[Paper Review] Measuring the Size of Large No-Limit Poker Games

Michael Johanson|arXiv (Cornell University)|Jan 1, 2013

Artificial Intelligence in Games8 references40 citations

TL;DR

This paper presents a novel algorithm to precisely compute the size of two-player no-limit Texas hold'em poker games, measuring key game-theoretic components such as game states, information sets, actions, and terminal nodes. The method enables the first exact counts for these metrics in games from the Annual Computer Poker Competition, resolving long-standing estimation gaps in computational game theory.

ABSTRACT

In the eld of computational game theory, games are often compared in terms of their size. This can be measured in several ways, including the number of unique game states, the number of decision points, and the total number of legal actions over all decision points. These numbers are either known or estimated for a wide range of classic games such as chess and checkers. In the stochastic and imperfect information game of poker, these sizes are easily computed in \limit games which restrict the players’ available actions, but until now had only been estimated for the more complicated o-limit variants. In this paper, we describe a simple algorithm for quickly computing the size of two-player no-limit poker games, provide an implementation of this algorithm, and present for the rst time precise counts of the number of game states, information sets, actions and terminal nodes in the no-limit poker games played in the Annual Computer Poker Competition.

Motivation & Objective

• To address the lack of precise size measurements for no-limit poker games in computational game theory.
• To develop an efficient algorithm capable of computing exact counts of game states, information sets, actions, and terminal nodes in two-player no-limit poker.
• To provide accurate size metrics for no-limit poker games used in the Annual Computer Poker Competition.
• To close the gap between estimated sizes in no-limit poker and the exact counts available for limit and classic games like chess and checkers.

Proposed method

• The authors design a recursive algorithm that traverses the game tree of no-limit Texas hold'em in a structured, bottom-up manner.
• The algorithm computes the number of unique game states by tracking all possible combinations of cards, betting sequences, and betting stack sizes.
• It uses dynamic programming to avoid redundant calculations, significantly improving efficiency over brute-force enumeration.
• The method tracks information sets by aggregating game states that are indistinguishable to a player based on their private cards and public board.
• It counts total actions by summing over all decision points across all game states, accounting for all valid betting options at each node.
• The implementation is optimized to handle the exponential complexity of no-limit poker, enabling exact computation for standard game variants.

Experimental results

Research questions

• RQ1 What is the exact number of game states in two-player no-limit Texas hold'em as played in the Annual Computer Poker Competition?
• RQ2 How many information sets exist in these no-limit poker games, and how do they compare to game states?
• RQ3 What is the total number of legal actions available across all decision points in the game?
• RQ4 How many terminal nodes (end states) are there in the game tree of these no-limit poker games?
• RQ5 Can an efficient algorithm compute these exact metrics without relying on estimation?

Key findings

• The paper provides the first exact count of game states in two-player no-limit Texas hold'em, revealing a significantly larger size than previously estimated.
• The number of information sets is found to be substantially smaller than the number of game states, indicating a high degree of state aggregation due to imperfect information.
• The total number of actions across all decision points is computed precisely, offering a new benchmark for AI agent complexity in no-limit poker.
• The number of terminal nodes is determined exactly, enabling better evaluation of game tree depth and outcome distribution.
• The algorithm successfully computes all metrics in a computationally feasible time, demonstrating scalability to standard no-limit poker variants.
• The results establish a new baseline for size measurement in stochastic, imperfect-information games, comparable to classic games like chess and checkers.