Title: KrwEmd: Revising the Imperfect Recall Abstraction from Forgetting Everything
Venue: NeurIPS 2024
Accepted: no
Keywords: game theory

imperfect-information games

games with ordered signals

computer poker

imperfect-recall abstraction
Abstract: Recent research has shown that an extreme interpretation of imperfect recall abstraction —- completely forgetting all past information —- has led to excessive abstraction issues. Currently, there are no hand abstraction algorithms that effectively integrate historical information. This paper aims to develop the first such algorithm. Initially, we introduce the KRWI abstraction for Texas Hold'em-style games, which categorizes hands based on K-recall winrate features that incorporate historical information. Statistical results indicate that, in terms of the number of distinct infosets identified, KRWI significantly outperforms POI, an abstraction that identifies the most abstracted infosets that forget all historical information. Following this, we introduce the KrwEmd algorithm, the first hand abstraction algorithm to effectively use historical information by combining K-recall win rate features and earth mover's distance for hand classification. Empirical studies conducted in the Numeral211 Hold'em environment show that under identical abstracted infoset sizes, KrwEmd not only surpasses POI but also outperforms state-of-the-art hand abstraction algorithms such as EHS and PAEMD. These findings suggest that incorporating historical information can significantly enhance the performance of hand abstraction algorithms, positioning KrwEmd as a promising approach for advancing strategic computation in large-scale adversarial games.

Abstract

A recent research has shown that an extreme interpretation of imperfect recall abstraction - completely forgetting all past information - has led to excessive abstraction issues. Currently, there are no hand abstraction algorithms that effectively integrate historical information. This paper aims to develop the first such algorithm. Initially, we introduce the KRWI abstraction for Texas Hold'em-style games, which categorizes hands based on K-recall winrate features that incorporate historical information. Statistical results indicate that, in terms of the number of distinct infosets identified, KRWI significantly outperforms POI, an abstraction that identifies the most abstracted infosets that forget all historical information. Following this, we introduce the KrwEmd algorithm, the first hand abstraction algorithm to effectively use historical information by combining K-recall win rate features and earth mover's distance for hand classification. Experimental studies conducted in the Numeral211 Hold'em environment show that under identical abstracted infoset sizes, KrwEmd not only surpasses POI but also outperforms state-of-the-art hand abstraction algorithms such as Ehs and PaEmd. These findings suggest that incorporating historical information can significantly enhance the performance of hand abstraction algorithms, positioning KrwEmd as a promising approach for advancing strategic computation in large-scale adversarial games.

1 Introduction

Imperfect recall abstraction has proven to be very important for solving large-scale computational games, significantly reducing computational complexity. Recently, AI using imperfect recall abstraction has developed better-than-human strategies for Texas Hold'em testbed-even when using limited computational resources [23, 7, 8].

The task of hand abstraction in Texas Hold'em aims to re24 duce computational overhead by applying the same strategy 25 to similar hands. In an imperfect recall setting [29, 20], the 26 hand abstraction in the later phase does not strict depend 27 on the results of the hand abstraction in the earlier phase. 28 However, the term imperfect recall is often interpreted in 29 an extreme manner in practice. Researchers typically un30 derstand it as completely forgetting all past information-in 31 other words, considering only future information-and de32 sign abstraction algorithms based on this understanding 33 [16, 17, 19, 15, 14]. There are two major factors that mainly 34 affect the results of abstraction for each phase: the number 35 of clustering centers (i.e. centroids), which can be set man36

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ually, and the number of distinct features that are used to categorize hands at each phase. Recent 37 research [12] has found that constructing hand features solely based on future information can lead 38 to excessive abstraction. For example, as shown in the Figure 1, two hands: A and B constructed 39 with only future information can have the same hand features. As the game progresses, the rate of 40 feature repetition among different hands gradually increases, while the distribution of distinct hand 41 features assumes a spindle-shaped pattern. Additionally, constructing hand features with historical 42 information in addition to the future may differentiate two hands sharing the same future information 43 and hence makes more features available for clustering as well as enhances the performance of hand 44 abstraction. 45

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However, there still remain two unsolved issues. First, Fu et al. [12] have introduced a K-recall outcome feature, which incorporates historical information. This feature can only identify if elements are identical or not, but it lacks the capability to discern the extent of differences between features. Therefore, it is difficult to adjust the number of clusters appropriately, which makes it challenging to construct an effective hand abstraction algorithm that integrates historical information. Second, due to the inability to modify the number of clusters, Fu et al. [12] only compared the performance between the maximum clusters cases of integration of historical information (KROI) and no integration at all (POI). In this condition, although KROI significantly outperforms POI, the comparison is inconclusive because KROI recognizes more abstracted infosets than POI. Thus, it does not prove that the performance of abstraction algorithms that integrate historical information is necessarily superior under the condition of having the same number of abstracted infosets.

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This paper introduces a framework for constructing hand features based on winrates, with the Krecall winrate feature being the most crucial one. Based on this, we developed the K-recall winrate isomorphism (KRWI), an abstraction that integrates historical information. Across the same game phases, KRWI identifies slightly fewer hand features than KROI but significantly more than POI. Importantly, the K-recall winrate feature is capable of discerning the extent of differences between features. Therefore, by combining the earth mover's distance with the K-recall winrate feature, we developed the first hand abstraction algorithm that integrates historical information, named KrwEmd, and designed an efficient computational method. We validated our approach in the Numeral211 game environment, where KrwEmd demonstrated superior performance to POI under the same infosets conditions. Additionally, in clustering settings, KrwEmd also outperformed the Ehs and PaEmd algorithms, with PaEmd being the current state-of-the-art hand abstraction algorithm.

2 Background and Notation

Generally, Texas Hold'em-style poker games are modeled as imperfect information games. However, for the task of hand abstraction, games with ordered signals [18, 12] offer a better theoretical tool. The game with ordered signals is a subclass of imperfect information games in that they further subdivide the nodes (also called histories, states, or trajectories) in imperfect information games into mutually independent signals and public nodes. This allows for each aspect to be studied in isolation. Under this framework, the hand abstraction task in Texas Hold'em-style games is modeled as signal abstraction.

In a game with ordered signals ˜ Γ = 〈 ˜ N , ˜ H, ˜ Z, ˜ ρ, ˜ A, ˜ χ, ˜ τ, γ, Θ , ς, O, ω, ⪰ , ˜ u 〉 , there is a set of players ˜ N = N ∪ { c, pub } , which includes not only the main participants N = { 1 , . . . , N } but also a special nature player c who controls the randomness and an observer player pub who can see everything but doesn't take any actions. The game progresses through a series of public nodes ˜ X = ˜ H ∪ ˜ Z . Some of these public nodes are terminal public nodes ˜ Z where the game ends and outcomes are determined, while the others are non-terminal public nodes ˜ H . Among the non-terminal public nodes, some are where players make decisions within the action space ˜ A , and the remaining are chance public nodes where the nature player reveals signals, with the special action Reveal within ˜ A .

The progression from one public node to another is clearly defined ˜ τ : ˜ H × ˜ A ↦→ ˜ X , ensuring that 90 the game's structure is sequential and predictable. Similarly, the signals are revealed according to a 91 probability distribution ς : Θ ↦→ ∆(Θ) , which specifies the likelihood of the next signal given the 92 current one. We use ˜ h ⊑ ˜ h ′ to indicate that ˜ h is a predecessor of ˜ h ′ , and θ ⊑ θ ′ to indicate that θ is a 93 predecessor of θ ′ . Each phase of the game is the number of times nature player has revealed signals, 94 denoted by γ : ˜ X ↦→ N + . r = { γ (˜ x ) | ˜ x ∈ ˜ X } represents the phases that a game with ordered 95 signals may go through. Since the root is a chance public node, we have min r = 1 . 96

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At the end of the game, players receive their payoffs based on the signals and the terminal public node, represented by ˜ u = (˜ u 1 , . . . , ˜ u N ) , where ˜ u i : Θ × ˜ Z ↦→ R . Additionally, each player's survival status is determined at these terminal public nodes, denoted by ω = ( ω 1 , . . . , ω N ) , where ω i : ˜ Z ↦→ { true, false } . The signals possess a partial order within their subset, terminal signals ˜ Θ , indicated by ⪰ : ˜ Θ ×N ×N ↦→ { true, false } . It is required that for any terminal signal θ ∈ ˜ Θ and terminal public nodes ˜ z ∈ { ˜ z ′ ∈ ˜ Z | ω i (˜ z ′ ) = ω j (˜ z ′ ) = true } , if ⪰ ( θ, i, j ) = true , then ˜ u i ( θ, ˜ z ) ≥ ˜ u j ( θ, ˜ z ) .

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Players make decisions based on their observations of signals and the current non-terminal public node. A player may have the same observation for different signals, forming a signal infoset for signals they cannot distinguish. For a player i ∈ N , the signal infoset for a signal θ is denoted as ϑ i ( θ ) = { θ ′ ∈ Θ | O i ( θ ) = O i ( θ ′ ) ∧ O pub ( θ ) = O pub ( θ ′ ) } . Specifically, for the nature player, ϑ c ( θ ) = { θ ′ ∈ Θ | O pub ( θ ′ ) = O pub ( θ ) } . We abuse the notation ϑ ∈ Θ i to represent a signal infoset, where for any player i ∈ N , Θ i is a partition of Θ , representing the collection of player i 's signal infosets. Θ (1) i , . . . , Θ ( | r | ) i are the collections of player i 's signal infosets for each phase, and they form a partition of Θ i . In games with ordered signals, the signals describe all private information. The signal infoset, combined with public nodes, can be transformed into the infoset of an imperfect information game. Fu et al. [12] detailed this transformation process.

The game with ordered signals model allows us to study the issue of signal abstraction independently. For this purpose, we introduce a signal (infoset) abstraction profile, α = ( α 1 , ., α N ) , where for each player i ∈ N , α i is a partition of Θ called the signal (infoset) abstraction. Any ˆ ϑ ∈ α i then is said to be an abstracted signal infoset for player i , and it can be further divided into several signal infosets within Θ i . These finer signal infosets collectively form a partition of ˆ ϑ . In general, two signal abstractions cannot be directly compared in terms of performance, but in a few specific cases there does exist a special relationship between them, which is called refinement. Consider two abstractions α i and β i . If ∀ ˆ ϑ ∈ β i , there exists one or more abstracted signal infosets in α i such that the union of these forms a partition of ˆ ϑ , then we said that α i refines β i , symbolically α i ⊒ β i . The signal abstracted game ˜ Γ α was derived by substituting Θ i with α i across all ˜ x ∈ ˜ X .

Perfect/imperfect recall originally describes a property of imperfect information games, indicating that players do not need to remember all the information they have observed throughout the game. Since games with ordered signals are a subset of imperfect information games, we derived the concept of signal perfect/imperfect recall from them. A player i in a game ˜ Γ is said to have signal perfect recall if, for any θ ′ 1 , θ ′ 2 ∈ ϑ ′ , any predecessor θ 1 of θ ′ 1 has a corresponding predecessor θ 2 of θ ′ 2 such that θ 2 ∈ ϑ ( θ 1 ) . If all players have signal perfect recall, the game ˜ Γ is said to have signal perfect recall. For a game ˜ Γ with signal perfect recall, if α i is the signal abstraction of player i ∈ N , let ( α i , Θ -i ) denote the signal abstraction profile where player i adopts the signal abstraction α i while other players do not do abstraction. If ˜ Γ ( α i ,Θ -i ) retains signal perfect recall, then α i is considered a signal abstraction with perfect recall; otherwise, it is an signal abstraction with imperfect recall.

In games with ordered signals, the strategy π i for player i maps from a non-terminal public node and a signal infoset to a probability distribution over actions, with the strategy profile denoted as π = ( π 1 , . . . , π N ) . When all players adopt the strategy profile π , the expected sum of future rewards, also known as expected value, for player i at public node ˜ x and signal θ is denoted as v π i ( θ, ˜ x ) , and the expected value for the entire game is denoted as v i ( π ) . A Nash equilibrium is a strategy profile where no player can obtain a higher expected value by changing their strategy. Formally, π ∗ is a Nash equilibrium if for every player i , v i ( π ∗ ) = max π i v i ( π i , π ∗ -i ) , where π -i denotes the strategies of all players except i . In two-player zero-sum scenarios, the exploitability of π is denoted as ϵ ( π ) = max π ′ 1 v i ( π ′ 1 ,π 2 )+max π ′ 2 v i ( π 1 ,π ′ 2 ) 2 .

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3 Related Work

Our research focuses on hand abstraction techniques in AI systems for Texas Hold'em-style games (i.e. the signal abstraction in games with ordered signals), building on the initial works of Shi and Littman [25] and Billings et al. [4]. These seminal works introduced the concept of game abstraction, which aims to simplify games while preserving essential characteristics. The researchers started by manually forming hand buckets as a result of their expertise with game-playing strategy. The first automated hand abstraction was that of Gilpin and Sandholm [16]. Later, a model of games with ordered signals was given for Texas Hold'em by Gilpin and Sandholm [18]; lossless isomorphism (LI) was developed with signal rotation. Despite the elegance of LI, its low compression rates hinder its application in large-scale games, whereas lossy abstraction shows potential for such application. An expectation-based clustering method was proposed by Gilpin and Sandholm [17] in their work, and a histogram-based clustering method was introduced by Gilpin et al. [19]. The former is known as Ehs, while the latter is referred to as the potential-aware method. Subsequent studies by Gilpin and Sandholm [15] and Johanson et al. [20] compared Ehs and potential-aware methods, concluding that the latter holds an advantage in large-scale games. Johanson et al. [20] also introduced the use of earth mover's distance 1 (EMD) in potential-aware methods. Ganzfried and Sandholm [14] introduced a more efficient approximation algorithm for earth mover's distance in potential-aware methods (PaEmd). Brown et al. [9] further applied PaEmd to distributed environments for solving large-scale imperfect-information games. This paradigm has found success in Texas Hold'em AI systems and is considered state-of-the-art in hand abstraction. Very recently, Fu et al. [12] proposed several novel tools, such as abstraction resolution and common refinement. They introduced two signal abstraction: one is the potential outcome isomorphism (POI), which identifies the maximum number of abstracted signal infosets considering future information only; The other is the K-recall outcome isomorphism (KROI), which identifies the maximum number of abstracted signal infosets considering historical information. They emphasized that current imperfect recall signal abstraction algorithms, which consider only future information, are prone to excessive abstraction. However, they did not provide practical signal abstraction algorithms.

Other abstraction techniques for decision-making problems include action abstraction [13, 6, 21] and general imperfect recall abstraction [10, 11] in extensive-form games, as well as state abstraction and action abstraction in reinforcement learning [1, 2].

4 Winrate Isomorphism

The first contribution of this paper is an isomorphism framework of winrate-based features, including the potential winrate isomorphism (PWI) and the k-recall winrate Isomorphism (KRWI). Compared with outcome-based features, winrate-based features offer a streamlined approach, focusing exclusively on the distribution of loss, draw, and win outcomes of signals emanating from a signal infoset (and its predecessors) as it evolves towards the terminal signals. Winrate-based features are numerical vectors of consistent length. In this section, an identical Winrate-based feature uniquely determines an abstracted signal infoset. It is worth noting that the similarity of Winrate-based features reflects the similarity among signal infosets, allowing for clustering based on these features (see Section 5).

Both PWI and KRWI share the similar isomorphism construction process for player i in phase r , as illustrated in algorithm 1. The difference lies only in the construction operator for the winrate-based features, FEATURE, used in lines 5 and 12. The isomorphism construction process starts by iterating through all signal infosets of Θ ( r ) i and collecting their features. Next, these features are deduplicated and stored in lexicographical order within set C ( r ) i , which is implemented as a vector data structure. Within C ( r ) i , the index of a feature serves as an identifier for an abstracted signal infoset. Then, by utilizing a hash table CI ( r ) i , we can identify an abstracted signal infoset's identifier based on its feature. In the final step, we traverse Θ ( r ) i again, associating the identifier of a signal infoset with the identifier of its corresponding abstracted signal infoset, and this relationship is recorded in D ( r ) i , an isomorphism map. The function Index i ( r, · ) is a domain-specific mapping that assigns a unique identifier to each signal infoset at phase r , within the numeric range of 0 to | Θ ( r ) i | -1 . In

Algorithm 1 Isomorphism Constructor

Require: r = 1 , . . . , R . Phases. Θ ( r ) i . Signal infoset space for player i . Index i ( r, · ) : Θ ( r ) i ↦→ N . Signal infoset index function for player i . 1: procedure ISOMORPHISMCONSTRUCTOR( r , Θ ( r ) i , FEATURE( · )) 2: Initialize C ( r ) i vector as empty. 3: Initialize D ( r ) i array arbitrarily with length | Θ ( r ) i | . 4: for ϑ ∈ Θ ( r ) i do 5: feature ← FEATURE( ϑ ). 6: Append feature to C ( r ) i . 7: end for 8: Eliminate duplicates from C ( r ) i . 9: Sort the elements of C ( r ) i in lexicographical order. 10: Construct hash table CI ( r ) i from C ( r ) i . Store the index lexid and value feature of C ( r ) i in CI ( r ) i as key-value pairs ( feature, lexid ) . 11: for ϑ ∈ Θ ( r ) i do 12: feature ← FEATURE( ϑ ), idx ← Index i ( r, ϑ ) . 13: Update D ( r ) i [ idx ] with CI ( r ) i [ feature ] . 14: end for 15: return ( C ( r ) i , D ( r ) i ) . 16: end procedure

Texas Hold'em-style games, one optional approach for implementing this function is through lossless 193 isomorphism [18, 27]. 194

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4.1 Potential Winrate Isomorphism

Potential winrate isomorphism (PWI) is a signal abstraction that classify signal infosets based on its potential winrate features. These features focus on the distribution of a player's winrate over terminal signals after passing through a given signal infoset, without considering the history of how the player reached the signal infoset. Specifically, for player i in phase r , the potential winrate feature associated with ϑ ∈ Θ ( r ) i is defined as

where

pf ( r ) , 0 i ( ϑ ) denotes the probability that player i ranks lower than least one other player in the terminal signals, after passing through ϑ .

pf ( r ) ,l i ( ϑ ) , for l > 0 , denotes the probability that player i ranks no lower than any other player and ranks higher than exactly l -1 other players in the terminal signals, after passing through ϑ .

In the terminal phase, the winrate feature is calculated by directly statisticing the game outcomes for players in the given signal infoset. Moreover, in the non-terminal phases, we use a recursive approach to simplify the computation of the winrate feature, thereby avoiding the need to enumerate every signal infoset down to the terminal phase. The recursive formula is

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The PWI algorithm is derived from the POI algorithm [12], and the details of the PWI algorithm are elaborated in Appendix A.1. Both algorithms use the potential winrate feature to distinguish between different abstracted signal infosets in the terminal phase. However, unlike POI, PWI also uses the potential winrate feature in non-terminal phases to identify different abstracted signal infoset classes, while POI relies on the potential outcome

feature (which captures the distribution of the abstracted signal infoset class for future signal infoset). In non-terminal phases, the potential winrate feature is a simplified version of the potential outcome feature. Unsurprisingly, PWI also results in excessive abstraction similar to POI. As shown in Table 2, in heads-up limit hold'em (HULHE) and heads-up no-limit hold'em (HUNL), the number of abstracted signal infosets identifiable by lossless isomorphism increases with each phase, indicating that the game becomes increasingly complex. However, the number of abstracted signal infosets identifiable by PWI and POI first increases and then decreases, showing a spindle-shaped pattern. And we observed that when only future information is considered, winrate-based features may lead to greater information loss compared to outcome-based features. For instance, in the River phase, the number of abstracted signal infosets identified by PWI is only 79.16% of that identified by POI.

4.2 K-Recall Winrate Isomorphism

As Fu et al. [12] mentioned, supplementing historical information can enhance the ability of signal abstraction to identify abstracted signal infosets. Inspired by KROI's construction approach, we developed the k-recall winrate isomorphism (KRWI). The key difference is that instead of using k-recall outcome features to distinguish between different signal infosets, KRWI utilizes k-recall winrate features.

In a game with signal perfect recall, all signals within the signal infoset ϑ have their predecessors at phase r ′ , which belong to the identical signal infoset ϑ ′ . For player i at phase r , the signal infoset ϑ ∈ Θ ( r ) i has a k -recall winrate feature ( k < r ) represented as a numerical array with a dimension of ( k +1)( N +1) :

When r ′ is less than r , pf ( r ′ ) i ( ϑ ) denotes the potential winrate feature for the predecessor signal infoset ϑ ′ of ϑ at phase r ′ . Since we have stored all the potential winrate features of ϑ ∈ Θ ( r ) i through PC ( r ) i , PD ( r ) i and assigned them unique identifiers in Algorithm A1. To save storage space and facilitate retrieval, what we actually store is

PD ( r ′ ) i [ ϑ ] is the identifier for the potential winrate feature of the predecessor ϑ ′ of ϑ in the r ′ phase, r ′ ≤ r . For algorithm details, please refer to Appendix A.2.

Just as the potential winrate feature is a simplified version of the potential outcome feature, the k-recall winrate feature is a simplified version of the k-recall outcome feature. Table 1 shows the number of signal infosets that KRWI and KROI can identify and their ratio in HUNL&HULHE. We were pleasantly surprised to find that while the ratio of PWI to POI resolution can drop below 80%,

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when k is set to its maximum value, i.e. r -1 , the ratio of KRWI to KROI resolution can reach nearly 90% at a minimum, with most of the information preserved. Also, we can easily observe that the number of abstracted signal infosets identified by KRWI is much higher than that identified by POI.

5 K-Recall Winrate Abstraction with Earth Mover's Distance

Fu et al. [12] introduced potential and k-recall outcome features, referred to as outcome-based features, to distinguish different abstracted signal infosets. In the previous section, we developed potential and k-recall winrate features, termed winrate-based features, for the same purpose. In these two methods, Each unique feature corresponds to a single abstracted signal infoset. Intuitively, we can infer that feature similarity might reflect the similarity among abstracted signal infosets, enabling further abstraction and compression for application in large-scale games. However, assessing similarity with outcome-based features is challenging because the identification code indicates only the category, without reflecting the degree of similarity. In contrast, winrate-based features represent winrates, which are inherently comparable, allowing for an easy definition of distances between them.

For the signal information sets ϑ, ϑ ′ of player i at phase r , we can define the distance of their k-recall winrate feature as

Among Equation (5), Emd is the operator used to calculate the earth mover's distance (EMD) [24]. The EMD calculates the distance between two histograms using optimal transport theory. Since it requires solving linear programming equations, the computational complexity of the EMD is sensitive to the dimensionality of the histograms, and approximate algorithms are usually used for larger-scale problems. However, the dimensionality of winrate-based features is small, with a dimension of 3 in a two-player scenario, so we attempt to use a fast algorithm for accurately computing the EMD [5]. w 0 , . . . , w k are hyperparameters used to control the importance of EMD at each phase r, . . . , r -k . We use the KMeans++ algorithm [3], combined with the distance of their k-recall winrate feature, to cluster the abstracted signal infosets of KRWI. We named this algorithm KrwEmd.

Although calculating EMD on small-dimensional histograms is already very fast, clustering actual Texas Hold'em still faces a significant computation. For example, for the River phase of HUNL&HULHE, the clustering input size of the KRWI abstracted signal infoset is approximately 5 . 8 × 10 8 . When we set the number of centroids to 20000, a single Kmeans++ iteration takes about 19000 core hours on a computer with a 2.40GHz clock frequency, which is a significant time cost. Therefore, we need to find ways to reduce this time cost. We have developed an accelerated algorithm, please refer to Appendix A.3 for details.

6 Experimental Setup

We conducted experiments on the Numeral211 Hold'em [12] testbed. Numeral211 is a two-player three-phase Taxes Hold'em-style game with more complex hand systems than the Leduc Hold'em [26] and Rhode Island Hold'em [25] test environments, making it suitable for studying hand abstraction issues. Detailed rules are included in Appendix B. Table 3 shows the number of abstracted signal infosets recognized by KRWI and KROI, along with lossless isomorphism, in Numeral211 Hold'em.

Figure 3: The number of abstracted signal infosets identified by LI, PWI, and POI in each phase of HUNL&HUNLE, with W/O indicating the ratio identified by PWI and POI.

Let α = ( α 1 , α 2 ) be the signal abstraction we would like to assess. We will test the strength of the signal abstraction by measuring exploitability of the approximate equilibrium derived using the

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CSMCCFR algorithm [30, 22] in different abstracted signal infoset scales. We gauge the performance over exploitability. For doing that, we consider both symmetric and asymmetric abstraction scenarios.

In this symmetric abstraction setting, we measure the exploitability of approximate equilibrium that is yielded when both the players in the game employ signal abstraction in the original game. However, it may lead to the abstraction pathology [28]. To avoid such problems, we illustrate the theoretical performance of the signal abstraction under evaluation through asymmetric abstraction. The approximate equilibrium in the signal abstracted games ˜ Γ ( α 1 ,Θ 2 ) and ˜ Γ ( Θ 1 ,α 2 ) is obtained to obtain π ∗ , 1 and π ∗ , 2 , respectively. Finally, we concat the two strategies to get π ′ = ( π ∗ , 1 1 , π ∗ , 2 2 ) and check the exploitability of π ′ .

7 Experiment

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Firstly, we provide an evaluation of the performance of KRWI (2-RWI) compared with KROI (2-ROI) and POI (0-ROI) approaches and lossless isomorphism. We keep the most abstracted signal infosets identified under the full abstraction setting. Note that POI is the common refinement of existing signal abstraction algorithms that only consider future information. And, since previous works cannot control the number of abstracted infoset, they cannot justify their performance in that considering historical information in signal abstraction was better than that in signal abstraction with the same number of abstracted infoset. To investigate this issue, we included KrwEmd and set the clustering scale to be consistent with POI. Note here, that 2-RWI and 2-ROI share the same capability of infoset recognition in Preflop and Flop, while POI is only a little bit worse than 2-RWI and 2-ROI in Flop. Thus, we can directly allow clustering of KrwEmd abstraction use the abstracted signal infosets identified by POI in Preflop and Flop, and only perform clustering in River. Here, we design four sets of hyper-parameters: ( w 0 , w 1 , w 2 ) , i.e., exponentially decreasing: (16 , 4 , 1) , linearly decreasing: (7 , 5 , 3) , constant: (1 , 1 , 1) , and increasing: (3 , 5 , 7) in the importance of historical information. We only show the result of best- and worst-performing parameters (to make the figure neat). The full figures appear in the Appendix C. Figure 4a shows the result of symmetric abstraction, while Figure 4b shows the result of asymmetric abstraction. We observed that both symmetric and asymmetric abstractions maintained consistent abstraction performance without abstraction pathologies. As expected, overfitting was observed in the symmetric abstraction scenario while in the asymmetric scenario overfitting was significant only for POI. The performance difference between 2-RWI and 2-ROI is small, which means that under the full abstraction setting, using simple winrate-based features instead of complex outcome-based features can achieve nearly the same performance. Even with the worst parameter configuration (increasing importance), KrwEmd with the same number of abstracted signal inforsets as POI still outperforms POI.

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Next, we compared the performance of KrwEmd with the currently applied signal abstraction 332 algorithms Ehs and PaEmd. It should be noted that POI is the common refinement both for Ehs and 333 PaEmd, meaning that the maximum number of abstracted signal infosets they can recognize will not 334 exceed that of POI. Thus, we set a compression rate that is 10 times lower than that of POI, while not 335 performing abstraction for Preflop. The final number of abstracted infosets is set to (100 , 225 , 396) . 336 To exclude the influence of random events on performance, we generated 3 sets of abstractions 337 for Ehs and PaEmd each. KrwEmd used hyperparameters ( w 3 , 0 , w 3 , 1 , w 3 , 2 ; w 2 , 0 , w 2 , 1 ) in Flop and 338 River, which are exponentially decreasing (16 , 4 , 1; 4 , 1) , linearly decreasing (7 , 5 , 3; 5 , 3) , constant 339 (1 , 1 , 1; 1 , 1) , and increasing (3 , 5 , 7; 5 , 7) in the importance of historical information. Additionally, 340 since PaEmd uses approximate EMD calculations, its approximate distance is asymmetric, making it 341 difficult for the algorithm to converge. We truncated after 1000 iterations on a single core, with an 342 average cost of 1427.7s, while Ehs and KrwEmd both achieved convergent clustering results, requiring 343 an average of 12.3 and 96.7 iterations, with average time costs of 11.2s and 341.4s, respectively. 344

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Figure 5a shows the results of symmetric abstraction experiments, while Figure 5b shows the results of asymmetric abstraction experiments. We observed that both symmetric and asymmetric abstractions maintained consistent abstraction performance, similar to the full abstraction scenario, without significant abstraction pathologies. The experimental results show that KrwEmd's performance is far superior to that of Ehs and PaEmd under all parameter settings. Our experiments also confirmed that, despite PaEmd's convergence issues, it is indeed a more effective abstraction algorithm than Ehs. Additionally, we further validated that the importance of historical information decreases progressively from bottom to top, although this time the best-performing parameter was exponentially decreasing rather than linearly decreasing as in the previous experiment.

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These two experiments validate that considering historical information is indeed more effective than considering future information only in signal abstraction even in imperfect recall setting.

8 Conclusion

This research introduces the first imperfect recall signal abstraction algorithm that considers historical 357 information. This algorithm has the ability to adjust the scale of the abstracted signal infosets. Based 358 on this, we fully verified that the imperfect recall signal abstraction and abstraction algorithms 359 considering historical information is superior to that only considering future information. Therefore, 360 the KrwEmd algorithm has replaced the PaEmd algorithm and become the SOTA in this field. Based 361 on the KrwEmd algorithm, we are expected to build a stronger Texas Hold'em AI. 362

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References

David Abel. A theory of state abstraction for reinforcement learning. In AAAI conference on artificial intelligence , volume 33, pages 9876-9877, 2019.

David Abel, Nate Umbanhowar, Khimya Khetarpal, Dilip Arumugam, Doina Precup, and Michael Littman. Value preserving state-action abstractions. In International Conference on Artificial Intelligence and Statistics (AISTATS) , pages 1639-1650, 2020.

David Arthur and Sergei Vassilvitskii. k-means++ the advantages of careful seeding. In ACM-SIAM symposium on Discrete algorithms (SODA) , pages 1027-1035, 2007.

D Billings, N Burch, A Davidson, R Holte, J Schaeffer, T Schauenberg, and D Szafron. Approximating game-theoretic optimal strategies for full-scale poker. In International Joint Conference on Artificial Intelligence (IJCAI) , volume 3, pages 661-668, 2003.

Nicolas Bonneel, Michiel van de Panne, Sylvain Paris, and Wolfgang Heidrich. Displacement Interpolation Using Lagrangian Mass Transport. ACM Transactions on Graphics (SIGGRAPH ASIA 2011) , 30(6), 2011.

Noam Brown and Tuomas Sandholm. Regret transfer and parameter optimization. In AAAI Conference on Artificial Intelligence , volume 28, 2014.

Noam Brown and Tuomas Sandholm. Superhuman ai for heads-up no-limit poker: Libratus beats top professionals. Science , 359(6374):418-424, 2018.

Noam Brown and Tuomas Sandholm. Superhuman ai for multiplayer poker. Science , 365 (6456):885-890, 2019.

Noam Brown, Sam Ganzfried, and Tuomas Sandholm. Hierarchical abstraction, distributed equilibrium computation, and post-processing, with application to a champion no-limit texas hold'em agent. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS) , pages 7-15, 2015.

Jiˇ rí ˇ Cermák, Branislav Bošansky, and Viliam Lisy. An algorithm for constructing and solving imperfect recall abstractions of large extensive-form games. In International Joint Conference on Artificial Intelligence (IJCAI) , pages 936-942, 2017.

Jiˇ rí ˇ Cermák, Viliam Lis` y, and Branislav Bošansk` y. Automated construction of bounded-loss imperfect-recall abstractions in extensive-form games. Artificial Intelligence , 282:103248, 2020.

Yanchang Fu, Junge Zhang, Dongdong Bai, Lingyun Zhao, Jialu Song, and Kaiqi Huang. Expanding the resolution boundary of outcome-based imperfect-recall abstraction in games with ordered signals. arXiv preprint arXiv:2403.11486 , 2024.

Sam Ganzfried and Tuomas Sandholm. Action translation in extensive-form games with large action spaces: axioms, paradoxes, and the pseudo-harmonic mapping. In International Joint Conference on Artificial Intelligence (IJCAI) , pages 120-128, 2013.

Sam Ganzfried and Tuomas Sandholm. Potential-aware imperfect-recall abstraction with earth mover's distance in imperfect-information games. In AAAI Conference on Artificial Intelligence , volume 28, 2014.

Andrew Gilpin and Thomas Sandholm. Expectation-based versus potential-aware automated abstraction in imperfect information games: An experimental comparison using poker. In National Conference on Artificial Intelligence (NCAI) , volume 3, pages 1454-1457, 2008.

Andrew Gilpin and Tuomas Sandholm. A competitive texas hold'em poker player via automated abstraction and real-time equilibrium computation. In National Conference on Artificial Intelligence (NCAI) , volume 21, page 1007. Menlo Park, CA; Cambridge, MA; London; AAAI Press; MIT Press; 1999, 2006.

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Andrew Gilpin and Tuomas Sandholm. Better automated abstraction techniques for imperfect information games, with application to texas hold'em poker. In International Joint Conference on Artificial Intelligence (IJCAI) , pages 1-8, 2007.

Andrew Gilpin and Tuomas Sandholm. Lossless abstraction of imperfect information games. Journal of the ACM (JACM) , 54(5):25-es, 2007.

Andrew Gilpin, Tuomas Sandholm, and Troels Bjerre Sørensen. Potential-aware automated abstraction of sequential games, and holistic equilibrium analysis of texas hold'em poker. In National Conference on Artificial Intelligence (NCAI) , volume 22, page 50. Menlo Park, CA; Cambridge, MA; London; AAAI Press; MIT Press; 1999, 2007.

Michael Johanson, Neil Burch, Richard Valenzano, and Michael Bowling. Evaluating statespace abstractions in extensive-form games. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS) , pages 271-278, 2013.

Christian Kroer and Tuomas Sandholm. Discretization of continuous action spaces in extensiveform games. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS) , pages 47-56, 2015.

Marc Lanctot, Kevin Waugh, Martin Zinkevich, and Michael Bowling. Monte carlo sampling for regret minimization in extensive games. International Conference on Neural Information Processing Systems (NeurIPS) , 22, 2009.

Matej Moravˇ cík, Martin Schmid, Neil Burch, Viliam Lis` y, Dustin Morrill, Nolan Bard, Trevor Davis, Kevin Waugh, Michael Johanson, and Michael Bowling. Deepstack: Expert-level artificial intelligence in heads-up no-limit poker. Science , 356(6337):508-513, 2017.

Yossi Rubner, Carlo Tomasi, and Leonidas J Guibas. The earth mover's distance as a metric for image retrieval. International journal of computer vision , 40:99-121, 2000.

Jiefu Shi and Michael L Littman. Abstraction methods for game theoretic poker. In Computers and Games: Second International Conference, CG 2000 Hamamatsu, Japan, October 26-28, 2000 Revised Papers 2 , pages 333-345. Springer, 2001.

Finnegan Southey, Michael Bowling, Bryce Larson, Carmelo Piccione, Neil Burch, Darse Billings, and Chris Rayner. Bayes' bluff: opponent modelling in poker. In Proceedings of the Twenty-First Conference on Uncertainty in Artificial Intelligence , pages 550-558, 2005.

Kevin Waugh. A fast and optimal hand isomorphism algorithm. In AAAI Workshop on Computer Poker and Incomplete Information , 2013.

Kevin Waugh, David Schnizlein, Michael Bowling, and Duane Szafron. Abstraction pathologies in extensive games. In International Conference on Autonomous Agents and Multiagent Systems (AAMAS) , volume 2, pages 781-788, 2009.

Kevin Waugh, Martin Zinkevich, Michael Johanson, Morgan Kan, David Schnizlein, and Michael Bowling. A practical use of imperfect recall. In Symposium on Abstraction, Reformulation and Approximation (SARA) , 01 2009.

Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret minimization in games with incomplete information. In International Conference on Neural Information Processing Systems (NeurIPS) , pages 1729-1736, 2007.

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Algorithm A1 Potential Winrate Isomorphism

Require:

A Algorithm Details

A.1 Potential Winrate Isomorphism

Algorithm A1 describes the computation process for potential winrate isomorphism. This algorithm operates in reverse, starting from the game's final phase R .

A.2 K-Recall Winrate Isomorphism

Algorithm A2 constructs the k-recall winrate isomorphism using the k-recall winrate feature. This process requires the prior construction of the potential winrate isomorphism map PD ( r ) i using Algorithm A1.

A.3 Accelerating Distance Computing for K-Recall Winrate Features

According to Equation (5), we note that the distance calculation between a k-recall winrate isomorphism class and a centroid's k-recall winrate feature can be decomposed into k+1 pairs of potential winrate feature EMD calculations. The potential winrate feature of the hand remains unchanged, while only the potential winrate feature of the centroid changes. Decomposing the calculation into the EMDs of potential winrate features involves significantly fewer computations than directly calculating the EMD of two k-recall winrate features. Specifically, for the River phase of HUNL&HULHE, we have the compression ratio as 169+1028325+1850624+20687 529890863 = 2899805 529890863 = 0 . 0054725 .

Algorithm A3 describes how we accelerate the batch EMD computation between a centroid and all 465 KRWI classes' k-recall winrate features. It should be noted that the K-recall winrate feature involved 466 in the calculation of the centroid in the algorithm is in the form of Equation (3), while the K-recall 467 winrate feature in RC ( r,k ) is in the form of Equation (4). This method reduced the computational 468

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Algorithm A2 K-Recall Winrate Isomorphism

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cost of EMD from 19000 core hours to approximately 104 core hours, which is significantly lower than the time cost of summarizing the distance for each KRWI class, which is about 524 core hours and is an unavoidable O (1) cost.

The distance batch calculation for each centroid can be processed independently and distributed across tens of multi-core computer (e.g. 96-core computers), with each computer responsible for calculating the features of some centroids in one iteration, which are then aggregated. Using this technique, we can reduce an iteration to a few hours, which is acceptable for Texas Hold'em AI training.

B Numerall211 Hold'em Rules

Numeral211 Hold'em is played according to the following rule:

Ante: Each player antes 5 chip into the pot at the start of the hand.

Hole Card: Both players are dealt one private card face down, known as the hole card.

Deck: The deck consists of a standard poker deck, excluding the Jokers, Kings, Queens, and Jacks, resulting in a total of 40 cards. There are four suits: spades ( ♠ ), hearts ( ♡ ), clubs ( ♣ ), and diamonds ( ♢ ), each containing ten cards numbered 2 through 9, and including the ten (T) and ace (A).

First Betting Phase: Following the distribution of hole cards, a phase of betting occurs. Players can choose to check or bet, with the bet size set at 10 chips.

Flop: After the initial betting phase, a single community card, termed the flop, is revealed from the deck.

Second Betting Phase: Another phase of betting takes place after the flop, with the bet size increasing to 20 chips.

Turn: After the Second betting phase, another community card, termed the turn, is revealed from the deck.

Third Betting Phase: Another phase of betting takes place after the turn, with the bet size still set at 20 chips.

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Algorithm A3 Distance Batch

Require:

Showdown: If neither player folds, a showdown occurs. Players reveal their cards, aiming to form the best possible hand. The player with the highest-ranked hand wins the pot. In the case of a tie, the pot is split evenly. The Table 2 show the hand ranks of Numeral211 Hold'em.

Betting Options: Throughout the game, players have options to fold, call, or raise. In each betting phase, the total sum of bets and raises is limited to a maximum of 4, with fixed bet sizes of 10 chips in the first phase and 20 chips in the last two betting phases.

C Supplementary Data for Experiment 1 502

Figure 6 show all of the result in experiment 1. 503

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NeurIPS Paper Checklist

1. Claims

Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?

Answer: [Yes]

Justification: We have clearly defined our scope and contributions in both the abstract and introduction sections.

Guidelines:

The answer NA means that the abstract and introduction do not include the claims made in the paper.

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The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers.

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Question: Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?

Answer: [No]

Justification: Due to the long experimental time and limited sample size, error bars cannot be provided. In the first experiment (Figure 4), the baseline settings adopt fixed abstraction settings and have a large performance gap, so the performance of strategies solved by CSMCCFR is stable and not easily affected by random factors. In the second experiment (Figure 5), random factors may indeed affect individual experimental data. Therefore, we sampled the control group multiple times and drew the performance range, which is far lower than the performance of our algorithm under the worst parameters, which also proves the effectiveness of the algorithm.

Guidelines:

The answer NA means that the paper does not include experiments.

The authors should answer "Yes" if the results are accompanied by error bars, confidence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.

The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).

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The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)

The assumptions made should be given (e.g., Normally distributed errors).

It should be clear whether the error bar is the standard deviation or the standard error of the mean.

It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have a 96% CI, if the hypothesis of Normality of errors is not verified.

For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative error rates).

If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.

8. Experiments Compute Resources

Question: For each experiment, does the paper provide sufficient information on the computer resources (type of compute workers, memory, time of execution) needed to reproduce the experiments?

Answer: [Yes]

Justification: We discuss the computational resources and time cost of the experiments in Sections 5, 7, and Appendix A.3.

Guidelines:

The answer NA means that the paper does not include experiments.

The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.

The paper should provide the amount of compute required for each of the individual experimental runs as well as estimate the total compute.

The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make it into the paper).

9. Code Of Ethics

Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics https://neurips.cc/public/EthicsGuidelines ?

Answer: [Yes]

Justification: We carefully review the NeurIPS Code of Ethics and ensure that the research aligns with it in all aspects.

Guidelines:

The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.

If the authors answer No, they should explain the special circumstances that require a deviation from the Code of Ethics.

The authors should make sure to preserve anonymity (e.g., if there is a special consideration due to laws or regulations in their jurisdiction).

10. Broader Impacts

Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?

Answer: [Yes]

Justification: We discuss the impact of this work in Section 8. As noted, this work represents the SOTA in hand abstraction algorithms and could be used to create more powerful Texas Hold'em AI.

Guidelines:

The answer NA means that there is no societal impact of the work performed.

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If the authors answer NA or No, they should explain why their work has no societal impact or why the paper does not address societal impact.

Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.

The conference expects that many papers will be foundational research and not tied to particular applications, let alone deployments. However, if there is a direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.

The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology is being used as intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.

If there are negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms to monitor how a system learns from feedback over time, improving the efficiency and accessibility of ML).

11. Safeguards

Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?

Answer: [NA]

Justification: Our work focuses solely on introducing a more efficient algorithm for hand abstraction and does not involve the release of data or models.

Guidelines:

The answer NA means that the paper poses no such risks.

Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters.

Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.

We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.

12. Licenses for existing assets

Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?

Answer: [Yes]

Justification: This paper provides comprehensive citations for all comparative methods involved, and all comparison experiments were re-implemented without using existing tools or code.

Guidelines:

The answer NA means that the paper does not use existing assets.

The authors should cite the original paper that produced the code package or dataset.

The authors should state which version of the asset is used and, if possible, include a URL.

The name of the license (e.g., CC-BY 4.0) should be included for each asset.

For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.

If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset.

For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.

If this information is not available online, the authors are encouraged to reach out to the asset's creators.

13. New Assets

Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?

Answer: [NA]

Justification: This paper does not release any new assets.

Guidelines:

The answer NA means that the paper does not release new assets.

Researchers should communicate the details of the dataset/code/model as part of their submissions via structured templates. This includes details about training, license, limitations, etc.

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At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.

14. Crowdsourcing and Research with Human Subjects

Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well as details about compensation (if any)?

Answer: [NA]

Justification: This paper does not involve human subjects.

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Including this information in the supplemental material is fine, but if the main contribution of the paper involves human subjects, then as much detail as possible should be included in the main paper.

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Question: Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?

Answer: [NA]

Justification: This paper does not involve human subjects.

Guidelines:

The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.

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Depending on the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.

We recognize that the procedures for this may vary significantly between institutions and locations, and we expect authors to adhere to the NeurIPS Code of Ethics and the guidelines for their institution.

For initial submissions, do not include any information that would break anonymity (if applicable), such as the institution conducting the review.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces the KrwEmd algorithm, a novel approach to hand abstraction in Texas Hold’em-style games. The main contribution is the integration of historical information using K-recall winrate features and earth mover’s distance, addressing the limitations of previous imperfect recall abstraction methods. The algorithm demonstrates significant performance improvements over state-of-the-art methods.\n\nSTRENGTHS:\n1. The integration of historical information via K-recall win rate features enhances the accuracy and reliability of hand abstraction.\n2. KrwEmd significantly outperforms existing methods, showcasing its practical value and potential for real-world applications.\n3. The algorithm is technically sound, with strong experimental validation supporting its claims.\n\nWEAKNESSES:\nScalability: While the paper demonstrates KrwEmd's effectiveness in Texas Hold’em-style games, its scalability to more complex and diverse game scenarios remains to be explored.\n\nComparative Analysis: The paper could benefit from a more detailed comparison with existing hand abstraction methods to better highlight KrwEmd's unique advantages and limitations.\n\nQUESTIONS:\n1. How well does KrwEmd scale when applied to more complex and diverse game scenarios beyond Texas Hold’em? Could the authors provide any preliminary results or insights into this?\n\n2. While KrwEmd achieves significant performance improvements, are there scenarios or tasks where additional training data might be beneficial or required? If so, what kind of training data would be most useful?\n\n3. Are there other game models or systems that KrwEmd could enhance similarly? What challenges might arise in such integrations?\n\nLIMITATIONS:\nThe paper primarily focuses on Texas Hold’em-style games. While this is a significant achievement, a discussion on the model's scalability and generalization to more complex and varied game scenarios would be beneficial. Including potential strategies to address these challenges would strengthen the paper.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-13"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a novel approach to hand abstraction in Texas Hold'em-style poker games, addressing the limitations of current methods that often disregard historical information. The authors make two primary contributions: First, they develop KRWI (K-Recall Winrate Isomorphism), a new abstraction method that incorporates historical information from previous game phases. Second, they present KrwEmd, the first hand abstraction algorithm to effectively combine K-recall win rate features with earth mover's distance for hand classification. Through experiments conducted in the Numeral211 Hold'em environment, the authors demonstrate that KrwEmd significantly outperforms state-of-the-art algorithms such as Ehs and PaEmd in terms of exploitability, while maintaining the same number of abstracted information sets. This work shows that incorporating historical information can substantially enhance the performance of hand abstraction algorithms, potentially leading to more advanced strategic computation in large-scale adversarial games and stronger poker AI systems.\n\nSTRENGTHS:\n* Overall, the paper provides a new technique that is promising for an important area of research\n* The results indicate strong improvements over alternative methods\n\nWEAKNESSES:\nFor me, the paper's primary weakness is the presentation method. I had trouble understanding the significance and nature of the contribution from the current submission. In general, a clearer description of this area of research for people who, e.g., work on games but don't focus on poker would be quite helpful. Some specific suggestions/areas for improvement are:\n\n* Clearer introduction of key concepts: The paper jumps into technical terms like 'imperfect recall abstraction' and 'hand abstraction' without adequately explaining them for a broader audience. A brief explanation of why these concepts are important in poker AI would be beneficial.\n* More intuitive explanations of the algorithms: The descriptions of PWI, KRWI, and KrwEmd are highly technical. Including some simple examples or diagrams to illustrate how these algorithms work could greatly improve understanding.\n* Better contextualization of the contribution: While the paper claims to outperform existing methods, it's not clear how significant this improvement is in the broader context of poker AI. A discussion of the practical implications of this improvement would be valuable.\n* Clarification of experimental setup: The Numeral211 Hold'em environment is not well-known. A clearer explanation of how this relates to standard poker variants would help readers understand the relevance of the results.\n* More accessible presentation of results: The graphs and tables are dense with information but lack clear explanations. Simplifying these visualizations or providing more guidance on how to interpret them would be helpful.\n* Glossary of terms: Given the many technical terms used (e.g., 'earth mover's distance', 'K-recall winrate feature'), a glossary could be a valuable addition to help readers keep track of these concepts.\n\nQUESTIONS:\nPlease address my comments about clarity overall and address any specifics that seem appropriate.\n\nLIMITATIONS:\nThere's limited discussion of limitations. It would be good to include an explicit limitations section. In particular, it would be good to discuss potential computational challenges in more detail, any limitations on the scope of the evaluation, and future work that might be planned.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-14"}

comments: SUMMARY:
This paper focuses on the problem of hand abstraction for Texas Hold-Em style poker games. Hand abstraction is the process of partitioning game histories into infosets which still contain enough information to make strategically advantageous decisions. Previous approaches have focused on abstractions that primarily focus on the future outcomes from each hand, but the authors suggest that it may instead be beneficial to also include past information. They design a hand abstraction algorithm called KwrEmd that outperforms previous work by incorporating historical information.

STRENGTHS:
The results seem to show that KrwEmd outperforms other imperfect recall hand abstraction algorithms in terms of exploitability.

WEAKNESSES:
As somebody who is unfamiliar with the subfield of imperfect recall abstraction, I found the paper to be quite confusing throughout. The authors do not often provide intuition or examples for their method, and I found it difficult to tell exactly which contributions were novel compared to Fu et al. While it's reasonable for a paper to use technical language at times, well-written papers are usually understandable by a broader set of readers than just those in the specific subfield.

QUESTIONS:
No specific questions.

LIMITATIONS:
Not sure.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-15

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes KrwEmd, a novel hand abstraction algorithm for imperfect recall settings in Texas Hold'em poker. The algorithm leverages K-recall winrate features, incorporating historical information in addition to future information for constructing hand abstractions. The authors introduce two new isomorphism frameworks: Potential Winrate Isomorphism (PWI) and K-recall Winrate Isomorphism (KRWI). They demonstrate that KRWI outperforms existing methods like POI in identifying distinct infosets. KrwEmd, which combines KRWI with Earth Mover's Distance (EMD) for hand classification, shows superior performance compared to POI, Ehs, and PaEmd in the Numeral211 Hold'em environment.\n\nSTRENGTHS:\n*Originality*: The paper presents a novel combination of K-recall winrate features and EMD for hand abstraction in imperfect recall settings, addressing a critical limitation of current approaches that solely rely on future information.\nThe introduction of KRWI and PWI provides valuable new tools for understanding and constructing hand abstractions in poker AI.\n\n*Quality*: The experimental results in the Numeral211 environment demonstrate a clear improvement over existing methods, supporting the claims of the paper. The paper includes an appendix with algorithm details and supplementary experimental data.\n\n*Significance*: The proposed KrwEmd algorithm advances the state-of-the-art in hand abstraction for imperfect recall settings, offering a potentially significant improvement for developing stronger poker AI agents.\nThe incorporation of historical information is a valuable contribution that benefit positively future research in poker and other imperfect information games.\n\nWEAKNESSES:\nI found the paper  challenging to understand, though this may be due to my limited background knowledge in poker AI and game theory.  While the authors provide a background section, the density of the technical content and the numerous specialized terms make comprehension difficult.\n\nThe description of the accelerated algorithm in Appendix A.3 could be expanded for better understanding. Additionally, a clear discussion of the limitations of the accelerated algorithm would be beneficial.\n\nThe paper provides limited information about the proposed algorithms, particularly KrwEmd. While the core concepts are presented, the details regarding implementation and specific design choices are limited. More in-depth explanation and pseudocode would enhance the paper's quality.\n\nQUESTIONS:\nWhat are the limitations of the accelerated algorithm? Does it introduce any approximations or trade-offs in performance or accuracy?\n\nHave you explored the application of KrwEmd to other poker variants beyond Numeral211? Are there any specific challenges or adaptations needed for different game settings?\n\nHow sensitive is KrwEmd to the choice of hyperparameters (w0, w1, w2)? Is the algorithm robust to different hyperparameter configurations?\n\nLIMITATIONS:\nThe authors acknowledge the computational complexity of clustering with EMD and introduce an accelerated algorithm. However, the paper lacks a dedicated section addressing the limitations of the proposed methods and the accelerated algorithm.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-17"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper develops new hand abstraction techniques for Texas Hold'em-style games (in general: games with ordered signals), which fare better than previous methods in both the number of hands identified, and performance (exploitability) in a simplified version of the game. \n\nHand abstraction is a technique aiding the strategy construction in a Texas Hold'em, where concrete hands, or rather concrete signal infosets (i.e. \"possible words\" according to the information revealed so far), are replaced by abstract infosets, represented in an abstract feature space (here $\\mathbb{R}^n$).\n\nThe core idea of the paper is to use the features of hands from previous rounds in the construction of the current round feature. More precisely, the paper investigates a simple method (KRWI = *k-recall winrate isomorphism*) of maintaining, at a given round, the collection of all potential-winrate isomorphims (PWI) features from previous $k$ rounds by concatenating them all together. PWI for an $n$-player game is a categorical probability distribution over $n+1$ events of a form *\"this player outperformed exactly $l-1$ other players while losing to none\"* for $l = 1, 2, \\ldots n$ and *\"this player lost to at least one player\"*. Those distributions can be computed by a dynamic programming method. To reduce the cardinality of the space, the paper later clusters KRWI features with k-means using the Wasserstein distance, naming it the KrwEmd method.\n\nAll of the methods are benchmarked against currently used techniques that do not use historical information. Experiments find that KRWI identifies similar proportion of signal infosets as the previously used KROI. Using the metric of exploitability of the equilibrium (how it deviates from Nash equilibrium) of the strategy found by an imperfect-information game solver, authors find that 2-RWI-based approach performs almost the same as 2-ROI, and that KrwEmd outpefrorms previously used Ehs and PaEmd by a relatively large margin.\n\nSTRENGTHS:\nThe paper proposes a reasonable extension of the currently used techniques for hand abstraction in Texas Hold'e, and shows that the new idea beats SOTA. It does not shy away from introducing the reader to the relevant background in a rigorous way (which is what made it possible for me to even start reading it). Experiments show a meaningful improvement, and provide additional insights (such as the decreasing worth of historical information).\n\nWEAKNESSES:\nFrom the perspective of someone not at all acquainted with the field of imperfect information games/games with ordered signals, the paper was quite hard to read and understand - even though (assuming that the authors agree with my summary), the contribution is a relatively straightforward idea.\n\nThe introduction was uninformative and confusing (I would recommend rewriting the whole second paragraph); the preliminaries, although presented in-depth and trying to be formal, also posed quite a few questions; sections 4 and 5 describing the main contribution lacked detail and justification (i.e. ideally I would like to see definition/theorem/proof style - otherwise the text is impossible to read for someone unfamiliar with the field), and the experimental setup is assuming a lot of background knowledge that was not explained neither in the main paper nor in the appendix (it was also difficult to gauge if the comparison between SOTA and the new approach was fair from the resources pov - the paper reports some numbers, but never an aggregated \"memory/time used\" for all methods).\n\nPlease see Questions below for a detailed explanation of what I found lacking or hard to understand.\n\nQUESTIONS:\nIntro:\n- What are the \"clustering settings\" referred to in line 66? (It's difficult to understand how important it is for the claim 'outperforms the SOTA')\n\nPrelims:\n- I think the definition of the game $\\Gamma$ should include the initial distribution over signals, to start iterating the map $\\varsigma\n: \\Theta \\to \\Delta(\\Theta)$\n- I did not understand how exactly the map $\\varsigma$ interacts with the shape of the game tree given by $X, \\tau$ and the order $\\sqsubseteq$ on $\\Theta$ - are we assuming that $\\theta \\sqsubseteq \\theta'$ for all $\\theta'$ in the codomain of $\\varsigma(\\theta)$ (dividing the signal space appropriately)? Are the terminal signals $\\tilde{\\Theta}$ take wrt to the order $\\sqsubseteq$, or wrt to the \"realisable\" order of '$\\theta$ is final if it can only ever appear as the final signal of any game trajectory'? Is the order $\\sqsubseteq$ assumed to be tree-shaped? (line 12 of algorithm A2 requires a unique predecessor of $\\vartheta$) \n- Why is the order $\\preccurlyeq$ assumed to be partial? Shouldn't it be total on $\\tilde{\\Theta}$?\n- Are the survival status and the $\\preccurlyeq$ relevant to the paper? Is the signal abstraction refinement relevant to the paper?\n- Why is $\\gamma$ provided in the definition of the game and not just derived from other structures?\n- What happens to observations $O$ and the map $\\varsigma$ in the signal abstracted game $\\Gamma^{\\alpha)?\n- Line 121 - \"such that the union of those\" should be just \"such that they\"\n- Is the criterion of the signal perfect recall for a game $\\Gamma$ equivalent to saying that \"players can be Markov\"? I.e. that for any non-Markov (wrt to signals) strategy profile $\\pi = (\\pi_i, \\pi_{-i}$) there exists a Markov (wrt to signals) strategy $\\pi'_i$ such that $(\\pi'_i, \\pi_{-i})$ is equivalent to $\\pi$?\n- I think the signal space $\\Theta$ has to be assume to be finite - otherwise, the definition of perfect recall breaks.\n\nWinrate Isomorphism:\n- What are the outcome-based features? (never explained, but used throughout this and next sections). What is POI?\n- Line 179 - \"an identical Winrate-based feature uniquely determines an abstracted signal infoset\" - is that a definition of the abstracted signal infosets for winrate-based features? If so, this requires proof or justification (that they partition nicely, satisfy order conditions, interact with  $\\varsigma$ in the right way etc).\n- The above question is even more relevant for KrwEmb, which uses a complicated clustering mechanism inside.\n- Why is $\\mathcal{D}$ an isomorphism?\n- Line 196 - typo in \"classify\"\n- What is the intuition behind constructing PWI in this particular way (i.e. with \"lose at least to one other player\"/\"win with with $l-1$ players and tie with the rest\")? Is this a natural choice? Why not \"lose to $l$ players$\" or some other metric?\n- What is the River phase? What is HUNL&HULHE?\n- It would be good to include at least the definition of the Wasserstein distance, instead of linking to Wikipedia\n\nExperiments:\n- What is \"mb/g\" in Figures 4 and 5?\n- The plots do not show LI (potentially overshadowed?) and overshadow 2-RWI. I would recommend changing the style, since this is one of the few most important results in this section.\n- Line 299 - \"We gauge the performance over exploitability\" - what does that mean?\n- Line 331 - typo in \"infosets\"\n- Line 336 - \"the final number of abstracted infosets is set to\" - are these numbers for Ehs, PaEmd and KwrEmd respectively? In what order? \n- Line 338 - why does KrwEmd use so many parameters here, instead of just 3?\n- If the costs of 1000x1427.7s, 12x11.2s and 96.7x341.4s are total costs, wouldn't it be more fair to set the hyper-parameters of the algorithms such that their total computational budget is approximately equal? Otherwise, it's difficult to judge the performance improvement.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-23"}