A Slave to Variance

I finally put up some equity tables that I’ve had laying around for some time.

• Preflop heads up hold’em equity table.
• Stud/8 third street rankings.

The first table is an exact calculation, and the second table was calculated using Monte Carlo simulations, so it may not represent the absolute ordering.

One of the most common criticisms of PokerStove equity is that it’s not information that the player wants. Equity is just one measure of a hands strength. There are often other measures that you’ll want to consider when playing a hand.

• Hand Value: This is basic every day hand evaluation result. At showdown, you need to determine which hand wins the pot. A hand’s value is usually expressed as “Two pair, Aces and Eights, Jack kicker”, or “Fives Full of Threes”. In software, these expressions are often encoded into a single comparable value. The source code that Steve Brecher has available at his site shows one approach to encoding hand strength into a comparable value.
• Linear Hand Strength: This is a measure which ranks your hand in the range of possible hands. Specifically, all possible hands are evaluated, and ordered, and the measure of the hand is the fraction of all hands which it beats or ties. By convention, this measure is mapped to a value in the range [0,1]. The evaluations are done on absolute hand strength, and don’t take into consideration draws. This is distinct from hand value because it is a relative measure. It is possible for two different hands to have the same linear strength. For example, given a board of [As Ac Kh], both [Ah Kd] and [Ah Ad] have a linear value of 1.0 (the nuts), but they represent different hand values.
• Draw Strength: This is a measure of how likely it is that your hand will become the winning hand. There are two basic approaches to computing this value. The usual method is to “count your outs”, and if you are sufficiently sophisticated possibly discount some of them. This generates a value somewhere in the range of [0,52], which can be normalized into a [0,1] value. The other method is to calculate your hands Positive Pot Potential (PPot) which is described by Denis Papp of the UA Games Group. The difference between these methods is that the counting of outs is more focused on absolute improvement (or improving to beat top pair), whereas PPot computes relative improvement. For example, PPot gives some draw strength to hands like [6c 7h] on a board of [As Ks Qs], because if a 6s or a 7s hits, you’ll improve over hands like [8h 5h] and [9c 2c], which may not be an improvement that you are interested in. But PPot also correctly accounts for outs when a player might overestimate them. With a board of [Kh Ts 3d 2h] a player might count ten outs for [Tc Ts], when in fact if they are behind, they only have one out.
• Preflop Hand Strength: During the preflop round the usual meaning of hand strength, and draw strength aren’t that useful. Without a board, the possibility of a draw is just that only a possibility. And the strength of your hand is greatly affected by the range of hands that your opponent will play. Issues of domination, implied odds, and position greatly affect how strong a hand is preflop. Because of this, creating measures of preflop hand strength has been a topic of great debate. Sklansky and Malmuth have developed the concept of hand groups, and Abdul Jalib created his preflop rankings through simulation.
• All-in Equity: If all betting has finished, and no one can fold, this measure represents the fraction of the pot that you will win on average. This value is independent of the size of the pot, and doesn’t take into consideration any future action, bets, or folds. It does consider all draws and redraws. And if you can put your opponent on a range of hands, then you can determine whether calling an all-in raise is correct or not. In some ways, equity can be viewed as a combination of both linear hand strength and draw strength, and it’s usefulness is directly linked to situations where every hand that is the best will be shown down, and every possible draw will be drawn to.
• Expected Value: This is the Holy Grail of hand measures. Given a game context, the expected value (EV) of specific action represents the average profit or loss over all possible scenarios. Everything you know about a context can be used as input for this measure. Because of this, it is almost impossible to compute with absolute accuracy. But the more information that you can integrate into an estimate of EV, the more precise that measure will be. In some limited situations, very good estimates of EV can be constructed. But with a nearly boundless amount of information (including general experience, specific opponent information, and external meta-game issues) it’s impossible to create a practical definition of EV for all cases. So, just as the Holy Grail will never be found, no computation can ever give you the true EV of a situation.

It takes a bit of work to grok the visual display of the data, but the results are interesting and a bit counter-intuitive. The vertical axis represents hand strength and ranges from .65 to .90. The horizontal axis represents the draw strength running from “flush draw” on the left to “no draw” on the right. If a cell is red, that means folding that hand is wrong in general. If a cell is black, then folding that hand is going to save you bets.

Obviously, you should never fold a flush draw, and that is pretty clear from giant vertical bar on the left. Keep them flush draws, folding them won’t save you any bets.

Notice the horizontal striations in the grid. This effect is directly attributable to linear values flipping from top pair to middle pair, along with kicker values. What this shows, is that when you are raised, it is preferable to hold a hand like second pair good kicker than a hand like top pair bad kicker. You’ll beat the same number of bluffs, but if you’re beat you’ll have more outs. In fact, if you are raised on the turn when holding top pair, bad kicker, you might consider folding. More to the point, if you hold second pair good kicker, you should also consider folding. Again, kicker issues are such that your effective outs are well below the five you might be counting on to make a call/call down profitable.

There are a lot of tough spots in limit hold’em. One of the toughest is dealing with river raises. Your hand is strong enough that you feel that you should bet it for value, but then you get raised. For a lot of players, the default behaviour here is to call the raise. You are usually being offered good odds, so you figure, even if you’re probably beat, the few times you catch a bluff will be enough to balance everything out. There are certainly other factors that come into play here besides the size of the pot. What kind of player the raiser is, as well as the texture of the board play a big role in the decision.

Below is a graph which describes the results of bet/calling the river over a sample of 464 hands. The two strategies compared were “Call” which is the strategy actually employed, and “Fold” which picks the hypothetical option of folding. The line tracks a moving average of the resulting net earn as a function of the hand strength.

One thing that struck me was just how strong your hand needs to be to make more money than folding. In the games sampled, it took a hand strength greater than .87 to make calling down preferable to folding. That corresponds roughly to top pair top kicker on a not-to-threatening board. AKo on a board of KJT32 has a linear strength of .877, beating or tying 87.7% of all possible hands.

When you are faced with a river raise, calling is clear when you have a hand which beats top pair top kicker. If you have a hand which is worse than TP/TK, then you need to strongly consider how likely the player is to be bluffing and pot size. Any legitimate hand that raises your medium hand will almost certainly beat you, so bluff beating becomes the primary facter in determining whether you should call.

I like to think that I’m smart, but one this for sure. A lot of people are a lot smarter than I am. A lotof people are also a lot more hard working than I am. Here are two links to work done by people who are both smarter than I, but also less inclined to laziness:

Optimal Preflop Hold’em: Alex Selby did this work years ago in 1999. I was aware of back then, but I’ve only recently decided to do more than just read the results. The included program generates optimal solutions for various preflop hold’em games which are defined by the command line options. The games are all modeled as a linear program and solved using the simplex method.

Equilibria Solutions: Troels Bjerre Sorensen’s work is much more recent, and I only became aware of it while searching the web for Selby’s. I haven’t had a chance to pick apart the papers (the recent one looking more interesting). It’s more theoretical, so be forewarned.