Title: Dominated Actions in Imperfect-Information Games
Venue: NeurIPS 2025
Accepted: no
Keywords: game theory

imperfect information
Abstract: Dominance is a fundamental concept in game theory. In strategic-form games dominated strategies can be identified in polynomial time. As a consequence, iterative removal of dominated strategies can be performed efficiently as a preprocessing step for reducing the size of a game before computing a Nash equilibrium. For imperfect-information games in extensive form, we could convert the game to strategic form and then iteratively remove dominated strategies in the same way; however, this conversion may cause an exponential blowup in game size. In this paper we define and study the concept of dominated actions in imperfect-information games. Our main result is a polynomial-time algorithm for determining whether an action is dominated (strictly or weakly) by any mixed strategy in n-player games, which can be extended to an algorithm for iteratively removing dominated actions. This allows us to efficiently reduce the size of the game tree as a preprocessing step for Nash equilibrium computation. We explore the role of dominated actions empirically in the "All In or Fold" No-Limit Texas Hold'em poker variant.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a novel concept of \"dominated actions\" in extensive-form games. In normal-form games, dominated strategies are strategies that one can eliminate from consideration because playing another strategy is always better. In extensive-form games, there has been no widely accepted definition of dominated actions. The paper is an effort to fill this gap by defining dominated actions in extensive-form games and providing a method to compute them. The authors first propose several \"flawed definitions\" of dominated actions, explain why they are flawed, and then present their own definition. Empirically, the authors show that in the \"All In or Fold\" No-Limit Texas Hold'em poker variant, a significant number of actions are dominated, demonstrating the potential of this concept in practical applications.\n\nSTRENGTHS AND WEAKNESSES:\n# Strengths\n\n- The idea of defining dominated actions in extensive-form games is novel and quite interesting.\n- The paper is written in a way that is accessible to a broad audience, including those who may not be experts in game theory.\n- The authors provide clear explanations with examples of why some definitions of dominated actions in extensive-form games are flawed.\n\n# Weaknesses\n\n- The contribution of the paper is novel, but slim. It seems to me that the proposal of the concept is the sole contribution. For example, more than half of this paper is dedicated to discussing flawed definitions of dominated actions and background information, which does not directly contribute to the main idea of the paper. As a result, the paper appears to be more of a position paper than a research paper.\n- The empirical results are limited to a single game variant (the \"All In or Fold\" No-Limit Texas Hold'em poker variant), which may not generalize to other extensive-form games. Thus, the empirical evaluation is not very convincing to me, and I remain skeptical about the exact impact of having dominated actions in extensive-form games that people care about.\n\nQUESTIONS:\n- Are the above weaknesses proper criticisms of the paper?\n\n## Minor Comments\n\n- In Lines 18 and 24, I believe the correct notation should be $u_i(\\sigma_i',s_{-i}) > u_i(\\sigma_i,s_{-i})$.\n\nLIMITATIONS:\nI think the authors adequately mentioned the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 4, "DATE": "2025-06-11"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers different notions of dominance in extensive-form games, going from too restrictive, to more useful, all while retaining computational efficiency. Experiments validate that the more permissive definitions indeed result in significant elimination of dominated actions.\n\nSTRENGTHS AND WEAKNESSES:\nI like the idea of the paper, and it was interesting to consider the different flawed definitions in the buildup. While I think the paper has an unfortunate typo around lines 114-116 (see below), I found the paper well organized.\n\nMy biggest reservation with the paper is along soundness and experimental setup. For soundness:\n1. On lines 224-226 you write \"We can construct a new two-player game where player 2 now controls all actions that\nwere previously controlled by any player other than player 1. Then we can run the above procedure in this new game.\" Are there concerns related to imperfect recall when setting up a metaplayer that controls any player other than player 1? In particular, the different players being controlled might now have asymmetric information, so a meta-player would struggle with imperfect recall.\n2. There doesn't seem to be a proof of the fact that the proposed definition of dominance indeed implies that the action can be removed safely from the tree. Is it immediately obvious that doing so is sound?\n\nFor the experimental setup:\n\n3. In the experiments, you write that \"In the two-player NLHE shove-or-fold game, each player has 169 strategically distinct hands.\" Even assuming taking suit isomorphism into account, the number of distinct hands would be much higher, because information about the multiplicity of suits, in relation to the public cards, must be taken into account. What am I missing?\n\nIn addition, I am confused about the authors' checklist:\n\n- 4a. The authors do not justify the lack of public/open source release (question 5 of the checklist). What is the reason for not providing code reproducing the experiments alongside the paper, especially when the paper's contribution is that of defining a notion of dominance that can be turned into an algorithm? The release of a reference implementation from the authors should be expected, in my view.\n\n- 4b. The authors claim that they discuss the type of compute workers, memory, time of execution (question 8 of the checklist), but I could not find any such discussion referenced in the body.\n\n- 4c. The authors claim to have \"described safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)\". However, no such risks were identified (and I don't think they exist).\n\n- 4d. The authors do not release code, but claim to have provided documentation for their assets (question 13 of the checklist). Was this filled out by mistake?\n\nOther comments:\n- L18, Introduction: typo in the mathematical definition of domination\n- L114-116. You say \"The problem with these definitions are that they are too strong; it is still possible for player 1 to strictly prefer to take action bi to ai regardless of the strategy used by player 2 even if Flawed Definition 1 holds.\" Did you mean to write \"even if Flawed Definition 1 does NOT hold\"?\n- L114: \"The problem with these definitions are that they are too strong\": It should be \"The problem ... is that\" (singular / plural)\n- L116: \"The is illustrated in the following game depicted\" -> I would remove \"following\" for clarity\n- Equation (8): missing period at the end of equation\n- Reference [9]: the reference appeared malformed.\n\nQUESTIONS:\n1. (Taken from above) On lines 224-226 you write \"We can construct a new two-player game where player 2 now controls all actions that\nwere previously controlled by any player other than player 1. Then we can run the above procedure in this new game.\" Are there concerns related to imperfect recall when setting up a metaplayer that controls any player other than player 1? In particular, the different players being controlled might now have asymmetric information, so a meta-player would struggle with imperfect recall.\n\n2. (Taken from above) There doesn't seem to be a proof of the fact that the proposed definition of dominance indeed implies that the action can be removed safely from the tree. Is it immediately obvious that doing so is sound?\n\n3. (Taken from above) In the experiments, you write that \"In the two-player NLHE shove-or-fold game, each player has 169 strategically distinct hands.\" Even assuming taking suit isomorphism into account, the number of distinct hands would be much higher, because information about the multiplicity of suits, in relation to the public cards, must be taken into account. What am I missing?\n\n4. The checklist contains problematic claims (see points 4a-4d above). Perhaps more importantly, echoing point 4a, the authors do not justify the lack of public/open source release (question 5 of the checklist). What is the reason for not providing code reproducing the experiments alongside the paper, especially when the paper's contribution is that of defining a notion of dominance that can be turned into an algorithm? \n\n5. Currently, in two-player zero-sum games, the cost of removing dominance is similar to the cost of computing a Nash equilibrium. What do you imagine the computational benefits of dominance elimination will be?\n\n6. I can see why the all-in or fold variant of poker would lend itself very well to the removal of dominated actions. In that sense, I think it was a very nice choice to highlight how your definition is better than the flawed definition. However, I am wondering how many actions are dominated in a more regular variant of poker (even something simple like fold-pot raise or limit hold'em). Do you have any sense for what the answer is? \n\n7. Just to confirm: on lines 114-116, discussing the flawed definitions and setting the stage for your proposed definition, you say \"The problem with these definitions are that they are too strong; it is still possible for player 1 to strictly prefer to take action bi to ai regardless of the strategy used by player 2 even if Flawed Definition 1 holds.\" Did you mean to write \"even if Flawed Definition 1 does NOT hold\"?\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper investigates the dominance action in imperfect-information extensive-form games. While dominated strategies are well-understood in strategic-form games, their impact in extensive-form games remains largely unexplored. To address this, the authors first consider several plausible definitions of dominated actions and identify their limitations. They then propose a new formal definition of dominated actions suitable for extensive-form games, addressing these limitations. Then they develop polynomial-time algorithms to identify strictly and weakly dominated actions and show that iterative removal of such actions can also be performed in polynomial time for n-player games. Finally, empirical results on a No-Limit Texas Hold’em poker variant illustrate that iteratively removing dominated actions can effectively reduce the size of the game.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper focuses on a very interesting topic: dominated actions in imperfect-information games. To the best of my knowledge, prior work has largely overlooked this specific problem, making the contribution both novel and relevant to the game theory community. The introduction of the definition of dominated action in Section 3 is clearly presented, and the discussion of several flawed definitions is helpful for motivating and understanding the final formal definition proposed by the authors.\n\nThe algorithm for identifying dominated actions involves solving a linear number of linear programs. Although this is polynomial in theory, its practical scalability to larger and more realistic imperfect-information extensive-form games remains uncertain. Furthermore, the experimental section is limited to relatively small-scale games, making it unclear whether the approach would remain computationally feasible in settings with thousands of information sets and actions.\n\nMirror Comment: In Lines 115–116, it seems the intended phrasing should be “even if Flawed Definition 1 does not hold” rather than “even if Flawed Definition 1 holds”, to match the logical argument of the surrounding discussion.\n\nQUESTIONS:\n1.\tIn the experimental section, to better evaluate the scalability of the proposed identification algorithm, could the authors provide the number of variables and constraints of the linear programming problem used in each identification step, as well as the total number of information sets and actions in the tested game, and the time required for each step?\n\n2.\tSince the proposed method is theoretically applicable to n-player imperfect-information extensive-form games, could the authors provide experimental results or at least a discussion for a simple multi-player setting to support the claimed generality? For example, even a small-scale three-player game would help illustrate the practical feasibility of extending the approach beyond the two-player case.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe goal of the paper is to introduce a new notion/definition of dominated actions in extensive form games and provide an algorithm for finding such actions and consequently for iterated elimination of dominated actions. Authors discuss potential candidates for the definition of dominated actions and show their weaknesses. Authors propose a new definition that does not have the weaknesses of the previous candidates. Then they derive a general procedure based on linear programming for finding dominated actions in EFGs (also for more than two players, and general sum). Authors evaluate their algorithm for iterative removal of dominated actions in \"all in or fold\" no-limit texas holdem poker and show that their algorithm successfully removes dominated actions according to their definition.\n\nSTRENGTHS AND WEAKNESSES:\nI like that the authors explained well the motivation behind the definitions, including the “flawed” ones. The “flawed” definitions really help to get the reasoning process across well, and make a good motivation for the final Definitions (1) and (2). The included examples/figures are also useful and help to explain the concepts.\n\nThe issue I have with the paper is that concepts and methods are all trivial and not particularly interesting or novel.\n\nThe final definition basically compares expected value of “taking an action a at infoset I” and “not taking an action a at infoset I”. Computing these expectations via a sequence form LP representation is trivial and mechanical - and that’s exactly what the equations in the later section do. All that one then needs for the action to be dominated is if one expectation is greater than the other, regardless of the agent’s policies (as long as they reach the infoset). This is again trivial and mechanical operation in the sequence form LP representation (as a small note, I don’t think there’s a need to go through Lagrangian and dual to get to a point where we have both player doing max - one can just argue that for the inequality to hold, it needs to hold for the extreme values and these are simply achieved when both players do max)\n\nFurthermore, the paper doesn’t include any discussion on prior work on this topic. My impression is that this work would classify more as an “exercise for a reader” in a game theory textbook rather than a submission to this venue.\n\nQUESTIONS:\nAs you assume you would construct and run LP in each infoset, wouldn't the construction and the final LP be as big/hard as solving the original game?\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 2, "REVIEWER_CONFIDENCE": 5, "CLARITY": 4, "IMPACT": 1, "ORIGINALITY": 1, "DATE": "2025-07-03"}