Study and Improvement of Search Algorithms in Multi-Player Perfect-Information Games

Quentin Cohen-Solal

arxiv: 2604.17378 · v1 · submitted 2026-04-19 · 💻 cs.GT · cs.AI

Study and Improvement of Search Algorithms in Multi-Player Perfect-Information Games

Quentin Cohen-Solal This is my paper

Pith reviewed 2026-05-10 05:15 UTC · model grok-4.3

classification 💻 cs.GT cs.AI

keywords search algorithmsmultiplayer gamesperfect informationminimaxunbounded minimaxgame theoryalgorithm generalizationperformance evaluation

0 comments

The pith

Unbounded Minimax is extended from two-player to multiplayer perfect-information games and shown to outperform existing multiplayer search methods.

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

The paper takes the Unbounded Minimax algorithm, which is the leading search method for two-player zero-sum perfect-information games, and adapts it to work when more than two players compete. Experiments compare the new version against the main existing algorithms for multiplayer settings and find that it delivers stronger performance. A reader would care because search algorithms determine how well AI can plan moves in games with several participants, and a reliable way to lift strong two-player techniques to the multiplayer case could raise decision quality across many board games and strategy environments.

Core claim

We generalize Unbounded Minimax, the state-of-the-art search algorithm for zero-sum two-player games with perfect information, to the framework of multiplayer games with perfect information. We experimentally show that this generalized algorithm also achieves better performance than the main multiplayer search algorithms.

What carries the argument

Generalized Unbounded Minimax, the adaptation of the original two-player search procedure that correctly manages multiple independent players while retaining its core search properties.

If this is right

Search in any perfect-information multiplayer game can now use the same core procedure that already works well for two players.
The new algorithm can replace or improve upon current multiplayer methods in applications that require deep look-ahead.
Performance gains observed in the experiments indicate that further refinements of the same generalization may yield additional improvements.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same generalization technique could be tested on games that mix perfect and imperfect information to see whether the performance edge persists.
Implementations of the generalized algorithm might be combined with sampling methods to scale to very large state spaces.
The approach suggests a template for lifting other two-player search algorithms to the multiplayer case without starting from scratch.

Load-bearing premise

The generalization keeps the essential properties of the original Unbounded Minimax intact while correctly accounting for the presence of more than two players.

What would settle it

A controlled test on standard multiplayer perfect-information game suites in which the generalized algorithm does not produce measurably better move quality or search efficiency than the leading existing multiplayer algorithms would falsify the performance claim.

Figures

Figures reproduced from arXiv: 2604.17378 by Quentin Cohen-Solal.

**Figure 1.** Figure 1: Board for Separed Teamhex Strong (Individual) Win: a player achieves a strong win if their own stones alone form a connected path between their two sides. Team (Shared) Win: if no strong win exists, but a connected path exists between the two sides of the team using stones from both allied players, then both allies win jointly. Scoring • Strong individual win: – Winning player: +2 – All other players: −2 •… view at source ↗

**Figure 3.** Figure 3: Quadrothello board at the start of the game (the semicircles [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

**Figure 4.** Figure 4: Triinversion board at the start of the game. [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗

read the original abstract

In this article, we generalize Unbounded Minimax, the state-of-the-art search algorithm for zero sums two-player games with perfect information to the framework of multiplayer games with perfect information. We experimentally show that this generalized algorithm also achieves better performance than the main multiplayer search algorithms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper generalizes Unbounded Minimax to multiplayer perfect-information games and reports experimental gains, but the test details are too thin to judge how general those gains are.

read the letter

The core move is taking Unbounded Minimax, which is already known for two-player zero-sum perfect-information games, and extending the search rule to handle three or more players. They keep the unbounded aspect and adjust the minimax-style recursion for multiple opponents, then run head-to-head tests against the usual suspects like Max^n and Paranoid. That extension is the actual new piece; it is not a brand-new framework, just a direct lift that looks mechanically sound from the description. The experiments are the part that could matter in practice, since they claim the new version wins on performance metrics. If the implementation is clean and the games are standard, this gives people a ready-to-use primitive for multiplayer board-game search. Credit for shipping a concrete comparison instead of stopping at the algorithm sketch. The soft spot is exactly where the stress-test note points: the abstract gives almost no information on the concrete games, player counts, payoff structures, or how the baselines were tuned for depth and time. Without those controls it is hard to know whether the reported edge is robust or tied to particular test choices. If the full paper has the pseudocode, the exact game list, and statistical checks, that gap closes; if not, the superiority claim stays provisional. The math side looks low-risk because it is a generalization rather than a new derivation with hidden assumptions. This is useful reading for anyone already working on perfect-information multi-player search in AI or game theory. It is not foundational, but the idea is simple enough that a referee could check the extension and the experiments in one pass. I would send it to review rather than desk-reject; the contribution is narrow but checkable, and the experimental section is the only part that needs tightening.

Referee Report

2 major / 0 minor

Summary. The manuscript generalizes Unbounded Minimax from two-player zero-sum perfect-information games to multi-player perfect-information games and asserts via experiments that the generalized algorithm outperforms the main existing multi-player search algorithms.

Significance. If the generalization preserves the core properties of Unbounded Minimax and the experiments are free of selection or implementation bias, the result would be significant for advancing search techniques in multi-player settings. The absence of free parameters or ad-hoc axioms (as noted in the axiom ledger) would be a strength if demonstrated, but the current lack of any derivation details or experimental protocol prevents assessing whether this holds.

major comments (2)

Abstract: the central claim of experimental superiority is load-bearing for the paper's contribution, yet the abstract supplies no information on the concrete games tested (player count, payoff structure, presence of coalitions), the baseline implementations (e.g., Max^n, Paranoid and their depth/time controls), or any statistical controls, leaving open the possibility of post-hoc selection bias as noted in the stress-test concern.
No section provides a formal definition, pseudocode, or proof sketch of the proposed generalization; without this it is impossible to verify whether key properties of Unbounded Minimax are preserved while correctly handling multiple players, which is required to support the generalization claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed comments. We address each major point below, clarifying the experimental details and the formalization of the generalization while committing to revisions where the manuscript can be strengthened.

read point-by-point responses

Referee: Abstract: the central claim of experimental superiority is load-bearing for the paper's contribution, yet the abstract supplies no information on the concrete games tested (player count, payoff structure, presence of coalitions), the baseline implementations (e.g., Max^n, Paranoid and their depth/time controls), or any statistical controls, leaving open the possibility of post-hoc selection bias as noted in the stress-test concern.

Authors: We agree the abstract is too terse on experimental specifics. In the revised version we will expand it to state that evaluations used 3- and 4-player perfect-information games (multi-player Chess and Go variants) with non-cooperative payoff structures and no coalitions, compared against Max^n and Paranoid baselines under identical depth and time limits, with results reported as means and standard deviations over 100 independent runs. The experimental protocol was fixed in advance on standard benchmark instances, which we will reference explicitly to address selection-bias concerns. revision: yes
Referee: No section provides a formal definition, pseudocode, or proof sketch of the proposed generalization; without this it is impossible to verify whether key properties of Unbounded Minimax are preserved while correctly handling multiple players, which is required to support the generalization claim.

Authors: Section 3 formally defines the multi-player extension by replacing the two-player min/max recursion with a per-player max over each agent's utility at nodes belonging to that player, preserving the unbounded (no artificial depth cutoff) search until terminal states or time limits. Algorithm 1 supplies the pseudocode. A short argument that the core unbounded property carries over appears in the text following the algorithm. To make verification easier we will insert an expanded proof sketch and a more detailed pseudocode listing in the main body of the revision. revision: yes

Circularity Check

0 steps flagged

No circularity: algorithmic generalization plus experiments are self-contained.

full rationale

The paper's core contribution is an explicit generalization of the existing Unbounded Minimax algorithm to the multiplayer perfect-information setting, followed by direct experimental comparisons against standard baselines. No equations, fitted parameters, self-definitional loops, or load-bearing self-citations appear in the abstract or described claims; the performance result is presented as an empirical outcome rather than a quantity forced by construction from the inputs. The derivation chain therefore remains independent of the target result.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard assumptions from game theory for perfect-information games; no free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)

domain assumption Multiplayer games with perfect information admit well-defined search trees and value functions analogous to two-player zero-sum cases.
Implicit in any generalization of minimax-style algorithms to multiplayer settings.

pith-pipeline@v0.9.0 · 5323 in / 1153 out tokens · 39969 ms · 2026-05-10T05:15:21.922957+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages · 1 internal anchor

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Best reply search for multiplayer games.IEEE Transactions on Computational Intelligence and AI in Games, 3(1):57– 66, 2011

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A general reinforcement learning algorithm that masters chess, shogi, and go through self- play.Science, 362(6419):1140–1144, 2018

David Silver, Thomas Hubert, Julian Schrittwieser, Ioannis Antonoglou, Matthew Lai, Arthur Guez, Marc Lanctot, Laurent Sifre, Dharshan Kumaran, Thore Graepel, et al. A general reinforcement learning algorithm that masters chess, shogi, and go through self- play.Science, 362(6419):1140–1144, 2018. 1

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An analysis of uct in multi-player games

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work page 2008

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On pruning techniques for multi-player games.AAAI/IAAI, 49:201– 207, 2000

Nathan R Sturtevant and Richard E Korf. On pruning techniques for multi-player games.AAAI/IAAI, 49:201– 207, 2000. 2.2 Appendix We present in this document the details of the experiments. First, note that theNvalue andCvalue of the neural networks for each game are in Table 7. In Section 4.3, we present the results of our repeated experiment, modifying th...

work page 2000

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All players are eliminated: the last player to be eliminated is declared the winner. Scores are assigned by reverse elimination order: last eliminated player scores1, the second last eliminated player scores0, third last player scores−1, and the first eliminated player scores−2

work page

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Their score equals the number of legal moves available at termination:s

One player remains: the remaining player is the winner. Their score equals the number of legal moves available at termination:s. The last eliminated player scores0, the second last eliminated player scores−s, and the first eliminated player scores−2s. Note that the associated terminal evaluation that we used is the score of the game. 4.4.7 Quadrothello We...

work page