General search techniques without common knowledge for imperfect-information games, and application to superhuman Fog of War chess
Authors: Brian Hu Zhang
Tuomas Sandholm
Venue: ICLR 2026 Poster (2026)
Keywords: imperfect-information games
subgame solving
game theory
Primary area: reinforcement learning

Abstract: Since the advent of AI, games have served as progress benchmarks. Meanwhile, imperfect-information variants of chess have existed for over a century, present extreme challenges, and have been the focus of decades of AI research. Beyond calculation needed in regular chess, they require reasoning about information gathering, the opponent’s knowledge, signaling, _etc_. The most popular variant, _Fog of War (FoW) chess_ (a.k.a. _dark chess_), has been a major challenge problem in imperfect-information game solving since superhuman performance was reached in no-limit Texas hold’em poker. We present _Obscuro_, the first superhuman AI for FoW chess. It introduces advances to search in imperfect-information games, enabling strong, scalable reasoning. Experiments against the prior state-of-the-art AI and human players---including the world's best---show that _Obscuro_ is significantly stronger. FoW chess is the largest (by amount of imperfect information) turn-based zero-sum game in which superhuman performance has been achieved and the largest game in which imperfect-information search has been successfully applied.

Published as a conference paper at ICLR 2026
GENERAL SEARCH TECHNIQUES WITHOUT COMMON KNOWLEDGE FOR IMPERFECT-INFORMATION GAMES, AND APPLICATION TO SUPERHUMAN FOG OF WAR
CHESS
Brian Hu Zhang
Tuomas Sandholm
MIT CSAIL zhangbh@csail mitaaiiiimmitt.... sit.edd
Carnegie Mellon University sandho cs cma. edu
ABSTRACT
Since the advent of AI, games have served progress benchmarks. Meanwhile, imperfect-information variants of chess have existed for over century, present ex- treme challenges, and have been the focus of decades of Al research. Beyond cal- culation needed in regular chess, they require reasoning about information gath- ering, the opponent's knowledge, signaling, etc. The most popular variant, Fog of War (FoW) chess (a.k.a. dark chess), has been a major challenge problem in imperfect-information game solving since superhuman performance was reached in no-limit Texas hold' em poker. We present Obxcuro, the first superhuman Al for FoW chess. It introduces advances to search in imperfect-informatio games, enabling strong, scalable reasoning. Experiments against the prior state-of-the-art Al and human players including the world's icantly stronger. FoW chess is the largest (by amount of imperfect information) turn-based zero-sum game which superhuman performance has been achieved and the largest zero-sum game in which imperfect-information search has been successfully applied.
show that Obscuro signif-
INTRODUCTION
The concept of breaking a large problem into subproblems and searching through them individually has been with us since time immemorial. In artificial intelligence (AI), search is core capability that required for strong performance in many applications. In game solving, this commonly takes the form of subgame solving. In games of perfect information, subgame solving is conceptually straight- forward, because every new state induces subgame that can be analyzed independently of the rest of the game. Subgame solving in perfect-informaatee Turing and David Champernowne wrote a chess engine Turochamp in 1948 using minimax search and hand-crafted function for evaluating nodes (Kasparov & Friedel, 20. In landmark results, subgame solving has played a key role in reaching superhuman level in chess (Campbell et al., 2002) and go (Silver al., 2016; 2017; 2018).
games is as old as computers themselves: Alan
In contrast to such perfect-information games, most real-world settings are imperfect-informace games. These include negotiation, business, finance, and defense applications. Thus is crucial for the field of AI to develop strong techniques for imperfect-informaae additional challenges not present in perfect-information games. For example, AIs for imperfect- information games might need to randomize their actions to prevent the opponent from learning too much information, and a player's 's optimal action in a state can depend on that same player's action in totally different state. Therefore, subgame solving in drastically more difficult. Methods for real-time subgame solving in imperfect-informatiin games have only been developed relatively recently (Gilpin & Sandholm, 2006; 2007; Waugh et al., 2009; Ganzfried & Sandholm, 2015; Burch et al., 2014; Moravcik et al., 2016; Brown & Sandholm, 2017; Moravčík et al., 2017; Brown & Sandholm, 2018; 2019; Brown al., 2020; Sokota et al., 2024), and they were key to achieving superhuman performance in no-limit Texas hold em poker (Brown & Sandholm, 2018: 2019; Moravčík et al., 2017). Strong Al performance has also been achieved
games. Such games involve
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in few imperfect-informan zero-sum games that even larger (Vinyals et al., 2019; Berner et al., 2019; Perolat et al., 2022). However, these were accomplished with learning alone and did not enjoy the further performance benefits that search could bring, due largely to the lack of scalability of subgame solving algorithms for imperfect-informatie games larger than poker.
In this paper we present dramatically more scalable general-purpose subgame solving techniques for imperfect-information games. We used these techniques to create Obscuro, an Al that achieved superhuman performance Fog of War (FoW) chess (a.k.a. dark chess), the most popular variant of imperfect-informatiinn chess. Over 120 games against humans of varying skill levels-including the #1-ranked human- and 1,000 games against the previous state-of- tethhheart art FoW chess Al (Zhang & Sandholm, 2021), we conclusively demonstrate that Obscuro is stronger than any other current agent-human or artificial for FoW chess. FoW chess now the largest (measured by amount of imperfect information) turn-based game in which superhuman performance has been achieved and the largest game which imperfect-information search techniques have been successfully applied.
In the next section we will introduce the game and discuss the challenges that players in these types of games must tackle. In the section after that, we present our AI agent Obscuro and the algorithms therein. In the section after that, we present our experiments. Finally we present conclusions and future research directions.
2 CHALLENGES IN IMPERFECT-INFORMATIONN GAMES SUCH AS FOG OF WAR CHESS
Imperfeectinnformatin versions of chess have captured the imagination of chess players and scien- tists alike for over century. To our knowledge, the first imperfect-informatio version of chess was Kriegspiel, invented in 1899 and based on the earlier game Kriegsspiel, a war game used by the Prussian army in the early 19th century for training (Pritchard, 1994). In the modern day, there variants of chess, including Kriegspiel, reconnaissance blind chess (RBC), and Fog of War (FoW) chess. Imperfect-infoomatior chess is recognized challenge prob- lem in AI. Although there has been Al research in Kriegspiel (Parker et al., 2005; Russell & Wolfe, 2005; Ciancarini & Favini, 2009) and RBC (Gardner al., 2023), strong performance has been achieved in Kriegspiel, and RBC is not played competitively by humans. By comparison, FoW chess has surged in popularity due its implementation on the major chess website chess.com and strong human experts have emerged among thousands of active players.² 2 It is the most popular variant of imperfect-informatie chess by far, and strong human experts exist who can serve as challenging benchmarks of progress.
multiple imperfect-informatio
FoW chess presents a unique combination of challenges that did not exist in prior superhuman AI milestones." First, chess itself is a highly tactical game often requiring careful lookahead, and FoW chess is no different: there are often positions where one player has perfect or near pear-perfect information and can execute sequence of moves that results in an advantage. Thus, strong agent must have solid lookahead capability. Lookahead in other games is usually accomplished by subgame solving. Thus it would be desirable to be able to conduct subgame solving in FoW chess too.
Second, private information is rapidly gained and lost. It is possible for the size of player's infor- mation set (infoset)-iee, set of indistinguishable positions given player's observations- rapidly increase and then decrease again, for example, from hundreds up to millions and then back down to hundreds, in a matter of a few moves. Thus, a strong agent must have the ability to reason about this rapidly-chame cpanniy- information.
Third, a strong agent must at least somewhat play mixed strategy-thar is, it must randomize its actions. Otherwise, an adversary who knows the strategy, or has learned the strategy from past observation, can easily exploit that knowledge.
Despite its similar name, Chinese dark chess has no private information, and thus does not require the types of reasoning that are required in FoW chess. 2 "s of April 2025, the Fog of War chess leaderboard on chess.com (Chess.com, 2025a) listed 19,150 active players.
'The complete rules of FoW chess can be found in Appendix A
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Finally, in games like FoW chess, reasoning about common knowledge is difficult. This is a key challenge because most algorithms for subgame solving-including those that led to breakthroughs in no-limit Texas hold em poker-rehh on the ability to reason about common knowledge, even the ability to enumerate the entire Hommon-knowledde C. with the property that it is common knowledge that the true history lies in C' (Brown & Sandholm, 2018; 2019). So, to prepare for solving a subgame, prior algorithms need to reason about what the agent knows about what the opponent knows about what the agent knows, and so on. This need can dramatically expand the set of states that need to be incorporated into the subgame solving algorithm, making such methods impractical for games much larger than no-limit Texas hold'em.
often set-that is, the smallest set of histories
For example, consider the two FoW chess positions in Fig. 1.* Although seemingly completely distinct, is possible to show (see Appendix E. that these two positions are connected by no fewer than nine levels of "I think that you think that..." reasoning. Prior techniques would require the ability to generate this complex connection before starting subgame solving from either of the two positions.
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Figure 1: Two FoW chess positions in the same common-knowledde set. (A) position after moves 1.
Ne3 g5 2. Nh3 d5; (B) position after moves 1. NI3 2. h3 Qh4. The boxed squares mark pieces visible to the opponent.
Such intricacies make it difficult to reason about common knowledge efficiently. For example, common--kooledde sets in FoW chess quickly grow prohibitively large, they cannot be held directly in memory (Zhang & Sandholm, 2021). In FoW chess, individual infosets often have size as large 10 and can have size 10". Common-kkowlese to be enumerated in reasonable time or space during search. (Detailed calculations for these lower bounds can be found in Appendix E.1.) Perhaps even more troubling is the fact that is not even clear that it is possible to efficiently decide whether two histories can be distinguished by common knowledge, so in some sense reasoning about common knowledge may require enumerating the common- common-knowledge knowledge set in the worst case.'
sets can have size 1028_far too large
This is in sharp contrast poker, which has special structure that has driven the success of past ef- forts in that game. First, at least in two-player (heads-up) Texas hold' em poker, common-knowledge. sets are not very large. They have size at most (52) ( x 1.6 x 108. and can thus easily be held in memory. Moreover, thanks to poker-specific optimizations (Johanson et al., 2011), subgame solv- ing in poker can be implemented in such way that its complexity depends not on the size of the common- -oomoon- set but merely on the size of the infoset, enabling feasible subgame solving even
'The sequence of moves in the figure is purely for the purpose of illustrating common knowledge, and does represent strong play. For example, Obscuro never plays 1... g5 2... Qh4.
'Solinas et al. (2023) formally state and study this and similar computational problems in general games, showing that they are intractable in the worst case, so it should perhaps not come as surprise that they appear be hard in FoW chess.
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when the commonn-knowlddee sets are large, is the case in multi-player poker." In more general games where these domain-spe speciiic techniques do not apply- splyy-such as FoW chess-the complexity games would scale with the
of traditional subgame solvving techniques for imperfect-informattee size of the common-knowledge set, which in our case renders such techniques totally infeasible.
DESCRIPTION OF OUR AI AGENT Obscuro AND THE NEW ALGORITHMS THEREIN
The technical innovations of Obscuro are in its search algorithms. At high level, they operate as follows. At all times, the program maintains the full set P possible positions' given the observations that has seen so far the game, as well as a partial game tree consisting of its calculations from the previous move. At the beginning of the game, P contains only the starting position Su, and I consists of single node $o, since the program has done no calculation. Although P is small enough to fit in memory (usually [P] S 10%). it is too large to feasibly allow nontrivial reasoning about every single position in P on every move. Therefore, the program instead samples a small subset I C P at random, whose size is no more than few hundred positions.
Given subset 7, the program at high level executes the following steps.
1. Construct an imperfect-information subgame I incorporating the saved computation from the previous move (I',, well the positions in the sampled subset I.
2. Compute an (approximately) optimal strategy profile (i.e., an approximate Nash equilib- rium) of r.
3. Use the Nash equilibrium to expand the game tree
4. Repeat the above two steps until a time budget is exceeded.
5. Select move.
We now elaborate on each step individually. Full detail about our techniques, including formal descriptions of all techniques, proofs, and comparisions to prior work, can be found in Appendix C.
3.1 STEP 1: GENERATING THE INITIAL GAME TREE AT THE BEGINNING OF A TURN
The imperfact-information subgame P is constructed from the old game tree and the sampled additional positions
according to new algorithm which we call knowledge-limteed unfrozen subgame solving (KLUSS). It is more effective than the knowledge-limiter subgame solving (KLSS) algorithm of Zhang and Sandholm (Zhang & Sandholm, 2021) (a comparison presented in Ap- pendix C). At high level, KLSS and KLUSS address the issue of reasoning about common knowl- edge by assuming that sufficiently high-order knowledge is essentially irrelevant to game play: there is position s in the old tree I such that we know that the opponent knows that we know that is not the true state, remove from I' it is assumed to be irrelevant. As an example, consider the game in Fig. There are two players, A and v. Suppose that we are A, and we have arrived at the circled node (which is alone in its infoset, i.e., this node, has perfect information).
The infosets (dotted lines) define a connectivity graph G among the five nodes in that layer of the tree: nodes and connected if there is an infoset connecting any descendant of (including u itself) to any descendant of u (including v itself). The nodes in that layer are labeled according to their distance from the circled node; the node labeled oo order of knowledge: if the true node is the circled node, then the distance is the smallest integer k for which the statement
connected. Distance corresponds to
everyone knows that everyone knows that ... everyone knows that the true node is not u
repetitions
"Specifically, Pluribus (Brown Sandholm, 2019) would not have been feasible without these poker- specific optimizations. A position descrihes where pieces well the castling and passant rights.
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Figure 2:
An example game tree, to illustrate KLUSS. The box (0) is chance node. Dotted lines connect nodes in the same infoset.
false. Thus the shaded red region corresponds to nodes that will be removed: everyone knows that these nodes are not the true nodes. This allows the game tree to be kept to a manageable size, even when the common-knowledge set (which the program never computes or uses) large.
Our approach has two important properties. First, it enables the agent to reason about the opponent's information in more powerful way than assuming something pessimistic, such as the opponent having perfect information (Parker et al., 2005; Russell & Wolfe, 2005). This allows behavior such as bluffing, which is important to strong play. Second, it accomplishes this while essentially only examining states that are relevant in the sense that, as far as our agent knows, the opponent might believe that is the true state. This ensures that, even when the common- common-knowledge knowledge set is large and poorly structured (e.g., even when the vast majority of states in the common-knowledd set are irrelevant), subgame solving is still possible and effective.
The difference between KLUSS and KLSS is that in KLSS, the strategy at the two-node infoset for in Fig. 2 (and more generally all A n-nodes at distance and their descendants) frozen to that node's strategy from the previous move. In KLUSS, it is unfrozen and will be thee-thherettccaly optimized together with the rest of the subgame that is not removed (i.e., not red).
KLSS and KLUSS are not always game-theorttically sound in theory because some of the removed (red in the figure) part of the game tree could be relevant to the decision, but they are often sound in practice (Zhang & Sandholm, 2021; Liu et al., 2023). They can be viewed as computationally feasible alternative to traditional game-theoretically sound subgame solving.
3.2 STEP 2: EQUILIBRIUM COMPUTATION
The remaining steps are inspired by the growing-tree counterfactual regret minimization (GT- CFR) algorithm (Schmid et al., 2023): game tree I' is simultaneously solved using an iterative equilibriun-fiddnng algorithm and expanded using an expansion policy.
For equilibrium finding we use state-of-tte-arr algorithm, predictive CFR+ (PCFR+) (Farina et al., 2021). PCFR+ an iterative, anytime algorithm for solving imperfect-inffomaaioo games, that can handle the fact that our game tree I' is changing over time. At all times t, PCFR+ maintains profile (4.))), where x' is our strategy and y² is the opponent's 's strategy.
PCFR+ has only been proven to converge in average strategies. That is, the empirical strategy ET_I"' y²) converges to Nash equilibrium t co. However,
profile (z', 7'') ( E Ei_ x. instead of computing the empirical average strategy, we circumvent this step and maintain only the last iterate (xf,y"). There are several reasons for this choice, which are detailed in Appendix C.7.
3.3 STEP EXPANDING THE GAME TREE
Nodes are selected for expansion by using carefully-designed expansion policies that balance ex- ploration and exploitation. Our program chooses node to expand by the following process. Fix one player to be the exploring player. (The choice of which player is exploring alternates: on odd- numbered iterations, P1 is the exploring player, on even numbered iterations, P2 is the exploring player.) For this exposition, will play according to its current strategy as computed by PCFR+, in this case y¹. The exploring player will play perturbed version J of its current strategy x!. The strategy 5 is designed to
will take Pl to be the exploring player. The non-exploring player
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balance between exploitation and exploration. Exploitation here means playing actions with high possible reward, that is, actions that have positive probability in x'. Exploration means assigning positive probability to every possible action, to hedge against the possibility that the current tree incorrectly estimates the value of the action due to lacking search depth. For this, we use a method based on the polynomial upper confidence bounds for trees (PUCT) algorithm (Silver et al., 2016). Finally, a leaf node of the current tree I' is selected for expansion according to the strategy profile (ō', y²).
One major difference between our algorithm and the GT-CER algorithm lies in having only one player use the exploring strategy 8 rather than both. Intuitively, this remains sound, because tree nodes that neither player plays to reach are irrelevant to equilibrium play, and thus do not need to be expanded. In Appendix C.4, we formally show that this variant, like GT-CFR, will find an exact equilibrium of any two-player zero-sum game given infinite search time. Thus, allowing one player to play directly from their equilibrium strategy (here, y²) allows the tree expansion to be more focused. We call this GT-CFR variant one-sided GT-CFR.
Once a leaf node z is chosen by the above process, its children are evaluated by node heuristic and added to the game tree. The node heuristic is an estimate of the perfect-information value of as evaluated by the chess engine Stockfish 14 (Stockfish). If z is the first node in its infoset that has been expanded, a local regret minimizer is created for PCFR+, and it is initialized to pick the action with highest value according to the node heuristic. Theoretically, the guarantees of PCFR+ do not depend on the initialization, which can be arbitrary. However, practically, we find that initializing to a "good guess" of a good action leads to faster empirical convergence to equilibrium. More details can be found in Appendix C.5.
3.4 STEP 4: REPEAT
The above two steps are repeated, in parallel using a multi- threaded implementation, until a time budget is exceeded. Our implementation uses one thread running CFR and two threads expanding the game tree, which is shared across all three threads. The node expansion threads use locks to avoid expanding the same node, but the equilibrium computation thread uses locks and only works on the already-expanded portion of the game tree. The time budget is set heuristically based on the amount of time remaining on the player's clock. Once the time budget is exceeded, the tree expansion threads (Step 3) are stopped first, and then, after a delay, the equilibrium computation thread (Step 2). The added time allocated to equilibrium computation is present so that a more precise equilibrium can be computed without the tree constantly changing.
3.5 STEP 5: SELECTING MOVE
After those computations have stopped, a move is selected based the (possibly mixed) strat- egy that PCFR+ has computed. Instead of directly sampling from this distribution, we first purify (Ganzfried al., 2012) that is, we limit the amount of randomness. In particular, we sample from only the m pronibbiitbbbbiiityy actions, where 1 < m < 3 is chosen based on the computed strategies. We only allow mixing (m > 1) when the algorithm believes that its computed strategy is safe-intuitively, this is when the algorithm's 's final strategy x" can guarantee expected value at least as good as what the algorithm thought to be possible before the turn. This purification technique made a significant difference in practice, detailed via an ablation test in Section 4.1.
EXPERIMENTAL EVALUATION
To evaluate our techniques, we conducted several experiments. The first was a 1,000-game match against the previous sfate-of-the-art AI for FoW chess (Zhang & Sandholm, 2021) (hereafter ZS2/). Our new AI scored 85.1% 1% (+834 =33 -133),, confidently establishing its superiority.
We then ran two experiments against human players. The first of these was series of games against human players of varying skill levels. Obscuro played a total of 117 games (with time control minutes 2 seconds per move). This time control was selected because was the most popular time
Thii notation means 834 wins, 33 draws, and 133 losses.

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control played on the most popular website for FoW chess (chess.com) at the time of the experiment. While in regular chess both fast and slow games are common, in FoW chess slow games are typically not played. The skill levels of the players, measured by their chess.com Fog of War chess ratings, ranged from 1450 to 2006. We excluded 17 of the games for various reasons such as disconnections, the opponent leaving before the game finished, or the opponent clearly losing purpose, leaving 100 completed games. Obscuro scored 97% (+97 =0 -3), establishing conclusively that it is stronger than humans of this level.
Finally, we invited the top FoW chess player to 20-game match (again at 3+2 time control). At the time of our match, i.e., of the rating list on August 16, 2024 (Chess.com, 2024), this player was rated 2318 and ranked #1 on the chess.com Fog of War blitz leaderboard. The games were played over the course of two days, 10 games per day, giving the human player an opportunity to analyze the first set of games overnight. In this match, Obscuro scored 80% 16 =0 -4, +241 Elo), a conclusive and statistically significant (p 0.0118 using an exact binomial test) victory against the world's strongest player. We thus conclude that Obscuro is superhuman.
The 20 games played against the top human are available through the following link: https: //1 //lichesss org/ study/ ja393U00
curated sample of particularly interesting games from our 100 games played against humans of varying skill levels, including all three games lost by Obscuro, is available through the following link: https: //lichess. org/ / study/ ynz
In both links, each game lists which side Obscuro played (Black or White) and the game result (Win or Loss). All games are shown from the perspective of Obscuro.
4.1 ABLATIONS
In addition, we conducted multiple ablations with Obscuro. In each of these experiments, we turned off one or more of the new techniques introduced in this paper in order to evaluate the contributions of the different techniques to the performance of Obscuro. All ablations were run a time control of 5 seconds per move. Unless otherwise stated, all ablations were Obscuro playing against version of Obscuro with the single stated technique turned off. Recall from above that Obscuro with all techniques turned scored 85.1 % against ZS21 and 80% against the top human.
1. Purification off. This version allowed mixing among all stable actions, even if the current margin is negative or there are more than three of them.
In 1,000-game match, Obscuro scored 70.2% (+662 =79-259)
2. KLUSS off. In this version, the strategies in infosets not touching our infoset were frozen, in 1-KLSS.
In a 1,000-game match, Obscuro scored 58.0% (+532 =96 -372).
3. One-sided GT-CFR off. In this version we use the two-sided node expansion algorithm proposed by the original GT-CFR paper (Schmid et al., 2023).
In 10,000-game -ame match, Obscuro scored 53.3% (+4535 =1583 -3882).
4. Non-uniform Resolve distribution off. In Appendix C.3, we describe a modification to the Resolve subgame solving (Burch al., 2014) algorithm that we made for Obscuro, in which the root nodes of the game tree are weighted using a nonuniform distribution, instead of the uniform distribution prescribed by Burch al. (2014). For this ablation, we turned this modification off, and instead used the uniform distribution, was done in prior papers on subgame solving including 7S21.
In 10,000-game match, Obscuro scored 53.3% (+4595 =1478 -3927).
5. Two-sided GT-CFR only, against ZS21. In this ablation, we turned off the above im- provements 1, 2, 3, and 4, and matched the resulting agent against that of ZS21. This serves to isolate the effect of using GT-CFR compared to using the LP-based equilibrium computation and iterative deepening node expansion as in 7S21.
In a 1,000 match, the two-sided GT-CFR version scored 72.6% (+711 =30 -259) against ZS21.
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6. Weaker evaluation function. To test the impact of the evaluation function, hand-crafted a simple evaluation function that takes into account only the material difference and num- ber of squares visible each player. We substituted this evaluation function in place of Stockfish 14's neural network-basse evaluation function, creating a new agent that we call simple-eval (SE) Obscuro. This evaluation function is very simple, and would not be well suited to regular chess, We tested simple-eval Obscuro against both Obscuro and ZS21. In a 1,000-game match, Obscuro scored 81.9% (+787 =63- -150) against SE Obscuro. In 10,000-game match, SE Obscuro scored 55.0% (+5258 =486 -4256) against ZS21. This experiment shows that the evaluation function has a significant impact on the perfor- mance of Obscuro. Yet, the search algorithm is also vital: even a simplistic evaluation function with our improved search techniques is enough be superior to ZS21.
All results in this and the next subsection are highly statistically significant (z > 5). The results suggest that each improvement played significant role in the improvement of Obscuro over the previous state-of tet-thheart art AI.
4.2 OTHER EXPERIMENTS
Finally, to test the effect of the time limit on the performance of Obscuro, we tested versions of Obscuro with different time limits against each other. The results were follows. All matches consisted of 10,000 games.
Obscuro with /s/move scored 56.4% (+5162
=943 (943-39955) against Obscuro with 16 s/move.
Obscuro with Şs/move scored 56.5% (+5031 =1231 -3738) against Obscuro with //move.
Obscuro with 1 s/move scored 56.7% (44923 =1503 -3574) against Obscuro with s/move.
Obscuro with ss/move scored 54.0% (+4617 =1566-116---0 -3817) against Obscuro with //move.
Obscuro with 2s/move scored 53.7% (+4589 =1561 -3850) against Obscuro with 1s/move.
Obscuro with 4s/move scored 52.3% (+4463
=1500 -4007) against Obscuro with 2s/move.
Obscuro with 8s/move scored 52.4% (+4501 =1482 -4017) against Obscuro with 4s/move.
Obscuro with 16s/move scored 52.3% (+4448 =1563-3989) against Obscuro with 8s/move.
These results, converted to the standard Elo scale, are visualized in Fig. 3. As expected and in line with known results for other settings (e.g., g. for regular chess (Silver et al., 2017)), increasing search time has a significant impact on playing strength, but with somewhat diminishing returns.
Finally as a sanity check, we also tested Obscuro against a random opponent. The only realistic way for Obscuro to lose to a random opponent is by not defending against Qa4+ or Qa5+ in the opening as previously discussed, which happens with only very small probability. As previously discussed, occasionally losing to weak (here, random) player would not in itself evidence that Obscuro playing suboptimally, since even an exact equilibrium player should lose to a random player with positive probability. Nonetheless, Obscuro won 1000 consecutive games against the random opponent.
CONCLUSIONS AND FUTURE RESEARCH
We presented the first superhuman agent for FoW chess, Obscuro. Our agent is completely based on real-time search. Thus, Obscuro ran regular consumer hardware, in contrast to most prior superhuman efforts involving search that we have discussed, which have run on large computing clusters with far more computing power at play time. This demonstrates the power of search alone. FoW chess is now the largest (measured by amount of imperfect information) turn-based game in which superhuman performance has been achieved and the largest game in which imperfect- information search techniques have been successfully applied.
Since FoW chess is somewhat similar to regular chess, it sufficient combine perfect- information evaluation function from regular chess (namely, that used by Stockfish) with our game- independent otate-of-the-art search algorithms for imperfect-informatii games. Also, Obscuro
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Playing strength (Elo relative 5s/move)
-50
-150
-200
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101
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Figure 3: Visualization of time scaling of Obscuro. The y-axis is relative to the playing strength of Obscuro with seconds per move.
stores at all times the entire set of possible states in memory. While these techniques were fea- sible for FoW chess due the similarity regular chess and the relatively small infosets one can imagine even more complex games on which they will not work directly.
Even more complex settings could be tackled by merging our techniques with deep reinforce- ment learning to learn the evaluation function, instead of using a perfect-informationn uation function (in our case, from Stockfish), and/or using continuation strategies (Brown Sand- holm, 2019) mitigate game-theoretic issues caused by using node-based evaluation functions in games. a different direction, further play strength and scalability could be achieved by sampling from infoset using model of opponent behavior instead of doing so uniformly.
eval-
imperfect-infe
ACKNOWLEDGMENTS
We thank Stephen McAleer for helpful discussions. This material is based on work supported by the Vannevar Bush Faculty Fellowship ONR 900014-23-1-28776 National Science Foundation grants RI-2312342 and RI-1901403, ARO award W911NF2210266 and NIH award A2401085001. B.H.Z. was also supported by the CMU Hans J. Berliner Graduate Fellowship in Artificial Intelli- gence.
REFERENCES
Christopher Berner, Greg Brockman, Brooke Chan, Vicki Cheung, Przemyslaw Debiak, Christy Dennison, David Farhi, Quirin Fischer, Shariq Hashme, Chris Hesse, et al. Dota 2 with large scale deep reinforcement learning. arXiv preprint 1012:0911006600 2019.
Noam Brown and Tuomas Sandholm. Regret- based pruning in extensive- games. In Neural Information Processing Systems (NeurIPS), 2015.
Noam Brown and Tuomas Sandholm. Safe and nested subgame solving for imperfect imperfecctinfoomaaton games. In Neural Information Processing Systems (NeuriPS), 2017. First published in AAAI-17 Workshop Computer Poker and Imperfect Information Games, submitted November 4, 2016.
Noam Brown and Tuomas Sandholm. Superhuman AI for heads-up no-limit poker: Libratus beats top professionals. Science, 359(6374):418 2018.

---

Published as a conference paper at ICLR 2026
Noam Brown and Tuomas Sandholm. Superhuman Al for multiplayer poker. Science, 365(6456): 885-890, 2019.
Noam Brown, Tuomas Sandholm, and Brandon Amos. Depth-limited solving for imperfect- information games. In Neural Information Processing Systems (NeurIPS), 2018.
Noam Brown, Anton Bakhtin, Adam Lerer, and Qucheng Gong. Combining deep reinforcement learning and search for imperfect-infomet information games. In Neural Information Processing Systems (NeurIPS), 2020.
Neil Burch, Michael Johanson, and Michael Bowling. Solving imperfect information games using decomposition. In AAAI Conference on Artificial Intelligence (AAAT), 2014.
Murray Campbell, A Joseph Hoane Jr, and Feng-hsiung Hsu. Deep Blue. Artificial Intelligence, 134 (1-2):57-83, 2002. 2002.
Chess.com. Fog of War Chess Leaderboards Play Chess Variants Online chess.con com. https: www. www.chess. com coomvvare //t//variantttttt foattt///ogoogoee fog- of weadr//eaerbbaaa 2024. [Online; accessed 16-August-2024).
Chess.com. Fog of War Chess Leaderboards- Play Chess Variants Online chess.ceessoooo https::/ www. chess. com com/vaaee vvari ants/ fog- of-wa weaerr//eeaeeberrrrrraao 2025a. [Online; accessed
13-April-2025j.
Chess.com. What is Fog Of War chess? Chess.com Help Center. https: / /sspport. chess. www.c/////////e.ecc art.iitsticceeee/eee line; accessed 13-April-2025).
owt.-ttt------------aa---------- 2025b. [On-
Chessgames.com Chess Statistics. https: http://wwwhhsgemeeee// 2024. [Online; accessed 16-Sep-2024).
html,
Paolo Ciancarini and Gian Piero Favini. Monte Carlo tree search techniques in the game of Kriegspiel. In International Joint Conference on Artificial Intelligence (IJAAI), 2009.
Gabriele Farina and Tuomas Sandholm. Fast payoff matrix sparsification techniques for structured extensive-form games. In AAAI Conference on Artificial Intelligence (AAAI), volume 36, pp. 4999-5007, 2022.
Gabriele Farina, Christian Kroer, and Tuomas Sandholm. Faster game solving via predictive Black- well approachability: Connecting regret matching and mirror descent. In AAAI Conference on Artificial Intelligence (AAAI), pp. 5363-5377 2021.
Gabriele Farina, Julien Grand-Clémen Christian Kroer, Chung- Wei Lee, and Haipeng Luo. Regret matching+: (in)stability and fast convergence in games. Neural Information Processing Systems (NeurIPS), 36, 2024.
Sam Ganzfried and Tuomas Sandholm. Endgame solving in large imperfect-informaaee games. In International Conference on Autonomous Agents and Multi-Agent Systems (AAMAS), 2015, Early version in AAAI-13 workshop Computer Poker and Imperfect Information.
Sam Ganzfried, Tuomas Sandholm, and Kevin Waugh. Strategy purification and thresholding: Ef- fective non-equilibrium approaches for playing large games. In International Conference Autonomous Agents and Multi-Agent Systems (AAMAS), 2012.
Ryan W Gardner, Gino Perrotta, Anvay Shah, Shivaram Kalyanakrishnan, Kevin A Wang, Gregory Clark, Timo Bertram, Johannes Fürnkranz, Martin Müller, Brady Garrison, et al. The machine reconnaissance blind chess tournament of neurips 2022. In NeurlPS 2022 Competition Track, pp- 119-132. PMLR, 2023.
Andrew Gilpin and Tuomas Sandholm. A competitive Texas Hold' em poker player via automated abstraction and real-time equilibrium computation. In National Conference on Artificial Intelli- gence (AAAI), pp. 1007-1013, 2006.
10

---

Published as a conference paper at ICLR 2026
Andrew Gilpin and Tuomas Sandholm. Better automated abstraction techniques for imperfect in- formation games, with application to Texas Hold' em poker. In International Conference on Au- tonomous Agents and Aultt-Agent Systems (AAMAS), pp. 1168-1175, 2007.
Michael Johanson, Kevin Waugh, Michael Bowling, and Martin Zinkevich. Accelerating best re- sponse calculation in large extensive games. In International Joint Conference on Artificial Intel- ligence (IJCAI), 2011.
Garry Kasparov and Frederic Friedel. Reconstructing turing's "paper machine". ICGA Journal, (2):1-8, 2018.
Levente Kocsis and Csaba Szepesvári. Bandit based Monte-Carlo planning. European Conference on Maching Learning (ECML), pp. 282-293. Springer, 2006.
Vojtëch Kovaří, Dominik Seitz, and Viliam Lisý. Value functions for depth-limited solving in imperfeectinnformation games. In AAAI Reinforcement Learning in Games Workshop, 2021.
Weiming Liu, Haobo Fu, Qiang Fu, and Yang Wei. Opponent-limited online search for imperfect information games. In International Conference on Machine Learning (ICML), pp. 21567-21585. PMLR, 2023.
Matej Moraveik, Martin Schmid, Karel Ha, Milan Hladik, and Stephen Gaukrodger. Refining subgames in large imperfect information games. In AAAI Conference on Artificial Intelligence (AAAI), 2016.
Matej Moravčík, Martin Schmid, Neil Burch, Viliam Lisy, 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(1667):508- May 2017.
Austin Parker, Dana Nau, and VS Subrahmanian. Game-tree search with combinatorially large belief states. In International Joint Conference on Artificial Intelligence (IJCAI), pp. 254-259, 2005.
Julien Perolat, Bart De Vylder, Daniel Hennes, Eugene Tarassov, Florian Strub, Vincent de Boer, Paul Muller, Jerome T Connor, Neil Burch, Thomas Anthony, et al. Mastering the game of stratego with model-free multiagent reinforcement learning. Science, 378(6623):990 996, 2022.
David Brine Pritchard. The encyclopedia of chess variants. Games & Puzzles Publications, 1994.
Stuart Russell and Jason Wolfe.
Efficient belief-state AND-OR search, with application to Kriegspiel. National Conference on Artificial Intelligence (AAAI), 2005.
Martin Schmid, Matej Moravčík, Neil Burch, Rudolf Kadlec, Josh Davidson, Kevin Waugh, Nolan Bard, Finbarr Timbers, Mare Lanctot, G Zacharias Holland, al. Student of games: A unified learning algorithm for both perfect and imperfect information games. Science Advances, 946): cadg3256, 2023.
David Silver, Aja Huang, Chris Maddison, Arthur Guez, Laurent Sifre, George Van Den Driessche, Julian Schrittwieser, loannis Antonoglou, Veda Panneershelvam, Marc Lanctot, et al. Mastering the game Go with deep neural networks and tree search. Nature, 529(7587):484, 2016.
David Silver, Julian Schrittwieser, Karen Simonyan, Ioannis Antonoglou, Aja Huang, Arthur Guez, Thomas Hubert, Lucas Baker, Matthew Lai, Adrian Bolton, et al. Mastering the game of go without human knowledge. Nature, 550(767)):344-559, 2017
David Silver, Thomas Hubert, Julian Schrittwieser, Ioannis Antonoglou, Matthew Lai, Arthur Guez, Mare 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.
Samuel Sokota, Gabriele Farina, David J Wu, Hengyuan Hu, Kevin A Wang, J Zico Kolter, and Noam Brown. The applate-equivalence framework for decision-tinn time planning. In International Conference Learning Representations (ICLR), 2024.

---

Published as a conference paper at ICLR 2026
Christopher Solinas, Doug Rebstock, Nathan Sturtevant, and Michael Buro. History filtering in imperfect information games: algorithms and complexity. Neural Information Processing Systems (NeurIPS), 08543634-43644 2023.
Stockfish. Stockfish Strong open source chess engine. httpe://stookkishhhessss download/.
Oriol Vinyals, Igor Babuschkin, Wojciech M Czamecki, Michaël Mathieu, Andrew Dudzik, Juny- oung Chung, David H Choi, Richard Powell, Timo Ewalds, Petko Georgiev, et al. Grandmaster level in StarCraft II using multi-agent reinforcement learning. Nature, 575(7782):350-354 2019.
Kevin Waugh, Nolan Bard, and Michael Bowling. Strategy grafting in extensive games. In Neural Information Processing Systems (NeurIPS), 2009.
Brian Hu Zhang and Tuomas Sandholm. Subgame solving without common knowledge. Neural Information Processing Systems (NeurIPS), 34:23993-24004 2021.
Martin Zinkevich, Michael Bowling, Michael Johanson, and Carmelo Piccione. Regret minimization in games with incomplete information. In Neural Information Processing Systems (NeurIPS), 2007.
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A
RULES OF FOW CHESS
FoW chess is identical regular chess, except for the following differences (Chess.com, 2025b).
A player wins by capturing the opposing king. There is no check or checkmate. Thus:
Moving into (or failing to escape) check is legal and thus results in immediate loss.
Castling into, out of, or through check is legal (though, of course, castling into check
loses immediately).
Stalemate is forced win for the stalemating player.
There is no draw by insufficient material. In particular, KN vs K is strong position for the KN, and even K K not an immediate draw (although vs K is drawn in equilibrium except in some literal edge cases where one king is the edge of the board and cannot immediately escape.)
After every move, each player observes all squares onto which her pieces can legally move.
If pawn is blocked from moving forward by an opposing piece (or pawn), the square on which the opposing piece/pawn sits is not observed. Thus, the player knows that the pawn is blocked, but not what is blocking (unless, of course, some other piece can capture it.)
If pawn can capture en passant, the pawn that can be captured en passant visible. In particular, the above rules imply that both players always know their exact set of legal moves.
Threefold repetition and 50-move-rule draws do need to be claimed. In particular, draw under either rule can happen without either player knowing for certain until it happens and the game ends.
B NOTATION AND PRELIMINARIES
The techniques used in Obscuro general, so in this section will formulate them in terms of general extensive-form games. To do this, we need to introduce some notation.
B. NOTATION AND PRELIMINARIES
A two two-player player zero-sum timeable extensive-form game (hereafter simply game) consists of:
1. tree of histories H. rooted the empty history Ø H. The set of leaves (terminal nodes) И is denoted Z. Each downward edge out of a non-leaf node h H \ Z is labeled with distinct action or move. The node reached by following the edge (action) at node h is denoted ha. The set of actions available at is denoted A(h).
2. payoff function Z + [-1, +1].
3. partition H Z Hc L H. Li Hr, denoting whose turn it is that is, for each i
(C, ) P. is the set nodes at which player i moves. Player is chance, who plays according fixed strategy p(-1h).²
4. for each player E {A, } and each H. an observation o;(h) that player receives upon reaching h. The observation uniquely determines whether h E P; (i.e., whether is player i's turn) and, it is, the set of legal actions. That is, if o, (h) = (h)), then h E P if and only if h' P.. and if so, A(h) A(h')
We will use - denote the precedence relation on tree. For example, if h, ht' are histories then an ancestor h:. If s, s' are sequences of player i, then s I s' means that is S means - s' for some s' E S. The downward closure of S is
< h means
prefix of s'. If S is S: (h: h S}. 10
"Fo chess contains no chance moves, but we include this in the interest of generality. In this paper, we visualize expanding from the top downwards, 5 is the set of descendants of
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We will distinguish between states and histories. A state is sufficient statistic for future play of the game. That is, all data about the subtree rooted at a history h is uniquely determined by the state at h. Multiple histories can have the same state.
The sequence of a player upon reaching a node h E H is the sequence of observations made and actions played by so far. Two nodes h, h' are indistinguishable to player i, written h ~e h², if they have the same sequence for player i. An equivalence class of H under my is an infoset, for player Throughout this paper, will denote a -innfoset, and J will denote Winnfoset. We will assume, without loss of generality, that the player sequence and opponent sequence together uniquely specify game tree node that is, | J 1 for every A-infoset I and v-infoset J.
By convention, information sets containing nodes at which player is not the acting player are typically not drawn (and often not even defined); in our paper, we will need them in order define the knowledge graph. Thus let I (resp. J) denote the set of infosets at which A (resp. v) is the acting player. If is a legal action at an infoset I E I. the sequence reached by playing (I, a). The set of nodes {ha h I will also be denoted (I, Let s;(h) denote the sequence of player i at h, as of the last time player played an action. Thus, 8A (h) (resp. 8 (h)) can be identified with pair (I, a) where I E I (resp. (J,a) where J E J).
is
(behavioral) strategy of (resp. V) selection of distribution of actions each infoset, X = Xiez A(A()) (resp. y E Y 'XJEJ A(A(J)). We will use the general notation cr("lu), where 3 u' to denote the probability that A plays all actions on the - u' path, where and u' are sequences, infosets, or nodes. Similarly, x<a/u) denotes the probability that æ takes action a at u (when I
H.). If the right half is omitted, e.g., x(u), it is understood to be Ø, e.g., æzl) = x(u]2)) In particular, z(h denotes the probability that A plays all actions on the e h path. Similar notation used for
The expected value for A in strategy profile ı = (a,, y) is u(T) == E-- u(z) where the expectation over terminal nodes z when plays x and plays y. (Since the game is zero sum, the value for
is (T.).
The conditional value u(x(S) the conditional expectation given that some node in the set is hit. The (conditional) best-response value u² [ar J, a) to -strategy æ E X upon playing action at infoset J is the best possible conditional value that against after playing a at
u (I) J+a)= yEY=)(2a)=1
min u(x, J..
Counterfactual values (CFVs), which we will denote by ut are defined similarly conditional values, but scaled by the probability of the other players playing to J:
(x, J, = ((,,)
Σ <<hh)p(h)= BEJ
Σ (=()p(())(ak))u(e)
The best-response value at infoset J is "f' (x|J) = min, n² (x|J, a). The best-response value u (x) minger u(x, y) = u" (x((2)) Analogous definitions hold when the players are swapped.
B.2 ORDER-A KNOWLEDGE, COMMON KNOWLEDGE, AND SUBGAME SOLVING
In this section, we present the mathematical notation we will use in the rest of the appendix, and relevant prior work on subgame solving (see, e.g., (Kovañík et al., 2021) for an overview).
The connectivity graph G of game I' is the graph whose vertices are the nodes of I, and whose edges connect nodes in the same infoset of any player. Now let I be any infoset." "I The order- l knowledge set 1 is the set of nodes at distance strictly less than k from some node in I. (In particular, / = I.) The common- haamonn-nowwedee set I is the set of all nodes a finite distance away from some node in I, i.e., it is the connected component in G containing I. Intuitively, the distance from I captures the level of common knowledge. If the true node is h E I, then
1. knows h E I,
TThe definition of 7 can also be applied to arbitrary sets of nodes I, but here will only need it for infosets.
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2. A knows
knows h I?
3. knows
knows
knows h E 13,
and so on. Hence the statement 1 common knowledge.
In the remainder of the paper, we take the perspective of the maximizing player A. Subgame solving starts with a blueprint strategy profile (ar y). In Obscuro, the blueprint strategy profile is simply the saved strategy from the computation on the previous move; on the first move, subgame solving does not require a blueprint.
Suppose that we reach infoset I during game. Before selecting move at Z, we would like to do some computation to compute a new strategy x' that we will use instead of æ. That is, we would like to perform subgame solving.
We will first describe two common variants of common-knowee konmlnn-nnowedge subgame solving: Resolve (Burch et al., 2014) and Maxmargin (Moravcik et al., 2016), both of which will use in Obscuro. Both variants begin by constructing a gadget game using common-knowledge set I and are based on the principle of searching for strategy x' that does not worsen the opponent's best response values. More formally, let M == u (x'l u (æ|J) be the margin at viinfoset J C x
The alternate value u² (xr () is the value which must restrict J in order ensure that exploitability does not increase.
Maxmargin and Resolve differ in how they aggregate the margins across the different information sets J E To. The Maxmargin objective is to maximize the minimum margin:
max min M(', a²' JEP
The Resolve objective to maximize the average margin truncated to zero:
max
Σ
[M
JJJ
where
[=] := min (0,
To
JEJa
and Jo = (J: is the set of possible root infosets for v in the subgame.
A subgame solving method is safe if applying it cannot increase exploitability of the overall agent compared to not applying it-ie., compared to playing the blueprint strategy. Both Maxmargin and Resolve are safe, assuming of course that subgames are solved exactly. 12
Subgame solving via Resolve and Maxmargin can also be performed using gadget games. In Re- solve, the following gadget game is played. First, chance chooses node h. E I with probability proportional to (h)) Then, v observes the infoset J > h, and decides whether to play or exit. If exits, the game ends immediately with utility equal to the alternate value u" (x)J). Otherwise, the game continues as normal from node h. In Maxmargin, v first selects the infoset J E To. and then chance samples node h E J with probability proportional to phh) Then, immediately receives utility -u" (z(J)..5 Chance then selects a node h E J with probability proportional to p(h)a(h), and the game continues from h.
Maxmargin and Resolve have very different behavior. When it is impossible to make all margins nonnegative (due to approximations), Maxmargin will make the pessimistic assumption that the op- ponent will play the worst infoset, whereas Resolve will, roughly speaking, assume that the opponent will play uniformly over all infosets with negative margin. On the other hand, when it is possible to make all margins nonnegative. there is set of subgame strategies that are maximizers of the Resolve objective, that is, equilibria of the Resolve gadget game. Resolve allows any one of these strategies to be selected, whereas Maxmargin enforces that the strategy be in particular the one that maximizes the minimum margin.
In the state-of-the-art common knowledge subgame-solving technique, reach subgame solv- ing (Brown & Sandholm, 2017), any gifts given to us by the opponent through mistakes in reaching the subgame can be given back the opponent within the subgame; this enlarges the strategy space
In our application, safety hard reason about: neither the blueprint strategy x nor the subgame-solved strategy are full-game strategies, so asking the question of which less exploitable is strange. This can he implemented, for example, by adding u" [z) to the value of every terminal node ł J in the subgame.
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that optimize safely and thus has been shown to yield stronger play (e.g., in poker games (Brown Sandholm, 2017: 2018; Brown et al., 2018; Brown & Sandholm, 2019)). This is done by adjusting the alternate values u" (z) in the case when provably made mistake(s) in playing to reach J. Reach subgame solving can be applied on top of either Resolve or Maxmargin. In particular, the value
9(J)
Σ J'a'zJ
(x; u (x; J']]
which an estimate of the gift using the current strategy profile (æ, y). added to the alternate value each infoset J To.
In those prior subgame-solving techniques and ours, the desired gadget game then replaces the full game, and its solution is used to select repeats, with the solution to the previous subgame taking the place of the blueprint.
T. When new infoset is reached, the process
C FURTHER DETAILS ABOUT Obscuro
We now give more details about Obscuro. Appendix C.I details the techniques that are identical to the prior SOTA (ZS21). The remaining subsections detail the improvements over ZS21 that we developed Obscuro. also include pseudocode the major components of Obscuro, in Figures 8-
C.I PRIOR STATE OF THE ART IN FoW CHESS
Obscuro decides between Maxmargin and Resolve by examining the current objective value in the subgame. If always chooses exit in the resolve gadget (i.e., the current strategy is safe), Max- margin is used. Otherwise, Resolve is used. This switch may happen, even multiple times, in the middle of the search process for move, if the subgame value is fluctuating. Intuitively, this choice prevents the agent from being too pessimistic when faced with novel situations that did not antic- ipate.
Between moves, Obscuro maintains the list of all possible states given its current sequence of ob- servations, well the search tree and current approximate equilibrium strategy profile (æ, y) from the previous search. This previous strategy profile (x, y) is used as the blueprint for subgame solving.
When it is Obscuro's 's turn, Obscuro first builds both the Maxmargin and Resolve gadget subgames. The gadget subgames share the same game tree memory after the subgame root layers. Thus, for example, node expansions and strategy updates for infosets beyond the subgame root layers apply both subgame gadgets. This allows the transition between the two subgames, if necessary, to be smoothly executed.
insufficiently many nodes exist in the sample of Obscuro's current infoset I, nodes are added by sampling at random without replacement from the set of possible states. At newly-added nodes ht the opponent is assumed to have perfect information, and the alternate value is set to min(((h) ""} where v was the expected value of Obscuro in the previous search, and ä(h) is Stockfish evaluation function evaluated at
C.2 BETTER ALTERNATE VALUES AND GIFT VALUES
For alternate values in both Resolve and Maxmargin, in Obscuro we use u(x,, v|J) instead of the best-response value u² (x] which is more typically used subgame solving as we described Ap- pendix B.2 (Brown & Sandholm, 2017). Similarly,
the counterfactual values of (zr y5 J, a) and (x, J) to define the gift instead of the counterfactual best responses u (r; J, a) and ue (x; J), resulting in the gift estimate
((J) Σ (x, (x, J'J+
These changes are for stability reasons: especially late in the tree, the current strategy x may be inaccurate, and the best-response value u² [x[[] may not be accurate reflection of the quality
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of the blueprint strategy especially near the top of the tree. Of course, if (I, y) is actually an equilibrium of the constructed subgame, then these values are the same.
C.3 BETTER ROOT DISTRIBUTION FOR Resolve
When using Resolve' 4 for the subgame solve in KLSS in games with chance actions, the standard algorithm for Resolve will choose an opponent infoset J uniformly at random from the distribution of possible infosets. In reality, the correctness of Resolve does not depend the distribution chosen, so long as it is fully mixed. To be more optimistic, we therefore use a different distribution. We choose an infoset J via even mixture of a uniformly random distribution and the distribution of infosets generated from the opponent strategy in the blueprint. That is, the probability of the subgame root being infoset J is
y(J)
+ )
a a(J) (J)==3 Er "r v(J)
where m is the number of V -infosets in the current subgame and the sum is taken over those same infosets. other words, the Resolve objective becomes
max
JEJJ
In this manner, more weight is given to those positions that were found to be likely in the previous iteration, while maintaining at least some positive weight on every strategy.
C.4 BETTER NODE EXPANSION VIA GT-CFR
Growing-tree CFR (GT-CFR) (Schmid et al., 2023) is general technique for computing good strate- gies in games. Intuitively, it works, like PUCT, by maintaining current game I' and simultaneously executing two subroutines: one that attempts to solve the game I", and one that expands leaf nodes of T. As mentioned in the body, we use PCFR+ for game solving.
For expansion, we use new variant of GT-CFR which we call one-sided GT-CFR, which, unlike PUCT and GT-CFR, may only expand a small fraction of nodes in the tree. As stated in the body, our one-sided GT-CFR algorithm selects the node expand according to the profile (#', ,²),, where y' is the non-expanding player's currení CFR strategy and j²' is an exploration profile constructed from the expanding player's current strategy." 15 As GT-CFR, the expanding player's strategy a' mixture of strategy M (a|L) derived from the player's current strategy x and exploration strategy # (a) [)) derived from PUCT (Silver et al., 2016). In particular, we define
Ta x 1 1( (a > 0}
to be the uniform distribution over the support of the current CFR strategy, and (a) = lfa argmax a{I,a)}
a'
where
VN'(T)
Q(I,a)= u(" (L, 1+ (*)
Here, C. is a tuneable parameter (which set to 1); (L, is the empirical variance of u(x", y² I a) over the previous times we have visited I during expansion (with two prior samples of -1 and +1 to ensure it is never zero); (T) is the number of times infoset has been visited during expansion; and (T( a) is the number of times action a has been selected. Finally, as in GT-CFR, define
1
1
aample (a) (a) , Shea (a)7) + # (a) 2
Unlike GT-CFR as originally described (Schmid et al., 2023), our one-sided GT-CFR works on the game tree itself, not the public tree. The public tree our setting would be difficult to work with since the amount of common knowledge is very low.
For Maxmargin, there no prior distribution because the adversary picks the distribution. in this presentation, is the expanding player. When V is the expanding player, the roles of and also flipped. As stated the body, the expanding player alternatss between and after every node expansion.
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A
B
G
H
Figure 4: The game tree from Fig. now with some nodes labeled, which will be referenced in the text.
Our one-sided GT-CFR, unlike PUCT and GT-CFR (Koesis & Szepesvári. 2006; Schmid et al., 2023), is not guaranteed our game I' is as in Fig. 4, and that both players are currently playing the strategy "always play left". Then Node F is reached by both players, nodes G and H are reached by only one of the two players (A and respectively), and Node J is reached by neither player. As such, Node J will not be expanded, and if the current strategy is equilibrium, this can be proven without knowing the details of any subtree that may exist J.
eventually expand the whole game tree. For example, suppose that
Nonetheless, we can still show an asymptotic convergence result:
Theorem 1. For any given e. >> O, the average strategy profile (ū., ū) in one-sided GT-CFR eventually converges to an e-Nash equilibrium of any finite two-player zero-sum 16
Proof. Since r is finite, eventually one-sided GT-CFR stops expanding nodes. this time, let i' be the expanded game tree. Since no more nodes are expanded, and CFR is correct, one-sided GT-CFR eventually converges to an approximate Nash equilibrium (ū, y) of I'. At this time, it is perhaps the case that there remain unexpanded nodes in the current tree r. However, any such nodes must have been played with asymptotic probability 0 by both players; otherwise, if (say) V plays to an unexpanded node h with asymptotically positive probability, then h would have been expanded some point when and therefore (ø. y) is also an approximate equilibrium in r.
the expander. Thus, best-response values in I are the same they in I,
C.5 EVALUATING NEW LEAVES
When a (non-terminal) leaf node of I' is selected, it is expanded. That is, all of its children are added to the tree. To assign utility values the children 2, we run the open-source engine Stockfish 14 (Stockfish). in MulriPV mode, at depth 1 on node 2, which gives evaluations for all children of in a single call," 17 and clamp result to [-1,+1] the same manner as done by 7S21 (Zhang & Sandholm, 2021).
When the children of = are added to the tree, becomes nonterminal node and hence will be placed in infoset. If = is the first node of its infoset to be expanded in I' we also need to initialize a new regret minimizer to be used by PCFR+ at this new infoset. Doing so naively would cause a sort of instability: the evaluation of will be (approximately) equal to the largest evaluation of any child of 2 (due to how regular perfect-informattee would initialize its strategy uniformly at random. Thus, the evaluation of would suddenly change to being the average of the evaluations of the children of z, which could be very different from the maximum (for example, if the move at is essentially forced). To mitigate this instability, we exploit
evaluation functions work), but PCFR+ normally
"Technically, (5, 3) is only partial strategy in T, since it does not specify how play after any unexpanded nodes. However, this is fine: any extension of (æ, y) will be an equilibrium of I' and unexpanded nodes reached by either player. For elarity, typical for extensive-fore games (see, e,g., Zinkevich et al. (Zinkevich et al., 2007)). the average of strategies is always taken in sequence form. That is, F is the strategy for which ((h) $ >T [0r,,x(hi (h).
17 Using single call has two minor advantages: first, it takes advantage of slight extensions that may be used in Stockfish at low depth; second, it reduces the overhead of calling Stockfish to one call per node being expanded, instead of one call per child of that node.
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the property that, in CFR (and all its variants, including PCFR+), the first strategy be arbitrary. Conventionally it is set to the uniform random strategy, but we instead set it by placing all weight on the best child of evaluated by Stockfish.
C.6 KNOWLEDGE-LIMITTH SUBGAME SOLVING
subgame solving (KLSS). KLSS results from two changes common common-knowledge knowledge subgame solving. Let / be the current infoset of As an example, consider the game in Fig. 4, and let 7 be the infoset at A. Let & be an odd positive integer. Then 7S22 (Zhang & Sandholm, 2021) defines K-KISS by making the following two changes.
ZS21 (Zhang & Sandholm, 2021) uses knowledge-linee
1. Nodes outside the downward closure I are completely removed from the game tree. In Fig. 4, this would amount to removing the subtrees rooted C, D, and
2. à A-ooes in I \ I are frozen to their strategies in the blueprint, i.e., they are made into chance nodes with fixed action probabilities. In Fig. 4, this amounts to making Node B a chance node.
ZS21 sets k = in their FoW chess agent. Freezing the -nodes in 12 A27 finding module, which based on linear programming, to scale more efficiently, since the nodes in that subtree are now only dependent on v's strategy, not A 's
7T allows their equilibrium-
KLSS, as implemented by ZS21, already lacks safety guarantees: they have an explicit counterexam- ple in which using KLSS may decrease the quality of the strategy relative to just using the blueprint. We make one simple change to KLSS for Obscuro: we allow à-nodes in I 7 to be unfrozen and hence re-optimized in the subgame. We may call this 2-kmowledge-limiee unfrozen subgame solving (KLUSS). since its complexity depends on the order-2 subgame l2, 2-KLUSS essentially amounts to pretending that T T
We now make few remarks about KLUSS.
1. Like I-KLSS, 2-KLUSS lacks safety guarantees in the worst case. However, KLSS is often safe in practice (Zhang & Sandholm, 2021), and KLUSS outperforms KLSS in FoW chess as we showed in the ablations in Section 4.1. There are two further considerations:
(a) Obscuro does not have a full-game blueprint: its blueprint is simply the strategy from the previous timestep, which is depth limited. Thus, must some form of sub- game solving to play the game. KL(U)SS is currently the only variation of subgame solving that is both somewhat theoreeiettt motivated for imperfect-informatioon games and computationally feasible in game like FoW chess.
(b) Although both KLSS and KLUSS are unsafe in the worst case, it should be heuris- tically intuitive that they should improve performance more when the blueprint itself is of low quality. Indeed, previous timestep) to have rather low quality, especially deep in the search tree where such strategies are based on very low-depth scarch! So, we believe heuristically that using KL(U)SS in this manner should usually be game-theoretically sound.
expect "blueprints" (strategies carried over from the
2. Since our cquilibrium- finddinngi module for Obscuro is based on CFR instead of linear programming-inn particular, uses the full game tree I' instead a sequence-form representation it does not benefit from freezing the -nodes in I IT, since those nodes would still need to be maintained. Thus, there is less reason for Further, with straightforward pruning techniques (namely, partial pruning (Brown & Sand- holm, 2015)). CFR iterations usually take sublinear time in the size of the game tree (unlike linear programming, which takes at least linear time in the representation size), reducing the need to optimize the size of the game representation.
freeze those nodes.
3. Again since we use CFR, the solutions that are computed by the equilibrium- suuiibrrum-finndin mod- ule are inherently approximate, and especially at levels deep in the tree, their approxima- tion can be relatively poor. As such, allowing these infosets to be unfrozen gives them the chance to learn better actions.
This he casily generalized k-KLUSS for any k.
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4. 1-KLSS removes the nodes in 1 IT, folding them into the sequence-font representation for efficiency. In contrast, our approach of maintaining these nodes allows them to be selected for expansion. This fixes a weakness of ZS21: ZS21 was only capable of searching for bluff opportunities "locally", since any A-node in 1 \ Il would cease be in the tree once the search horizon was passed. In contrast, Obscuro is capable of maintaining nodes in 7 \ 7I for a long time, allowing deeper bluff opportunities.
5. Liu al. (2023) introduced a safe variant of KLSS, which they call safe KLSS, in which the subgame solver attempts to find subgame strategy x' that maintains at least the same value for every opponent strutegy y,, instead of against every infoset J. This a much stricter condition that is much more difficult satisfy and thus substantially constrains the strategy to be close to the blueprint. Therefore, the safety requirement significantly decreases the power and value of subgame solving, especially when the blueprint is bad. Moreover, safe KLSS drops all nodes outside I', which once again introduces the problem of the previously-lige -isttdd item: if we were to use safe KLSS in our setting, our Al would not be capable of exploiting long bluff opportunities.
C.7 SELECTING AN ACTION
As mentioned in the body, Obscuro selects its action using the last iterate of PCFR+, rather than the average iterate which is known to converge to a Nash equilibrium. We do this for two reasons.
1. The stopping time of the algorithm, due to the inherent randomness of processor speeds, is already slightly randomized. Thus, stopping on the last iterate does actually stop the same timestep 7 every time: it in effect mixes among the last few strategies. Thus, we do not need to actually randomize ourselves to gain the benefit of randomization.
2. PCFR+ is conjectured (e.g., Farina et al. (2024)) to exhibit last-iterate convergence as well. Indeed, we measured the Nash gap of the last iterate (x:², "²" (in the expanded game f') and the typical Nash gap was approximately equivalent half a pawn- much less than the reward range of the game. This suggests that assuming last-iterate convergence is not unreasonable for our setting.
C.8 STRATEGY PURIFICATION
As mentioned in the body, we partially purify strategy before playing, When Maxmargin is used as the subgame solving algorithm (i.e., when the margins are all nonnegative), we allow mixing between k = 3 actions; when Resolve is used, deterministically play the top action. Moreover, we only allow mixing among actions other than the tighest-probabiigg action if they have appeared continuously in the support of x for every iteration
T1/2, where Ti/2 is chosen be the iteration
number when half the time budget elapsed. 19 We call such actions "stable". These restrictions reduce the chance that transient fluctuations in the strategy of the player, which occur commonly during game solving especially with an algorithm like PCFR+, would affect the final action that is played. Any probability mass that was assigned to actions that are excluded in the above manner is shifted to the action with highest probability.
D HARDWARE
Obscuro, for its human matches, ran on a single desktop machine with 6-core Intel i5 CPU, Abla- tions and further matches were run on an AMD EPYC 64-core server machine using 10 cores (5 per side). We now report statistics about the computational performance of Obscuro. These statistics were collected over the course of 1,000-game sample, at a time control of 5 seconds per move." 20
Average game length: 116.6 plies (58.3 full moves)
It will almost always be the case that Ti/2 T/2. This is because, the game tree grows larger, PCFR+ iterations, whose time complexity scales with the size of the game tree, get slower.
For this and all other Al-vs-Al matches in this paper, the stated time control, usually seconds per move, is the time limit allocated to the main search loop, and does not include the time it takes to enumerate the set of all legal positions.
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Average search depth: 10.7 plies
Average search tree size: 1,070,552 nodes, 14,404 infosets
Average search tree size carried over from previous search: 181,421 nodes, 3,162 infosets
Average number possible positions: 17,264
E OBSERVATIONS ABOUT FOW CHESS
E. SIZE INFOSETS AND HOMMON-KNOWLEDGE SETS
Here we elaborate on the discussions about common-knowlede sets and infosets, alluded to introduction.
Consider the family positions in which both sides have spent the first eight moves playing a4 a5 2. b4 b5 8. h4 h5, and subsequently shuffle all their remaining pieces around their first three ranks. An example of such a position is Fig. 5, Each player must have one bishop on a light square (12 ways), one bishop
dark square (12 ways), one queen, one king, two knights, and two rooks (22-21 20-20-999 18 17/2² ways). When multiplied, this gives a total of approximately M = 2 x 10 ways. This is lower bound the maximum size of infoset. For common- -nonnond---wwedgeeee sets, both players can arrange their pieces arbitrarily along the first three ranks, yielding approximately M2 x 4 1018 different arrangements, which provides lower bound the maximum size of common Hommon-knowledee set."
8
7
6
5
4
8
3
2
1
b
d
9
Figure 5: FoW chess position illustrating the existence of large infosets and common-knowledge sets. A full explanation is given in the text.
Although infosets can get this large, they almost never do in practical games, because both sides are making effort to obtain information.
We now elaborate on Fig. 1. particular, we will show that the two positions in that figure are the same common-knowledee set. Consider the sequence of positions in Fig. 6, read in order from top- left to bottom-right. The positions marked A and B are the same as those in in Fig. Each position is connected the next one by infoset of one of the players: the first pair by White infoset, the second pair by a Black infoset, and so on. computer search showed that the depicted path, which
21 These common- common-kkowledd knowledge sets are measured with respect states, not histories. Measuring common- knowledge sets with histories would result in a significantly larger number, because the order of the moves would matter.
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has length 9, is the shortest path between these two positions. 22 Hence, if the true position is A, then the statement "The true position is not B" 8th-order knowledge for both players. That is, true that
everyone knows everyone knows everyone knows Y
repetitions
yet the same statement would be false if there were 9 repetitions, so is not common knowledge.
99
Ne3 g5 2. Nh3 d5
Na3 N.3 d5
Na3
N5
1. Na3 Rb1
1. Na3 a5 2. Rb1
à
t
X
&
BAAH
4
A
4
2
A
Na3 2.
Na3 2.
Ne3
h3
Ne3
NB e5
Figure 6: Sequence of positions illustrating the connectivity between the two positions in Fig. 1. Circles mark squares that the opponent knows are occupied by some piece, but not by which piece. A full explanation is given in the text.
E.2 MIXED STRATEGIES
Playing a mixed strategy is fundamental part of strong play almost any imperfect information game, and it is particularly important in games like FoW chess where there is no private information assigned by chance, such as private cards in poker. Indeed, in small poker endgames, deterministic strategies exist for playing near-optimally (Farina & Sandholm, 2022). However, in FoW chess, if a player plays a pure strategy that the opponent knows, the opponent would essentially be playing regular chess, because the opponent predict with full certainty what the player would play. This a significant disadvantage that will result in a rapid loss against any competent opponent.
Consider, for example, the position in Fig. 7(A). White win almost full pawn (in expectation) by mixing between the moves 2. Qa4 with low probability and 2. Ne3 with high probability. No move for Black simultaneously defends the threats against both the king and the pawn. (2... c6 may look like it does, but after 3. exd5, Black cannot recapture the pawn without risking hanging a king queen.)
This necessity of playing mixed strategy explains why we do not adopt full purification of our strategy and instead opt to allow mixing.
E.3 FIRST-MOVER ADVANTAGE
We evaluated the first-mover advantage in FoW chess by running 10,000 games with Obscuro play- ing against itself time control of 5 seconds per move. Of these games, White scored 57.5% (+4935 =1623 -3422). This is, with statistical significance (z 5), larger than the empirical first- move advantage in regular chess, which is about 55% (Chessgames.com, 2024), We believe that the
A similar computer search shows that this is nearly the longest possible shortest path between any pair of nodes after two moves each side: there a shortest path length 10, but no shortest paths longer than that.
Obscuro prefers also include 3. NE3 and 3. e3 in its mixed strategy dissuade 2... d4.
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é
A
A
AAAAA
MAA
A
B
D
Figure 7: FoW chess positions from actual gameplay illustrating common themes. (A) Opening position after the common trap 1. e4 d5?! (B) An early-game bluff. White bluffs that its attacking bishop is defended by the queen on d1. (C) A highly-risky queen maneuver from a losing position. (D) An endgame position which the disadvantaged side sacrifices material for chance at the opposing king. Details can be found in the text.
White
Black
d4
66.4%
Ne6
32.5%
c4
29.6%
c6 25.1% %
e4 1.9%
e6 20.7%
Ne3 1.4%
NI6
15.9%
c3 0.4%
c5 4.8%
Nf3
0.2%
d5
0.9%
Table Distribution of first moves played by Obscuro both White and Black, over a 10,000-game sample. Percentages may not add up to 100% due to rounding.
fundamental reason for this discrepancy is the weakness of the a4-e8 diagonal, as already exhibited in Fig. 7(A), discussed above. This risk presents Black from developing in natural manner against c4 or 1. d4, allowing White healthy opening lead.
Indeed, our 10,000-game sample included 10 games with length 12 ply (6 moves from each player) or fewer; all 10 of these games ended with either Black failing to cover Qa4+ or White failing to cover Qa5+:
1. c4 d5 2. Qa4+ d4 1-0
1. c4 c6 2. d4 d5 3. exd5 Qa5+ 4. Qa4 0-1
1. c4 Ne6 2. d4 d5 3. Qa4 dxe4 4. d5 Nh8 1-0
1. d4 c6 2. c4 d5 3. exd5 Bí5 4. Qa4 exd5 1-0 (This play-through occurred three times.)
1. d4 c6 2. c3 e6 3. e4 d5 4. e5 c5 5. Qa4+ cxd4 1-0
1. c4 e 2. d4 c5 3. d5 Qa5+ 4. Nd2 Ní6 5. e4 Nxe4 6. Nxe4 0-1
1. d4 c6 2. Ne3 d5 3. Qd3 Nf6 4. e4 dxe4 5. Nxe4 Qas+ 6. Nxf6+ O-1
1. c4 d5 2. Qa4+ e6 3. exd5 Nf6 4. dxc6 Nxc6 5. Nf3 e5 6. Nxe5 Nxe5 1-0
These games may seem like they contain major mistakes, but that not so. It is rather likely that most or all of these play-throughs are part of optimal play; after all, bluff' must sometimes get called!
In Table I we give Obscuro's mixed strategy the first move for both White and Black, over the 10,000-game sample. The above observation about the a4-e8 diagonal has a large effect on opening choices. We believe that this explains why White strongly prefers opening with d4 and e4 rather than which is equally favored in regular chess, and why Black almost never opens with d5 and instead prefers immediately close the dangerous diagonal by moving something c6.
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E.4 BLUFFS
Obscuro bluffs. An example bluff is in Fig. 7(B), which is from the aforementioned 10,000-gama sample. White knows that d6 is defended (in fact, White knows the exact position). Black does not know the location of the white queen (for example, it could be on d1 instead). This allows White to play Bxd6, exploiting the fact that Black cannot recapture without risking losing the queen.
E.5 PROBABILISTIC TACTICS AND RISK-TAKING
The existence of hidden information in FoW chess allows tactics that would not work in regular chess. An example of this phenomenon as early move has already been described above, where mixing allows White to win a pawn after 1. c4 d5. We now give additional examples.
Fig. 7(C) depicts position encountered during Obscuro (White) was in losing position, down a minor piece. It decided to play the highly risky queen maneuver Qg8-gl -a--g--ll---aaa- leaving its own king exposed in order to attempt to hunt the opposing king. This risky tactic worked: the game played out 68. Qgl Qe7 69. Qal Nb8 70. Qa7 Nd7 71. Qa8+ Nxb6 1-0. 24 This sequence of moves heavily exploits the opponent's imperfect information: if Black knew that White was attempting this attack, Black could easily either defend the attack or launch a counterattack on the completely undefended white king.
20-game match against the top-rated human.
For another example, consider the position in Fig. 7(D), again from the aforementioned 10,000- game sample, and suppose for the sake of the example that White has perfect information. White faces slight material disadvantage in an endgame. However, Obscuro as White finds the tactical blow 1. Rxe6! Kxe6 upon which mixing evenly between 2. Be4+ and 2. Bh3+ wins on the spot with 50% probability.
E.6 EXPLOITATIVE VS. EQUILIBRIUM PLAY
The position in Fig. 7(D) is also example of the difference between exploitative play and equi- librium play in FoW chess. The above tactic has expected value at least 50% against any player, because it wins the spot with probability at least 50%. is likely the best move if playing against a perfect opponent. However, against substantially weaker player, it may be far from the best move: against a weak player, one can argue that the endgame is probably a win even with the slight material disadvantage, whereas the tactic will lead to a significant disadvantage (down three points of material) if it fails win. Therefore, if one knew the strength of one's opponent, one may opt to not go for this tactic and instead attempt to win the endgame in a "safer" manner. Another example of this phenomenon was also seen above. Obscuro, with small probability, can lose in two moves (1. c4 d5 2. Qa4+ d4). Any player, no matter how weak, can therefore beat Obscuro with posi- tive probability as White by simply playing the above move sequence. However, against opponents below a certain level, playing the above moves as Black may be considered a needless risk.
Obscuro does not know or attempt to model the opponent. It will simply play what believes to be near-equilibrium strategy. Therefore, it may not do well against weak players as an agent designed specifically to exploit weak players. This design choice was intentional, and follows other efforts in superhuman game taaee-playig AI such as those mentioned in the introduction, most of which attempted to find and play equilibria rather than to exploit a particular opponent.
E.7 VOLATILITY
FoW chess is a highly volatile, highly stochastic game. Indeed, the previous two observations re- garding risk taking and exploitative play evidence of this. Most games, including majority of our 20 games against the world #1 player, are ultimately decided by one side outright "blundering material" because of lack of knowledge of the opponent's position. We emphasize, however, that this is not a sign of poor quality of play: rather, we believe that strong play in FoW chess involves calculated risk-taking that, with nontrivial probability, leads to such "blunders". More skilled play-
The immediate 70. Qa8 would have worked in this position well, but it was not played, likely because it would have risked losing the queen in case the king were on b8.
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are better at taking calculated risks while restricting the probability of losing material, and at foreing their opponents into more risky situations.
E.8 KING Vs KING
To make some of the above discussions about mixing, volatility, and equilibrium play more concrete, we include here a partial analysis of the king-vs -ogg-v--ae ti-gvvg endgame, assuming the starting position of the kings is common knowledge. While this endgame is an immediate draw the rules of regular chess (because lone king cannot checkmate), FoW chess allows such endgames to play out, and not all such endgames are immediately drawn; fact, the analysis turns out rather intricate already. In the below discussion, 0 is draw, +1 is a certain win for White, and -1 is certain win for Black.
Claim 1. Suppose that there are two legal moves for the black king that are 1) guaranteed to be safe (i.e., guaranteed to not immediately lose), and 2) adjacent to each other (orthogonally or diag- onally). Then Black secures at least a draw.
Proof. Black randomly moves to one of them on their first move, and shuffles between them forever thereafter. The white king cannot approach without being captured half of the time.
Thus, it remains only to discuss the case where one king is on the edge of the board. Assume, without loss of generality, that this is the black king, and that is on the 8th rank.
Claim 2. If the white king prevents the black king from immediately moving off the back rank (e.g.. a6 and a8), the equilibrium value is strictly positive, regardless of which side is to move.
Proof. We will show that Black has no strategy that achieves expected value Consider two cases.
Case 1. Black's strategy involves attempting to move off the back rank with positive probability on some move & (but not earlier). Then consider the following strategy for White. Let x7 (for {a, b, .... h}) be the square the 7th rank with maximal probability after moves. White places its king on square 26 before Black's 's ith move. With probability p. White wins immediately. Otherwise, White runs away downwards, executing the strategy from Claim 1, forcing a draw.
0 for the black king
Case 2. Black's strategy is to always stay the back rank. Then consider the following strategy for White. Let 28 be the square on the back rank with minimal probability S 1/4 for the black king, at the time when White makes its 8th move. White places its king on x7 on its Sth move, then moves left and right on the 7th rank until it wins. We claim that White has expected value at least 1 2q 1/2 with this strategy. To see this, note that, since Black always stays on the back rank, the parity of its rank alternates between moves; therefore, if the black king is not on 28, then White will not lose on its 8th move. Further, also by a parity argument, White will eventually chase down the black king and win the game.
If the black king is on the edge of the board, it is always the case that either White can force the kings be squares apart with common knowledge (Claim 2) Black has safe pair of adjacent moves (Claim 1), so this completes the analysis.
We complete this section by pointing the corner (a8), the white king starts on either b6, c7, or c6, and it is White to move, then White can secure value strictly larger than 1/2: randomize between Kb6, Kc7, Ka6, or Kc8 (whichever are legal moves) on the first move. This wins with probability 1/2 immediately, and otherwise immediately forces the kings to be two squares apart (Claim 2).
an interesting special case: If the black king starts in
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1:
maintain
2:
current game tree I' (initially containing only the root Ø)
3:
current strategy profile (x: y)
4:
current expected value v*
5:
current of possible positions P (initially containing only the root Ø))
6:
procedure MovE(observation sequence 0)
7:
CONSTRUCTSUBBAMMIOO
8:
in parallel
9:
RUNSOLVERTIIREADO
10:
MUNEXPANDEKTHREADDO & or multiple parallel copies of expansion thread
11:
I + our current infoset
a" +- argmax A(T)) <{a[T) ties broken arbitrarily
if Pmax 0 then If using Resolve, just play a", i.e,, purify completely.
14:
S t set of stable actions U {a" { "Stable" is defined in the text. set to 3.
if MAXSUPPORT then The parameter MAXSUPPORT remove all but the top MAXSUPPORT most likely actions in S
16:
17:
plyyy (-11) + T--
18:
for action a E A(h) \ S do Shift all mass of such actions onto a".
Tlayy (a" (<)) < Tplay (a plly [ (a]z)
20:
Tlaay (a) /) 4 0
21:
sample π (-11)
22:
play action a
Figure 8: Pseudocode, Part 1.
procedure CONSTRUCTSSTRRAAACCCeeevinnn sequence o)
2: The set of possible positions is updated on every move by simply enumerating
3:
all possibilities
4:
P < all positions consistent with
5: I + set of nodes consistent with o
6: Construct KLUSS subgame:
7:
for each opponent infoset J C 72 do
8: set alternate value (J) t u(x, J) O(J)
9:
while I| min MININFOSETSIZEE] do Add more states to if there not enough.
10:
The parameter MININFOSETTIZE is set to 256.
11:
get random state E P P\
12:
o Assume has perfect information at newly-sampled states:
13:
add -infosse and -iinfoset J {s}
14:
set alternate value الهي + min [*)
15:
Fix prior probabilities:
16:
To + (J C 12)
17:
for each opponent infoset J J T do set prior probability a(J)
v(J)
+
18:
create new root node where selects infoset To, reaching node h, (h) is the probability that all other players play all actions on the path to h in I. for each J E To do
19:
20:
21:
make h chance node where J is selected w.p. (h)
22:
make new regret minimizer R2 with strategy space [0, 1] using PRM+
23:
Regret minimizer R2 controls the probability with which enters / in Resolve.
24:
r < game tree with root Ø
25:
delete all game tree nodes not reachable from Ø
Figure 9: Pseudocode, Part 2.
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1: procedure RUNSOLVERTHREAD
2:
while time permits do Run for longer than expander threads.
3:
RUNCFR ATERATION
4: RUNCFRITERATION
5: Special case, must be done separately: RM+ updates for the Resolve subgame: for each J E To do perform regret minimizer update at Ry with utility ut (J)
6:
7:
D Transition between Resolve and Maxmargin smoothly, based on whether Resolve chooses enter at any infoset resolvee (J) is the probability that Resolve (Ry) enters at J. # maxmarrgin (J) is the probability Maxmargin picks J.
8:
9:
10:
11: Pmax f- maxJEJo
" [esoovve (J)
12:
for each J E Jo do (J) Pmax a(J) T + (1 Pmax)
makmargin
Note: is possible for EJEo. (J) if Resolve is being used!
14:
15:
procedure RUNCFRRIEERAIIONCCPlloii
player i) MAKEUTILITIESS, Ø) MAKEUTILITIES will mark some infosets VISITED. (J) is v's CFV for picking J at the root.
17:
18:
if then
19:
for each J E To do (J) + (T( + et (J) for each VISITED infoset in bottom-up order do
20:
21:
D n. is player i's strategy. (() is the parent sequence of infoset I.
22:
CFR value backpropagation:
23:
" (a(()) + uf (o())) + EeeA()) (z(a([)u;((I,)
perform regret minimizer update using counterfactual values uff (I,-)
25:
mark / not VISITED u (1, + > reset
26:
Figure 10: Pseudocode, Part 3.
1: procedure MAKEUTII.TIESSexplloii player i, node h)
2: mark h NEW
3: if f is not EXPANDED or h is terminal then
4:
(1,a) (h)
5: mark I as VISITED
6:
) is the Stockfish evaluation terminal node value of from i's perspective.
7:
"T, a) stores the CFV at sequence (T, Initialized to O.
8:
(1, + (1, a) + (h)()))
9:
else
10:
> No need to explore nodes to which the opponent does not play. No locks needed: all EXPANDED nodes are safe access. for each legal action at h do
12:
13:
plays at or π T- (ha) > 0 then MAKEUTILITISSÚ, ha)
Figure 11: Pseudocode, Part 4.
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procedure RUNEXPANDERTHREAD
2:
while time permits do DOEXPANSIONSSTER
3:
4:
DOEXPANSIONSTER
5:
6: procedure ODEXPANSIONNSTEPPCPPI
player i)
7: h <- root node of current subgame r
8: while h EXPANDED do Find leaf to expand.
9:
Terminal nodes cannot be expanded.
10: D Also, we should expand nodes that CFR has not yet iterated on.
if h is terminal h is NEW then return
12:
> Select action:
D n; the expansion strategy of player i as defined in the text.
14:
for action E A(h) do
15:
h belongs to then ((ahh) + (a/h))
16:
else (a(fh) T=<(a/h)
h and is using Resolve, #(-hh may not be distribution sample E A(h) w.p. x ((-hh)
18:
ha
20:
Expand h:
21:
< active player at ha
22:
add all children of h to P
23:
let be the infoset that & should be in if is not created then
24:
25:
create /
initialize current strategy as (z)) where a" argmax A(h) (ha)
27:
add h to I
28: mark h as EXPANDED
Figure 12: Pseudocode, Part 5.
28


Reviews:
[{"review_id": "j9I4RZNhYG", "reviewer": "Reviewer_74mV", "summary": "This paper contributes a Fog-of-War chess bot that is ostensibly state of the art. It performs real-time search without trying to untangle unwieldly \"I know that you know...\" loops that usually choke imperfect-information solvers.", "questions": "- What is the broader significance?\n- How could the methods generalize?", "limitations": "", "rating": 2, "confidence": 4, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "Obscuro seems to be the first superhuman AI in the fog of war chess variant.\nThe paper does a good job of explaining the difficulties associated with developing AI in this kind of setting.", "weaknesses": "- My biggest concern is the significance of the contribution. Even if all of the claims in the paper are completely accurate (and see below for concerns on that front), it's not clear to me that applying mostly known tricks to develop superhuman AI in this niche chess variant constitutes enough for acceptance. Despite being a lifelong chess fan, and even a chess variants fan, I've never heard of fog of war chess. I don't think the abstract's claim that FoW chess has been \"the main challenge problem in imperfect information games\" is accurate. \n\n- Building on the previous point, the methods used are interesting but not super exciting, and don't seem to be very generalizable. The work would be stronger if the authors could demonstrate or at least argue why their innovations are useful beyond FoW chess.\n\n- The results section is quite short, and I'm not convinced that the authors have done everything they can to put Obscuro up to the toughest tests.\n\n- The paper is somewhat unprofessionally written (e.g. section 4.1; the 4-line title).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "WLZeUNz7fJ", "reviewer": "Reviewer_eLzR", "summary": "This paper introduces *Obscuro*, a superhuman Fog of War (FoW) chess agent.\nIt improves the SOTA by using better SOTA algorithms for the individual steps. The most significant improvement results from replacing LP-based equilibrium computation with GT-CFR. Further improvements are generated by careful engineering.", "questions": "- Can you motivate the choices of your adaptations?\n- What are the implications of your work in the field? I.e., how can your work contribute to solving other tasks?\n- The approach uses advanced search techniques and does not utilize any learning algorithm or learnable parameters. What makes this paper a reinforcement learning paper?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 4, "presentation": 4, "contribution": 2, "strengths": "- very good motivation\n- great outcomes, producing the first superhuman FoW agent\n- overall well-written paper", "weaknesses": "- low originality by mainly engineering the current SOTA\n- contributions only clearly stated in the ablation\n- use of crafted, not learned value-function\n- no learning at all, only search\n- overuse of footnotes decreases readability", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "TuzSORClSj", "reviewer": "Reviewer_qYdi", "summary": "This paper presents Obscuro, a search-based AI agent achieving superhuman performance in Fog of War chess, introducing general subgame-solving algorithms that do not rely on explicit common-knowledge reasoning. The authors propose the KLUSS framework that prunes high-order belief states considered strategically irrelevant. This is integrated into algorithms like PCFR+ for efficient tree expansion. Expirical results show that Obscuro surpasses the previous SOTA system and human experts.", "questions": "See weaknesses", "limitations": "", "rating": 8, "confidence": 4, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The paper addresses a central limitation of existing imperfect-information search methods. KLUSS provides a pragmatic and scalable alternative that extends the reach of search-based techniques.\n- The system combines ideas from recent developments in counterfactual regret minimization, tree expansion policies, and real-time planning, demonstrating a coherent and well-engineered design.\n- The empirical results are strong. The performance can be achieved with relatively small computation, underscoring the method's practical efficiency.", "weaknesses": "- The interaction between PCFR+, one-sided GT-CFR, and KLUSS lacks a unified theoretical analysis. Each component has individual convergence guarantees under specific settings, but their concurrent use in a dynamically expanding and pruned search tree leaves correctness unproven.\n- Although framed as a general search method, Obscuro’s performance critically depends on Stockfish’s perfect-information evaluation function. This component embeds extensive domain knowledge from conventional chess, potentially limiting the generality of the claimed framework.\n- The paper can also benefit from adding tests on other imperfect-information domains such as poker or Stratego, which strengthens the empirical basis for calling the approach “general-purpose.”", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "qQrAEnEx6A", "reviewer": "Reviewer_3cmb", "summary": "This paper introduces Obscuro, an AI system to achieve superhuman performance in Fog of War (FoW) Chess, a long-standing benchmark in imperfect-information game solving. The key technical contribution is a suite of scalable real-time search techniques—most notably Knowledge-Limited Unfrozen Subgame Solving (KLUSS) and one-sided GT-CFR—that circumvent the need to compute or enumerate common-knowledge sets, which are intractable in FoW Chess (up to ~10¹⁸ states). The authors demonstrate Obscuro’s superiority through extensive experiments: it wins 85.1% of 1,000 games against the prior SOTA (ZS21) and 80% of 20 games against the world’s top-ranked human player (rating 2318 on chess.com).", "questions": "Stockfish is designed for chess with perfect information. Using it as an evaluation function for the game tree in FoW chess is not entirely reasonable, since the winning conditions of the two games differ. Please provide an explanation.", "limitations": "", "rating": 6, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 3, "strengths": "Significant Technical Advance: The paper directly addresses a fundamental scalability bottleneck in imperfect-information game solving—the reliance on common knowledge—and proposes practical, theoretically motivated approximations (KLUSS) that enable real-time search in previously intractable settings.\n\nStrong Empirical Validation: The evaluation is comprehensive:\nLarge-scale AI-vs-AI matches (1,000–10,000 games) with statistical significance (z > 5).\nHuman evaluation across skill levels (100 games vs. players rated 1450–2006, 97% win rate).\nA decisive 16–4 victory over the #1 human player, with p = 0.011 (binomial test).\n\nExtensive ablation studies isolating the contribution of each component (e.g., KLUSS alone improves win rate from 58% to 85% vs. ZS21).\nAlgorithmic Innovation: The combination of one-sided GT-CFR, last-iterate strategy selection, and purification is well-motivated and empirically validated. The use of Stockfish as a node evaluator is pragmatic and effective.\n\nReproducibility & Transparency: Private game links are provided; hardware requirements are modest (consumer CPU); pseudocode and detailed appendices support reproducibility.", "weaknesses": "The article is not clearly articulated and lacks proper formatting. It fails to explicitly highlight the core contributions of the KLUSS algorithm and one-sided GT-CFR (which may represent the most significant departure from prior methods).\n\nStockfish is designed for chess with perfect information. Using it as an evaluation function for the game tree in FoW chess is not entirely reasonable, since the winning conditions of the two games differ. Please provide an explanation.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Dp6Ytxhjdy", "reviewer": "Reviewer_abjR", "summary": "The paper presents Obscuro, the first AI to reach a superhuman level in the chess variant Fog of War chess (FoW), and reaches a new state-of-the-art for this game type.\nThe main innovation lies in its new search adaptations.\nIn fact, the agent is entirely based on real-time search, rather than training a new neural network policy or value function; it relies on classical Stockfish 14 for standard chess for their node evaluation.\nRelated to search, they introduce knowledge-limited unfrozen subgame solving (KLUSS) search as an improvement over the KLSS algorithm. Next, they utilize growing-tree counterfactual regret minimization (GT-CFR), modifying it to one-sided GT-CFR. Here, they use predictive CFR+ (PCFR+) for equilibrium finding. To expand the game tree, they utilize the polynomial upper confidence bounds for trees (PUCT) algorithm.\nThe authors also provide game footage of their agents, along with an elaborate appendix that explains the search techniques in detail.", "questions": "\"4. Two-sided GT-CFR only, against ZS21. In this ablation, we turned off all the above im-\nprovements 1, 2, 3, and 4, and matched the resulting agent against that of ZS21. This\nserves to isolate the effect of using GT-CFR compared to using the LP-based equilibrium\ncomputation and iterative deepening node expansion as in ZS21.\nIn a 1,000-game match, the GT-CFR version scored 72.6% (+711 =30 -259).\"\n\n-> This is a bit unexpected to score so highly here, when turning off all 3 previous improvements, right?\nI likely misunderstood what was done here.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "+ The engine runs on regular consumer hardware (the details are provided in Appendix B.5.).\n(Preferably, refer to Appendix B.5 directly in the main paper.)\n\n+ The presented agent Obscuro wins both against the previously best FoW engine and the strongest human player on chess.com by a significant margin (+302 +/- 29 Elo over ZS21, +241 +/- 274 Elo over top human).\n\n+ The paper presents numerous ablation studies that highlight the benefits of their individual modifications to existing algorithms.\n\n+ The paper is well formulated and guides the reader throughout the content.\n\n+ It includes a detailed appendix with pseudocode that provides a detailed explanation of the search techniques.\n\nRelevancy:\nThe paper appears to be highly relevant for a major audience.\n\nOverall Assessment\nOverall, I believe it is a great paper that pushes the state-of-the-art in the field of imperfect information games to a new level.", "weaknesses": "- \"required no large-scale computation to learn a value function or blueprint strategy\"\n-> This is true, but the evaluation function was taken from Stockfish 14, which underwent a long period of development. The authors also show in their ablation studies that the evaluation function plays a significant role in their playing strength.\nMaybe clarify this statement a bit.\n\n- The authors do not mention whether they will make their source code and/or engine publicly available.\n\n- It was not directly clear to me at first sight in which perspective the published games were, i.e. if Obscuro was White or Black. -> Please add a comment in the paper that the games are always in the view of Obscuro.\n\n- The sample size (20 games  (+16 =0 -4), +241 +/- 274 Elo) against the top human was not large enough to 100% claim superiority over the top human. You may also provide the relative Elo superiority with error bounds in the main paper. The top human could also try to find fundamental weaknesses in its play over time. This seems, however, not likely as it attempts to find a near-equilibrium strategy.\nI also consider 20 games to be a good sample size, given it's hard to find top players playing an engine. \n\n- The paper would benefit from an additional evaluation environment despite FoW, to highlight the broader applicability of their search agent. This alternative environment could be poker or Stratego for example.\n\nMinor things:\n\"Liu et al. (Liu et al., 2023) introduced a safe variant of KLSS\"\n-> Here, Liu et al. is mentioned twice. You can avoid this by using \\citet{}", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "CfBRJaYvwr", "reviewer": "Reviewer_uPu3", "summary": "The paper presents a game-playing agent, named Obscuro. Obscuro is the first artificial agent that has achieved superhuman performance in Fog-of-war Chess, an imperfect information variant of Chess. Obscuro is a fully search-based agent that builds on the previous state-of-the-art agent in FoW Chess. The main algorithm behind Obscuro, Knowledge-limited Unfrozen Subgame Solving (KLUSS), is a generalization of KLSS. The paper reports good results against the previous state-of-the-art agent, human players of various skill levels, as well as the number one-ranked human player. Finally, the proposed improvements and their importance to the final performance of Obscuro are validated in a series of ablation experiments.\n\nThe paper is clearly written overall, with a well-motivated introduction, detailed algorithmic description, and a comprehensive experimental section that includes strong evaluations against both prior state-of-the-art methods and human players. The ablation studies effectively validate the importance of the proposed improvements. However, the clarity is weakened by missing or inconsistent details, such as incorrectly labeled figures, missing descriptions of the subgame solving algorithm in the main text, and ambiguities in the ablation setup. Some claims, like FoW Chess being the largest imperfect-information game where search has been applied, seem unsupported.", "questions": "* Is the set $P$ from Section 3 an infoset of the acting player?\n* Are the ablations run against the previous state-of-the-art algorithm for FoW Chess?\n* Do you have any intuition why two-sided GT-CFR scored so low as compared to one-sided GT-CFR in the ablations?\n\nMinor comments:\n* Line 53: A missing citation to DeepStack\n* Lines 147-150: The problem of deciding whether two histories belong to the same common-knowledge closure has been studied by Solinas et al. [1]\n\n[1] Solinas, Christopher, et al. \"History filtering in imperfect information games: algorithms and complexity.\" Advances in Neural Information Processing Systems 36 (2023): 43634-43645.", "limitations": "", "rating": 8, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "**Originality.**\nThe experiments convincingly support the claim that Obscuro is the first AI agent to achieve superhuman performance in FoW Chess.\n\n**Clarity**\nThe whole paper is nicely and concisely written. The introduction motivates the need for scalable search in imperfect information games and the reasons it’s substantially harder than in perfect information games.\nThe agent’s description is clearly and thoroughly written, providing detailed explanations of each component.\n\n**Soundness.**\nThe experimental section is comprehensive, including detailed evaluations against both previous state-of-the-art methods and human players of different skill levels, up to the top-rated player on chess.com, providing strong empirical evidence for the quality of the proposed method.\nThe ablation studies consistently demonstrate performance improvements of the proposed modifications, confirming that the improvements are both well-justified and impactful.", "weaknesses": "**Clarity.**\nSeveral parts of the paper suffer from unclear or inconsistent presentation:\n* Lines 216–220 introduce a connectivity graph with nodes labeled by their distance from the circled node and one node marked with an asterisk, yet no such labeling appears in the corresponding figure, making it difficult to follow the explanation.\n* The description of the main algorithm omits details about the subgame solving algorithm, describing it only in the appendix.\n* It is not clear which opponent was used in the ablation experiments and what the “above four improvements” in the fourth ablation (GT-CFR Only) refer to\n* It is also not clear which components were and were not used in some of the ablations\n\n**Soundness.**\nThe claims that FoW Chess is the largest (measured by the amount of hidden information) turn-based game and the largest imperfect information game where search has been successfully applied are not based on any concrete numbers comparing the game to other such games.\nThe first sentence in the conclusion is misleading, as Obscuro requires a strong value function to reach such a strong play (as shown by one of the ablation experiments).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)