alex Selby had performed based on Hankin's prompting. A slightly modified version is archived at http://www.archduke.org/simplex/art, and this is what the citations point to. The original Usenet thread can still be found at Google Groups in case anyone is interested. However, I'll use the one at Selby's archduke.org web site in this discussion.

Introduction

So, what is this posting about? It's an optimal solution to a restricted form of limit Texas hold'em. In this game, we have two players, they post $1 and $2 blinds, the small blind (SB) acts first, both blinds are live, and multiple raises are allowed. The only way in which this game is different from "real" heads-up limit hold'em is that in Selby's game, there's no betting action beyond the flop.

So, yes this is another "toy" poker game for mathematicians, in that it doesn't completely correspond to real poker, but the solution to this toy game has some obvious value for those who play limit hold'em.

Strategy

Selby derives a game theoretical optimum strategy for this restricted form of Texas hold'em. He lists this strategy in three charts. The first one lists the small blind's strategy. The second one lists the big blind's (BB) strategy if the SB calls. The third one lists the BB strategy if the SB raises. The charts themselves are pretty self-explanatory once one examines the key. Note that sometimes the SB initially calls and then reraises if the BB raises. This is the only situation where a call doesn't close the action.

The software used to compute this strategy is still available online. I'd also suggest that interested persons might want to examine the README file referenced at the same location. It contains some good notes about the software and answers some questions about the methodology used to generate the strategy.

The code solves the game by use of the simplex algorithm, a method commonly used in linear programming that is frequently applied to game theory problems. In this case, its use greatly reduces the complexity of the game. The README file referenced above provides a moderately complex mathematical explanation of how and why it is used, and why doing a standard minimax-type matrix would be intractable.

Observations on the Results

Some of the observations I'll mention here were made by the authors, some were made by commentators in the original Usenet thread, and some are my own. In any case, I believe there's a great deal an attentive poker player can learn from this exammple. Here are some of those lessons.

In this game it's correct to play loose.
In this optimum strategy, the big blind never folds to a raise. Why is this? It's because the small blind has a wide range of raising hands, and no hand is a 3 to 1 underdog to this range.

Second, the small blind rarely, but sometimes, folds preflop. Why is it correct for the small blind to play tighter than the big blind? It is because the small blind's call doesn't close the betting. Still, a fold very seldom happens. Only ten out of 169 possible starting hands warrant a pre-flop fold by the small blind. Also, once the small blind at least completes the bet, it is never correct to fold after that point. This is because if the big blind raises, the small blind is facing the same pot odds as before the blind was completed. If it was correct to call then, knowing that there was a good cance the small blind would be facing a raise, it is correct to call the second bet.

What can we learn from this? Well, the more bets (from raises or additional betting rounds) we will face, the more likely we are to want to fold marginal hands early on. Similarly, the more betting rounds, the worse off bad starting hands are. This basic result was also arrived at in Chris and Thomas Ferguson's paper, discussed in a previous article in this series. If we were to extend the game to account for multiple betting rounds, I expect it would be correct to fold a larger number of starting hands at the outset.

There are few mixed strategy cases.
For many games, optimal strategy dictates that we vary our strategy randomly in order to be deceptive. A classical example is rock-paper-scissors. The optimum strategy is to choose each selection 1/3 of the time. In this pre-flop hold'em example, out of over 28,000 possible hand vs. hand situations, in only nine of these is it correct to adopt a mixed strategy.

In our book, Winning Strategies for No-limit Hold'em, Russ Fox and I note that there are two types of deception in poker: (1) Selecting between multiple ways to play a single hand, and (2) Playing different hands in the same way. We suggest that poker players should try to emphasize the latter. This is consistent with the findings of the Shelby posting. However, as the number of possible hands with which we might adopt a given strategy decreases, the more important it is to start randomizing play. This is why most of the cases where a mixed strategy is correct in the Selby game are relatively rare situations, namely, those where it is sometimes appropriate to make multiple raises.

Sandbagging is a part of the optimal strategy.
The small blind sometimes adopts a "call-reraise" strategy. Sandbagging doesn't show up as part of an optimum strategy in many other toy games, especially those with only one betting round. Of course, since reraises are permitted, this game does have the possibility of multiple bets, even if there is technically only one betting round.

Another reason why sandbagging is part of the optimum strategy is that this is one of the rare toy poker games where a call doesn't always close the betting. In order for sandbagging to ever be correct, you need at least one of the following conditions: (1) at least three players, (2) multiple betting rounds, (3) the ability to sometimes call without closing the action. Keep this in mind when examining toy poker games where sandbagging is not part of the optimum strategy. The reason may very well be because none of these three conditions hold.

This strategy includes no obvious bluffing.
The small blind never raises with his worst hands, and the big blind never reraises with her worst hands. Why is this? Well, in part it's because the BB never folds to a raise, and the SB never folds to a raise after there is any action. That doesn't mean there isn't any bluffing, though. Note that it's correct for the SB to raise with a hand like 86off, a hand that has about 43% equity vs. a random hand. That's a bluff.

In poker we want to raise as a bluff with our marginal hands, when we'd call a reraise anyway. However, we want to bluff with our weakest hands when we wouldn't be able to call a reraise. We want to see showdowns with our marginal hands, but we know we're doomed with our weakest hands. Because all hands have significant equity in this game, it's best to bluff with our marginal hands.

The advantage in this game goes to the big blind.
Despite the fact that the big blind has more money invested at the start of the game, the BB has the edge. This is for two closely related reasons, the big blind gets to act second, and the big blind is live. These two factors outweigh the fact that the big blind has to put in more money pre-flop.

How much of the big blind's advantage is due to each of the two factors from which it benefits? I don't know, but one could fairly easily download the software and modify it so that calling by the small blind would close the betting action, then rerun the simulation and look at the value of the modified game. This would be an interesting result if someone were motivated to attempt it.

There are never four raises.
This isn't a restriction that was imposed on the game, it's a result of the optimum strategy. The competitors in this game are allowed to raise as many times as they wish, it's just never proper to go beyond three raises with any hand other than aces. So, once an optimal player puts in the fourth raise, the other player would know that he must have aces.

Conclusions

The posting I have discussed here is a Usenet article consisting of about 350 words not counting the charts. Nonetheless, this examination of a toy version of Texas hold'em contains a lot of wisdom about poker. This posting was made at a time when sophisticated analysis of even restricted forms of a real poker was rarely practiced. I would claim that this was one of the key moments in the history of the application of professional level mathematics to real world poker. I feel confident that most of the pioneers in this endeavor, people such as Darse Billings, Bill Chen, Jerrod Ankenman, Chris Fergusson, and many others were heavily influenced by this particular message. Further, I think it still has something worthwhile to teach us about the game of poker.