Pith. sign in

REVIEW 2 major objections 3 minor 41 references

ICM Out! Better Tournament Strategy from Computed Continuations, vs. Solvers and LLMs

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read ICM, the standard poker tournament equity model, reads only stack sizes and misses seat, blind, and elimination-pressure effects; a policy built on computed continuations beats a fixed-ICM policy by $214.33 per hand in a three-player…

desk verdict Solid matched-cost census showing ICM loses money as a controller; the result survives robust checks, but the $938.03 rollout figure is per tournament, not per hand, and the abstract/conclusion need that fix. read the letter →

arxiv 2608.09586 v1 pith:VXV5OFSX submitted 2026-08-10 cs.AI cs.GTcs.MA

Boning Li , Longbo Huang This is my paper · ORCID
classification cs.AIcs.GTcs.MA
keywords IndependentChipModeltournamentpokerstrategycontinuationvaluesjam/foldequilibriumpolicyevaluationprizeequityvalue-basedcontrolmatchedcomparison
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the Independent Chip Model, the standard translator from tournament chips to prize money, is reliable enough as a description of what a stack is worth but fails as an objective for choosing actions, because it sees only stack sizes and misses seat rotation, blind obligations, and the threat a covering stack exerts on short stacks. To show the failure is real, the authors build Strategic-Continuation Optimization (SCO), which prices successor states with continuation values computed from the finite tournament model, and compare the frozen policy it induces with a policy built the same way but priced by analytic ICM. In a three-player all-in/fold tournament with a $1,000,000 pool, the SCO policy earns $214.33 more prize equity per hand across all 2,838 state-owner units, and $938.03 per hand when the tournament is played out to a finish. A seat-symmetrization decomposition attributes $130.43 of the per-hand gain to knowing which seat holds which stack and $83.90 to the tournament's own chip dynamics even when seats are averaged away. The intended takeaway is that a value model used to drive decisions should be judged by the decisions it induces, not by how close its numbers land.

What carries the argument

The load-bearing object is the continuation table: a lookup that maps each successor stack vector to the expected prize money each surviving player will collect, standing in for all future hands. SCO optimizes the current-hand jam/fold game against the strategic continuation table $C_{SC}$ computed by value-iteration with fictitious play from the finite tournament model, while the comparison policy $\pi_{\mathrm{ICM}}$ solves the identical current-hand game against the analytic Malmuth-Harville table $C_{\mathrm{ICM}}$, which reads only the stack vector. The matched-evaluation protocol replaces only the focal player's frozen policy rows, holds opponents and the common evaluator fixed, and prices both arms with the same $C_{SC}$, so the paired difference $\Delta PE$ in Equation (4) isolates the pricing difference. A seat-symmetrization operator that averages the strategic table over the six seat assignments of each chip multiset splits the gain exactly into a positional term and a level term. The formal boundary is Proposition 1: as long as $\pi_{SCO}$ is $\varepsilon$-optimal in the current-hand game priced by $C_{SC}$, its matched advantage over any comparison policy is bounded below by $-\varepsilon$; the magnitude and coverage are then empirical.

What would settle it

Re-run the matched comparison in a four-player jam/fold game with the same protocol at comparable depth and blinds; if the SCO-minus-ICM per-hand gain collapses to zero or reverses, or the level term turns negative, the claim that ICM is systematically inadequate as a tournament objective would fail to generalize beyond the three-player model. A second, sharper check: add a post-flop action to the three-player game and recompute the census; if the policy difference no longer favors SCO in a majority of state-owner units, the result depends on the all-in/fold restriction.

Watch

Extended reading notes

Core claim

The central discovery is that ICM's value error is a control error, not just an estimation error: averaging over the complete census of 946 states and three seats, fixing the policy optimizer and changing only the continuation pricing makes the SCO policy overcome the fixed-ICM policy in 2,433 of 2,838 matched units, with a mean gain of $214.33 per hand. The error chain is measured layer by layer: analytic ICM deviates from the frozen strategic-continuation benchmark by $9,036 mean absolute value error; those value differences move the induced jam frequency by an average of 14.08% relative to each decision point's own ICM jam range, and by 32.42% at the button; and the resulting matched prize-equity difference is $214.33. Because only the focal player's policy changes between the two arms while opponents and the evaluator stay fixed, the difference is attributable to the pricing, not to the scoring. The paper also shows that a seat-blind repair is not enough: a symmetrized strategic table that averages over seat assignments still beats ICM by $83.90 per hand, so the problem is not only position blindness but how chips are discounted by the tournament's own dynamics.

Load-bearing premise

The three-player, all-in/fold, 45-chip, 169-hand-class tournament is a faithful enough stand-in for real tournament poker that the measured ICM disadvantage transfers beyond the model; the paper's own appendix flags that more players, more streets, or richer action sets could change the seat/level split.

Editorial extensions

If this is right

  • Fixed-ICM strategies leave prize equity on the table in the modeled three-player jam/fold tournament: $214.33 per hand in the matched one-hand census and $938.03 per hand when the tournament is played out to a finish.
  • Two separable defects in ICM are priced: seat and position blindness costs $130.43 per hand, and a seat-blind but tournament-computed table still beats ICM by $83.90 per hand, so fixing only position misses a real part of the error.
  • The advantage holds when opponents are replaced by two LLM-driven profiles and by a family of non-modeling threshold players, crossing zero only against very tight opponents that jam 10-30% of hands, which places the benefit in the competent-opponent regime.
  • Value functions used as controllers should be compared by induced policy differences and matched downstream cost, not by mean absolute value error, because a large common-level error can be action-irrelevant while a small successor-contrast error flips a decision.
  • The full-tournament rollout raises the gain to $938.03 per hand and flattens the per-seat asymmetry, since seats rotate over a tournament, even though the per-unit sign is less stable than the aggregate mean.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension suggested by the level term: add a fourth player or a post-flop street and re-run the same matched design; because ICM's proportional-elimination approximation and its blindness to future blind obligations are structural, the same positive sign should persist, but the paper's specific numbers cannot be assumed to transfer.
  • The seat/level decomposition is a general diagnostic recipe: for any static value function driving a controller in a sequential game, symmetrize over the state variable the model omits and price the remainder, converting 'how wrong is the value' into 'how much does acting on it cost.'
  • The sign reversal when both arms are scored with analytic ICM shows that an evaluator aligned with one arm biases the comparison; a neutral, simulation-based evaluator is the honest common ground for policy comparison.
  • Because the full-rollout gain is 4.38 times the one-hand census gain and the per-seat spread disappears, the one-hand matched protocol likely understates the practical value of the improvement over a complete tournament.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 3 minor

Summary. The manuscript proposes Strategic-Continuation Optimization (SCO), which freezes a current-hand jam/fold policy obtained by optimizing against a continuation table computed from a finite three-player tournament model, and compares it with a fixed-ICM policy built with the same optimizer and analytic ICM. In a complete census of the 946 ordered stack states at T = 45, it reports $9,036 mean absolute ICM value error, a 14.08% average relative jam-range change, and a matched mean prize-equity gain of $214.33 per hand for SCO over ICM across 2,838 state-owner units. Robustness experiments include six continuation endpoints, a shared-deck Monte Carlo evaluator, two LLM opponent profiles, a family of threshold opponents, three chip depths, two prize ladders, and a full-tournament rollout. The paper's central empirical claim is that ICM's value errors change policy and that the policy change costs prize equity.

Significance. If the measurements hold, the paper makes a useful contribution by separating value-model accuracy from control quality, and by giving an exact, verifiable decomposition of the policy cost into positional and level terms. The complete-domain census, the machine-checked identity in Eq. (11), the reproducibility artifacts, and the multiple independent evaluators are notable strengths. The finite three-player jam/fold setting is small, and the authors correctly acknowledge in Appendix B.9 that generalization to larger games is an open empirical question. My concerns are limited to two load-bearing presentation issues: the full-rollout unit error and the overstatement about threshold opponents in the abstract.

major comments (2)
  1. [Abstract; Section 9; Table 11; Conclusion; Contribution 2] The headline claim that the full-tournament rollout advantage is "$938.03 per hand" is a unit error. Section 9 reports a mean difference of $938.03 per tournament (per paired replicate starting from a state-owner unit), not per hand. The paper itself reports that the three-player stage lasts 4.37 hands on average and that the rollout mean is 4.38 times the one-hand census mean; dividing $938.03 by 4.37 gives approximately $214.6 per three-player hand, essentially the $214.33 census mean. The 4.38 factor is the average number of three-player hands per tournament, not a multiplicative gain in per-hand advantage. The Abstract, Contribution 2, Section 9, Conclusion, and Table 11 caption need to state the unit explicitly and correct the per-hand label.
  2. [Section 7 and Table 7; Abstract] The abstract's statement that the ordering "survives replacing the solver-built opponent with ... a family of non-modeling threshold players" is not supported for the full family. Table 7 reports negative mean gains of -$100.80, -$92.63, and -$25.25 per hand for the jam-top-10%, jam-top-20%, and jam-top-30% anchors, with the crossover between 30% and 40%. The main text is honest about this, but the abstract and conclusion should be qualified to say the ordering survives for the solver, LLM, and loose-to-moderate threshold opponents, with a documented negative region against tight mechanical opponents.
minor comments (3)
  1. [Section 5] In the paragraph "The mean is small; the underlying gap is not," the paper states that $214.33 is 0.0257% of the $333,333 equal-seat value; the correct percentage is approximately 0.0643% (214.33 / 333,333).
  2. [Section 4, Eq. (4) discussion] The primary census is scored with the same continuation table CSC used to construct the SCO arm, and Proposition 1 makes the sign of the matched difference a near-mathematical consequence of ε-optimality under that continuation. The paper acknowledges this in the Proposition 1 discussion, but the main-text narrative would benefit from an explicit sentence before the headline results stating that the independent external evidence is the shared-deck evaluator, the alternative endpoints, and the full-rollout experiment.
  3. [Table 11] The caption should give the unit of the mean gain (per tournament) and provide the per-three-player-hand conversion so that the table cannot be misread as per-hand.

Circularity Check

1 steps flagged · score 4.0 of 10

Primary census sign is self-definitional: the winning arm optimizes the same continuation objective used to score both arms, though magnitude and external rollouts keep the central claim empirically grounded.

  1. self definitional [Section 3 (Eq. 4, Proposition 1) and Section 4 ('Matched evaluation')]
    "Strategic-Continuation Optimization (SCO) maps a strategic continuation table to a current-hand policy: for every full state, it solves the jam/fold game whose terminal outcomes are priced by that table, then freezes the resulting policy. ... Both profiles are then evaluated with the same strategic continuation CSC. ... Then ΔPE(s, i; o, C) ≥ −ε for every comparison policy πICM_i ∈ Πhand_i, and exact optimality gives ΔPE ≥ 0."

    Eq. (4) defines the headline gain using the same continuation C=CSC that πSCO was constructed to optimize, and Proposition 1 then proves ΔPE≥−ε, i.e. nonnegative up to solver tolerance. Therefore the positive sign of the headline advantage is a theorem of the experimental design, not a measured property: πICM is feasible in the optimization that defines πSCO, so the census 'gain' is the gap between πICM and the optimum of the scoring objective. The $214.33 magnitude is empirical, and the full rollout, shared-deck evaluator, LLM opponents, and alternative continuation endpoints are external checks; but the sign of the primary census is self-definitional.

full rationale

The paper's value-error→policy→cost chain is an empirical measurement, not a derivation, and much of the support is genuinely external: the full-tournament rollout, the shared-deck Monte Carlo evaluator, information-restricted LLM opponents, six continuation endpoints, and the matched-tolerance sweep all avoid the same-objective construction. However, the primary census is scored under CSC, the same continuation table used to build πSCO, and Proposition 1 makes the nonnegative sign of each matched unit a matter of definition rather than discovery. The headline magnitude remains measured rather than derived, and the independent rollout and LLM checks keep the central claim from being fully circular, so a moderate score is appropriate. The abstract's '$938.03 per hand' rollout figure appears to be a unit-labeling error, since Section 9 reports a mean difference of $938.03 that is 4.38 times the census mean while the three-player stage averages 4.37 hands; this is a correctness issue, not a circularity issue, and does not change the score.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on modeling choices and solver tolerances rather than fitted parameters. No number is fit to make the target result appear; the gain is measured, not constructed. The main assumptions are the fidelity of the jam/fold abstraction, the accuracy of the VI-FP continuation table, and the representativeness of frozen opponents. All three are acknowledged and stress-tested in the paper.

free parameters (3)
  • Solver continuation tolerances = $500 local acceptance / $250 inner tolerance (main grid)
    Hand-chosen stopping rules for the VI-FP strategic table and the fixed-ICM current-hand solves. The SCO arm's ex-post worst one-hand deviation gain is $364.68, larger than the headline $214.33; the matched-tolerance sweep (Table 8) is the paper's direct test that the gain is not slack-driven, and it passes with 87.8% of the gain retained at $62.50.
  • Stable-subset mask thresholds = $7,500 aggregate-reference ICM bias / $2,500 cross-start spread
    Define the secondary 1,314-unit stable subset sensitivity row only; not used for the headline 2,838-unit census. The mask was defined in an earlier start-sensitivity experiment, so it is not post hoc for this paper.
  • Payoff-table Monte Carlo sample sizes = 2,000 deals per hand-class pair; 1,024 shared three-way boards
    Chosen sample sizes for estimating the two-player and three-way payoff tables. The paper quantifies the resulting noise via jackknife and multinomial resampling (sigma_table = $1.04, SNR 206), so the choice is not an unrecognized source of error.
assumptions (6)
  • domain assumption The finite three-player jam/fold game with 169 preflop classes and all-in/fold-only actions adequately represents the strategic structure of the tournament being analyzed.
    Invoked throughout (Section 3 and all experiments). Appendix B.9 concedes that the seat/level split may not survive more players, streets, or action sets.
  • domain assumption The strategic continuation table CSC, computed by VI-FP, is a sufficiently accurate approximation of the continuation value of future play in this finite model.
    CSC is the frozen benchmark for value errors and the evaluator for the primary census (Eq. 4, Section 5). The paper reports solver tolerance and endpoint sensitivity rather than a formal accuracy proof.
  • domain assumption Holding opponents fixed at a frozen anchor (pi-SCO or an LLM/threshold profile) is a valid way to measure the focal policy's prize-equity value.
    The matched protocol (Section 4) isolates the focal policy. The gain is negative against tight mechanical opponents, so the anchor choice is load-bearing.
  • domain assumption One-hand prize equity with a continuation lookup is a meaningful proxy for full-tournament outcome.
    The headline census uses this estimand; Section 9 removes the lookup and finds a larger gain ($938.03), so the proxy is conservative in direction.
  • domain assumption The 169-class preflop abstraction and the Monte Carlo estimates of showdown payoffs are accurate enough for the comparisons.
    Used for all tables; the paper provides a noise decomposition (sigma_table = $1.04) and a shared-deck evaluator that bypasses the class-level independence approximation.
  • standard math Harville's ICM formula (Eq. 1) is the correct definition of the comparison baseline.
    CICM is the standard analytic ICM used in the poker literature; the paper's claim is about this baseline, not a strawman.

how reviews work

0 comments
Cite this review

Pith. "Pith review of ICM Out! Better Tournament Strategy from Computed Continuations, vs. Solvers and LLMs." pith.science (2026). https://pith.science/paper/VXV5OFSX

@misc{pith2026260809586,
  author       = {Pith},
  title        = {Pith review of: ICM Out! Better Tournament Strategy from Computed Continuations, vs. Solvers and LLMs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VXV5OFSX}},
  note         = {Machine review of arXiv:2608.09586}
}
abstract

The Independent Chip Model (ICM) converts tournament chips into reference prize equity, and policies are routinely constructed against those values. Because ICM reads only stack sizes, it omits action order, blind obligations, and seat rotation, and it does not price the elimination pressure a big stack puts on the short stacks it can bust. Those omissions can alter the successor-state contrasts that determine a move. We introduce Strategic-Continuation Optimization (SCO), a policy-construction method that enumerates current-hand outcomes, maps them to successor states, prices those states with continuation values computed from the finite tournament model, and optimizes and freezes the resulting current-hand policy. The fixed-ICM comparison policy changes one thing only: the same optimizer solves the same game with successor states priced by analytic ICM, so the two policies differ only through that pricing. We evaluate the resulting policies in a three-player jam/fold tournament with a \$1M prize pool. Relative to the frozen strategic-continuation benchmark, analytic ICM has \$9{,}036 mean absolute value error across all 2,838 state--seat entries. That value error rewrites the ranges it prices: measured against each decision point's own fixed-ICM jam range, SCO moves the jam frequency by an average of 14.08\%. To price those different moves, we compare all 946 states and three policy owners while changing only the focal policy and holding both opponents and the continuation evaluator fixed. The policy produced by SCO earns \$214.33 more prize equity per hand on average and is favored in 2,433 of 2,838 matched units. The ordering survives replacing the solver-built opponent with two LLMs and with a family of non-modeling threshold players. This value-to-policy-to-cost chain shows directly when ICM becomes an inadequate objective for tournament strategy construction.

Figures

Figures reproduced from arXiv: 2608.09586 by the authors.

Figure 1
Figure 1. The two primary arms are solved against their own continuation table, then frozen [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. The complete census, per state and per seat. Panels (a)–(c) place each of the 946 chip [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

41 extracted references · 25 canonical work pages

  1. [1]

    Using counterfactual regret minimization to create competitive multiplayer poker agents

    Nicholas Abou Risk, Duane Szafron, et al. Using counterfactual regret minimization to create competitive multiplayer poker agents. InAAMAS, pages 159–166, 2010

    work page 2010

  2. [2]

    Qwen technical report.arXiv preprint arXiv:2309.16609, 2023

    Jinze Bai, Shuai Bai, Yunfei Chu, Zeyu Cui, Kai Dang, Xiaodong Deng, Yang Fan, Wenbin Ge, Yu Han, Fei Huang, et al. Qwen technical report.arXiv preprint arXiv:2309.16609, 2023

    arXiv 2023

  3. [3]

    Opponent modeling in poker.Aaai/iaai, 493(499):105, 1998

    Darse Billings, Denis Papp, Jonathan Schaeffer, and Duane Szafron. Opponent modeling in poker.Aaai/iaai, 493(499):105, 1998

    work page 1998

  4. [4]

    Heads-up limit hold’em poker is solved.Science, 347(6218):145–149, 2015

    Michael Bowling, Neil Burch, Michael Johanson, and Oskari Tammelin. Heads-up limit hold’em poker is solved.Science, 347(6218):145–149, 2015. 18

    work page 2015

  5. [5]

    George W. Brown. Iterative solution of games by fictitious play. In Tjalling C. Koopmans, editor,Activity Analysis of Production and Allocation, number 13 in Cowles Commission Monograph, pages 374–376. John Wiley & Sons, New York, 1951

    work page 1951

  6. [6]

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

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

    2018

  7. [7]

    Superhuman ai for multiplayer poker.Science, 365(6456): 885–890, 2019

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

    2019

  8. [8]

    An investigation into tournament poker strategy using evolutionary algorithms

    Richard G Carter and John Levine. An investigation into tournament poker strategy using evolutionary algorithms. In2007 IEEE Symposium on Computational Intelligence and Games, pages 117–124. IEEE, 2007

    work page 2007

Show all 41 references
  1. [9]

    Gambler’s ruin and the icm.Statistical Science, 37(3): 289–305, 2022

    Persi Diaconis and Stewart N Ethier. Gambler’s ruin and the icm.Statistical Science, 37(3): 289–305, 2022

    2022

  2. [10]

    Computing an approximate jam/fold equilibrium for 3-player no-limit texas hold’em tournaments

    Sam Ganzfried and Tuomas Sandholm. Computing an approximate jam/fold equilibrium for 3-player no-limit texas hold’em tournaments. InProceedings of the 7th international joint conference on Autonomous agents and multiagent systems-Volume 2, pages 919–925, 2008

    2008

  3. [11]

    Computing equilibria in multiplayer stochastic games of imperfect information

    Sam Ganzfried and Tuomas Sandholm. Computing equilibria in multiplayer stochastic games of imperfect information. InProceedings of the 21st International Joint Conference on Artificial Intelligence, pages 140–146, 2009

    2009

  4. [12]

    Successful nash equilibrium agent for a three-player imperfect-information game.Games, 9(2):33, 2018

    Sam Ganzfried, Austin Nowak, and Joannier Pinales. Successful nash equilibrium agent for a three-player imperfect-information game.Games, 9(2):33, 2018

    2018

  5. [13]

    Regret minimization in multiplayer extensive games

    Richard Gibson and Duane Szafron. Regret minimization in multiplayer extensive games. In IJCAI Proceedings-International Joint Conference on Artificial Intelligence, volume 22, page 2802, 2011

    2011

  6. [14]

    On strategy stitching in large extensive form multiplayer games.Advances in Neural Information Processing Systems, 24, 2011

    Richard Gibson and Duane Szafron. On strategy stitching in large extensive form multiplayer games.Advances in Neural Information Processing Systems, 24, 2011

    2011

  7. [15]

    Lossless abstraction of imperfect information games

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

    2007

  8. [16]

    The llama 3 herd of models.arXiv preprint arXiv:2407.21783, 2024

    Aaron Grattafiori, Abhimanyu Dubey, Abhinav Jauhri, Abhinav Pandey, Abhishek Kadian, Ahmad Al-Dahle, Aiesha Letman, Akhil Mathur, Alan Schelten, Alex Vaughan, et al. The llama 3 herd of models.arXiv preprint arXiv:2407.21783, 2024

    2024 arXiv

  9. [17]

    Suspicion-agent: Playing imperfect information games with theory of mind aware GPT-4

    Jiaxian Guo, Bo Yang, Paul Yoo, Bill Yuchen Lin, Yusuke Iwasawa, and Yutaka Matsuo. Suspicion-agent: Playing imperfect information games with theory of mind aware GPT-4. CoRR, abs/2309.17277, 2023

    2023 arXiv

  10. [18]

    Are chatgpt and GPT-4 good poker players? - A pre-flop analysis.CoRR, abs/2308.12466, 2023

    Akshat Gupta. Are chatgpt and GPT-4 good poker players? - A pre-flop analysis.CoRR, abs/2308.12466, 2023

    2023 arXiv

  11. [19]

    Harville

    David A. Harville. Assigning probabilities to the outcomes of multi-entry competitions.Journal of the American Statistical Association, 68(342):312–316, 1973. doi: 10.1080/01621459.1973. 10482425. 19

    1973

  12. [20]

    Effective short-term opponent exploitation in simplified poker

    Bret Hoehn, Finnegan Southey, Robert C Holte, and Valeriy Bulitko. Effective short-term opponent exploitation in simplified poker. InAAAI, volume 5, pages 783–788, 2005

    2005

  13. [21]

    Pokergpt: An end-to-end lightweight solver for multi-player texas hold’em via large language model.CoRR, abs/2401.06781, 2024

    Chenghao Huang, Yanbo Cao, Yinlong Wen, Tao Zhou, and Yanru Zhang. Pokergpt: An end-to-end lightweight solver for multi-player texas hold’em via large language model.CoRR, abs/2401.06781, 2024

    2024 arXiv

  14. [22]

    Empirical validation of the independent chip model

    Juho Kim. Empirical validation of the independent chip model. In2025 IEEE Conference on Games (CoG), pages 1–4. IEEE, 2025

    2025

  15. [23]

    Leslie and Edmund J

    David S. Leslie and Edmund J. Collins. Generalised weakened fictitious play.Games and Economic Behavior, 56(2):285–298, 2006. doi: 10.1016/j.geb.2005.03.005

    2006 doi

  16. [24]

    Efficient online pruning and abstraction for imperfect information extensive-form games

    Boning Li and Longbo Huang. Efficient online pruning and abstraction for imperfect information extensive-form games. InThe Thirteenth International Conference on Learning Representations, 2025

    2025

  17. [25]

    Agents that certify their own exploits: Confidence-scheduled restricted responses for safe opponent exploitation, 2026

    Boning Li and Longbo Huang. Agents that certify their own exploits: Confidence-scheduled restricted responses for safe opponent exploitation, 2026. URL https://arxiv.org/abs/2607. 28520

    2026

  18. [26]

    Effective, efficient, and general information abstraction for imperfect-information extensive-form games.arXiv preprint arXiv:2605.10900, 2026

    Boning Li and Longbo Huang. Effective, efficient, and general information abstraction for imperfect-information extensive-form games.arXiv preprint arXiv:2605.10900, 2026

    2026 arXiv

  19. [27]

    Real-time parallel counterfactual regret minimization.arXiv preprint arXiv:2605.19928, 2026

    Boning Li and Longbo Huang. Real-time parallel counterfactual regret minimization.arXiv preprint arXiv:2605.19928, 2026

    2026 arXiv

  20. [28]

    Rl-cfr: improving action abstraction for imperfect information extensive-form games with reinforcement learning

    Boning Li, Zhixuan Fang, and Longbo Huang. Rl-cfr: improving action abstraction for imperfect information extensive-form games with reinforcement learning. InProceedings of the 41st International Conference on Machine Learning, pages 27752–27770, 2024

    2024

  21. [29]

    Av-aivat: 74x cheaper agent evaluation with certified anytime-valid stopping in imperfect-information games, 2026

    Boning Li, Yu Chen, and Longbo Huang. Av-aivat: 74x cheaper agent evaluation with certified anytime-valid stopping in imperfect-information games, 2026. URL https://arxiv.org/abs/2608. 06362

    2026

  22. [30]

    Correlated chance sampling for monte carlo counter- factual regret minimization, 2026

    Boning Li, Yu Chen, and Longbo Huang. Correlated chance sampling for monte carlo counter- factual regret minimization, 2026. URL https://arxiv.org/abs/2607.27035

    2026 arXiv

  23. [31]

    Pokerskill: Llms can play expert-level poker without training or solvers.arXiv preprint arXiv:2605.30094, 2026

    Boning Li, Baoxiang Wang, and Longbo Huang. Pokerskill: Llms can play expert-level poker without training or solvers.arXiv preprint arXiv:2605.30094, 2026

    2026 arXiv

  24. [32]

    Two Plus Two Publishing LLC, 1999

    Mason Malmuth.Gambling Theory and Other Topics. Two Plus Two Publishing LLC, 1999

    1999

  25. [33]

    A near-optimal strategy for a heads-up no-limit texas hold’em poker tournament

    Peter Bro Miltersen and Troels Bjerre Sørensen. A near-optimal strategy for a heads-up no-limit texas hold’em poker tournament. InAAMAS, page 191. IF AAMAS, 2007

    2007

  26. [34]

    Deepstack: Expert-level artificial intelligence in heads-up no-limit poker.Science, 356(6337):508–513, 2017

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

    2017

  27. [35]

    The state of solving large incomplete-information games, and application to poker.Ai Magazine, 31(4):13–32, 2010

    Tuomas Sandholm. The state of solving large incomplete-information games, and application to poker.Ai Magazine, 31(4):13–32, 2010. 20

    2010

  28. [36]

    Robert Scott, Mikhail Sher, and Michael Thomas Paz. In the money: An analysis of monetary value of chips and player outcomes in no limit texas hold’em poker tournaments.The Journal of Gambling Business and Economics, 17(1):75–98, 2024

    2024

  29. [37]

    Two Plus Two Publishing LLC, 2007

    David Sklansky.Tournament poker for advanced players. Two Plus Two Publishing LLC, 2007

    2007

  30. [38]

    Two Plus Two Publishing, 1994

    David Sklansky and Mason Malmuth.Hold’em Poker for Advanced Players. Two Plus Two Publishing, 1994

    1994

  31. [39]

    Bayes’ bluff: opponent modelling in poker

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

    2005

  32. [40]

    A parameterized family of equilibrium profiles for three-player kuhn poker

    Duane Szafron, Richard G Gibson, and Nathan R Sturtevant. A parameterized family of equilibrium profiles for three-player kuhn poker. InAAMAS, volume 13, pages 247–254, 2013

    2013

  33. [41]

    Regret mini- mization in games with incomplete information.Advances in neural information processing systems, 20, 2007

    Martin Zinkevich, Michael Johanson, Michael Bowling, and Carmelo Piccione. Regret mini- mization in games with incomplete information.Advances in neural information processing systems, 20, 2007. 21 A Protocol and Evaluation Definitions A.1 State space and policy estimand A sta...

    2007

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.