Body: <p><strong>What is the number of game states/information sets in 6-players, no limit, Texas Holdem?</strong></p>
  <p>A year ago, <a href="https://science.sciencemag.org/content/365/6456/885#:%7E:text=Developing%20a%20superhuman%20AI%20for,poker%20format%20in%20the%20world." rel="nofollow noreferrer">Pluribus</a> reached a super-human level in 6-players no limit Holdem Poker.  I am interested in the size of poker because it is a simple heuristic method to compare the complexity of different games.</p>
  <hr />
  <p>In the paper <a href="https://poker.cs.ualberta.ca/publications/2013-techreport-nl-size.pdf" rel="nofollow noreferrer">Measuring the Size of Large No-Limit Poker Games (2013)</a>, they write</p>
  <blockquote>
  <p>The <strong>size of a game</strong> is a simple heuristic that can be used to describe
  its complexity and compare it to other games, and a game’s size can be
  measured in several ways. The most commonly used measurement is to
  count the number of <strong>game states</strong> in a game.</p>
  <p>...</p>
  <p>In <em>imperfect information games</em>, an alternate measure is to count the number of decision points, which are more formally called <strong>information sets</strong>.</p>
  </blockquote>
  <p>Here's the definition of <strong>game states</strong> and <strong>information sets</strong>.</p>
  <blockquote>
  <p><strong>Game states</strong> are the number of possible sequences of actions by the players or by chance, as viewed by a third party that observes all of the players' actions. In the poker setting, this would include all of the ways that the players private and public cards can be dealt and all of the possible betting sequences.</p>
  <p><strong>Information sets:</strong> When a player cannot observe some of the actions or chance events in a game, such as in poker when the opponent’s private cards are unknown, many game states will appear identical to the player. Each such set of indistinguishable game states forms one information set, and an agent's strategy or policy for a game must necessarily depend on its information set and not on the game state: it cannot choose to base its actions on information it does not know.</p>
  </blockquote>
  <hr />
  <p>Here are the number of game states of certain variants of Poker.</p>
  <ul>
  <li>2-players <strong>no limit</strong> Holdem - un-abstracted game has approximately <span class="math-container">$10^{75}$</span> game states.  <a href="https://poker.cs.ualberta.ca/publications/2013-techreport-nl-size.pdf" rel="nofollow noreferrer">Measuring the Size of Large No-Limit Poker Games (2013)</a></li>
  <li>2-players <strong>limit</strong> Holdem - approximately  <span class="math-container">$10^{18}$</span> game states. Abstracted game has <span class="math-container">$10^7$</span> game states.  <a href="http://webdocs.cs.ualberta.ca/%7Edarse/Papers/IJCAI03.pdf" rel="nofollow noreferrer">Approximating Game-Theoretic Optimal Strategies for Full-scale Poker (2003)</a>. </li>
  <li><strong>6-players no limit</strong> Holdem - ?  -  in the publication of Pluribus that won professional poker players they do not state the size of the game. <a href="https://science.sciencemag.org/content/365/6456/885#:%7E:text=Developing%20a%20superhuman%20AI%20for,poker%20format%20in%20the%20world." rel="nofollow noreferrer">Superhuman AI for multiplayer poker (2019)</a></li>
  </ul>
  
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-01-23T19:04:09.870Z
Comments.Text: one or two points - (1) the paper you cite considers a number of versions of Poker. It stands to reason that the number of possible bets per turn (mediated by stack size and blinds) would be important regarding branching factor. Hence with $20k stack size the game has >10^164 states. (2) Maybe I'm being stupid but is it possible that the generalization for game states is quite close to simply (Num Game States for 2-Players)^(6/2)? (If we assume the vast majority of cards remain in the deck unused on a single hand)
Comments.UserDisplayName: Mobeus Zoom
Comments.ContentLicense: CC BY-SA 4.0
Comments.CreationDate: 2022-01-24T15:14:53.900Z
Comments.Text: (1) Pularius discretized betting into buckets (bet size of 1/2 pot, pot, 2 pots...).
  (2) I think more.  Since each of the 6 players can raise and force another round where each of the other 5 players face again with all the 10K betting options (fold, call, rerise(X)).
Comments.UserDisplayName: Cohensius
Tags: <game-ai><game-theory><games-of-chance><poker><state-space-complexity>
Title: What is the size of 6-players no limit Texas holdem Poker?
UserDisplayName: Cohensius
ThreadText: # What is the size of 6-players no limit Texas holdem Poker?
  
  (Asked by: username_0 on 2021-03-02)
  
  **What is the number of game states/information sets in 6-players, no limit, Texas Holdem?**
  
  A year ago, [Pluribus](https://science.sciencemag.org/content/365/6456/885#:%7E:text=Developing%20a%20superhuman%20AI%20for,poker%20format%20in%20the%20world.) reached a super-human level in 6-players no limit Holdem Poker. I am interested in the size of poker because it is a simple heuristic method to compare the complexity of different games.
  
  ---
  
  In the paper [Measuring the Size of Large No-Limit Poker Games (2013)](https://poker.cs.ualberta.ca/publications/2013-techreport-nl-size.pdf), they write
  
  > The **size of a game** is a simple heuristic that can be used to describe
  > its complexity and compare it to other games, and a game’s size can be
  > measured in several ways. The most commonly used measurement is to
  > count the number of **game states** in a game.
  >
  > ...
  >
  > In *imperfect information games*, an alternate measure is to count the number of decision points, which are more formally called **information sets**.
  
  Here's the definition of **game states** and **information sets**.
  
  > **Game states** are the number of possible sequences of actions by the players or by chance, as viewed by a third party that observes all of the players' actions. In the poker setting, this would include all of the ways that the players private and public cards can be dealt and all of the possible betting sequences.
  >
  > **Information sets:** When a player cannot observe some of the actions or chance events in a game, such as in poker when the opponent’s private cards are unknown, many game states will appear identical to the player. Each such set of indistinguishable game states forms one information set, and an agent's strategy or policy for a game must necessarily depend on its information set and not on the game state: it cannot choose to base its actions on information it does not know.
  
  ---
  
  Here are the number of game states of certain variants of Poker.
  
  * 2-players **no limit** Holdem - un-abstracted game has approximately $10^{75}$ game states.  [Measuring the Size of Large No-Limit Poker Games (2013)](https://poker.cs.ualberta.ca/publications/2013-techreport-nl-size.pdf)
  * 2-players **limit** Holdem - approximately $10^{18}$ game states. Abstracted game has $10^7$ game states.  [Approximating Game-Theoretic Optimal Strategies for Full-scale Poker (2003)](http://webdocs.cs.ualberta.ca/%7Edarse/Papers/IJCAI03.pdf).
  * **6-players no limit** Holdem - ?  -  in the publication of Pluribus that won professional poker players they do not state the size of the game. [Superhuman AI for multiplayer poker (2019)](https://science.sciencemag.org/content/365/6456/885#:%7E:text=Developing%20a%20superhuman%20AI%20for,poker%20format%20in%20the%20world.)
  
  **Tags:** game-ai, game-theory, games-of-chance, poker, state-space-complexity
  > **Comments:**
  > **username_1**: one or two points - (1) the paper you cite considers a number of versions of Poker. It stands to reason that the number of possible bets per turn (mediated by stack size and blinds) would be important regarding branching factor. Hence with $20k stack size the game has >10^164 states. (2) Maybe I'm being stupid but is it possible that the generalization for game states is quite close to simply (Num Game States for 2-Players)^(6/2)? (If we assume the vast majority of cards remain in the deck unused on a single hand)  
  > **username_0**: (1) Pularius discretized betting into buckets (bet size of 1/2 pot, pot, 2 pots...). (2) I think more.  Since each of the 6 players can raise and force another round where each of the other 5 players face again with all the 10K betting options (fold, call, rerise(X)).  
  
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  ## 0 answers
  
  