Flexible Texas Holdem Poker algorithm for calculating winning probabilities using fast Monte Carlo Simulation.

Hopefully comming soon ...

• % Distributions for each of the 10 possible hands
• Improving random hand simulation w/ symmetry
• Python extension w/ Pybind11?

Currently an exculsively Texas Holdem probability calculator, using random (MonteCarlo) simulation. Takes advantage of a comprehensive, but compact (10MB), lookup table for fast simulation speed. Accepts any number of player hands, including unknown hands, as well as a community hand. A more detailed explanation of the inner-workings is at the bottom.

Computational time spent running simulations increases linearly with total number of hands. Including more variable cards (less community cards or more random hands) will require more random card generation, creating additional time delay.

Reasonable inputs for 100k simulations take between 1-2 sec. for my (Macbook Pro 2.3 GHz Dual-Core Intel Core i5)

Accuracy of the algorithm has been confirmed through comparisons with other open source projects.

Load the Simulator (Ensure lookup_tablev3.bin is in root, or change path in tools.cpp/read_vect)

#include "simulator.h"
Simulator sim;

Provide poker hand pairs for each player

vector<vector<string>> known_hands = {{"Ad", "Kh"}, {"2c", "7d"}};

Provide between 0-5 cards available in the community hand

vector<string> comm_hand = {"9c"};

Number of MonteCarlo simulations to run

int N = 100000;

Last parameter is optionally including more players, whose hands are unknown (random)

vector<vector<int>> results = sim.compute_probabilities(N, comm_hand, known_hands, 3);

Simulating...

^ Last row represents an average of the unknown hands (theoretically have identical probabilities)

Time elapsed: 1.77971s

Fast poker simulations depend on two variables 1) time spent generating random samples 2) time spent evaluating winner of each sample. MonteCarlo-poker acheives both at linear time.

For a single simulation, random cards fill in the remaining community cards and random hands. Thus, it boils down to being able to efficiently generating random n (52 - cards from community and holes) choose k (# of random cards needed) combinatorials with no replacement.

Unfortunately I found no online solution, so I developed my own. The procedure involves taking enough repitition of each of the n available cards and shuffling them into a random sequence. MonteCarlo Poker then iterates along the sequence to generate the samples, keeping track of the placement of each of the n unique cards to ensure each samples contains unique cards.

The efficiency of this sequence-based algorithm in this use-case is mostly thanks to the fact that the sequence contains at most ~100 excess cards that are not used by the simulation, which is negligble for large N. In particular, if L = ceil(N x k) /n, MonteCarlo-Poker generates a n x (L + 1)-length random sequence.

MonteCarlo poker performs 21 vector lookups (7 choose 5 cards) to determine the strength of a hand. Transitions between each 5 card combination are also optimized, with inspiration from Heap's algorithm.

The lookup table used is a single array of size 2.6M (52 choose 5 cards), mapping every 5 card combination to a unique "score". A combinatorial number system is leveraged, allowing for a dense one-to-one mapping of each combinations to a unique index in the table from 0 - 2598960 (52 choose 5).

Each score is calculated such that the lowest score represents the worst hand, all the way ranging to the highest score representing the best hand. Thus, these scores can be used to quickly determine the best 5-card combination of a 7-card hand. The maximum score is approximately 260M, so all scores can be stored in an int32.