Introduction
Thirty hands an hour.
When you're dealing a real-money game of No Limit Texas Hold'em, that's what you're shooting for. Thirty hands an hour, or roughly one hand every two minutes, like clockwork. Two minutes to wash, shuffle, cut the cards, deal them out in two concentric rings. Two minutes to gently coax a table of beer-drinking adults through four rounds of betting, keeping your sanity in the midst of two and three-way all-ins, split pots and side pots, pot bets and string bets, kills, half-kills, straddles, and other inventions of bored-out-of-their-mind poker players.
"Run it twice, dealer."
After a while, it becomes automatic. You get to where you can eyeball a stack of chips and tell it's one chip short of twenty. You get to where you can cut three, five, eight chips from a stack in one clean motion. Your dealing improves, as your hands learn the curt yet unhurried, almost lazy rhythm of high-throughput dealing. Wash, shuffle, shuffle, shuffle, cut, don't forget the deck guard, go. Two months before you were sending cards flying over the table rails in curlicues, now you're tossing them with pinpoint accuracy.
Like shuriken.
The rituals of the poker table are a kind of language, and as a fluent speaker of this language, you don't stop and think about what the individual words mean, or whether the grammar is correct. The words and the meaning they convey are one and the same. To hear the words is to understand them, but at the poker table, your "words" are bets, actions, the casual tossing of two useless cards into the muck, the spill of meaningful color we call the flop, the turn, the river, whatever. Symbols. Grammar. Inflection. Punctuation.
Language.
Think of a Texas Hold'em hand as a seven-letter word built from a 52-card alphabet. Consonants and vowels. In combination, they have meaning. One of your jobs as a poker dealer is to figure out when one person's "word" is better than another person's, and push the pot accordingly. As a poker player, you have to take things a step further. You have to understand the word. Not just its specific meaning, its denotation; but its connotations, associations, nuance, history. And you have to do all of this in the blink of an eye, automatically, without thinking.
It's almost robotic.
Speaking of robots, this is the kind of stuff at which poker robots excel:
- Hand Recognition
- Hand Comparison
- Hand Enumeration
- Hand Evaluation
These terms are often used interchangeably, but I like to give them specific meanings.
- Hand Recognition. Figuring out what a poker hand "is": what hand category it belongs to, the number of outs it has, it's equivalence class, and so forth.
- Hand Comparison. Taking two or more poker hands and determining the winner.
- Hand Enumeration. Enumerating all possibly combinations of cards, whether the cards in a player's hand, or the cards on the board, or both.
- Hand Evaluation. Computing the win/loss percentage or pot equity for two or more hands or hand ranges.
And of course, it goes without saying that robust hand evaluation is the stepping-stone to a mature poker A.I. If your poker bot can't do this...
...it's fighting with one arm tied behind its back.
That, by the way, is a screenshot of Andrew Prock's PokerStove, a poker calculator capable of performing hand vs. hand-range analysis across multiple opponents, using exhaustive exploration or Monte Carlo simulation. There are many such tools out there, but PokerStove is the most popular, and probably the most functional, of the lot. PokerStove is free, so if you haven't already done so, I suggest downloading it and getting familiar with how it works.
Because we're going to teach our poker bots and related tools how to do what PokerStove does.
Subjective All-In Equity (SAIE)
Notice I said hand vs. hand-range analysis. The "range" part is crucial, because in poker, rarely are we able to put our opponents on specific hands. Instead, we assign them a range of hands, and narrow that range as the hand progresses. In order to compute our equity, we have to figure out how well our known hand fares not against another known hand, but against the range of hands we believe our opponent has. This is what's referred to as subjective all-in equity or SAIE.
I like Michael Maurer's definition:
SAIE is a player's pot equity given particular beliefs about the possible hands of the opponent(s) and assuming no further betting.
Provided we follow it with Andrew Prock's note in the PokerStove FAQ:
This is not the chance that a hand will win the pot. Rather it is the fraction of the pot that a hand will win on average over many repeated trials, including split pots. The equity for a hand is calculated by dividing the number of "pots" that the hand won by the number outcomes considered. Because two players can split a pot, a player can win fractional pots. Thus, it is possible for a hand to have non-zero equity despite the fact that it cannot win.
For example, in the above screen shot of PokerStove, we can see that the equity of pocket Jacks versus two opponents is around 36%, if we assign each of those opponents a range of [any pair; any AQ or better]. If you took this situation and played it out a million times, you'd find that on average, you'd win 36% of the pot (assuming no further betting, and assuming nobody folds).
This is a useful number to have.
It takes into account the fact that, if one of your opponent has Aces, and the other has Kings, you're equity is lower—only 15%. It takes into account the fact that, if one opponent has AK (any AK) and the other has AQ (any AQ), your equity is higher—about 44%. SAIE provides a sort of weighted average of how your hand fares against each of the hands in your opponents' range, taking into account the likelihood of each specific opponent holding. You see, it's quite a bit easier for an opponent to have AK (12 combinations) than it is for him to have AA (only 6 combinations). And the mathematics need to allow for that.
Once again: this is not the same thing as your chance of winning the pot! Another example should make this clear.
Here, Player 0's chance of winning the pot is only 11%, but his equity is around 57%. This is because most of the time—whenever a 4 or higher falls on the river—the hand is going to be tied. But when a 3 or 2 falls on the river, Player 0 wins with Four Aces and a Four kicker, which beats Player 1's Four Aces and a Three Kicker.
And although Player 1's pot equity is 44%, he has zero chance of winning the pot outright.
That's a contrived situation, but an illustrative one. By thinking in terms of pot equity, rather than chance of winning the pot, we neatly fold all those messy tie-game situations into our mathematics.
The Need For Speed
Believe it or not, calculating SAIE is easy. Calculating it efficiently may not be.
It all hinges on hand comparison: the ability to compare two or more poker hands and determine the winner. If you can do that, all you have to do is programmatically deal out large numbers of hands and start tallying wins, losses, and ties. When it comes to dealing those hands out programmatically, you have two options:
- Exhaustive Exploration. Enumerate all possible outcomes: all the combinations of cards fitting your opponents' range times all the possibly combinations of board cards. Keep a tally of how many hands each player wins, loses, and draws. Afterwards, when you've enumerated every possibly combination, express these as a percentage.
- Monte Carlo Simulation. Randomly generate hands fitting your opponents' ranges and board cards, keeping a tally of wins, losses, and draws for each player. Over many trials (hands or games), your results will start to converge.
Typically in the game of Hold'em, we use Monte Carlo simulation for preflop matchups, and exhaustive exploration for matchups on the flop or later. You'll notice PokerStove has a radio button allowing you to specify which technique to use although, as Andrew Prock notes:
Monte Carlo simulations work very well for most practical problems, and enumeration driven software is only going to be of interest to the true poker geeks who must have all of their i's dotted and t's crossed. Of course, I'm one of those geeks :)
Whether you're a geek or not, you're going to need throughput: the ability to evaluate large numbers of hands quickly. If you're going to be randomly generating, or consecutively enumerating, millions of hands, you can't afford to do four bubble sorts, nine redundant array copies, a Google search, and send off 3 emails every time you evaluate one of those millions of hands.
You've got to optimize.
(Remember, this is something poker bots and other tools need to do in real time, at multiple tables, on various and sundry hardware.)
So it's got to be fast.
I don't mean 2,000 hands per second fast. That's not fast, that's slow. I mean fast like a bat out of hell. Fast like the wind. Fast like the Flash. Speedy Gonzales. Run Forest Run! fast. Or even—dare I say it—as fast as that paragon of speed and grace, the Roadrunner.
MEEP-MEEP!
And the best part?
This is not a wheel you have to reinvent. Highly-optimized hand evaluation code is already written, and is publically available in multiple languages (C, C++, C#, Java, and Python, among others) with support for multiple poker variants (Hold'em, Omaha, Stud) provided you know where to look.
A Small Army of Poker Hand Evaluators
Some people collect butterflies, leaves, flowers; others collect coins, marbles and shot-glasses.
Me? I collect open-source poker code.
If it's code, and if it's about the game of poker, and if it's freely available, I want a copy. Even if the code is written in a language I'll never use. Even if it's slower, messier, or buggier than my existing code. A lot of clever people have devoted a lot of time and energy to building robust poker hand evaluators, you see, and I want to leverage their hard work.
So every time I stumble across a new specimen, I collect it. I haul it out to the garage, open my toolbox, and start hammering away. I clean up the code, throw it into a Visual Studio project or who knows, I might even crack open my Eclipse IDE and do some Java work. Whatever.
There is no "one size fits all" poker hand evaluator, so I want a hand evaluation library which incorporates multiple evaluators within it, all of them written by other people.
Here are just a few (all of which we'll be covering in detail):
- The Pokersource Poker-Eval library
- The Java enumeration library built on top of Pokersource
- Keith Rule's C# port of the Poker-Eval library
- Cactus Kev's 5-Card Evaluator
- Paul Senzee's Improved 5-Card Evaluator
- Paul Senzee's 7-Card Evaluator
- Moritz Hammer's Directed Acyclic Graph-based 7-Card Evaluator
- Steve Brecher's HandEval routines
- The Two Plus Two 7-Card Evaluator
So instead of thinking of your poker hand evaluator as a single tool, think in terms of building a poker hand evaluation toolset; a collection of code which can be applied to the specific problems you're facing. Even within the relatively narrow context of an online poker bot, there's a lot of wiggle room. Are you building a multi-table online Texas Hold'em poker bot? If so, you need a fast 7-card evaluator. Are you building something which needs to support Hold'em as well as Omaha and 7-Card Stud. You need a good generic evaluator. And so forth.
If you think of poker hand evaluation as a smorgasbord, rather than a single component, you'll be doing yourself a favor.
Conclusion
Now, I know what you're thinking: where's the code? Where are the algorithms?
Well, it turns out that poker hand recognition, comparison, enumeration, and evaluation are such complex topics, it's impossible to fit them all into a single post, and at the same time, provide and explain the source code for a dozen different pre-existing evaluators. So I've broken these off into a set of background articles, to be published shortly:
- Poker Hand Analysis, Part 1: Theory
- Poker Hand Analysis, Part 2: 3rd-Party Evaluators and the XPokerEval library
- Poker Hand Analysis, Part 3: Hand Ranges and Multiple Opponents
Today, I just wanted to introduce the subject, and tell you how you can make some money by writing hand evaluation code.
Coding the Wheel, in conjunction with a site to be named shortly, will be presenting an award of X dollars (we're not quite sure what X will be yet, but it will be hundreds of dollars, rather than thousands or tens) to the person or persons who can produce a substantial improvement to the hand evaluation code we'll be presenting.
In addition, that person's solution will be highlighted, with full credit, or anonymously if preferred, in a dedicated post on Coding the Wheel and possibly some other sites as well.
There are only a few requirements:
- The solution must be unpublished
- The solution must evaluate 7-card poker hands
- The solution must be fast (noticeably faster than current evaluator implementations)
Full rules to follow. Stay tuned!
Posted by James Devlin 70 comment(s)
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