More Texas Holdem Analysis in C#: Part 1

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Introduction

In December 2005, I published "Fast, Texas Holdem Hand Evaluation and Analysis" which was my first Texas Holdem Analysis article. Since then, I've done a lot of enhancements and extensions to the code. I've also had several folks more skilled in poker analysis offer their advice. This update to the poker library offers many new and unique features that make analysis of Texas Holdem much easier. In this article, you will find the following topics covered:

• Pocket Query Language - This offers a straightforward set of methods that allow you experiment with different pocket hand combinations without rewriting your code.
• Outs and draw methods - Determining outs and draws is key to analyzing a situation. This section discusses techniques for doing just that.

Part 2 will cover these topics:

• Monte Carlo analysis - Sometimes quick estimates are good enough. Monte Carlo Analysis makes it possible to make high quality estimates in a very short time.
• Multi-player hand analysis - More often than not, you are playing against multiple players. Here's how to calculate win odds against multiple players in a reasonable amount of time.
• Multiple core support - Using multiple cores to decrease calculation time. The speed ups using this technique are nearly linear when adding cores.

Pocket Query Language

If you are reading this article, I'm sure you've read a book or two on Holdem. One of the things I've noticed while reading poker books is that there is a de facto standard for describing pocket cards. Most books use some variant of the following to describe pocket hands.

Most poker software that parses string representations of pocket hands only supports the first item in the de facto standard. I've implemented a much richer query language. I support all of the common poker book syntax, plus some extensions.

I've also added some operators.

Why a Pocket Query Language

One of the things I've done too many times has been to write code to analyze specific match-ups. For example, have you ever wondered what the advantage is to having a suited connector versus a non-suited connector? I'd guess that you've probably wondered, but weren't enough of a masochist to write the code. On the other hand, I am enough of a masochist to write the code for many, many match-ups. After awhile, I decided I'd had enough of that and wrote a query language so that I could write my match-up analysis once and just put in query strings. The following is an example of the result of using query strings rather than hard coded match-ups. Oh, and there is about a 5% advantage for suited connectors.

Using Pocket Queries in code

I've attempted to make it trivial to write analysis code that utilized Pocket Queries. Here's an example.

using System;
using System.Collections.Generic;
using System.Text;
using HoldemHand;

namespace ConsoleApplication1
{
    class Program
    {
        static void Main(string[] args)
        {
            // A Pocket Query Returns an array of all
            // hands that meet the criterion.
            ulong[] player1 = PocketHands.Query("Connected Suited");
            ulong[] player2 = PocketHands.Query("Connected Offsuit");

            // Holds stats            
            long player1Wins = 0, player2Wins = 0,
                ties = 0, count = 0;

            // Iterate through 10000 trials.            
            for (int trials = 0; trials < 10000; trials++)
            {
                // Pick a random pocket hand out of 
                // player1's query set                
                ulong player1Mask = Hand.RandomHand(player1, 0UL, 2);

                // Pick a random pocket hand for player2                
                ulong player2Mask = Hand.RandomHand(player2, player1Mask, 2);

                // Pick a random board                
                ulong boardMask 
                    = Hand.RandomHand(player1Mask | player2Mask, 5);

                // Create a hand value for each player
                uint player1HandValue = 
                    Hand.Evaluate(boardMask | player1Mask, 7);
                uint player2HandValue = 
                    Hand.Evaluate(boardMask | player2Mask, 7);

                // Calculate Winners                
                if (player1HandValue > player2HandValue)
                {
                    player1Wins++;
                }
                else if (player1HandValue < player2HandValue)
                {
                    player2Wins++;
                }
                else
                {
                    ties++;
                }
                count++;
            }
            // Print results
            Console.WriteLine("Player1: {0:0.0}%",
                (player1Wins + ties / 2.0) / ((double)count) * 100.0);
            Console.WriteLine("Player2: {0:0.0}%",
                (player2Wins + ties / 2.0) / ((double)count) * 100.0);
        }
    }
}

Notice that I've added a new class to the Holdem library. It's called PocketHands. Here are some simple examples of how to use this class:

The simplest way to use this class is to iterate through all possible pocket hands for the specified Pocket Query.

// This will iterate through all the possible "connected suited" pocket hands
foreach (ulong pocketmask in PocketHands.Query("Connected Suited"))
{
    // Insert calculation here.
}

Another way to use this class is to iterate through a Pocket Query given a specific hand match-up.

// Looks at an AKs match up (specifically As Ks) against all possible 
// opponents hands that are connected and suited.
ulong mask = Hand.Evaluate("As Ks"); // AKs
foreach (ulong oppmask in PocketHands.Query("Connected Suited", mask))
{
    // Insert calculation here.
}

This example loops exhaustively through two specific Pocket Query match-ups.

// Iterates through all possible "Connected Suited" versus
// "Connected Offsuit" match ups.
foreach (ulong playermask in PocketHands.Query("Connected Suited"))
{
    foreach (ulong oppmask in PocketHands.Query(
        "Connected Offsuit", playermask))
    {
        foreach (ulong board in Hand.Hands(0UL, playermask | oppmask, 5))
        {
            // Insert Calculation Here
        }
    }
}

It's also possible to use pocket queries while doing random samples of match-ups.

// Randomly selects 100000 possible hands when player starts with a
// suited connector
ulong[] masks = PocketHands.Query("Connected Suited");
for (int trials = 0; trials < 100000; trials++)
{
    // Select a random player hand from the list of possible 
    // Connected Suited hands.
    ulong randomPlayerHandMask = Hand.RandomHand(masks, 0UL, 2);
    // Get a random opponent hand
    ulong randomOpponentHandMask = Hand.RandomHand(randomPlayerHandMask, 2);
    // Get a random board
    ulong boardMask = 
        Hand.RandomHand(randomPlayerHandMask | randomOpponentHandMask, 5);
    // Insert evaluation here
}
    

Outs and draws

Most Hold'em players know what outs are. According to Wikipedia:

[A]n out is any unseen card that, if drawn, will improve a player's hand to one that is likely to win

The question for the programmer is, "Is this definition sufficient to write a function that returns the cards that are outs?" Let's look at the two key points made by Wikipedia. They are:

1. The hand must improve.
2. The new hand is likely to win.

Let's start by writing a function that meets the first criterion.

using System;
using HoldemHand;

// A first try at calculating outs
namespace ConsoleApplication1
{
    class Program
    {
        static void Main(string[] args)
        {       
            string pocket = "As Ac";
            string board = "Kd 8h 9c";
            // Calcuate the outs
            ulong outsmask = 
                Outs(Hand.ParseHand(pocket), Hand.ParseHand(board));

            Console.WriteLine("[{0}] {1} : Outs Count {2}",
                    pocket, board, Hand.BitCount(outsmask));

            // List the cards
            foreach (string card in Hand.Cards(outsmask))
            {
                Console.Write("{0} ", card);
            }
            Console.WriteLine();
        }

        // Return a hand mask of the cards that improve our hand
        static ulong Outs(ulong pocket, ulong board)
        {
            ulong retval = 0UL;
            ulong hand = pocket  board;

            // Get original hand value
            uint playerOrigHandVal = Hand.Evaluate(hand);

            // Look ahead one card
            foreach (ulong card in Hand.Hands(0UL, hand, 1))
            {
                // Get new hand value
                uint playerNewHandVal = Hand.Evaluate(hand  card);

                // If the hand improved then we have an out
                if (playerNewHandVal > playerOrigHandVal)
                {
                    // Add card to outs mask
                    retval = card;
                }
            }

            // return outs as a hand mask
            return retval;
        }
    }
}

Passing this starting hand A♠ A♣, K♦ 8♥ 9♣ into our new method returns the following outs:

1. K♠, 9♠, 9♥, 8♠, K♥, 9♦, 8♦, K♣, 8♣ - Two pair
2. A♥, A♦ - Trips
3. Q♠, J♠, T♠, Q♥, J♥, T♥, Q♦, J♦, T♦, Q♣, J♣, T♣ - Improves Kicker

I think most people would agree that super-sizing your kicker probably doesn't help much here. I think most people would also agree that improving the board doesn't help either. So, let's add two more rules:

1. The hand must improve.
2. The new hand improvement must be better than an improved kicker.
3. The new combined hand must be stronger than the just the board.
4. The new hand is likely to win.

The following example handles these new rules and allows opponent hands to be added.

using System;
using HoldemHand;

// A first try at calculating outs
namespace ConsoleApplication1
{
    class Program
    {
        static void Main(string[] args)
        {       
            string pocket = "As Ac";
            string board = "Kd 8h 9c";
            // Calcuate the outs
            ulong outsmask = 
                Outs(Hand.ParseHand(pocket), Hand.ParseHand(board));

            Console.WriteLine("[{0}] {1} : Outs Count {2}",
                    pocket, board, Hand.BitCount(outsmask));

            // List the cards
            foreach (string card in Hand.Cards(outsmask))
            {
                Console.Write("{0} ", card);
            }
            Console.WriteLine();
        }

        // Return a hand mask of the cards that improve our hand
        static ulong Outs(
            ulong pocket, ulong board, params ulong [] opponents)
        {
            ulong retval = 0UL;

            // Get original hand value
            uint playerOrigHandVal = Hand.Evaluate(pocket  board);

            // Look ahead one card
            foreach (ulong card in Hand.Hands(0UL, board  pocket, 1))
            {
                // Get new hand value
                uint playerNewHandVal = Hand.Evaluate(pocket  board  card);
               
                // Get new board value
                uint boardHandVal = Hand.Evaluate(board  card);

                // Is the new hand better than the old one?
                bool handImproved = 
                    playerNewHandVal > playerOrigHandVal &&
                    Hand.HandType(playerNewHandVal) > Hand.HandType(
                    playerOrigHandVal);

                // This compare ensures we move up in hand type.
                bool handStrongerThanBoard = 
                    Hand.HandType(playerNewHandVal) > Hand.HandType(
                    boardHandVal);

                // Check against opponents cards
                bool handBeatAllOpponents = true;
                if (handImproved && handStrongerThanBoard && 
                    opponents != null && opponents.Length > 0)
                {
                    foreach (ulong opponent in opponents)
                    {
                        uint opponentHandVal = 
                            Hand.Evaluate(opponent  board  card);
                        if (opponentHandVal > playerNewHandVal)
                        {
                            handBeatAllOpponents = false;
                            break;
                        }
                    }
                }

                // If the hand improved then we have an out
                if (handImproved && handStrongerThanBoard && 
                    handBeatAllOpponents)
                {
                    // Add card to outs mask
                    retval = card;
                }
            }

            // return outs as a hand mask
            return retval;
        }
    }
}

The problem with outs calculation is that you often don't know the opponent cards you are up against. That makes this calculation a bit subjective. I've had many discussions with different folks about this. One of the more interesting discussions was with Matt Baker. He rewrote my outs function (shown above) to include a heuristic that tries to more accurately account for opponent cards. I won't go into that here, but you can use his code. He generously provided OutsDiscounted and OutsMaskDiscounted.

static int OutsDiscounted(ulong player, ulong board, params ulong[] opponents)

The OutsDiscounted function returns the number of outs for the specified player's pocket hand, the current board and optionally the opponent pocket hands.

ulong OutsMaskDiscounted(ulong player, ulong board, params ulong[] opponents)

The OutsMaskDiscounted function returns a card mask of all of the cards that are outs. You can turn this mask into a string by calling MaskToString().

Draw variation

I've had several folks contact me about other functions that focus on specific outs calculations. These are referred to as "draw" calculations. Wesley Tansey requested several variants of draw functions and offered to test them for me. The result is the following set of functions.

StraightDrawCount

The StraightDrawCount method returns the number of straight draws that are possible for the player, board and dead card configuration. It also filters the results so only player hand improvements are counted.

public static int StraightDrawCount(ulong player, ulong board, ulong dead)

IsOpenEndedStraightDraw

The IsOpenEndedStraightDraw function returns true if the combined mask is an open-ended straight draw. Only straight possibilities that improve the player's mask are considered in this method.

public static bool IsOpenEndedStraightDraw(ulong pocket, 
    ulong board, ulong dead)

IsGutShotStraightDraw

The method IsGutShotStraightDraw returns true if the combined cards contain a gut shot straight draw.

public static bool IsGutShotStraightDraw(ulong pocket, 
    ulong board, ulong dead)

IsStraightDraw

The method IsStraightDraw returns true if the combined cards contain a straight draw.

public static bool IsStraightDraw(ulong pocket, ulong board, ulong dead)

IsOpenEndedStraightDraw

The method IsOpenEndedStraightDraw returns true if the combined cards contain an open-ended straight draw.

public static bool IsOpenEndedStraightDraw(ulong pocket, 
    ulong board, ulong dead)

FlushDrawCount

This method counts the number of hands that are a flush with one more drawn card. However, only flush hands that improve the board are considered.

public static int FlushDrawCount(ulong player, ulong board, ulong dead)

IsFlushDraw

This method returns true if there are 4 cards of the same suit.

public static bool IsFlushDraw(ulong pocket, ulong board, ulong dead)

IsBackdoorFlushDraw

This method returns true if there are three cards of the same suit. The pocket cards must have at least one card in that suit.

public static bool IsBackdoorFlushDraw(ulong pocket, ulong board, ulong dead)

DrawCount

This method returns the number of draws that are possible for the specified HandType. It only returns the counts that improve the player's mask, rather than just the board.

public static int DrawCount(ulong player, ulong board, 
    ulong dead, Hand.HandTypes type)

Demo programs

I've included several demo programs and examples in the downloadable project. This is a quick summary of each of the demo applications:

MultiOddsApp

This application graphs several interesting values to allow hand/board scenarios to be scenarios against 1 through 9 opponents. This demo application accepts Pocket Query descriptions for the Player Pocket field. These values include:

• Win - These are the odds that the given specified hand will end up the winning hand.
• Hand Strength - These are the odds that the given hand -- not including any future dealt cards -- is currently the best hand.
• Positive Potential - These are the odds that a currently losing hand will improve to be a winner.
• Negative Potential - These are the odds that a current winning hand will become a loser.

Hand Odds

This application allows a player pocket hand and an opponent pocket hand definition -- using the Pocket Query Language -- to be entered along with a board definition. The resulting odds are returned for a single random opponent. Since this takes the Pocket Query Language as input, it makes it very easy to try all kinds of pocket hand scenarios that would otherwise require a fair amount of programming to determine the answer.

Acknowledgements

• Wesley Tansey - For a lot of help with functions determining different variations of outs.
• Matt Baker - For many discussions on outs and for providing the DiscountedOuts functions.
• Scott Turner - For using and providing feedback on enhancements.
• poker-eval - The core to my evaluation engine.
• ZedGraph - I've used ZedGraph in several of the examples that graph results.
• Malcolm Crowe - Author of the lexer generator/parser generator tools for C# that I used for the Pocket Query Language.

Disclaimer

I could easily have spent another 6 months editing and tweaking the code and examples. I decided that I had spent enough time on this and had a "critical mass" of features available. So, here is my current code, warts and all.

I don't claim to be a great poker player or even a good one, for that matter. To paraphrase: those who can do, those who can't code. I don't recommend taking any poker advice I may have given (or implied to have given) in this article without independent verification.

I'd also like to encourage feedback. Poker programmers appear to be a somewhat secretive lot, for good reason I'm sure. I've learned a lot from the feedback I've been given over the last year or so. Please keep it coming.

History

First released May 2007