Title: AlphaBeta is not as good as you think: a simple class of synthetic games for a better analysis of deterministic game-solving algorithms
Venue: NeurIPS 2025
Accepted: yes
Keywords: Algorithmic Game Theory

Probabilistic Models

Minimax

Planning

Synthetic Game

Complexity Analysis
Abstract: Deterministic game-solving algorithms are conventionally analyzed in the light of their average-case complexity against a distribution of random game-trees, where leaf values are independently sampled from a fixed distribution. This simplified model enables uncluttered mathematical analysis, revealing two key properties: root value distributions asymptotically collapse to a single fixed value for finite-valued trees, and all reasonable algorithms achieve global optimality. However, these findings are artifacts of the model’s design: its long criticized independence assumption strips games of structural complexity, producing trivial instances where no algorithm faces meaningful challenges. To address this limitation, we introduce a class of synthetic games generated by a probabilistic model that incrementally constructs game-trees using a fixed level-wise conditional distribution. By enforcing ancestor dependencies, a critical structural feature of real-world games, our framework generates problems with adjustable difficulty while retaining some form of analytical tractability. For several algorithms, including AlphaBeta and Scout, we derive recursive formulas characterizing their average-case complexities under this model. These allow us to rigorously compare algorithms on deep game-trees, where Monte-Carlo simulations are no longer feasible. While asymptotically, all algorithms seem to converge to identical branching factor (a result analogous to that of independence-based models), deep finite trees reveal stark differences: AlphaBeta incurs a significantly larger constant multiplicative factor compared to algorithms like Scout, leading to a substantial practical slowdown. Our framework sheds new light on classical game-solving algorithms, offering rigorous evidence and analytical tools to advance the understanding of these methods under a richer, more challenging, and yet tractable model.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose a new probabilistic model for constructing game trees to evaluate deterministic solvers. In the standard model, leaf values are independently sampled from a fixed distribution and propagated up the tree according to minimax constraints. The authors argue that this strips the games of structural complexity and results in less meaningful analysis. The proposed forward model instead samples a game value at the root and recursively samples child values from a conditional distribution on the parent's value, while still ensuring minimax constraints are respected. This leads to more realistic and harder games for average-case analysis. This analysis leads to the interesting result that AlphaBeta search incurs a large multiplicative constant on its expected branching factor, which seems to have significant practical performance implications.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n* The paper is well-written and generally very easy to follow.\n* The forward model is simple to understand, implement, and seems to offer convenient analysis for very large trees. I see it as a significant contribution to the area.\n* The numerical analysis offers a hypothesis which validates long-standing observations about the practical advantages to using Scout over AlphaBeta.\n* Though I have some questions/comments about limitations and I’m not completely familiar with all of the cited literature, I think the authors have somewhat accomplished the title goal of providing a new model that enables better analysis of deterministic game-solving algorithms.\n\n**Weaknesses**\n* The major weakness I feel this works has is its lack of discussion related to the limitations of the forward model. Similar to how the standard model does not enforce “ancestor dependency”, are there important structural game tree properties not present under the forward model? I feel some discussion on this could be added to the paper.\n* One area where this shows up is the experiments in Section 6. The parameterization of the forward model starts with, in my opinion, a relatively unrealistic set of distributions. Fairly or not, this may have a significant effect on AlphaBeta, because the distribution choice may influence how frequently and rapidly AlphaBeta degenerates to the null window. I think this is worth investigating further as I noticed that AlphaBeta seems to suffer less in the bimodal-uniform case.\n* The paper would be stronger if the analysis was extended to include a real game, even just to justify parameterization choices for the forward model.\n* Though this work seems to make progress toward answering important questions about deterministic solvers, it is worth mentioning that its applicability to the broader community at NeurIPS may be somewhat limited.\n\nQUESTIONS:\n1. What are “realistic” parameterizations for the forward model?\n2. In light of 1, can the authors please justify their parameterizations?\n3. Are there $\\mu$ where AlphaBeta outperforms the other algorithms?\n3. Can the authors elaborate on the modifications or additional assumptions needed for the continuous case?\n\nLIMITATIONS:\nThe authors discuss certain limitations about the work related to the analysis of the forward model and the discrete setting. However, as mentioned in the Weaknesses section, the authors do not discuss the potential limitations of the forward model itself.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-06-20"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a forward model for analyzing deterministic game-solving algorithms, addressing the limitations of the standard model (e.g., independence assumption) by enforcing ancestor dependencies in game-tree construction. The authors derive recursive complexity equations for algorithms like AlphaBeta and Scout, showing that while all algorithms converge to the same asymptotic branching factor, finite-depth analysis reveals AlphaBeta incurs a larger constant factor, making it less efficient than Scout in practice. The model enables adjustable problem difficulty and rigorous comparisons via recursive formulas, validated by Monte Carlo simulations.\n\nSTRENGTHS AND WEAKNESSES:\n**Strength**\n\n- Theoretical Innovation: The paper introduces a novel forward model that addresses the long-criticized independence assumption in standard game-tree analysis. By enforcing ancestor dependencies, the authors claim that the model captures structural complexities of real-world games (e.g., chess, Go) more realistically, enabling adjustable difficulty calibration for algorithm evaluation.\n\n- Rigorous Analysis: The recursive complexity equations for algorithms like AlphaBeta and Scout under the forward model provide a solid framework for theoretical comparison. The numerical validation via Monte Carlo simulations enhances the credibility of results.\n\n**Weakness**\n\n- The authors claim the forward model better represents real-world games by introducing ancestor dependencies, but fail to Provide concrete examples or case studies (e.g., in chess endgame search) to illustrate how ancestor dependencies mirror actual game dynamics. Thus, there no idea whether the introducing of the forward model is important.\n\n- The literature review focuses on classic studies (mostly pre-2000) and the authors need to check if there are new works in this area. More discussion and comparison may need to be conducted.\n\nQUESTIONS:\nSince this paper introduces a new forward model, which is claimed to be more \"realistic\", how does its tree distribution differ from that of standard random models?\n\nLIMITATIONS:\nThe authors have discussed the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new model for analyzing average-case complexity of determinsitic game-solving algorithms.\nPreviously, the \"standard\" model sampled leaf values to a tree of branching factor $b$ and depth $d$ from a fixed distribution.\nThe paper proposes a new \"forward\" model, where intermediate values are sampled progressively, level by level, until leaf nodes are reached.\n\nThe standard model is too simplistic, as shown by previous works, as the value asymptotically collapses to a single fixed value for all distributions. The new forward model allows for asymptotic analysis on deep trees much more efficiently than Monte-Carlo simulations.\nAuthors show in the asymptotic regime a result analogous to those of the standard model.\nHowever, Alpha-Beta incurs a multiplicative constant proportional to the game’s value range, compared to other algorithms like Scout.\n\nSTRENGTHS AND WEAKNESSES:\nI am not very familiar with the existing literature, so I can't comment on the novelty of the result.\n\nStrenghts\n\n- The paper is very clearly written. It confirms superiority of Scout over alpha-beta as predicted in multiple numerical studies.\n\nWeaknesses\n\n- I believe the results are significant as-is, but they could be greatly improved if there was analysis of MCTS, as that is the leading algorithm for large deterministic games. The lack of MCTS analysis is pointed out in the related works section.\n\nQUESTIONS:\nDoes the level dependence in the forward model increase probability of leaf-disjoint strategies (as defined in [1]), compared to to the standard model?\n\n[1] Lorenz, U. and Monien, B. (2002). The secret of selective game tree search, when using random-error evaluations. \nIn STACS 2002: 19th Annual Symposium on Theoretical Aspects of Computer Science Antibes-Juan les Pins, France, March 14–16, 2002 Proceedings 19, pages 203–214. Springer.\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces a new method for constructing a virtual game tree and compares the standard Alpha–Beta algorithm with several alternatives, including SCOUT. The results confirm the empirically observed advantage of SCOUT over Alpha–Beta.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n- The proposed \"forward model\" seems new.\n\nWeaknesses:\n- Lack of novelty. It has long been established that algorithms more efficient than Alpha–Beta exist—most notably MTD(f)—and that SCOUT generally outperforms the standard Alpha–Beta search. Demonstrating this well-known result again does not, by itself, constitute a new contribution.\n\n- Unclear analysis in Section 5.1 (lines 178–180). The manuscript states:\n\n> “… but under the assumption that the special child—which inherits the root’s value owing to the negamax formulation—has already been identified.”\n\nI do not see how the **special child** can be identified **during** the search.\nWhat exactly is being assumed in the subsequent analysis? For example, does the study first run a complete search to discover which move is the special child, and then use that information to analyse node counts? Please clarify the procedure and the underlying assumptions.\n\n- Missing citations on forward-model tree generation. The paper does not reference earlier work that generates non-pathological game trees using techniques closely related to the forward model. A well-known example is the P-game described by Stephen J. J. Smith and Dana S. Nau, “An Analysis of Forward Pruning,” AAAI 1994. The same approach is also employed in Levente Kocsis and Csaba Szepesvári, “Bandit-Based Monte-Carlo Planning,” ECML 2006. Moreover, if the edge costs to child nodes are constrained to be non-positive, with the edge to the optimal move set to zero, the analysis becomes much simpler—a common trick whose original source, however, appears undocumented.\n\nQUESTIONS:\nI raised a question in the weaknesses section (second bullet). Could you please clarify this point so I can better understand your analysis?\n\nLIMITATIONS:\nSimply listing MTD(f) and PVS in the checklist is insufficient; a full discussion should be in the main text of the paper.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-03"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper critiques standard analyses of game-solving algorithms that assume game-tree leaf values are independent, arguing that this simplification makes problems unrealistically easy. To address this, the authors introduce a new probabilistic model that enforces ancestor dependency, better capturing the structure of real-world games while remaining analytically tractable. Using this model, they derive average-case complexity formulae for classical algorithms like AlphaBeta and Scout, enabling rigorous comparisons, even on deep trees where simulations fail.\n\nTheir key finding is that although all algorithms share the same asymptotic branching factor, finite-depth analysis reveals major differences: AlphaBeta suffers from a large multiplicative constant, slowing it down compared to alternatives like Scout. This suggests that prior claims of universal optimality were artifacts of an oversimplified model, and that more realistic frameworks are crucial for meaningful algorithm evaluation.\n\nSTRENGTHS AND WEAKNESSES:\nThis paper re-examines how deterministic two-player zero-sum games, represented as uniform min/max trees, are used to evaluate game-solving algorithms. The authors introduce the forward model, which builds trees level-by-level using conditional distributions that preserve ancestor dependency. This better reflects the strategic coherence and the structural dependencies found in real games, and allows difficulty to be tuned.\n\n\nThis seems intuitive, but it is not always clear exactly how this model results in more realistic games. This is somewhat subjective, and more examples would help.\n\nA big positive is that, using this model, they derive recursive formulas for the average-case complexity of algorithms like AlphaBeta and Scout. These formulas offer both asymptotic insight and tractable simulation of deep trees (up to height ~5000), unlike expensive Monte Carlo methods. However, I'd like more precise theoretical characterization, ideally in terms of distribution properties and abstract characterizations of games under their model, rather than empirical plots of limited ranges with specific distributions.\n\nQUESTIONS:\n54: Real-world games, from chess to Go, exhibit intricate dependencies between positions: the value of a leaf is inherently tied to the sequence of moves leading to it.\n\nI'm not sure I follow this criticism, and it's not clear to me what independence means here. \nIt seems to me that in games like chess and go, it doesn't matter how a position is reached, once we are there, we are there. So in that sense, the value of a leaf is not a function of the sequence of operations that were used to reach it. \nAre you referring to independence in a statistical sense? It seems later that you want to propose sampling in a way that the destination leaf depends on some structure higher up in the tree. This seems totally reasonable, but the term “independence” is still quite confusing to me. Exactly which sense did you mean? I suppose it would help me to clarify whether we are talking about mathematical or functional independence of a quantity on another quantity, as opposed to independence of random variables.\n\nLIMITATIONS:\nOne of the big claims is that for games with $n$ distinct utility values, alpha-beta search has a factor $O(n)$ overhead as compared to scout.\n\nSpecifically,\n70: “However, finite-depth analysis, in Section 6, reveals critical practical differences masked by asymptotics. Specifically, alpha-beta incurs a multiplicative constant proportional to the game’s value range, causing significant slowdowns compared to other algorithms like scout.”\n\nThis makes it seem like a theorem is proved that shows that, in some sense, alpha-beta is less efficient than scout. Perhaps it's difficult, because we have to consider the asymptotics of multiple variables (depth and range), but I would like to see this theorem formally shown, rather than just plots and empirical results.\n\n\nThe discussion of a game's “value range”, beginning on line 72, is quite confusing, and notation seems to change between discrete {-n, …, n} and continuous [-n, n]. \n\nFrom the name, it sounds like we are talking about the difference between maximum and minimum value, which would be perfectly fine with continuous value sets. But then, on line 149, you use a discrete value range of 2n + 1 elements. If it’s inherently about discreteness, I’d use terms like “cardinality” or “size” instead of range. So far, this is section 5 rather than section 6, but section 5 doesn't yet distinguish between alpha-beta and scout.\n\n\n\nThen in section 6, e.g., on line 314, we are no longer in the discrete world, as we sample continuous random variables. It seems like there is some discretization happening here in the test brute force and test by section algorithms, but it's not clear that alpha beta or scout are also being discretized. \n\nIt seems to me like with this discretization, these are actually approximation algorithms, and you implicitly set the approximation parameter to one. Is this the case? Are you identifying $n/\\epsilon$ overhead? I'm reading a lot into the paper here, it would be very helpful if this were made explicit.\n\nTheorem 5.1 also seems to mix continuous and discrete notation.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-07"}