Solver-Guided Reasoning for Mixed-Equilibrium Strategies

# Solver-Guided Reasoning for Mixed-Equilibrium Strategies
Source: [https://arxiv.org/html/2608.06741](https://arxiv.org/html/2608.06741)
Han Wang1Philippe Beardsell2Boning Li3 Aaron Sasmita4Shuai Li1Hongyuan Zha4Baoxiang Wang4,5 1Shanghai Jiao Tong University2GTO Wizard3Tsinghua University 4The Chinese University of Hong Kong, Shenzhen5Vector Institute xwanghan@sjtu\.edu\.cnphil@gtowizard\.comli\-bn22@mails\.tsinghua\.edu\.cn 121040033@link\.cuhk\.edu\.cnshuaili8@sjtu\.edu\.cnzhahy@cuhk\.edu\.cn bxiangwang@cuhk\.edu\.cn

\(August 4, 2026\)

###### Abstract

Reasoning in large language models \(LLMs\) is often grounded in human text, human demonstrations, and human\-generated rationales\. For equilibrium reasoning in complex games, however, relying on human data can be suboptimal\. In fact, human play is often guided by intuition and heuristics and can deviate substantially from game equilibrium\. This discrepancy is amplified in games with mixed\-strategy equilibria, where human data is heavily biased toward pure strategies\. Consequently, conditioning LLMs on this data yields weak game strategies\. To grant LLMs the reasoning capacity in games, in this work, we study how to elicit equilibrium play using solver output\. We propose Mixed\-Strategy Decision Tree \(MDT\), which articulates the silent optimality of the equilibrium into sparse strategic rules that both humans and LLMs could understand\. Using solver output rather than human annotation allows us to extend the input to arbitrarily new states and continuations\. We instantiate this study on No\-Limit Texas Hold’em by querying a solver oracle for over250 million mixed\-strategy decisions; MDT together with other techniquesreduces theℓ1\\ell\_\{1\}distance to the equilibrium by52\.6%52\.6\\%across88different LLM configurations\. A Route\-only ablation tests the incremental contribution of the shadow\-based contrast, while complete River\-endgame and Liar’s Dice experiments evaluate strategic fidelity and portability beyond the original NLH communication setting\.

## 1Introduction

Large language models are trained and evaluated with an extensive amount of human data, including demonstrations, solutions, rationales, and reasoning tracesWeiet al\.\([2022](https://arxiv.org/html/2608.06741#bib.bib25)\); Achiamet al\.\([2023](https://arxiv.org/html/2608.06741#bib.bib43)\)\. For reasoning in complex games, however, learning from human data is fundamentally limited\. On the one hand, human play and commentary are selective\. Many human strategies are exploitative, largely deviating from game\-theoretically optimal equilibrium strategies\. Even players considered “good” by human standards utilize strategies that are mostly effective against weaker humans but underperform against AI solversBrown and Sandholm \([2019b](https://arxiv.org/html/2608.06741#bib.bib4)\); Silveret al\.\([2018](https://arxiv.org/html/2608.06741#bib.bib44),[2017](https://arxiv.org/html/2608.06741#bib.bib18)\)\. On the other hand, human data is mostly presented as pure strategies\. It is intuitive for humans and LLMs to treat decisions as prediction tasks, attempting to figure out which single action yields the best outcome\. But for games with imperfect information, the equilibrium strategy is often mixed\. It is difficult to find such mixed strategies in human data, making it equally difficult for LLMs to acquire them\.

Recent evaluations of LLMs on No\-Limit Texas Hold’em \(NLH\) poker make this gap concrete\. The GTO Wizard Benchmark reports that the strongest evaluated model, GPT\-5\.3 Extra High reasoning, performs substantially below its approximate equilibrium strategy, at roughly−16±3\.0\-16\\pm 3\.0bb/100 under its luck\-adjusted evaluation\(Provostet al\.,[2026](https://arxiv.org/html/2608.06741#bib.bib41)\)\. LLMs also make fundamental errors in card representation and hand\-strength evaluation, as well as strategic errors in range\-level mixing and frequency allocation\. PokerBench similarly observes that LLMs fail to predict the highest\-frequency actions at certain decision points even after task\-specific fine\-tuning\(Zhuanget al\.,[2025](https://arxiv.org/html/2608.06741#bib.bib42)\)\. These failures are not simply a lack of poker vocabulary\. LLMs can often discuss pot odds, blockers, equity, and bluffing, yet still miss the equilibrium logic that couples those concepts across hidden states and action frequencies\.

This fundamental gap stems from a mismatch in the underlying objectives\. LLMs are optimized for linguistic predictions and reasoning, whereas game solvers compute policies by minimizing exploitability across complex, hidden\-state game trees\(Zinkevichet al\.,[2007](https://arxiv.org/html/2608.06741#bib.bib1); Moravčíket al\.,[2017](https://arxiv.org/html/2608.06741#bib.bib15); Brown and Sandholm,[2018](https://arxiv.org/html/2608.06741#bib.bib3)\)\. Consequently, an LLM can produce fluent, conceptually accurate poker commentary while entirely failing to execute the precise frequency allocations required for equilibrium\. Relying on human\-generated examples or LLM self\-rationalization cannot bridge this gap, as neither reliably captures full mixed\-strategy policies\(Linet al\.,[2026](https://arxiv.org/html/2608.06741#bib.bib45)\)\. Conversely, while AI solvers naturally generate optimal mixed strategies, their outputs consist of raw numerical distributions rather than generalized, textual reasoning\. To endow LLMs with game\-theoretic optimality, we must translate these raw numbers into a linguistic format\. We define this task as*solver articulation*:

How can we extract articulate, verifiable reasoning from the silent optimality of solver\-generated data?

We study this problem in imperfect\-information extensive\-form games, using No\-Limit Texas Hold’em \(NLH\) decision points where two players are left in the game\. We introduce Mixed\-Strategy Decision Tree \(MDT\), which converts solver\-implied decision logic into an inspectable and readable form\. MDT represents each decision point with its solver\-derived public state and range\-level and hand\-level summaries\. It uses sparse hierarchical routing to assign probability mass to pure\-action leaf prototypes\. The hierarchy avoids forcing all strategic interactions into a single dense mapping\. In this way, coarse public and range conditions select a local strategic regime, while sparse node\-level summaries expose the hand\-specific boundary that changes the action mixture\.

Motivated by counterfactual methods for game solving\(Wachteret al\.,[2017](https://arxiv.org/html/2608.06741#bib.bib17); Agarwalet al\.,[2021](https://arxiv.org/html/2608.06741#bib.bib10)\), we propose Scenario\-Constrained Counterfactual Sampling \(SCCS\) that provides additional reasoning paths on top of MDT\. SCCS selects shadow hands that share the same public context but exhibit clear solver\-policy divergence, route to different MDT leaves, and differ along a salient summary\. By isolating such contrastive pairs, SCCS exposes the local boundary at which a hand changes its role inside the mixed strategy\. In this way, it converts an inspectable tree route into a transferable strategic statement\.

Our method is applied to two\-player NLH postflop game play with over 250 million decision points accessed\. To ensure the quality of the articulation, we used one of the best available commercial poker solvers in the world\. The solver enjoys a Nash Distance less than 0\.3% of the current pot and gives solutions to arbitrary spots in the game tree\. The stream of data therefore spans all 1,755 NLH flops and their turn and river continuations\. The obtained MDT is tested across 8 different LLM configurations, where the sparse rules by the MDT are given to LLMs before they are asked to reason the equilibrium strategy\. The LLMs have theirℓ1\\ell\_\{1\}distance to the solver target reduced from0\.2110\.211to0\.1000\.100\. The argmax\-action agreement, defined by the highest\-probability action in each distribution, improves from57\.2%57\.2\\%to76\.1%76\.1\\%\.

Beyond the immediate improvements in game\-theoretic reasoning, our framework sheds light on a fundamentally new regime in artificial intelligence: endowing LLMs with complex reasoning capabilities entirely through synthetic, AI\-generated data\. This paradigm is especially appealing in the current landscape, where the supply of high\-quality human data is rapidly becoming depleted\. By demonstrating that the implicit optimality of an AI solver can be systematically extracted and translated into readable linguistic rules, we provide a concrete pathway to bypass the human data bottleneck\. As envisioned bySilver and Sutton \([2025](https://arxiv.org/html/2608.06741#bib.bib16)\), the future of artificial intelligence relies not on mimicking human demonstrations, but on learning directly from ground\-truth interactions and solver\-backed experience\.Our work represents an effort toward realizing this new era of experiences\.

## 2Related Work

#### Solver\-based game reasoning\.

CFR, subgame solving, and self\-play search have enabled strong imperfect\-information game agents, including superhuman poker systems\(Zinkevichet al\.,[2007](https://arxiv.org/html/2608.06741#bib.bib1); Brown and Sandholm,[2018](https://arxiv.org/html/2608.06741#bib.bib3),[2019b](https://arxiv.org/html/2608.06741#bib.bib4); Moravčíket al\.,[2017](https://arxiv.org/html/2608.06741#bib.bib15); Brownet al\.,[2020](https://arxiv.org/html/2608.06741#bib.bib5)\)\. These systems compute mixed policies and values, but their outputs are primarily numerical prescriptions rather than communicable reasoning\.

#### Interpretable policy distillation\.

Prior work distills learned policies into trees, programs, or concept\-based representations\(Bastaniet al\.,[2018](https://arxiv.org/html/2608.06741#bib.bib11); Vermaet al\.,[2018](https://arxiv.org/html/2608.06741#bib.bib12); Frosst and Hinton,[2017](https://arxiv.org/html/2608.06741#bib.bib9); McGrathet al\.,[2022](https://arxiv.org/html/2608.06741#bib.bib14)\)\. Our setting differs because the target is a mixed equilibrium policy in an imperfect\-information game, where local decisions depend on range\-level coupling and action\-frequency balance\.

#### LLMs and poker reasoning\.

Recent poker benchmarks show that LLMs struggle with solver\-level poker decisions despite fluent strategic language\(Gupta,[2023](https://arxiv.org/html/2608.06741#bib.bib6); Zhuanget al\.,[2025](https://arxiv.org/html/2608.06741#bib.bib42); Provostet al\.,[2026](https://arxiv.org/html/2608.06741#bib.bib41)\)\. Rather than training an LLM to play poker directly, we study how solver\-derived mixed\-strategy distinctions can be converted into contrastive rules that independent LLMs can use\. Additional related work is discussed in Appendix[A](https://arxiv.org/html/2608.06741#A1)\.

## 3Preliminaries

#### Imperfect\-information extensive\-form games\.

No\-Limit Texas Hold’em \(NLH\) is a zero\-sum extensive\-form game with imperfect information; Appendix[D](https://arxiv.org/html/2608.06741#A4)summarizes the rule structure and domain vocabulary used throughout\. Postflop decisions occur after public community cards are revealed: the flop is the betting round after three public cards, and the turn is the round after the fourth public card\. An extensive\-form game is defined by𝒢=\(𝒩,ℋ,𝒵,𝒜,P,u,ℐ\)\\mathcal\{G\}=\(\\mathcal\{N\},\\mathcal\{H\},\\mathcal\{Z\},\\mathcal\{A\},P,u,\\mathcal\{I\}\), where𝒩=\{1,2\}\\mathcal\{N\}=\\\{1,2\\\}is the player set,ℋ\\mathcal\{H\}is the set of finite histories,𝒵⊂ℋ\\mathcal\{Z\}\\subset\\mathcal\{H\}is the set of terminal histories,𝒜​\(h\)\\mathcal\{A\}\(h\)is the set of legal actions after a non\-terminal historyhh,P​\(h\)∈𝒩∪\{c\}P\(h\)\\in\\mathcal\{N\}\\cup\\\{c\\\}specifies whether a player or chance acts athh, andui​\(z\)u\_\{i\}\(z\)is playerii’s payoff at terminal historyzz\. The game is zero\-sum, sou1​\(z\)\+u2​\(z\)=0u\_\{1\}\(z\)\+u\_\{2\}\(z\)=0\. In poker, a history contains public events such as betting actions and community cards, together with private cards dealt by chance\.

Imperfect information is represented by information sets\. For playerii,ℐi\\mathcal\{I\}\_\{i\}partitions the decision histories at whichiiacts\. Historiesh,h′∈I∈ℐih,h^\{\\prime\}\\in I\\in\\mathcal\{I\}\_\{i\}are indistinguishable to playerii: they share the same public betting/card history and the same private hand forii, but may differ in the opponent’s private hand\. A behavioral strategy is therefore a distribution over actions at each information set,πi\(⋅∣I\)∈Δ\(𝒜\(I\)\)\\pi\_\{i\}\(\\cdot\\mid I\)\\in\\Delta\(\\mathcal\{A\}\(I\)\)\. Throughout the paper,hhdenotes a generic non\-terminal decision history, whilezzis reserved for terminal histories\.

#### Nash equilibrium and GTO strategies\.

A strategy profileπ=\(π1,π2\)\\pi=\(\\pi\_\{1\},\\pi\_\{2\}\)induces an expected utilityui​\(π\)u\_\{i\}\(\\pi\)by integrating terminal utilities over chance outcomes and both players’ randomized actions\. A Nash equilibrium \(NE\) is a profileπ∗\\pi^\{\*\}such that no player can improve by unilateral deviation\(Nash,[1951](https://arxiv.org/html/2608.06741#bib.bib37)\):

ui​\(πi∗,π−i∗\)≥ui​\(πi′,π−i∗\)∀i∈𝒩,∀πi′\.u\_\{i\}\(\\pi\_\{i\}^\{\*\},\\pi\_\{\-i\}^\{\*\}\)\\geq u\_\{i\}\(\\pi\_\{i\}^\{\\prime\},\\pi\_\{\-i\}^\{\*\}\)\\qquad\\forall i\\in\\mathcal\{N\},\\ \\forall\\pi\_\{i\}^\{\\prime\}\.\(1\)In poker terminology, a Game\-Theoretic Optimal \(GTO\) strategy is an approximate NE strategy computed by a solver\. Modern poker solvers are commonly based on regret minimization, search, public\-belief reasoning, and subgame solving\(Zinkevichet al\.,[2007](https://arxiv.org/html/2608.06741#bib.bib1); Bowlinget al\.,[2015](https://arxiv.org/html/2608.06741#bib.bib2); Moravčíket al\.,[2017](https://arxiv.org/html/2608.06741#bib.bib15); Brown and Sandholm,[2018](https://arxiv.org/html/2608.06741#bib.bib3)\)\. Solver outputs are commonly reported as mixed action distributions together with action values or counterfactual values\. For a decision historyhh,π∗\(⋅∣h\)\\pi^\{\*\}\(\\cdot\\mid h\)denotes the computed action distribution, andQ∗​\(h,a\)Q^\{\*\}\(h,a\)denotes the value associated with legal actiona∈𝒜​\(h\)a\\in\\mathcal\{A\}\(h\)\. The notationQ∗​\(I,a\)Q^\{\*\}\(I,a\)is used when discussing the corresponding information\-set formalism\.

#### Public belief states and ranges\.

Solvers do not reason about a single fully observed state\. Given a public statessconsisting of the public board and betting sequence, there are many private\-card assignments consistent with what has been observed\. The public belief state \(PBS\)\(Brownet al\.,[2020](https://arxiv.org/html/2608.06741#bib.bib5)\), also called a range representation in poker, is the conditional distribution over these private assignments:βs​\(c1,c2\)=Pr⁡\(c1,c2∣s\)\\beta\_\{s\}\(c\_\{1\},c\_\{2\}\)=\\Pr\(c\_\{1\},c\_\{2\}\\mid s\), wherecic\_\{i\}denotes playerii’s private hand\. A player’s range is the corresponding marginal distribution over that player’s possible private hands\. The solver’s equilibrium policy can be viewed as a high\-dimensional mapping from this public belief state and a particular private hand to a mixed action distribution:f∗:\(s,ci,βs\)↦πi∗\(⋅∣Ii\(s,ci\)\)f^\{\*\}:\\ \(s,c\_\{i\},\\beta\_\{s\}\)\\mapsto\\pi\_\{i\}^\{\*\}\(\\cdot\\mid I\_\{i\}\(s,c\_\{i\}\)\)\. This mapping is high\-dimensional because the action distribution for one private hand can depend on the public state, the player’s range, the opponent’s possible range, and continuation values induced by future play\.

#### Indifference and mixed strategies in Nash equilibrium\.

Mixed strategies are often necessary in imperfect\-information games because deterministic action patterns can reveal exploitable information\. An observed action changes what an opponent can infer about hidden private states\. If an action is associated too strongly with a narrow class of private states, the opponent may be able to respond profitably to that revealed structure\. Equilibrium mixing helps prevent such profitable deviations by distributing probability mass across actions so that no player can improve unilaterally\.

Indifference is a consequence of this equilibrium condition, not an independent assumption\. If two actions are both used with positive probability at an information set, then neither can have strictly higher counterfactual value against the opponent’s equilibrium strategy; otherwise probability could be shifted toward the better action\. Thus supported actions have equal value up to approximation error, while unsupported actions have no higher value\(Nash,[1951](https://arxiv.org/html/2608.06741#bib.bib37); Osborne and Rubinstein,[1994](https://arxiv.org/html/2608.06741#bib.bib38)\)\. In games with perfect recall, behavioral strategies represent a player’s randomized choices at information sets\(Kuhn,[1953](https://arxiv.org/html/2608.06741#bib.bib39)\), and CFR\-style algorithms approach equilibrium by minimizing counterfactual regret\(Zinkevichet al\.,[2007](https://arxiv.org/html/2608.06741#bib.bib1)\)\. Formally, for an information setIIand any actionaain the support of the equilibrium strategy,

Qi∗​\(I,a\)≈Vi∗​\(I\)for​a​with​πi∗​\(a∣I\)\>0,Q\_\{i\}^\{\*\}\(I,a\)\\approx V\_\{i\}^\{\*\}\(I\)\\qquad\\text\{for \}a\\text\{ with \}\\pi\_\{i\}^\{\*\}\(a\\mid I\)\>0,\(2\)withQi∗​\(I,a\)≤Vi∗​\(I\)Q\_\{i\}^\{\*\}\(I,a\)\\leq V\_\{i\}^\{\*\}\(I\)for actions outside the support\.

#### Global dependence in imperfect\-information games\.

The preceding definitions still leave the central difficulty: in imperfect\-information games, a local decision generally cannot be interpreted as an isolated choice at a fully observed state\. The value of a local strategy can depend on the belief over hidden states and on constraints imposed by the full\-game strategy, rather than only on the visible public state\. This is a standard obstacle in imperfect\-information subgame solving: unlike perfect\-information games, an optimal strategy for a reached subgame may depend on strategies in other, unreached parts of the game, so the subgame cannot be solved independently of the full\-game strategy\(Brown and Sandholm,[2017](https://arxiv.org/html/2608.06741#bib.bib40)\)\.

![Refer to caption](https://arxiv.org/html/2608.06741v1/1.png)Figure 1:Why equilibrium strategies in imperfect\-information games are hard to interpret\.\(a\) The same public situation corresponds to many hidden private\-card states inside one information set, so a visible action cannot be explained from public context alone\. \(b\) Equilibrium mixing is constrained by indifference and belief\-dependent continuation values, so action frequencies cannot be reduced to a single best action\. \(c\) A solver table exposes the numerical policy and continuation values, but not the contrastive rule that makes strategically similar hands diverge; our goal is to compress this latent continuation logic into communicable strategic summaries\.

## 4Why Equilibrium Mixing in Poker Is Difficult

NLH is difficult for language models not only because the game tree is large, but because the strategic object to be learned is a*mixed equilibrium policy*over imperfect\-information states\. The relevant target is not a single best action for a visible hand\. It is a range\-level allocation of action frequencies that remains hard to exploit after the opponent updates beliefs from the observed betting line\. This distinction is central to modern poker AI: superhuman systems such as DeepStack, Libratus, and Pluribus rely on equilibrium\-oriented search, self\-play, abstraction, and subgame reasoning in hidden\-information games rather than on human explanations alone\(Moravčíket al\.,[2017](https://arxiv.org/html/2608.06741#bib.bib15); Brown and Sandholm,[2018](https://arxiv.org/html/2608.06741#bib.bib3),[2019b](https://arxiv.org/html/2608.06741#bib.bib4)\)\. These results suggest that solver\-generated equilibrium behavior is a more appropriate source of strategic targets than human verbal heuristics alone\.

#### Publicly available poker discourse provides language, not equilibrium logic\.

Publicly available poker text is abundant but structurally mismatched to the object we need to learn\. Forum posts, coaching examples, and hand histories are selective: they usually explain memorable or exploitative decisions, not the full support of a balanced range at an information set\. This is consistent with recent LLM poker evaluations\. PokerBench reports that strong pretrained LLMs substantially underperform on curated GTO decision spots, with GPT\-4 reaching only53\.55%53\.55\\%accuracy before task\-specific fine\-tuning\(Zhuanget al\.,[2025](https://arxiv.org/html/2608.06741#bib.bib42)\)\. Earlier work similarly finds that ChatGPT and GPT\-4 can discuss starting\-hand value, position, and GTO concepts while still failing to play game\-theoretic optimal poker\(Gupta,[2023](https://arxiv.org/html/2608.06741#bib.bib6)\)\. This gap does not mean that verbal heuristics are useless: conservative advice such as calling rather than raising in marginal aggressive nodes may reduce immediate losses\. The limitation is that such advice is local, whereas equilibrium requires solver\-level frequency allocation across the whole range\. Our own pilot experiments with scraped Two Plus Two \(2\+2\) forum text and SFT on an open Qwen model point in the same direction: next\-token imitation can teach poker vocabulary and local heuristics, but it does not reliably recover basic range\-level decision logic\. We therefore treat publicly available poker discourse as a useful source of terminology and surface\-level heuristics, but not as ground\-truth evidence for equilibrium decision logic\. Appendix[E](https://arxiv.org/html/2608.06741#A5)provides additional details\.

#### Equilibrium mixing is functional, not incidental\.

In imperfect\-information games, randomization is not noise around an underlying pure decision\. It is a mechanism for controlling information leakage\. A bet must often contain value hands, bluffs, protection hands, and blocker\-driven candidates in proportions that prevent profitable counter\-strategies\. Consequently, a hand may bet not because it is locally strong, but because it occupies a necessary role inside the betting range; conversely, a stronger hand may check because its showdown value realizes well and the betting line needs weaker bluff candidates for balance\. The decision boundary is therefore determined jointly by private cards, public board texture, both players’ ranges, blocker effects, and continuation values\. This is exactly the kind of coupled hidden\-state reasoning that CFR\-style and search\-based poker solvers are designed to approximate\(Zinkevichet al\.,[2007](https://arxiv.org/html/2608.06741#bib.bib1); Brown and Sandholm,[2017](https://arxiv.org/html/2608.06741#bib.bib40); Brownet al\.,[2020](https://arxiv.org/html/2608.06741#bib.bib5)\)\.

#### Solver outputs are precise but not communicable\.

Solver outputs provide the desired equilibrium target, but not in a directly communicable form\. A solver table gives action frequencies and continuation values for many private hands under a fixed public state; it does not state the compact contrastive rule explaining why two similar hands diverge\. An LLM can therefore produce a plausible one\-hand rationale while still missing the frequency allocation that makes the whole range balanced\. The core task in this paper is to bridge this gap: first distill solver behavior into a sparse strategic representation, and then articulate local counterfactual distinctions that an independent LLM can transfer to unseen hands\. This motivates our use of MDT as a solver\-grounded intermediate representation and SCCS as the rule\-extraction mechanism\.

## 5Methodology

Our goal is to achieve solver\-guided articulation, which is a conditional prediction problem with an explicit intermediate representation\. At a decision pointhh, letπ∗\(⋅∣h\)\\pi^\{\*\}\(\\cdot\\mid h\)be the computed mixed action distribution, let\{Q∗​\(h,a\)\}a∈𝒜​\(h\)\\\{Q^\{\*\}\(h,a\)\\\}\_\{a\\in\\mathcal\{A\}\(h\)\}be the corresponding action\-value summaries, and let𝐱\\mathbf\{x\}denote a compact representation of public context and continuation summaries\. The solver\-side object is

𝒪\(h\)=\(π∗\(⋅∣h\),\{Q∗\(h,a\)\}a∈𝒜​\(h\),𝐱\),\\mathcal\{O\}\(h\)=\\bigl\(\\pi^\{\*\}\(\\cdot\\mid h\),\\\{Q^\{\*\}\(h,a\)\\\}\_\{a\\in\\mathcal\{A\}\(h\)\},\\mathbf\{x\}\\bigr\),\(3\)and the articulation procedure produces a ruler=A​\(𝒪​\(h\),𝒟\)r=A\(\\mathcal\{O\}\(h\),\\mathcal\{D\}\), where𝒟\\mathcal\{D\}is a reference collection used to locate matched public contexts and policy\-divergent comparisons\. This collection can be expanded by querying additional solver states, for example by extending action continuations or adding private\-hand assignments\. The rule works as an intermediate representation supplied to a downstream predictor\.

The rule is constrained to use a small set of quantities from𝐱\\mathbf\{x\}and comparisons drawn from matched public contexts in𝒟\\mathcal\{D\}\. This rules out explanations that simply restate the full mixed policy table\. It also distinguishes articulation from dominant\-action labeling: the target remains the full distributionπ∗\(⋅∣h\)\\pi^\{\*\}\(\\cdot\\mid h\), and the intermediate rule must preserve a local mixed\-policy distinction rather than only identify the largest\-probability action\.

For evaluation, the target handhtesth\_\{\\mathrm\{test\}\}is excluded from the displayed reference examples, and its solver policy is masked from the downstream predictor\. SCCS uses the target on the solver side to identify its MDT route and select policy\-separated shadow hands, but never displays the target policy\. A predictor receives the public scenario, the target hand, and optionally the rulerr, then outputs a distributionπ~\(⋅∣htest\)\\tilde\{\\pi\}\(\\cdot\\mid h\_\{\\mathrm\{test\}\}\)\. The primary metric is the distance betweenπ~\\tilde\{\\pi\}and the masked solver targetπ∗\(⋅∣htest\)\\pi^\{\*\}\(\\cdot\\mid h\_\{\\mathrm\{test\}\}\), compared against direct prompting and prompting with raw summaries alone\. Under this formulation, a rule is useful only if it improves distributional prediction on an unseen target rather than copying a displayed policy\.

### 5\.1Mixed\-Strategy Decision Tree

The main technique is a sparse mixed\-strategy distillation model, which we call the Mixed\-Strategy Decision Tree \(MDT; Figure[2](https://arxiv.org/html/2608.06741#A0.F2)\)\. For each solver\-labeled decision point, we write the MDT input as𝐱\\mathbf\{x\}\. It contains public context \(board, action history, position, stacks, and pot information\) together with line\-conditioned range\-level and hand\-level continuation summaries derived from solver outputs\. These summaries include EV and EQ quantities and action\-gap quantities under available lines\. In this way,𝐱\\mathbf\{x\}captures decision\-relevant continuation information in a compact form for solver\-policy articulation\.

Given strategic\-summary input𝐱\\mathbf\{x\}, each leafl∈ℒl\\in\\mathcal\{L\}stores a pure actional∈𝒜a\_\{l\}\\in\\mathcal\{A\}\. The router induces a probability mass over leaves,

ρl​\(𝐱\)=∏\(n,c\)∈Path​\(l\)pn,c​\(𝐱\),\\rho\_\{l\}\(\\mathbf\{x\}\)=\\prod\_\{\(n,c\)\\in\\mathrm\{Path\}\(l\)\}p\_\{n,c\}\(\\mathbf\{x\}\),\(4\)and the distilled mixed policy is obtained by aggregating the mass of leaves assigned to each action:

π^​\(a∣𝐱\)=∑l∈ℒρl​\(𝐱\)​𝟏​\[al=a\]\.\\hat\{\\pi\}\(a\\mid\\mathbf\{x\}\)=\\sum\_\{l\\in\\mathcal\{L\}\}\\rho\_\{l\}\(\\mathbf\{x\}\)\\mathbf\{1\}\[a\_\{l\}=a\]\.\(5\)Thus, MDT represents mixed strategies through probabilistic routing over pure\-action leaves, rather than by storing a mixed action distribution inside each leaf\. A leaf never carries a full action\-frequency vector; it names one action prototype\. The leaves remain intentionally simple, while the routing structure encodes when each pure strategic action should receive probability mass\. The final model uses hard sparse local routers; implementation details and router ablations are provided in Appendix[F\.1](https://arxiv.org/html/2608.06741#A6.SS1)\.

We train the MDT against solver policy labels using a composite objective\. Letπ∗\(⋅∣h\)\\pi^\{\*\}\(\\cdot\\mid h\)denote the oracle action distribution and letQ∗​\(h,a\)Q^\{\*\}\(h,a\)denote solver\-provided action values\. We useℒtask=λπ​ℒL1\+λev​ℒEV\\mathcal\{L\}\_\{\\mathrm\{task\}\}=\\lambda\_\{\\pi\}\\mathcal\{L\}\_\{L\_\{1\}\}\+\\lambda\_\{\\mathrm\{ev\}\}\\mathcal\{L\}\_\{\\mathrm\{EV\}\}, with

ℒL1=1\|𝒜\|∑a∈𝒜\|π∗\(a∣h\)−π^\(a∣h\)\|,\\mathcal\{L\}\_\{L\_\{1\}\}=\\frac\{1\}\{\|\\mathcal\{A\}\|\}\\sum\_\{a\\in\\mathcal\{A\}\}\|\\pi^\{\*\}\(a\\mid h\)\-\\hat\{\\pi\}\(a\\mid h\)\|,\(6\)ℒEV=V∗​\(h\)−∑a∈𝒜π^​\(a∣h\)​Q∗​\(h,a\),V∗​\(h\)=∑a∈𝒜π∗​\(a∣h\)​Q∗​\(h,a\)\.\\mathcal\{L\}\_\{\\mathrm\{EV\}\}=V^\{\*\}\(h\)\-\\sum\_\{a\\in\\mathcal\{A\}\}\\hat\{\\pi\}\(a\\mid h\)Q^\{\*\}\(h,a\),\\qquad V^\{\*\}\(h\)=\\sum\_\{a\\in\\mathcal\{A\}\}\\pi^\{\*\}\(a\\mid h\)Q^\{\*\}\(h,a\)\.\(7\)HereℒL1\\mathcal\{L\}\_\{L\_\{1\}\}measures fidelity to the oracle mixing frequencies, whileℒEV\\mathcal\{L\}\_\{\\mathrm\{EV\}\}measures the oracle\-conditioned EV gap under solver\-provided action values\. We treat this strictly as a local fidelity measure, not as a full\-game exploitability estimate\.

There are two optimization strategies\. The first strategy keeps the router differentiable and adds sparsity\-inducing regularization:ℒsoft=ℒtask\+λ1​‖𝐰‖1\+λ2​ℋ​\(𝐩\)\+λ3​ℒortho\\mathcal\{L\}\_\{\\mathrm\{soft\}\}=\\mathcal\{L\}\_\{\\mathrm\{task\}\}\+\\lambda\_\{1\}\\\|\\mathbf\{w\}\\\|\_\{1\}\+\\lambda\_\{2\}\\mathcal\{H\}\(\\mathbf\{p\}\)\+\\lambda\_\{3\}\\mathcal\{L\}\_\{\\mathrm\{ortho\}\}\. This regime generally achieves better fidelity, but in practice it leaves a long tail of weak summaries with non\-zero influence\. As a result, the visible top summaries do not fully explain the final decision\. The second strategy progressively converts the router into a strict top\-KKsparse structure\. We use a teacher\-assisted curriculum: a dense teacher first absorbs the raw mapping, then a structured student inherits topology and is finally locked into hard summary selection withK=5K=5active local summaries per node\. The goal is not maximal fidelity, but explicitness: the displayed sparse summaries are the variables used to compute the router probabilities\. We compare both approaches in the experiments\.

### 5\.2Scenario\-Constrained Counterfactual Sampling

The MDT exposes which sparse summaries are used along a route, but a routed path alone is not yet a communicable rule\. A path can indicate that a hand depends on draw strength, kicker quality, hand EV, or range\-relative equity, but it does not identify which local change would move the hand across a strategic boundary\. This distinction matters in poker because strategically nearby hands can share the same public context and similar raw equity while occupying different roles inside the equilibrium range\.

SCCS addresses this gap by explaining a target hand contrastively\. Instead of describing the hand in isolation, it selects a shadow hand from a reference collection of solver queries in the same public scenario whose solver policy diverges and whose MDT route crosses a critical branching decision\. For example, under the same board and betting line, a weak\-kicker draw may be used as a semi\-bluff, while a higher\-showdown\-value version may check because it realizes enough equity without building the pot\. By fixing the public scenario and varying only the private hand, SCCS isolates the strategic quantity that changes the hand’s role\.

For a target statehh, SCCS identifies its trained\-tree route, finds policy\-divergent shadow hands under the same public context, localizes the routing boundary where the target and shadow diverge, and converts the active\-summary contrast into a natural\-language rule\. The procedure is targeted at policy\-divergent samples rather than nearest visual or lexical neighbors; details are given in Algorithm[1](https://arxiv.org/html/2608.06741#alg1)and Appendix[F\.2](https://arxiv.org/html/2608.06741#A6.SS2)\. The resulting rule also defines a communicability test: an independent reasoner should be able to apply the extracted local distinction to an unseen target state with matched public context\.

## 6Experiments and Results

The NLH distillation and communicability evaluation aims to answer three questions\. First, can the proposed MDT articulate the solver output to improve LLM reasoning? Second, do the rules extracted by SCCS help independent LLMs use solver\-derived distinctions on unseen target hands in matched public contexts? Third, what kind of strategic distinction does the rule expose in an individual case?

### 6\.1Solver Oracle Interface and Evaluation Samples

Our experiments use over 250 million solver\-labeled postflop decision samples obtained by querying a commercial, high\-end NLH solver oracle, including mixed policies and continuation\-value quantities\. These samples comprise approximately 16M flop decisions and 235M turn decisions under a 6\-player 100BB no\-rake NLH configuration\. Decision points involve postflop spots where two players are left in the game\. We include five preflop configurations: SB vs BB single\-raised pot and 3\-bet pot, BTN vs BB single\-raised pot and 3\-bet pot, and BTN vs SB 3\-bet pot\. The same pipeline can continue to generate additional labeled states for new board textures, action branches, private\-hand assignments, and configuration choices\.

To ensure coverage of strategically distinct public states, flop data is sampled from 1,755 strategic board textures\. For each board we record multiple canonical action nodes, including the root, check line, bet line, bet\-call line, and bet\-bet line\. Turn data is generated by extending representative flop branches such as check\-check, bet\-call, and check\-bet\-call, then sampling five turn cards per branch\. Bet sizes follow the solver’s abstraction; all\-in actions are folded into the generic “bet” action category\.

For communicability, we construct matched\-context tests with unseen target hands\. Each test fixes the public context and asks an independent LLM to predict the solver\-equilibrium mixed strategy for an unseen target hand\. We compare four prompting conditions:Direct, which provides only the public state and hand;Direct\+Summaries, which additionally provides the numerical summaries;Route\-only, which displays the target hand’s MDT route trace but omits every reference/shadow hand and cross\-hand comparison; andSCCS Rule, which adds a contrastive rule extracted from policy\-divergent shadow hands in the trained MDT\. Route\-only therefore controls for exposing the tree computation without the matched counterfactual contrast\. To prevent the SCCS rule from serving as a near\-label lookup, the target hand is never included among the rule\-displayed reference/shadow hands, and its solver policy remains hidden from the LLM\. Its solver policy must also differ from the policy of every rule\-displayed hand by at least0\.200\.20under the same action\-averagedL1L\_\{1\}metric\. Thus, the rule cannot be applied by copying a displayed strategy from a near\-duplicate hand; improvement requires transferring the extracted strategic distinction to the held\-out target\. We reportL1L\_\{1\}both to the solver target and to the distilled MDT policy\. Additional construction details, including the SCCS matching criteria and prompt format, are reported in the appendix[M](https://arxiv.org/html/2608.06741#A13)\.

### 6\.2Training Loss and Distillation Fidelity

Table 1:Training loss and distillation fidelity on the NLH solver\-labeled evaluation set\.LowerL1L\_\{1\}and oracle\-conditioned EV gap are better\. The oracle row is zero under this metric by construction\. Oracle EV Gap is a local fidelity metric under solver\-provided action values, not full\-game exploitability\. The final hard MDT is the model used for communicable rule extraction\.The distillation results motivate MDT as an articulation layer rather than only a predictor\. Within the same strategic\-summary input, tree\-structured routing fits the solver policy substantially better than flat MLPs at comparable or smaller parameter counts in our experiments, suggesting that mixed\-equilibrium decisions benefit from a hierarchical representation: coarse public\-state and range\-level conditions first select a strategic region, while hand\-level summaries define local action\-frequency boundaries\. The hard sparse MDT sacrifices some fidelity by allowing only a small set of summaries at each node, but this constraint makes the routing boundaries explicit enough for SCCS rule extraction\. Full training\-loss comparisons and EV\-gap analysis are reported in Appendix[B](https://arxiv.org/html/2608.06741#A2)\.

### 6\.3Communicability on Unseen Target Hands

Table[2](https://arxiv.org/html/2608.06741#S6.T2)is the main quantitative result\. Across eight LLM configurations, SCCS rules reduce averageL1L\_\{1\}to the solver target from0\.2110\.211to0\.1000\.100, a52\.6%52\.6\\%relative improvement over direct prompting\. The distance to the distilled MDT policy falls from0\.2040\.204to0\.1140\.114, a44\.0%44\.0\\%relative improvement\. The SCCS columns also show low variation across the evaluated LLM configurations, suggesting that the effect is not specific to one model setting\. SCCS also improves on Route\-only:L1L\_\{1\}falls from0\.1730\.173to0\.1000\.100relative to the solver and from0\.1720\.172to0\.1140\.114relative to MDT, reductions of42\.2%42\.2\\%and33\.7%33\.7\\%, respectively\. The SCCS point estimate is lower than Route\-only in all eight configurations under both targets\. Argmax\-action agreement, defined as whether the highest\-probability predicted action matches the highest\-probability solver action, also rises from57\.2%57\.2\\%to76\.1%76\.1\\%from Direct to SCCS\.

The summaries\-only condition is intentionally included as a negative control\. Direct\+Summaries has worse averageL1L\_\{1\}to the solver target \(0\.2560\.256\) than the direct prompt, suggesting that raw continuation quantities are not automatically communicable to an LLM\. Without a contrastive rule, the model may not know which summary changes are decision\-relevant in the current scenario; the extra quantities can be treated as noise or can reinforce a locally conservative interpretation\. SCCS improves performance because it organizes those quantities around a policy\-divergent boundary under the same public context\.

Table 2:Communicability on unseen target hands in matched public contexts\.LowerL1L\_\{1\}is better\. Each model entry reports mean±\\pmstandard error over unseen target cases\. The Mean row reports mean±\\pmsample standard deviation across eight LLM configurations\.The table supports a specific interpretation of the result\. SCCS does not make an LLM a standalone poker agent; it makes a local solver distinction communicable enough for an independent model to transfer the extracted rule to an unseen target hand in a similar strategic neighborhood\. The Route\-only comparison further shows that exposing the target route is useful, but the matched shadow contrast communicates additional information about how probability mass moves across the boundary\. In a post\-hoc matched diagnostic restricted to predictions where Direct, Route\-only, and SCCS all select the correct solver argmax, SCCS still lowers Route\-onlyL1L\_\{1\}by38\.8%38\.8\\%to the solver and33\.0%33\.0\\%to MDT\. Thus, the difference is not explained only by correcting the dominant action; it includes the remaining probability allocation\. Appendix[H](https://arxiv.org/html/2608.06741#A8)gives the construction, paired diagnostics, and parse\-success accounting\.

### 6\.4Case Study: From Over\-Folding to Draw\-Aware Mixing

To qualitatively inspect the strategy acquired by the LLM, we inspect one of the most difficult spots for humans: SB vs BB single raised pot\. Table[5](https://arxiv.org/html/2608.06741#A3.T5)in the appendix reports the case where the public state is a8s6h5dboard after the bet–raise actions\. Now the out\-of\-position player \(SB\) is facing pressure withTd9c\.

Directly prompting an LLM identifies the hand as an offsuit gutshot with poor equity realization and assigns most mass to Fold\. Despite that the LLM is able to correctly identify the hand bucket, the solver disagrees with its action and assigns no fold mass and mixes between Call and the Raise action\. A studied human player understands that at relatively deep stack depths, it is crucial to continue to the turn withT9to ensure that the absolute nuts remain in their range on7turn or river runouts\. This holds true even if a naive calculation of immediate pot odds suggests the call is unprofitable\. At the same time, with the wide range of SB vs BB, the overcardsJandTprovides additional equity even without the backdoor flush draw\. SCCS makes this distinction communicable by contrasting a folding prototype \(Jc4c\) with the same gut shot bucket\. Their differences in draw strength, hand EV, and range\-relative equity moveTd9cinto a continuing draw class with some aggressive raise frequency\.

This example illustrates why conservative heuristics are not enough\. The direct and summaries\-only prompts over\-fold, while the MDT rule exposes the local boundary between weak folding draws and continuing draw candidates\. With that boundary exposed, GPT\-5\.5 high shifts toward the solver’s Call/Bet mixture\. Appendix[C](https://arxiv.org/html/2608.06741#A3)provides the full case details\.

### 6\.5Exact Strategic Evaluation in Complete River Endgames

The localL1L\_\{1\}and oracle\-conditioned EV gap measure fidelity at sampled decisions\. To complement them with a complete\-policy evaluation, we construct two full heads\-up NLH River endgames initialized from Subgames 3 and 4 released byBrown and Sandholm \([2019a](https://arxiv.org/html/2608.06741#bib.bib46)\)\. The two public states begin with root pots of55and37\.537\.5big blinds, respectively\. From each state, our two\-size action tree proceeds through fold or showdown\. Soft and Hard MDT use the same depth\-4 ternary probabilistic\-routing architecture with pure\-action leaves; Soft may use all eligible summaries, while Hard locks each router to five summaries\. We then compute exact exploitability within each River tree, defined here as the average additional payoff available when each player independently switches to its best response\.

Table 3:Exact strategic evaluation in two complete local River endgames\.Values are milli\-big\-blinds per game \(mbb/g\), followed by exploitability as a percentage of the root pot\. Hard–Soft reports the observed difference associated with strict Top\-5 sparsity\. Deep CFR is a same\-tree finite\-training reference\. Lower is better\.The larger absolute values in Subgame 4 reflect its7\.5×7\.5\\timeslarger root pot\. After normalization, the two states agree closely: Soft costs0\.13380\.1338–0\.1467%0\.1467\\%of the root pot and Hard Top\-5 costs0\.34350\.3435–0\.3548%0\.3548\\%, making the measured Hard–Soft difference approximately0\.210\.21percentage points in both endgames\. For scale, we train Deep CFR\(Brownet al\.,[2019](https://arxiv.org/html/2608.06741#bib.bib47)\)on the identical trees with10,00010\{,\}000sampled traversals per player per iteration; Table[3](https://arxiv.org/html/2608.06741#S6.T3)reports its linearly weighted average strategy at iteration 800\. This is a matched finite\-compute reference rather than a convergence limit\. The MDT policies are trained and strategically selected on these same trees, so this experiment measures in\-domain strategic compression rather than cross\-game generalization\. Appendix[J](https://arxiv.org/html/2608.06741#A10)provides the states, fixed\-target value losses, policy errors, training and selection details, and metric definition\.

### 6\.6Beyond NLH: Liar’s Dice

We additionally instantiate the solver–MDT–SCCS pipeline in two\-player Liar’s Dice, with two three\-faced dice per player and bids up to quantity three\. Each player has six unordered private dice types\. A full\-tree CFR\+ solver supplies the approximate\-Nash target, and six\-fold cross\-fitting holds out one private type per public context\. MDT uses 17 summaries of the public history, posterior belief, private dice, and equity; per\-action continuation values and Q\-gaps are neither MDT inputs nor shown in the prompts\.

Table 4:Exploratory Liar’s Dice communicability results\.Entries are unweighted means across eight LLM configurations; each configuration mean is computed over its successfully parsed outputs\. No missing output is imputed\. LowerL1L\_\{1\}is better\. Full per\-model results and parse coverage are in Appendix[I](https://arxiv.org/html/2608.06741#A9)\.On this exploratory surface, SCCS lowers solverL1L\_\{1\}by39\.7%39\.7\\%relative to Direct and has the lowest descriptive mean under both targets\. Its incremental improvement over Route\-only is smaller than in NLH\. One plausible reason is the much sparser within\-context comparison space: Liar’s Dice has only six private types per public context, while an NLH range contains hundreds of feasible private hands\. After imposing the same\-context, route\-divergence, and policy\-divergence constraints, fewer nearby shadows remain, so the target route already conveys more of the available local information\. This explanation is consistent with the observed setting but is not a causal game\-size ablation\.

We also assemble one held\-out MDT prediction for every Liar’s Dice information set from the six cross\-fitting folds and evaluate the resulting policy against the fixed approximate\-Nash opponent over the complete tree\. The seat\-averaged value loss is0\.0580\.058per game on the\[−1,\+1\]\[\-1,\+1\]terminal payoff scale\. Together with the NLH result, this experiment shows that the computational interface is not tied to poker cards, poker vocabulary, or a commercial poker solver\. Appendix[I](https://arxiv.org/html/2608.06741#A9)records the exploratory selection, partial\-response and parse\-success accounting, per\-configuration results, and fully reproducible game/solver setup\.

## 7Scope and Broader Applications

The empirical claim of this paper is deliberately specific: solver\-derived rules improve an independent LLM’s inference of held\-out mixed\-Nash policies in imperfect\-information games\. SCCS is especially natural in this setting because it can hold the public situation fixed while contrasting feasible private states\. The compact MDT representation is a human\-readable communication layer over a large solver state, rather than an assumption that the underlying task is intrinsically low\-dimensional\.

The broader solver\-to\-LLM paradigm requires three conditions\. First, a reliable solver, planner, search procedure, or simulator\-backed optimizer must produce decisions and evaluations that are trustworthy or independently verifiable\. Second, the task must be difficult enough that human demonstrations are incomplete, unreliable, or systematically below the desired solution quality\. Third, the solver must generate an abundant stream of solved states, trajectories, alternatives, or evaluations\. Under these conditions, solver\-generated supervision can expose structure that is absent from human records rather than merely imitate the human\-data ceiling\.

The closest direct extensions are other solver\-backed imperfect\-information games\. In Bayesian security and cyber\-defense games, equilibrium solvers can vary private threat signals or attacker types under the same public constraints; articulated contrasts could strengthen an LLM’s reasoning about how hidden\-risk beliefs change mixed defense allocations\. In discretized auctions or bargaining games, private valuations or reservation values play the role of hidden types; solver\-derived rules could teach how those types change equilibrium bid or offer distributions\. These settings retain the public/private decomposition used by SCCS, although their domain summaries would differ from poker’s\.

The paradigm is potentially broader than the present SCCS construction\. In chess and Go, superhuman engines can generate effectively unlimited positions, policies, and values; extracted rules could strengthen long\-horizon planning and transfer of engine\-discovered strategic patterns to unseen positions\(Silveret al\.,[2017](https://arxiv.org/html/2608.06741#bib.bib18),[2018](https://arxiv.org/html/2608.06741#bib.bib44)\)\. In weather forecasting, validated numerical and ensemble systems generate trajectories under controlled initial\-condition changes\(Lamet al\.,[2023](https://arxiv.org/html/2608.06741#bib.bib49)\); their outputs could strengthen LLMs’ physical, causal, and probabilistic reasoning, including tracing how interacting pressure, moisture, and wind patterns alter extreme\-event risk and expressing calibrated uncertainty\. In chip and circuit design, electronic\-design\-automation optimizers, simulators, and formal verification explore designs that humans cannot enumerate\(Mirhoseiniet al\.,[2021](https://arxiv.org/html/2608.06741#bib.bib50)\); solver\-derived rules could strengthen LLMs’ constraint\-aware multi\-objective reasoning, failure diagnosis, and verifiable design refinement across timing, power, area, and stability\. These are prospective applications of the broader paradigm, not claims that SCCS transfers unchanged: other domains may require different intermediate representations and counterfactual constructions\.

## 8Conclusion

This paper studies how to make solver\-computed mixed strategies usable by LLMs through communicable rules\. Human play, commentary, and human\-generated rationales provide weak supervision for complex games because they are selective, heuristic, and biased toward pure actions, whereas equilibrium play in imperfect\-information games requires precise mixed\-strategy frequency allocation\. Solver outputs provide the desired optimality signal, but only as numerical policy distributions and continuation values\.

We introduced Mixed\-Strategy Decision Tree \(MDT\) to articulate solver\-implied decision logic into sparse strategic rules, and Scenario\-Constrained Counterfactual Sampling \(SCCS\) to expose local contrastive boundaries between hands in the same public context\. In NLH, using over 250 million solver\-labeled mixed\-strategy decisions, the resulting rules reduce LLM action\-averagedL1L\_\{1\}distance to the solver target by52\.6%52\.6\\%across eight LLM configurations\. These results suggest that solver\-generated data can serve not only as supervision for policy prediction, but also as a source of readable reasoning traces for LLMs\. The Route\-only ablation, complete River\-endgame evaluations, and fully releasable Liar’s Dice game/solver pipeline with archived outputs further measure the contribution of the contrastive rule, the strategic cost of strict Top\-5 sparsity, and portability to another imperfect\-information mixed\-equilibrium game\.

## References

- J\. Achiam, S\. Adler, S\. Agarwal, L\. Ahmad, I\. Akkaya, F\. L\. Aleman, D\. Almeida, J\. Altenschmidt, S\. Altman, S\. Anadkat,et al\.\(2023\)GPT\-4 technical report\.arXiv preprint arXiv:2303\.08774\.Cited by:[§1](https://arxiv.org/html/2608.06741#S1.p1.1)\.
- Neural additive models: interpretable machine learning with neural nets\.Advances in Neural Information Processing Systems34,pp\. 4699–4711\.Cited by:[§1](https://arxiv.org/html/2608.06741#S1.p6.1)\.
- O\. Bastani, Y\. Pu, and A\. Solar\-Lezama \(2018\)Verifiable reinforcement learning via policy extraction\.Advances in neural information processing systems31\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px2.p1.1)\.
- M\. Bowling, N\. Burch, M\. Johanson, and O\. Tammelin \(2015\)Heads\-up limit hold’em poker is solved\.Science347\(6218\),pp\. 145–149\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px2.p1.8)\.
- N\. Brown, A\. Bakhtin, A\. Lerer, and Q\. Gong \(2020\)Combining deep reinforcement learning and search for imperfect\-information games\.Advances in neural information processing systems33,pp\. 17057–17069\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px1.p1.1),[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px3.p1.5),[§4](https://arxiv.org/html/2608.06741#S4.SS0.SSS0.Px2.p1.1)\.
- N\. Brown, A\. Lerer, S\. Gross, and T\. Sandholm \(2019\)Deep counterfactual regret minimization\.InProceedings of the 36th International Conference on Machine Learning,Proceedings of Machine Learning Research, Vol\.97,pp\. 793–802\.External Links:[Link](https://proceedings.mlr.press/v97/brown19b.html)Cited by:[§6\.5](https://arxiv.org/html/2608.06741#S6.SS5.p2.7)\.
- N\. Brown and T\. Sandholm \(2017\)Safe and nested subgame solving for imperfect\-information games\.Advances in neural information processing systems30\.Cited by:[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px5.p1.1),[§4](https://arxiv.org/html/2608.06741#S4.SS0.SSS0.Px2.p1.1)\.
- N\. Brown and T\. Sandholm \(2018\)Superhuman ai for heads\-up no\-limit poker: libratus beats top professionals\.Science359\(6374\),pp\. 418–424\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.06741#S1.p3.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px1.p1.1),[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px2.p1.8),[§4](https://arxiv.org/html/2608.06741#S4.p1.1)\.
- N\. Brown and T\. Sandholm \(2019a\)Solving imperfect\-information games via discounted regret minimization\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.33,pp\. 1829–1836\.External Links:[Document](https://dx.doi.org/10.1609/aaai.v33i01.33011829)Cited by:[Appendix J](https://arxiv.org/html/2608.06741#A10.SS0.SSS0.Px1.p1.8),[§6\.5](https://arxiv.org/html/2608.06741#S6.SS5.p1.3)\.
- N\. Brown and T\. Sandholm \(2019b\)Superhuman ai for multiplayer poker\.Science365\(6456\),pp\. 885–890\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.06741#S1.p1.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px1.p1.1),[§4](https://arxiv.org/html/2608.06741#S4.p1.1)\.
- N\. Frosst and G\. Hinton \(2017\)Distilling a neural network into a soft decision tree\.arXiv preprint arXiv:1711\.09784\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px2.p1.1)\.
- S\. Ganzfried and T\. Sandholm \(2014\)Potential\-aware imperfect\-recall abstraction with earth mover’s distance in imperfect\-information games\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.28\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1)\.
- S\. Gunasekar, Y\. Zhang, J\. Aneja, C\. C\. T\. Mendes, A\. Del Giorno, S\. Gopi, M\. Javaheripi, P\. Kauffmann, G\. de Rosa, O\. Saarikivi,et al\.\(2023\)Textbooks are all you need\.arXiv preprint arXiv:2306\.11644\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px3.p1.1)\.
- J\. Guo, B\. Yang, P\. Yoo, B\. Y\. Lin, Y\. Iwasawa, and Y\. Matsuo \(2024\)Suspicion\-agent: playing imperfect information games with theory of mind aware gpt\-4\.InConference on Language Modeling,External Links:[Link](https://openreview.net/forum?id=F2yGbwXJAi)Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px4.p1.4)\.
- A\. Gupta \(2023\)Are chatgpt and gpt\-4 good poker players? a pre\-flop analysis\.arXiv preprint arXiv:2308\.12466\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px3.p1.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px3.p1.1),[§4](https://arxiv.org/html/2608.06741#S4.SS0.SSS0.Px1.p1.1)\.
- G\. Hinton, O\. Vinyals, and J\. Dean \(2015\)Distilling the knowledge in a neural network\.arXiv preprint arXiv:1503\.02531\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1)\.
- C\. Huang, Y\. Cao, Y\. Wen, T\. Zhou, and Y\. Zhang \(2024\)PokerGPT: an end\-to\-end lightweight solver for multi\-player texas hold’em via large language model\.arXiv preprint arXiv:2401\.06781\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px3.p1.1),[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px4.p1.4)\.
- R\. A\. Jacobs, M\. I\. Jordan, S\. J\. Nowlan, and G\. E\. Hinton \(1991\)Adaptive mixtures of local experts\.Neural computation3\(1\),pp\. 79–87\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1)\.
- P\. W\. Koh, T\. Nguyen, Y\. S\. Tang, S\. Mussmann, E\. Pierson, B\. Kim, and P\. Liang \(2020\)Concept bottleneck models\.InInternational Conference on Machine Learning,pp\. 5338–5348\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1)\.
- H\. W\. Kuhn \(1953\)Extensive games and the problem of information\.InContributions to the Theory of Games II,H\. W\. Kuhn and A\. W\. Tucker \(Eds\.\),Annals of Mathematics Studies, Vol\.28,pp\. 193–216\.Cited by:[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px4.p2.2)\.
- R\. Lam, A\. Sanchez\-Gonzalez, M\. Willson, P\. Wirnsberger, M\. Fortunato, F\. Alet, S\. Ravuri, T\. Ewalds, Z\. Eaton\-Rosen, W\. Hu,et al\.\(2023\)Learning skillful medium\-range global weather forecasting\.Science382\(6677\),pp\. 1416–1421\.External Links:[Document](https://dx.doi.org/10.1126/science.adi2336)Cited by:[§7](https://arxiv.org/html/2608.06741#S7.p4.1)\.
- B\. Li and L\. Huang \(2025\)Efficient online pruning and abstraction for imperfect information extensive\-form games\.InThe Thirteenth International Conference on Learning Representations,Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1)\.
- M\. Lin, E\. Dai, H\. Liu, X\. Tang, Y\. Yan, Z\. Dai, J\. Zeng, Z\. Zhang, F\. Wang, H\. Gao, C\. Luo, X\. Zhang, Q\. He, and S\. Wang \(2026\)How far are llms from professional poker players? revisiting game\-theoretic reasoning with agentic tool use\.InThe Fourteenth International Conference on Learning Representations,External Links:[Link](https://openreview.net/forum?id=vV54ShHvGi)Cited by:[§1](https://arxiv.org/html/2608.06741#S1.p3.1)\.
- T\. McGrath, A\. Kapishnikov, N\. Tomašev, A\. Pearce, M\. Wattenberg, D\. Hassabis, B\. Kim, U\. Paquet, and V\. Kramnik \(2022\)Acquisition of chess knowledge in alphazero\.Proceedings of the National Academy of Sciences119\(47\),pp\. e2206625119\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px2.p1.1)\.
- A\. Mirhoseini, A\. Goldie, M\. Yazgan, J\. W\. Jiang, E\. Songhori, S\. Wang, Y\. Lee, E\. Johnson, O\. Pathak, A\. Nova,et al\.\(2021\)A graph placement methodology for fast chip design\.Nature594,pp\. 207–212\.External Links:[Document](https://dx.doi.org/10.1038/s41586-021-03544-w)Cited by:[§7](https://arxiv.org/html/2608.06741#S7.p4.1)\.
- M\. Moravčík, M\. Schmid, N\. Burch, V\. Lisỳ, D\. Morrill, N\. Bard, T\. Davis, K\. Waugh, M\. Johanson, and M\. Bowling \(2017\)Deepstack: expert\-level artificial intelligence in heads\-up no\-limit poker\.Science356\(6337\),pp\. 508–513\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.06741#S1.p3.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px1.p1.1),[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px2.p1.8),[§4](https://arxiv.org/html/2608.06741#S4.p1.1)\.
- J\. Nash \(1951\)Non\-cooperative games\.Annals of Mathematics54\(2\),pp\. 286–295\.Cited by:[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px2.p1.3),[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px4.p2.2)\.
- M\. J\. Osborne and A\. Rubinstein \(1994\)A course in game theory\.MIT Press\.Cited by:[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px4.p2.2)\.
- M\. Provost, N\. Ilenic, C\. Solinas, and P\. Beardsell \(2026\)GTO wizard benchmark\.External Links:2603\.23660,[Link](https://arxiv.org/abs/2603.23660)Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px3.p1.1),[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px4.p1.4),[§1](https://arxiv.org/html/2608.06741#S1.p2.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px3.p1.1)\.
- A\. A\. Rusu, S\. G\. Colmenarejo, C\. Gulcehre, G\. Desjardins, J\. Kirkpatrick, R\. Pascanu, V\. Mnih, K\. Kavukcuoglu, and R\. Hadsell \(2015\)Policy distillation\.arXiv preprint arXiv:1511\.06295\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1)\.
- N\. Shazeer, A\. Mirhoseini, K\. Maziarz, A\. Davis, Q\. Le, G\. Hinton, and J\. Dean \(2017\)Outrageously large neural networks: the sparsely\-gated mixture\-of\-experts layer\.arXiv preprint arXiv:1701\.06538\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1)\.
- D\. Silver, A\. Huang, C\. J\. Maddison, A\. Guez, L\. Sifre, G\. Van Den Driessche, J\. Schrittwieser, I\. Antonoglou, V\. Panneershelvam, M\. Lanctot,et al\.\(2016\)Mastering the game of go with deep neural networks and tree search\.nature529\(7587\),pp\. 484–489\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1)\.
- D\. Silver, T\. Hubert, J\. Schrittwieser, I\. Antonoglou, M\. Lai, A\. Guez, M\. Lanctot, L\. Sifre, D\. Kumaran, T\. Graepel,et al\.\(2018\)A general reinforcement learning algorithm that masters chess, shogi, and go through self\-play\.Science362\(6419\),pp\. 1140–1144\.External Links:[Document](https://dx.doi.org/10.1126/science.aar6404)Cited by:[§1](https://arxiv.org/html/2608.06741#S1.p1.1),[§7](https://arxiv.org/html/2608.06741#S7.p4.1)\.
- D\. Silver, J\. Schrittwieser, K\. Simonyan, I\. Antonoglou, A\. Huang, A\. Guez, T\. Hubert, L\. Baker, M\. Lai, A\. Bolton,et al\.\(2017\)Mastering the game of go without human knowledge\.nature550\(7676\),pp\. 354–359\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.06741#S1.p1.1),[§7](https://arxiv.org/html/2608.06741#S7.p4.1)\.
- D\. Silver and R\. S\. Sutton \(2025\)Welcome to the era of experience\.Google AI1,pp\. 11\.Cited by:[§1](https://arxiv.org/html/2608.06741#S1.p8.1)\.
- A\. Verma, V\. Murali, R\. Singh, P\. Kohli, and S\. Chaudhuri \(2018\)Programmatically interpretable reinforcement learning\.InInternational Conference on Machine Learning,pp\. 5045–5054\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px2.p1.1)\.
- S\. Wachter, B\. Mittelstadt, and C\. Russell \(2017\)Counterfactual explanations without opening the black box: automated decisions and the gdpr\.Harv\. JL & Tech\.31,pp\. 841\.Cited by:[§1](https://arxiv.org/html/2608.06741#S1.p6.1)\.
- J\. Wei, X\. Wang, D\. Schuurmans, M\. Bosma, F\. Xia, E\. Chi, Q\. V\. Le, D\. Zhou,et al\.\(2022\)Chain\-of\-thought prompting elicits reasoning in large language models\.Advances in neural information processing systems35,pp\. 24824–24837\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px3.p1.1),[§1](https://arxiv.org/html/2608.06741#S1.p1.1)\.
- E\. Zelikman, Y\. Wu, J\. Mu, and N\. Goodman \(2022\)Star: bootstrapping reasoning with reasoning\.Advances in Neural Information Processing Systems35,pp\. 15476–15488\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px3.p1.1)\.
- R\. Zhuang, A\. Gupta, R\. Yang, A\. Rahane, Z\. Li, and G\. Anumanchipalli \(2025\)Pokerbench: training large language models to become professional poker players\.InProceedings of the AAAI Conference on Artificial Intelligence,Vol\.39,pp\. 26175–26182\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px3.p1.1),[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px4.p1.4),[§1](https://arxiv.org/html/2608.06741#S1.p2.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px3.p1.1),[§4](https://arxiv.org/html/2608.06741#S4.SS0.SSS0.Px1.p1.1)\.
- M\. Zinkevich, M\. Johanson, M\. Bowling, and C\. Piccione \(2007\)Regret minimization in games with incomplete information\.Advances in neural information processing systems20\.Cited by:[Appendix A](https://arxiv.org/html/2608.06741#A1.SS0.SSS0.Px1.p1.1),[§1](https://arxiv.org/html/2608.06741#S1.p3.1),[§2](https://arxiv.org/html/2608.06741#S2.SS0.SSS0.Px1.p1.1),[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px2.p1.8),[§3](https://arxiv.org/html/2608.06741#S3.SS0.SSS0.Px4.p2.2),[§4](https://arxiv.org/html/2608.06741#S4.SS0.SSS0.Px2.p1.1)\.

![Refer to caption](https://arxiv.org/html/2608.06741v1/2.png)Figure 2:Distilling solver policies into communicable strategic rules\.Public context and continuation outputs form the compact MDT input𝐱\\mathbf\{x\}, while the solver policy supplies the supervised target\. MDT uses sparse routing over the input summaries to assign probability mass to pure\-action leaf prototypes; the mixed strategy arises from the routing distribution, not from mixed leaves\. SCCS compares shadow hands with matched public context but clear solver\-policy divergence, then converts the routing contrast into a rule that an independent reasoner can apply to unseen target hands\.## Appendix AAdditional Related Work

#### Superhuman solvers as silent oracles\.

The resolution of imperfect\-information games has been driven by equilibrium\-finding algorithms like Counterfactual Regret Minimization \(CFR\) and its variants\(Zinkevichet al\.,[2007](https://arxiv.org/html/2608.06741#bib.bib1); Bowlinget al\.,[2015](https://arxiv.org/html/2608.06741#bib.bib2); Brown and Sandholm,[2018](https://arxiv.org/html/2608.06741#bib.bib3),[2019b](https://arxiv.org/html/2608.06741#bib.bib4); Moravčíket al\.,[2017](https://arxiv.org/html/2608.06741#bib.bib15); Brownet al\.,[2020](https://arxiv.org/html/2608.06741#bib.bib5)\)\. In other strategic domains, superhuman systems have become more than competitors: AlphaGo and AlphaZero changed how strong players and researchers study Go and chess\(Silveret al\.,[2016](https://arxiv.org/html/2608.06741#bib.bib27),[2017](https://arxiv.org/html/2608.06741#bib.bib18)\), and later analyses recovered human\-understandable chess concepts from AlphaZero play\(McGrathet al\.,[2022](https://arxiv.org/html/2608.06741#bib.bib14)\)\. Poker has undergone a parallel shift toward solver\- and GTO\-guided study, as reflected by solver\-based benchmarks and training resources\(Zhuanget al\.,[2025](https://arxiv.org/html/2608.06741#bib.bib42); Provostet al\.,[2026](https://arxiv.org/html/2608.06741#bib.bib41)\)\. Yet poker solvers remain largely silent oracles: they provide exact frequency prescriptions \(e\.g\., “bet 33\.4%”\) but not communicable rationales for those frequencies\.

#### Concept discovery and interpretable policy distillation\.

A growing body of work aims to expose structure inside learned or optimized policies\. Concept\-based analysis has been effective for perfect\-information agents such as AlphaZero\(McGrathet al\.,[2022](https://arxiv.org/html/2608.06741#bib.bib14)\), while policy distillation compresses cumbersome teacher models into lightweight students\(Hintonet al\.,[2015](https://arxiv.org/html/2608.06741#bib.bib8); Rusuet al\.,[2015](https://arxiv.org/html/2608.06741#bib.bib24)\)\. Interpretable variants\(Kohet al\.,[2020](https://arxiv.org/html/2608.06741#bib.bib13); Jacobset al\.,[1991](https://arxiv.org/html/2608.06741#bib.bib30); Shazeeret al\.,[2017](https://arxiv.org/html/2608.06741#bib.bib31)\), including Tree Imitation Learning, VIPER, and Programmatically Interpretable RL, project policies into trees or programs\(Bastaniet al\.,[2018](https://arxiv.org/html/2608.06741#bib.bib11); Vermaet al\.,[2018](https://arxiv.org/html/2608.06741#bib.bib12)\); soft decision trees provide another differentiable route to tree\-structured explanations\(Frosst and Hinton,[2017](https://arxiv.org/html/2608.06741#bib.bib9)\)\. In imperfect\-information games, abstraction and bucketing methods cluster strategically similar hands for computational tractability\(Ganzfried and Sandholm,[2014](https://arxiv.org/html/2608.06741#bib.bib22); Li and Huang,[2025](https://arxiv.org/html/2608.06741#bib.bib23)\)\. These methods expose useful structure, but their objectives are usually concept probing, compression, or efficient solving; they do not directly address the range\-level coupling and action\-frequency mixing that make poker equilibrium policies hard to communicate\.

#### LLMs, poker benchmarks, and synthetic reasoning\.

LLMs have recently been tested as poker decision makers, but existing results show substantial gaps\.Gupta \([2023](https://arxiv.org/html/2608.06741#bib.bib6)\)evaluate ChatGPT and GPT\-4 on preflop poker decisions, whileHuanget al\.\([2024](https://arxiv.org/html/2608.06741#bib.bib7)\)explore LLM\-based poker agents trained from online poker data; both lines highlight the difficulty of obtaining reliable strategic behavior from language models alone\. PokerBench provides a broader benchmark and training set over curated preflop and postflop spots, using dominant\-action and bet\-size labels with action\-accuracy and exact\-match metrics\(Zhuanget al\.,[2025](https://arxiv.org/html/2608.06741#bib.bib42)\)\. The GTO Wizard Benchmark evaluates frontier LLMs against a superhuman poker agent and finds persistent failures in card representation, range construction, and solver\-level action selection\(Provostet al\.,[2026](https://arxiv.org/html/2608.06741#bib.bib41)\)\. These works primarily evaluate or train LLM poker play\. By contrast, structured reasoning and synthetic\-data work suggests that explicit rationales can improve model behavior\(Weiet al\.,[2022](https://arxiv.org/html/2608.06741#bib.bib25); Gunasekaret al\.,[2023](https://arxiv.org/html/2608.06741#bib.bib26); Zelikmanet al\.,[2022](https://arxiv.org/html/2608.06741#bib.bib35)\), but poker requires such rationales to be grounded in solver\-computed mixed strategies rather than in fluent human commentary alone\.

#### Relation to action\- and agent\-level evaluations\.

PokerGPT reports action prediction, aggregate behavioral statistics, and play against Slumbot; Suspicion\-Agent reports finite\-match chip outcomes against named agents in Leduc Hold’em\(Huanget al\.,[2024](https://arxiv.org/html/2608.06741#bib.bib7); Guoet al\.,[2024](https://arxiv.org/html/2608.06741#bib.bib48)\); and PokerBench scores one dominant action category or an exact bet size and compares agents in head\-to\-head play\(Zhuanget al\.,[2025](https://arxiv.org/html/2608.06741#bib.bib42)\)\. These are useful evaluations of realized actions and opponent\-specific performance, but they do not compare every probability in a solver mixture at matched information sets\. The recent GTO Wizard Benchmark further reports, under its separate HUNL protocol, that its approximate\-Nash agent beats Slumbot by194±41194\\pm 41mbb/hand and that GPT\-4 loses1,362±2561\{,\}362\\pm 256mbb/hand to that agent\(Provostet al\.,[2026](https://arxiv.org/html/2608.06741#bib.bib41)\)\. Because the games and protocols differ, these numbers are not directly comparable with ourL1L\_\{1\}results; they illustrate why success against a particular imperfect opponent and fidelity to a Nash mixture are distinct questions\. Likewise, action accuracy, exact match, and our argmax agreement all discard the frequencies among supported actions, whereas action\-averagedL1L\_\{1\}evaluates the complete local distribution\.

## Appendix BDistillation Fidelity Details

Table[1](https://arxiv.org/html/2608.06741#S6.T1)reports the training objective components used to evaluate solver\-policy distillation: action\-distributionL1L\_\{1\}and oracle\-conditioned EV gap\. Dense high\-capacity models obtain the lowest numerical loss, but their decisions are not directly inspectable\. The raw\-PBS baselines use a 1482\-dimensional input and 512\-wide hidden layers, whereas the summary\-input baselines use a 156\-dimensional input and 256\-wide hidden layers\. In the tree variants, the soft\-sparse model keeps dense routers over the 156 summaries and relies on regularization rather than hard feature selection; Teacher 2 uses higher\-capacity two\-layer MLP routers; and the final hard MDT selects its top five summaries at each node through a mask over the original 156\-dimensional input\. Within the summary\-input comparison, tree\-structured routing improves substantially over the flat MLP, supporting the use of MDT for context\-dependent policy structure\. The final hard MDT has higher loss because hard sparsification removes small corrective effects used by dense routers\. This fidelity cost is intentional: the displayed node\-local summaries are exactly the variables used by the model, which avoids explanations that omit many low\-weight contributors\.

## Appendix CCase Study Details

This appendix expands the draw\-aware transfer case study from the main text using the GPT\-5\.5 high visible\-rationale diagnostic run\. The example is chosen because all methods receive the same public state and target hand, and the main error is strategic rather than notational:Td9cis recognized as an offsuit gutshot, but the question is whether it belongs to the folding part of the range or to a continuing draw class\. We report action distributions in the order shown in the public context, and report the averageL1L\_\{1\}distance to the solver target\.

Table 5:Draw\-aware rule transfer on an unseen out\-of\-position target hand\.The SCCS rule exposes the boundary between weak folding draws and stronger continuing draw candidates\.The visible rationales clarify the failure mode\. Direct prompting describesTd9cas a gutshot with two overcards but poor out\-of\-position equity realization, and therefore assigns most of its probability mass to Fold\. Adding summaries does not change the qualitative decision: the model notes some relevant card\-removal effects, but still treats the hand as low\-ranked air facing pressure and views the large aggressive action as too ambitious\. SCCS changes the evidence structure by presenting a matched folding prototype together with continuing prototypes in the same public context\. This contrast highlights whyTd9cbelongs to a stronger continuing\-draw class: a7on the turn completes the nut straight, aJproduces an open\-ended straight draw, and aTor9adds pair equity\. Despite its low current showdown value, the hand therefore has sufficient draw quality, future\-street potential, and range\-relative value to continue, with suit differences playing a secondary role\. The SCCS\-guided model consequently removes the fold mass and recovers the solver’s Call/Bet mixture up to a small error\. This behavior exemplifies the local range\-boundary transfer measured in the communicability experiment; it should not be interpreted as an evaluation of live exploitability\.

## Appendix DNLH Rules and Poker Terminology

This appendix provides a compact reference for the NLH rule structure and poker terminology used throughout the paper\. The descriptions are intended to fix notation and vocabulary for the experiments, not to introduce new modeling assumptions\.

### D\.1Rules of No\-Limit Texas Hold’em

#### Game format\.

No\-Limit Texas Hold’em \(NLH\) is a form of Texas Hold’em in which a player may bet up to the full remaining stack\. Our experiments use a six\-player configuration, while each evaluated postflop decision has two players remaining in the hand\. Each player receives two private cards, usually called*hole cards*\. Up to five public*community cards*are then revealed on the board\. At showdown, each remaining player forms the best five\-card poker hand using any combination of their two private cards and the five community cards\.

#### Blinds and positions\.

Each hand begins with forced bets called the small blind \(SB\) and big blind \(BB\)\. The button marks the dealer position and moves clockwise between hands; in a six\-player game, the two players immediately to its left post the SB and BB\. Before the flop, action begins with the active player to the left of the BB\. On each postflop street, action begins with the first active player to the left of the button\. In the two\-player postflop situations studied here, the player acting first is out of position \(OOP\), while the player acting second is in position \(IP\)\.

#### Betting streets\.

A hand proceeds through four betting rounds, also called*streets*\. The*preflop*round occurs after private cards are dealt and before any community card appears\. The*flop*reveals three community cards, the*turn*reveals a fourth community card, and the*river*reveals the fifth and final community card\. In this paper, the dataset and evaluations focus on postflop decisions, especially flop and turn states\.

#### Legal actions\.

At a decision point, the legal actions depend on the previous betting sequence\. A player may*check*if no bet is currently faced,*bet*to put chips into the pot,*call*to match an opponent’s bet,*fold*to surrender the pot, or*raise*to increase an existing bet\. In no\-limit poker, a bet or raise can be any legal size up to the player’s remaining stack; an all\-in action commits the full remaining stack\. Our action abstraction groups available aggressive actions under the generic bet/raise category when reporting solver mixtures\.

#### Pots, stacks, and bet sizes\.

The*pot*is the number of chips currently contested\. A player’s*stack*is their remaining chips\. Stack\-to\-pot ratio \(SPR\) is the remaining effective stack divided by the pot and measures how much future betting leverage remains\. Bet sizes are often written as fractions or multiples of the pot, e\.g\., a0\.5×0\.5\\timespot bet or a1\.79×1\.79\\timespot raise\. The experiment configuration uses 100BB starting stacks and no rake\.

#### Hand ranking\.

Texas Hold’em uses the standard poker hand order: high card, one pair, two pair, three of a kind, straight, flush, full house, four of a kind, and straight flush\. A*kicker*is a side card used to break ties between otherwise similar made hands, such as top pair with an ace kicker versus top pair with a weaker kicker\.

### D\.2Common Poker Terms Used in the Paper

TermMeaning in this paperAction line / historyThe sequence of previous betting actions and public card events leading to the current decision point\.All\-inA bet or raise that commits a player’s entire remaining stack\.AirA hand with little or no current showdown value and limited immediate equity\.Backdoor drawA draw that needs favorable cards on both later streets to complete, such as needing both turn and river to make a flush\.Bet sizeThe amount placed into the pot, often normalized by the current pot size\.BlockerA card in a player’s hand that removes combinations from the opponent’s possible range, often reducing the chance that the opponent holds strong continuing hands\.BoardThe public community cards visible to both players\.Board textureStrategic properties of the board, such as pairedness, connectedness, straight potential, and flush potential\.BluffAn aggressive action with a hand that is unlikely to be best if called, used to make better hands fold\.CallMatching the current bet to continue in the hand\.CheckPassing the action when no bet is faced\.Continuation betA postflop bet made by the player who was the previous aggressor, commonly abbreviated as c\-bet\.DrawA hand that is not currently strong but can improve to a strong hand on later community cards\.Equity \(EQ\)The probability, or solver\-computed share, that a hand or range wins at showdown under the relevant future\-card distribution\.Expected value \(EV\)The expected payoff of a hand, range, or action under the solver’s continuation strategy\.Fold equityThe value gained from the probability that an opponent folds to an aggressive action\.Flush drawA draw to five cards of the same suit\.GutshotAn inside straight draw that can complete with one specific rank\.GTOGame\-Theoretic Optimal; in this paper, an approximate Nash\-equilibrium poker strategy computed by a solver\.HandUsually the player’s two private cards, and sometimes the resulting best five\-card category depending on context\.In position \(IP\)The player who acts second on postflop streets\.MDFMinimum defense frequency, a pot\-odds\-derived threshold describing how often a range must continue to avoid being immediately exploitable by a bet\.Mixed strategyA probability distribution over legal actions at a decision point\.NutsThe strongest possible hand, or class of strongest hands, for the current board\.Nut advantageA range\-level advantage in the frequency or equity of nut\-class hands\.OffsuitA two\-card private hand whose cards have different suits\.Open\-ended straight draw \(OESD\)A straight draw that can complete with a card on either end of the sequence\.Out of position \(OOP\)The player who acts first on postflop streets\.OvercardA private card higher than every card on the board\.PostflopAny decision after the flop has been dealt; includes flop, turn, and river\.Pot oddsThe price offered by the pot for calling a bet, usually expressed as a required equity threshold\.Protection betA bet with a vulnerable made hand or semi\-made hand intended to deny equity to hands that can improve\.RangeThe probability distribution over private hands a player can hold after conditioning on public cards and betting history\.Range advantageA range\-level equity or EV edge for one player over the other in the current public state\.RiverThe fifth community card and final betting street\.Semi\-bluffA bluffing bet or raise with a hand that can improve to a strong hand on later streets\.SetThree of a kind made with a pocket pair and one matching board card\.Showdown valueThe ability of a hand to win if betting stops and the hand reaches showdown\.Single\-raised potA pot where the preflop action contains one raise and no 3\-bet\.SuitedA two\-card private hand whose cards share the same suit\.Thin valueA value bet with a hand that is ahead of some calling hands but not strong enough to be clearly dominant\.Three\-bet pot / 3\-bet potA pot where the preflop action contains a raise and then a re\-raise\.TrapA passive action with a very strong hand, used to keep weaker hands or bluffs in the opponent’s range\.TurnThe fourth community card and the betting street after it is dealt\.UnblockerA card property that leaves the opponent’s folding range relatively intact, which can improve bluff quality in some contexts\.Value betA bet made with a hand expected to be called by worse hands often enough to profit\.

## Appendix EPilot Study on Public Poker Discourse

This appendix gives additional context for the pilot study mentioned in Section[4](https://arxiv.org/html/2608.06741#S4)\. Two Plus Two \(2\+2\) refers to the Two Plus Two poker forum111[https://forumserver\.twoplustwo\.com/](https://forumserver.twoplustwo.com/), a long\-running public discussion forum for poker strategy, theory, and community discussion\. Such forum text is useful for exposing a language model to poker vocabulary, common strategic concepts, and the informal reasoning style used by human players\.

In our pilot experiment, we scraped public 2\+2 forum discussions and used them to perform supervised fine\-tuning \(SFT\) on an open Qwen model with the standard next\-token prediction objective\. The goal was not to build the final system in this paper, but to test whether imitation of naturally occurring poker discourse could by itself teach equilibrium\-relevant poker reasoning\. The resulting model learned to use many terms that appeared frequently in the forum text, such as range, blocker, equity, pot odds, bluff, value bet, and GTO\. It also produced more fluent local hand explanations than the base model\.

However, the improvement was mostly linguistic and heuristic\. In qualitative evaluations, the fine\-tuned model still showed shallow understanding of many poker\-theoretic concepts that appeared in the training text\. It could often repeat the vocabulary of range advantage, blockers, or mixed strategy, but it did not reliably apply these concepts to make decisions close to solver\-computed GTO strategies\. In particular, it frequently reduced mixed\-equilibrium decisions to single\-hand narratives, over\-relied on visible hand strength, and failed to reason about how one private hand’s action frequency is constrained by the rest of the range\.

These observations support the distinction made in the main text\. Public poker discourse is valuable as a source of terminology and human\-readable explanation style, but it does not provide ground\-truth equilibrium targets\. Therefore, our main pipeline uses solver\-derived mixed policies and continuation summaries as the strategic target, while using language only as the medium for articulation\.

## Appendix FAdditional Method Details

### F\.1Router Parameterization

Each internal nodenncomputes a branch distribution from a small set of active summaries\. Let𝐦n∈\{0,1\}D\\mathbf\{m\}\_\{n\}\\in\\\{0,1\\\}^\{D\}denote the node\-local hard mask and let‖𝐦n‖0≤K\\\|\\mathbf\{m\}\_\{n\}\\\|\_\{0\}\\leq K\. The branch logits are

𝐠n​\(𝐱\)=rn​\(𝐱⊙𝐦n\),𝐩n​\(𝐱\)=softmax​\(𝐠n​\(𝐱\)\),\\mathbf\{g\}\_\{n\}\(\\mathbf\{x\}\)=r\_\{n\}\(\\mathbf\{x\}\\odot\\mathbf\{m\}\_\{n\}\),\\qquad\\mathbf\{p\}\_\{n\}\(\\mathbf\{x\}\)=\\mathrm\{softmax\}\(\\mathbf\{g\}\_\{n\}\(\\mathbf\{x\}\)\),\(8\)wherernr\_\{n\}is a small local router\. In the ablations in Table[1](https://arxiv.org/html/2608.06741#S6.T1),rnr\_\{n\}can be either a sparse linear map,

𝐠n​\(𝐱\)=Wn​\(𝐱⊙𝐦n\)\+𝐛n,\\mathbf\{g\}\_\{n\}\(\\mathbf\{x\}\)=W\_\{n\}\(\\mathbf\{x\}\\odot\\mathbf\{m\}\_\{n\}\)\+\\mathbf\{b\}\_\{n\},\(9\)or an additive one\-dimensional router,

𝐠n​\(𝐱\)=∑j:mn,j=1𝐡n,j​\(xj\),\\mathbf\{g\}\_\{n\}\(\\mathbf\{x\}\)=\\sum\_\{j:m\_\{n,j\}=1\}\\mathbf\{h\}\_\{n,j\}\(x\_\{j\}\),\(10\)with each𝐡n,j\\mathbf\{h\}\_\{n,j\}implemented as a small scalar\-to\-logit network\. These choices are implementation variants for fitting the tree; the final hard MDT exposes the same object in either case: a node\-local set of at mostKKsummaries and the resulting branch probabilities\.

### F\.2SCCS Sampling and Verification

SCCS is designed to extract a rule for a held\-out target without showing the target policy to the downstream LLM\. The sampling procedure first fixes the public context, then looks for private\-hand changes that both cross an MDT routing boundary and induce a clear solver\-policy change\. The resulting prompt displays only the contrastive hands used to form the rule; the evaluation target is required to be policy\-separated from every displayed hand\.

Algorithm 1SCCS: Scenario\-Constrained Contrastive Rule Extraction0:Target hand

htesth\_\{\\mathrm\{test\}\}, trained MDT

ℳ\\mathcal\{M\}, reference solver set

𝒟\\mathcal\{D\}, policy\-divergence threshold

τπ\\tau\_\{\\pi\}, prompt\-separation threshold

τsep\\tau\_\{\\mathrm\{sep\}\}
0:SCCS prompt containing contrastive rule evidence, with the target policy masked

1:Fix the public scenario

c=Context​\(htest\)c=\\mathrm\{Context\}\(h\_\{\\mathrm\{test\}\}\)\.

2:Compute target summaries

𝐱test\\mathbf\{x\}\_\{\\mathrm\{test\}\}, target route

Ptest=Routeℳ​\(𝐱test\)P\_\{\\mathrm\{test\}\}=\\mathrm\{Route\}\_\{\\mathcal\{M\}\}\(\\mathbf\{x\}\_\{\\mathrm\{test\}\}\), and target policy

π∗\(⋅∣htest\)\\pi^\{\*\}\(\\cdot\\mid h\_\{\\mathrm\{test\}\}\)\.

3:Initialize candidate set

𝒞←∅\\mathcal\{C\}\\leftarrow\\emptyset\.

4:Scenario\-constrained candidate sampling

5:For each solver\-labeled hand

h′∈𝒟h^\{\\prime\}\\in\\mathcal\{D\}with

Context​\(h′\)=c\\mathrm\{Context\}\(h^\{\\prime\}\)=c:

6:compute

𝐱′\\mathbf\{x\}^\{\\prime\},

P′=Routeℳ​\(𝐱′\)P^\{\\prime\}=\\mathrm\{Route\}\_\{\\mathcal\{M\}\}\(\\mathbf\{x\}^\{\\prime\}\), and

π∗\(⋅∣h′\)\\pi^\{\*\}\(\\cdot\\mid h^\{\\prime\}\)\.

7:discard

h′h^\{\\prime\}if

P′=PtestP^\{\\prime\}=P\_\{\\mathrm\{test\}\}\.

8:discard

h′h^\{\\prime\}if

L¯1\(π∗\(⋅∣h′\),π∗\(⋅∣htest\)\)<τπ\\bar\{L\}\_\{1\}\(\\pi^\{\*\}\(\\cdot\\mid h^\{\\prime\}\),\\pi^\{\*\}\(\\cdot\\mid h\_\{\\mathrm\{test\}\}\)\)<\\tau\_\{\\pi\}\.

9:otherwise add

h′h^\{\\prime\}to

𝒞\\mathcal\{C\}\.

10:Boundary localization

11:For each candidate

h′∈𝒞h^\{\\prime\}\\in\\mathcal\{C\}, identify the earliest node

n​\(h′\)n\(h^\{\\prime\}\)where

P′P^\{\\prime\}and

PtestP\_\{\\mathrm\{test\}\}diverge\.

12:At

n​\(h′\)n\(h^\{\\prime\}\), collect the active summaries selected by the hard MDT router\.

13:Rank candidates by policy divergence, routing\-boundary clarity, and sparsity of the active\-summary contrast\.

14:Select one or more shadow hands

𝒮\\mathcal\{S\}from the top\-ranked candidates\.

15:Rule construction

16:For each shadow hand

hs∈𝒮h\_\{s\}\\in\\mathcal\{S\}, extract the diverging node, route branch, active summaries, summary values, MDT policy, and solver policy\.

17:Convert these contrasts into a short natural\-language rule that names the public scenario, the shadow\-hand contrast, and the active summary boundary\.

18:Held\-out verification prompt

19:Hide

π∗\(⋅∣htest\)\\pi^\{\*\}\(\\cdot\\mid h\_\{\\mathrm\{test\}\}\)from the downstream LLM\.

20:Require

L¯1\(π∗\(⋅∣htest\),π∗\(⋅∣hs\)\)≥τsep\\bar\{L\}\_\{1\}\(\\pi^\{\*\}\(\\cdot\\mid h\_\{\\mathrm\{test\}\}\),\\pi^\{\*\}\(\\cdot\\mid h\_\{s\}\)\)\\geq\\tau\_\{\\mathrm\{sep\}\}for every displayed hand

hs∈𝒮h\_\{s\}\\in\\mathcal\{S\}\.

21:Query the LLM with the public scenario, target hand, target summaries, and SCCS rule evidence; evaluate its predicted distribution against the masked target policy\.

HereL¯1\\bar\{L\}\_\{1\}denotes the same action\-averagedL1L\_\{1\}distance used in the main evaluation\. In the communicability experiments, we useτsep=0\.20\\tau\_\{\\mathrm\{sep\}\}=0\.20to prevent the displayed SCCS examples from serving as near\-label lookups for the held\-out target\.

Input \(SCCS Prompt \- Abbreviated\)Scenario:BTN vs BB 3\-Bet\.Board:4​♠​3​♡​3​♢4\\spadesuit 3\\heartsuit 3\\diamondsuit\.Hand:T​♡​8​♡T\\heartsuit 8\\heartsuit\(Hero\)\.Decision Trace:Node \#0 \(Filter\)→\\toNode \#21 \(Split\)→\\toLeaf \#X \(Strategy: Bet 100%\)\.SCCS Comparison:Shadow HandK​♡​9​♡K\\heartsuit 9\\heartsuit\(Shadow1\) diverges at Node \#21 toCheck\.Critical Summary Contrast:HeroPhys:Kicker= 0\.57 \(TT\-High\)\. Shadow1Phys:Kicker= 0\.64 \(KK\-High\)\.HeroEQ \- Nut MDF= 0\.03\. Shadow1EQ \- Nut MDF= 0\.20\.Model Output \(Hard MDT Logic \- Generated by Gemini\-3\.1\-Pro\)1\. Strategy Summary:The model executes a Polarized Continuation Bet strategy\. It aggressively semi\-bluffs hands with low Showdown Value but high potential \(TT\-high flush draws\) to deny equity, while checking back strongerKK\-high hands to realize equity as \(S\.D\. Value\)\.2\. Path Logic & Divergence:•Node \#21 \(The Polarization Split\):This is the pivot point separating Semi\-Bluffs from S\.D\. Value\.•Hero \(T​♡​8​♡T\\heartsuit 8\\heartsuit\)→\\toBet:Hero has a weakerPhys:Kicker\(0\.57\)\. The model identifies this hand as having insufficient Showdown Value to check\-call\. It bets to fold out better hands \(likeQQ\-high\) and deny their equity realization\.•Shadow1 \(K​♡​9​♡K\\heartsuit 9\\heartsuit\)→\\toCheck:Shadow1 possesses a strongerPhys:Kicker\(0\.64\)\. Despite similar raw equity, theKK\-high acts as a blocker and has enough Showdown Value to beat missed draws\. Checking maximizes EV by avoiding isolation against the villain’s calling range\.3\. Test Case Solution \(Unseen Hand:K​♢​9​♢K\\diamondsuit 9\\diamondsuit\)•Predicted Action:Check\(Frequency∼\\sim60%\)\.•Reasoning:The target hand follows the same routed strategic pattern as Shadow1 \(K​♡​9​♡K\\heartsuit 9\\heartsuit\)\. With a Kicker of 0\.64 \(KK\-High\), it falls into the S\.D\. Value bucket\. It is too strong to turn into a pure bluff \(like theT​♡​8​♡T\\heartsuit 8\\heartsuit\) but too weak to value bet, confirming the model’s preference for Range Protection over Polarization for this specific kicker class\.Figure 3:Example of SCCS Rule Extraction\.The SCCS pipeline successfully distills the subtle strategic boundary between aSemi\-BluffandShowdown Value\. The Hard MDT uses the summaryPhys: Kickeras a precise cut\-off: on a4​♠​3​♡​3​♢4\\spadesuit 3\\heartsuit 3\\diamondsuitboard,KK\-High \(Kicker 0\.64\) is strong enough to Check, whereasTT\-High \(Kicker 0\.57\) must Bet to deny equity\. The LLM correctly generalizes this rule to the unseen target handK​♢​9​♢K\\diamondsuit 9\\diamondsuit\.

## Appendix GDetailed Optimization Curriculum

Here we detail the three\-phase curriculum for theHierarchical Hard Distillation\. This curriculum is designed to distill knowledge from a dense Oracle into a strict symbolic structure without performance collapse\.

#### Phase 1: The Global Teacher \(Denoising\)\.

We first train a “Global\-Gated ResNet” \(Teacher 1\)\. It is restricted to use onlyKg=50K\_\{g\}=50summaries globally but allows unrestricted non\-linear interactions\. This phase acts as a filter, removing strictly irrelevant summaries while preserving high\-dimensional correlations inherent in the solver data\.

#### Phase 2: The Structural Teacher \(Topology\)\.

We then distill Teacher 1 into a “Deep Tree” \(Teacher 2\)\. This model adopts the target tree topology but uses higher\-capacity local routers\. This step establishes the correct decision hierarchy \(e\.g\., branching on Board Texture before Kickers\) without being constrained by limited routing capacity\.

#### Phase 3: Hard Student Locking \(Logic Extraction\)\.

Finally, we distill Teacher 2 into the target “Hard MDT”\. Crucially, we switch from soft gating to “Hard Gating” using the Straight\-Through Estimator \(STE\)\. We physically lock the summary set to the Top\-KK\(e\.g\., 5\) at each node:

𝐦h​a​r​d=TopK​\(𝐦l​o​g​i​t​s,k=5\)\\mathbf\{m\}\_\{hard\}=\\text\{TopK\}\(\\mathbf\{m\}\_\{logits\},k=5\)This forces the student to find the optimal strategy that existsstrictly withinthe 5\-summary subspace, mathematically eliminating the residue problem found in soft optimization\.

#### Objective scaling and checkpoint selection\.

In the reported configuration, the distribution and oracle\-conditioned EV terms use a5:15\{:\}1coefficient ratio\. The distribution term is primary: every training phase selects its retained checkpoint by the true validation action\-averagedL1L\_\{1\}\. The EV term is an auxiliary safeguard against moving probability toward an action with a large local value cost; it is not used as the primary model\-selection metric\.

## Appendix HRoute\-only Component and Mixed\-Frequency Diagnostics

#### Component definition\.

Route\-only uses the same held\-out target, public context, legal actions, and displayed MDT route trace as SCCS\. It includes the visited nodes, branches, routing/reach probabilities, node\-local summary impacts, and reached pure\-action prototype\. It removes every reference or shadow hand, every displayed reference policy, and every directional cross\-hand rule\. The target solver policy and the aggregate MDT mixed policy remain hidden in all prompting conditions\. Route\-only therefore asks whether the tree trace alone is sufficient; SCCS adds the feasible same\-context comparison and the policy change across the learned boundary\.

This control is also the closest measured comparison to several generic explanation alternatives\. Feature attribution can identify influential quantities for one prediction, but does not by itself construct a feasible private state under the same public history, verify that its solver mixture differs, or express the direction in which probability moves across the learned boundary\. Nearest\-neighbor retrieval need not preserve the public scenario or cross a policy\-relevant route boundary, while a human heuristic or compressed strategy table need not be tied to the summaries that actually determine MDT’s computation\. Direct\+Summaries tests whether the unstructured numerical inputs alone are sufficient, and Route\-only tests whether the displayed hierarchy alone is sufficient\. Our experiments therefore isolate the incremental value of the matched shadow\-based contrast over these two implemented controls; they do not claim a comparison with unimplemented attribution, retrieval, or hand\-authored systems\.

Table[7](https://arxiv.org/html/2608.06741#A8.T7)distinguishes the two aggregation conventions used in our analyses\. The main Table[2](https://arxiv.org/html/2608.06741#S6.T2)follows the submitted convention, in which each prompt condition is averaged over its own parse\-success set and the eight configuration means are then averaged without weighting\. The paired comparison instead restricts Route\-only and SCCS to the same parsed target IDs within each model configuration\.

Table 7:Route\-only versus SCCS under two aggregation conventions\.Values are action\-averagedL1L\_\{1\}; parenthesized values are relative reductions from Route\-only to SCCS\.Route\-only has 100 parsed targets in seven configurations and 99 in Gemini\-3\.1 Flash\. The retained SCCS outputs parse on 100, 100, 15, 100, 99, 68, 100, and 29 targets for Gemini\-3\.1 Flash, Gemini\-3\.1 Pro low/high, DeepSeek\-V4 Flash/Pro, and GPT\-5\.4/5\.5 low/high, respectively\. The main table reports these parse\-conditional condition means; the paired diagnostics never compare different target IDs\. For the three GPT configurations, the Route\-only calls used a separate API wrapper from the saved SCCS calls while retaining the same named model and reasoning setting\.

#### Holding the dominant action fixed\.

We perform two post\-hoc, parse\-conditional diagnostics\. First, among 342 matched model–target cases where Direct already selects the solver argmax, Route\-only and SCCS have nearly identical argmax agreement \(81\.3%81\.3\\%and80\.1%80\.1\\%\), but SCCS lowers Route\-only solverL1L\_\{1\}from0\.156330\.15633to0\.100820\.10082, a35\.5%35\.5\\%reduction\. SCCS also lowers Direct’s remaining solverL1L\_\{1\}from0\.119480\.11948to0\.100820\.10082, a15\.6%15\.6\\%reduction\.

Second, we restrict further to the 230 model–target cases where Direct, Route\-only, and SCCS all select the correct solver argmax\. SCCS lowers Route\-only solverL1L\_\{1\}from0\.125060\.12506to0\.076480\.07648\(38\.8%38\.8\\%\) and MDTL1L\_\{1\}from0\.149080\.14908to0\.099840\.09984\(33\.0%33\.0\\%\)\. The configuration\-level point estimate favors SCCS in all eight configurations under both targets\. Because dominant\-action correctness is fixed, these differences measure the remaining probability allocation rather than an argmax\-action correction\. They are diagnostic subset analyses, not a formal probability\-calibration study, and repeated target states across model configurations are not treated as independent game samples\.

## Appendix ILiar’s Dice Experimental Details

#### Game and protocol\.

The game has two players, two three\-faced dice per player, and bids up to quantity three\. Because a player’s two dice are unordered and repetitions are allowed, there are six private types:11,12,13,22,23,3311,12,13,22,23,33\. A full\-tree CFR\+ solver produces the approximate\-Nash policy\. The MDT interface contains 17 public\-history, posterior\-belief, private\-state, and equity summaries; per\-action continuation Q\-values, Q\-gaps, and the target policy are excluded from both MDT inputs and model\-facing prompts\. Six cross\-fitting folds each hold out one private type at every public context\. The communication surface contains 31 held\-out target information states and four matched prompts per target\.

This 31\-target surface is a post\-result exploratory expansion of an earlier pilot\. Its eligibility and anti\-copy filters, prompts, and admissible reference/shadow ranking were frozen before collecting outputs on the expanded surface\. Metrics are computed over successfully parsed responses, and unavailable responses are not imputed\. Table[8](https://arxiv.org/html/2608.06741#A9.T8)reports the complete released configuration\-level results\.

Table 8:Per\-configuration Liar’s Dice communicability\.Model entries are mean±\\pmstandard error over parse\-success cases\. Asterisks mark incomplete provider runs; missing responses are not imputed\.Six configurations contain all 124 expected responses\. Gemini\-3\.1 Pro high contains 117 of 124, and GPT\-5\.5 high contains 123 of 124\. In the order Direct, Summaries, Route\-only, and SCCS, the parse counts out of 31 are: Gemini\-3\.1 Flash\(24,27,31,15\)\(24,27,31,15\); Gemini\-3\.1 Pro low\(31,31,31,31\)\(31,31,31,31\); Gemini\-3\.1 Pro high\(29,28,31,29\)\(29,28,31,29\); DeepSeek\-V4 Flash\(30,31,31,30\)\(30,31,31,30\); DeepSeek\-V4 Pro\(31,31,30,29\)\(31,31,30,29\); GPT\-5\.4 and GPT\-5\.5 low\(31,31,31,31\)\(31,31,31,31\); and GPT\-5\.5 high\(30,31,31,31\)\(30,31,31,31\)\. The unweighted Mean row therefore summarizes configuration\-level, parse\-conditional point estimates rather than eight complete replications\. On paired parse\-success targets, the expanded\-surface point estimate favors SCCS over Direct and Direct\+Summaries in every listed configuration, whereas the smaller SCCS–Route\-only margins are heterogeneous and should be interpreted as a component diagnostic\.

#### Complete fixed\-opponent value evaluation\.

The six held\-out folds provide exactly one MDT prediction for each of the game’s 3,072 information sets\. We assemble these predictions into a fixed behavioral policy and traverse the complete tree against the approximate\-Nash solver, once in each player position\. Relative to solver self\-play, the MDT value losses are0\.0807230\.080723as player 0 and0\.0351290\.035129as player 1, for a seat average of0\.0579260\.057926\(reported as0\.0580\.058\) per game on the terminal payoff scale\[−1,\+1\]\[\-1,\+1\]\. This measures the complete policy’s value against a fixed reference opponent; it does not recompute an adaptive best response\.

## Appendix JComplete River\-Endgame Evaluation Details

#### Public states and action trees\.

The experiment starts from the released HUNL River states used for Subgames 3 and 4 byBrown and Sandholm \([2019a](https://arxiv.org/html/2608.06741#bib.bib46)\)\. Subgame 3 uses board4s 8h Tc 9h 2s, a55bb root pot, and197\.5197\.5bb remaining per player\. Its two root bet sizes are3\.4/7\.53\.4/7\.5bb and its two bet sizes after a check are2/3\.42/3\.4bb\. Subgame 4 uses boardJs Ks 5c Qs 7d, a37\.537\.5bb root pot, and181\.25181\.25bb remaining per player\. Its corresponding sizes are9\.4/37\.59\.4/37\.5bb at the root and56\.3/112\.556\.3/112\.5bb after a check\. We retain the released root ranges and build complete local trees through fold or showdown\. These are our own two\-size trees initialized from the published states, not the betting abstraction used in the original benchmark curves\.

Local DCFR solves provide high\-precision supervised targets\. Their residual exact exploitabilities are0\.0090340\.009034mbb/g on Subgame 3 and0\.0099070\.009907mbb/g on Subgame 4\. Both MDT variants use depth\-4 ternary probabilistic routing and pure\-action leaves, with one model per acting position shared across that position’s decision states\. The inputs exclude action Q/value features, target\-policy probabilities, and exact private\-card indicators\. Soft can use all eligible summaries; Hard uses exactly five summaries at each router\. After supervised fitting, the deployments are fine\-tuned and selected using the exact same\-tree strategic metric\. The result is therefore an in\-domain strategic\-fit evaluation of the compressed policies\.

Table 9:Additional River\-endgame diagnostics\.L1L\_\{1\}is equally averaged over decision states\. Fixed\-target value loss keeps the local target opponent fixed; exact exploitability recomputes each player’s best response\. All value entries are mbb/g\.For the matched learning\-based reference, Deep CFR uses10,00010\{,\}000sampled game\-tree traversals per player per iteration\. We evaluate its linearly weighted empirical\-average strategy, rather than fitting a separate final actor, with the same exact best\-response computation\. At iteration 800 it reaches43\.07543\.075mbb/g on Subgame 3 and133\.495133\.495mbb/g on Subgame 4\. These checkpoints give scale on the same trees and metric; they are neither convergence limits nor a general comparison between MDT and Deep CFR outside these two in\-domain endgames\.

## Appendix KMetric Interpretation and Fixed\-Opponent NLH Evaluation

#### L1L\_\{1\}is primary and local EV is auxiliary\.

The action\-averagedL1L\_\{1\}directly measures fidelity to the solver’s complete local mixture\. Its unaveraged form is twice total variation, so probability\-mass displacement changes the metric linearly\. By contrast, the oracle\-conditioned EV gap evaluates the predicted mixture using fixed solver continuation values\. At an exact equilibrium information set, every supported solver action is value\-maximizing; under that condition, the one\-decision EV gap equals the predicted mixture’s one\-step regret relative to the best action\. Our targets are approximate solver solutions, so we use the more limited term*local value gap*\. It is not CFR counterfactual regret or a complete\-policy best\-response metric\.

EV alone is insufficient for the articulation objective\. Multiple supported equilibrium actions can be nearly indifferent against the fixed equilibrium opponent, so an almost\-pure prediction can have low local EV loss while remaining far from the intended mixture\.L1L\_\{1\}therefore determines checkpoint selection, while the EV term discourages locally costly probability shifts among predictions with similar distributional fidelity\.

#### Corpus\-scale fixed\-opponent comparison\.

As a complementary NLH diagnostic, we evaluate Soft\-sparse Tree and final Hard MDT against the fixed approximate\-Nash opponent on the solver\-labeled postflop distribution\. Their value losses are7\.8367\.836and20\.68820\.688mbb per evaluated postflop hand, respectively, a descriptive difference of12\.85212\.852mbb per evaluated postflop hand\. This evaluation keeps the opponent and solver\-derived summaries fixed\. The checkpoint families also differ in more than the Top\-5 mask, so the controlled same\-architecture River results in Section[6\.5](https://arxiv.org/html/2608.06741#S6.SS5)provide the cleaner measurement of strict\-sparsity cost\.

## Appendix LLimitations and Future Work

While our framework successfully distills articulate reasoning from solver data, we identify several current limitations regarding its scope and deployment\.

#### Dependence on Mixed\-Strategy Equilibria

The Mixed\-Strategy Decision Tree \(MDT\) is architecturally specialized for imperfect\-information games characterized by mixed Nash equilibria\. The model’s inductive bias, specifically its decomposition of strategy into probabilistic routing over pure\-action leaf prototypes, is designed to capture the delicate frequency balancing required in games like No\-Limit Texas Hold’em\. Consequently, this approach may yield diminishing returns in perfect\-information domains \(e\.g\., chess\) or games dominated by pure strategies, where such complex mixing specifications are unnecessary\.

#### Dependence on Solver\-Derived Summaries

Currently, our system operates as an offline analytical agent rather than a standalone poker agent\. Because the MDT relies on solver\-derived strategic summaries, it cannot yet function in a live setting where such ground\-truth solver information is unavailable\. Bridging this gap, potentially by training a separate state\-estimation module to approximate these summaries from raw history, remains a critical direction for future work to enable live\-agent deployment\.

#### Licensed Data and Reproducibility

The commercial NLH solver corpus cannot be redistributed in bulk\. We can release the solver\-interface schema, MDT/SCCS implementation, configuration, prompt and evaluation code, aggregate results, and legally distributable examples, but exact regeneration of the 250\-million\-decision corpus requires access to a compatible licensed solver\. The Liar’s Dice experiment addresses this reproducibility boundary with a releasable game solver, data generator, solver targets, trained cross\-fitted MDTs, prompts, outputs, and scoring code\.

#### Action Abstraction and Coverage

The NLH targets use response sizes selected by the solver abstraction\. A non\-standard earlier bet size can be represented as part of the observed history and changes the state summarized for MDT, but the current prediction remains a mixture over the solver\-supported response actions\. We do not claim robustness to arbitrary histories outside this interface\. SCCS also requires a feasible same\-context shadow satisfying both route\- and policy\-divergence constraints\. If no such shadow exists, the system can expose the Route\-only trace but cannot construct a supported contrastive rule\.

#### Inference\-Time Evaluation

The downstream LLM parameters are fixed in all reported communicability experiments\. The demonstrated result is that an independent LLM can use solver\-derived rules at inference time to improve prediction of a hidden mixed policy\. Training language models on a large corpus of such rules is a natural extension, but is not evaluated here\.

## Appendix MPrompt Templates

We use a structured prompt to ground the LLM’s generation in the routed logic exposed by MDT\. The template below is populated dynamically by the SCCS engine\. We also provide example output of Gemini\-3\.1\-Pro\.

Template 1:

`Answer 1: LLM Response 1: Template 2: Answer 2: LLM Response 2: Template 3: Answer 3: LLM Response 3:`