LAMIR: Look-Ahead Reasoning in Imperfect Games

• LAMIR is a learned algorithm for test-time look-ahead reasoning in imperfect-information games, using state abstraction to create tractable subgames.
• It integrates a learned simulator, state abstraction, and CFR+ based subgame solving to efficiently approximate equilibrium strategies under limited capacity.
• Empirical evaluations on games like Goofspiel demonstrate LAMIR’s ability to recover near-optimal strategies and outperform baseline methods.

Searching arXiv for the LAMIR paper and closely related work on MuZero, CFR, and Regularized Nash Dynamics. LAMIR, introduced in “Look-ahead Reasoning with a Learned Model in Imperfect Information Games” (Kubíček et al., 6 Oct 2025), is an algorithm for test-time look-ahead reasoning in two-player zero-sum extensive-form games with imperfect information. It learns an abstracted model of the game directly from agent–environment interaction and uses that model at inference time to build tractable subgames for theoretically principled reasoning. The method is designed for settings in which explicit environment models are unavailable or too complex, and it combines a learned simulator with a learned state abstraction that caps the number of abstract information sets per public state. In the reported experiments, LAMIR recovers the exact underlying game structure when capacity is sufficient and, under limited capacity, still learns abstractions that improve the performance of pre-trained agents in both small and large imperfect-information games (Kubíček et al., 6 Oct 2025).

1. Formal setting and problem formulation

LAMIR is formulated for a two-player zero-sum extensive-form game with imperfect information, equivalently viewed as a simultaneous-move game $G = (N, W, A, T, R, O)$, where $N = \{1,2\}$ is the player set, $W$ is the set of world states with unique initial state $w^0 \in W$, $A = A_1 \times A_2$ is the set of joint actions, $T: W \times A \to W$ is the deterministic transition function, $R: W \times A \to \mathbb{R}$ gives the reward to player 1, and $O: W \times A \times W \to O$ is the observation function, factored into public and private components as $O = O^0 \times O_1 \times O_2$ (Kubíček et al., 6 Oct 2025).

A history $h = w^0, a^0, \ldots, a^{l-1}, w^l$ is any sequence consistent with $N = \{1,2\}$0. For player $N = \{1,2\}$1, the private information set at $N = \{1,2\}$2 is

$N = \{1,2\}$3

The set of all information sets for player $N = \{1,2\}$4 is denoted $N = \{1,2\}$5, while public information sets partition histories by public observations alone (Kubíček et al., 6 Oct 2025).

Strategies are behavioral: $N = \{1,2\}$6 assigns a distribution over legal actions at each information set, and a joint strategy $N = \{1,2\}$7 induces reach probabilities $N = \{1,2\}$8 and expected utilities $N = \{1,2\}$9. In the two-player zero-sum case, exploitability of $W$0 is measured by

$W$1

where $W$2 is player $W$3’s best-response value (Kubíček et al., 6 Oct 2025).

This setup places LAMIR within the standard apparatus of imperfect-information game solving, but its central intervention is to replace a hand-specified look-ahead model with a learned and compressed one. A plausible implication is that LAMIR targets the same class of strategic planning problems as CFR-based solvers, while relaxing the assumption that the full game structure is known explicitly at test time.

2. Learned simulator and abstraction mechanism

LAMIR trains from self-play data sampled by a baseline policy and learns both a simulator of the imperfect-information game and a state abstraction that bounds each public-state subgame to at most $W$4 abstract infosets per player (Kubíček et al., 6 Oct 2025). The paper presents this as a direct response to the difficulty of applying model-based search, in the style of MuZero, to imperfect-information settings where reasoning depends on information states rather than fully observed states. MuZero itself is a canonical learned-model search method for perfect-information domains (Schrittwieser et al., 2019).

For each player $W$5, the learned model contains three parameterized components. First, a representation map

$W$6

maps a real information set to a fixed-size latent vector. Second, a dynamics model

$W$7

predicts next latent states, immediate reward, and a termination bit from the two latent states and a joint action. Third, a legal-actions head

$W$8

predicts which actions are legal from a latent state (Kubíček et al., 6 Oct 2025).

Given a ground-truth trajectory $W$9, the model is unrolled over multiple start times and depths. For $w^0 \in W$0, one initializes

$w^0 \in W$1

and for successive steps $w^0 \in W$2,

$w^0 \in W$3

Legal-action logits are also predicted from the initial latent states (Kubíček et al., 6 Oct 2025).

The abstraction module is tied to public states. For a given public state $w^0 \in W$4, a public-state encoder $w^0 \in W$5 proposes $w^0 \in W$6 abstract centers $w^0 \in W$7. An infoset-to-abstract selector

$w^0 \in W$8

assigns a real infoset $w^0 \in W$9 with public state $A = A_1 \times A_2$0 to a soft distribution over those centers, while training and test-time use the argmax center to enforce a many-to-one mapping. A further network $A = A_1 \times A_2$1 predicts ancillary features referred to as a “clustering property,” which drive cluster formation (Kubíček et al., 6 Oct 2025).

This factorization is the defining structural choice in LAMIR. Rather than learn a single latent state for planning, it learns player-specific latent information states and then compresses them into a bounded abstract subgame at each public state.

3. Optimization objectives and training signals

The model-learning objective combines supervision on legal actions, termination, rewards, and latent consistency across rollout steps. The per-trajectory loss $A = A_1 \times A_2$2 is

$A = A_1 \times A_2$3

where $A = A_1 \times A_2$4 is binary cross-entropy and $A = A_1 \times A_2$5 is mean-squared error (Kubíček et al., 6 Oct 2025). The stated purpose of this construction is to force $A = A_1 \times A_2$6 to simulate both belief-state transitions and reward or termination behavior.

The abstraction objective is defined in terms of features

$A = A_1 \times A_2$7

In a public state $A = A_1 \times A_2$8, if the $A = A_1 \times A_2$9 centers are $T: W \times A \to W$0, then the soft-clustering loss is

$T: W \times A \to W$1

and the assignment-matching loss is defined by choosing

$T: W \times A \to W$2

then

$T: W \times A \to W$3

All parts are trained jointly by minimizing

$T: W \times A \to W$4

with gradients from $T: W \times A \to W$5 updating only $T: W \times A \to W$6, gradients from $T: W \times A \to W$7 updating $T: W \times A \to W$8, and gradients from $T: W \times A \to W$9 updating $R: W \times A \to \mathbb{R}$0 (Kubíček et al., 6 Oct 2025).

The choice of clustering property is empirically important. The reported experiments vary $R: W \times A \to \mathbb{R}$1 among “legal actions,” “legal+RNaD strategy,” and “legal+strategy+action-history” (Kubíček et al., 6 Oct 2025). Regularized Nash Dynamics, used as the baseline policy generator, is a recent policy-learning framework for zero-sum games (Tang et al., 2023). Within LAMIR, these features operationalize which aspects of an information set are preserved by abstraction. This suggests that the abstraction is not purely geometric in latent space; it is explicitly task-shaped by strategically relevant signals.

4. Test-time continual resolving in the abstract subgame

At test time, LAMIR assumes access only to the acting player’s real infoset $R: W \times A \to \mathbb{R}$2, hence its public state $R: W \times A \to \mathbb{R}$3. It then applies continual resolving with depth-limited CFR in an abstract subgame learned from data (Kubíček et al., 6 Oct 2025). The abstract subgame is built by first computing the $R: W \times A \to \mathbb{R}$4 abstract centers for each player from the public-state encoder and selecting the center matched to $R: W \times A \to \mathbb{R}$5 via the argmax of $R: W \times A \to \mathbb{R}$6. A depth-$R: W \times A \to \mathbb{R}$7 game tree is then constructed in the abstracted model. Its states are pairs of abstract infoset indices $R: W \times A \to \mathbb{R}$8 together with latent memory, transitions are induced by $R: W \times A \to \mathbb{R}$9, and legal actions are provided by $O: W \times A \times W \to O$0 (Kubíček et al., 6 Oct 2025).

At depth $O: W \times A \times W \to O$1, the construction adds an extra joint “transformation” layer of $O: W \times A \times W \to O$2 heuristic actions leading to terminal payoffs given by a learned multi-valued state value function $O: W \times A \times W \to O$3 (Kubíček et al., 6 Oct 2025). The abstract subgame is then solved with CFR+, which is a refinement of counterfactual regret minimization for extensive-form games (Tammelin, 2014). LAMIR runs CFR+ for $O: W \times A \times W \to O$4 outer iterations and obtains an approximate equilibrium policy $O: W \times A \times W \to O$5 over abstract infosets in the root public state (Kubíček et al., 6 Oct 2025).

Action selection maps the real infoset $O: W \times A \times W \to O$6 to its abstract center $O: W \times A \times W \to O$7 and samples a real action from $O: W \times A \times W \to O$8. After observing the opponent’s action and the next public observation, LAMIR updates the public state, extracts new abstract infosets through $O: W \times A \times W \to O$9, and reuses pertinent subtree values, specifically counterfactual values and reach probabilities, in a new subgame. This repeats until a terminal history is reached (Kubíček et al., 6 Oct 2025).

The reported pseudocode summarizes this process as a recursive ACT(x) routine that constructs the subgame, solves it with CFR+, plays according to the equilibrium policy at the selected abstract center, and then resolves again after the next observation (Kubíček et al., 6 Oct 2025). In conceptual terms, LAMIR combines model-based rollouts with public-state subgame solving. That makes it closer to continual-resolving traditions in imperfect-information search than to one-shot policy inference.

5. Theoretical properties and representational limits

LAMIR’s principal tractability claim is combinatorial. By enforcing at most $O = O^0 \times O_1 \times O_2$0 abstract infosets per public state, each depth-$O = O^0 \times O_1 \times O_2$1 subgame has at most

$O = O^0 \times O_1 \times O_2$2

nodes (Kubíček et al., 6 Oct 2025). The stated significance is that the resulting search space is polynomial in $O = O^0 \times O_1 \times O_2$3 and action-set cardinalities rather than exponential in the number of real hidden states.

The paper also states a capacity-based recovery result. As $O = O^0 \times O_1 \times O_2$4 for every public state $O = O^0 \times O_1 \times O_2$5, as the feature network $O = O^0 \times O_1 \times O_2$6 becomes infinitely expressive, and as the latent-state dimension tends to infinity, the learned abstraction and learned model converge in capacity to the true game, so LAMIR’s look-ahead recovers the exact public-state CFR solution (Kubíček et al., 6 Oct 2025). This is not presented as a finite-sample guarantee, but as an asymptotic representational statement.

A crucial caveat concerns imperfect recall. General CFR convergence requires perfect recall, but LAMIR’s abstraction may violate it. The paper states that if the abstraction satisfies the A-loss recall condition, specifically “public observations and last round actions only,” CFR still converges; Goofspiel and Oshi-Zumo are cited as games meeting this condition, while other games may not admit formal guarantees (Kubíček et al., 6 Oct 2025). This limitation aligns LAMIR with a broader literature in imperfect-information abstraction, where practical compression often outruns available convergence theory. A plausible implication is that empirical success may depend substantially on how benign the recall loss induced by abstraction is in a given domain.

6. Empirical evaluation and reported performance

The empirical study separates small domains, where exploitability can be computed exactly, from larger domains, where evaluation relies on head-to-head win rates (Kubíček et al., 6 Oct 2025).

In the small-domain setting, the paper studies Imperfect-Information Goofspiel with $O = O^0 \times O_1 \times O_2$7 and Oshi-Zumo with $O = O^0 \times O_1 \times O_2$8. The baseline is concurrently trained Regularized Nash Dynamics without test-time search, and the main metric is exploitability $O = O^0 \times O_1 \times O_2$9. For abstraction limits $h = w^0, a^0, \ldots, a^{l-1}, w^l$0 and clustering properties $h = w^0, a^0, \ldots, a^{l-1}, w^l$1legal actions, legal+RNaD strategy, legal+strategy+action-history$h = w^0, a^0, \ldots, a^{l-1}, w^l$2, the method is trained for 10 seeds over 100 K episodes, and every 1 K episodes exploitability is computed by building depth-1 subgames in each public state, solving them via CFR+, and mapping back to the real game (Kubíček et al., 6 Oct 2025). The reported Goofspiel-5 result is that $h = w^0, a^0, \ldots, a^{l-1}, w^l$3 with $h = w^0, a^0, \ldots, a^{l-1}, w^l$4 “legal+strategy+history” yields exploitability approximately $h = w^0, a^0, \ldots, a^{l-1}, w^l$5, described as near zero, and that even $h = w^0, a^0, \ldots, a^{l-1}, w^l$6 produces significantly lower exploitability than RNaD alone (Kubíček et al., 6 Oct 2025).

In the large-game setting, the domains are II-Goofspiel with $h = w^0, a^0, \ldots, a^{l-1}, w^l$7, with the largest public-state sizes reported as up to $h = w^0, a^0, \ldots, a^{l-1}, w^l$8 real histories. The baseline remains RNaD with the same training budget of 3 M episodes. The metric is LAMIR’s win rate against RNaD over more than 100 K matches, averaged over $h = w^0, a^0, \ldots, a^{l-1}, w^l$9 seed pairings with $N = \{1,2\}$00 error bars (Kubíček et al., 6 Oct 2025).

Configuration	Domain	Result
LAMIR ($N = \{1,2\}$01 = legal actions)	Goofspiel 10	$N = \{1,2\}$02
LAMIR ($N = \{1,2\}$03 = legal actions)	Goofspiel 13	$N = \{1,2\}$04
LAMIR ($N = \{1,2\}$05 = legal actions)	Goofspiel 15	$N = \{1,2\}$06
LAMIR ($N = \{1,2\}$07 = RNaD strategy)	Goofspiel 10	$N = \{1,2\}$08
LAMIR ($N = \{1,2\}$09 = RNaD strategy)	Goofspiel 13	$N = \{1,2\}$10
LAMIR ($N = \{1,2\}$11 = RNaD strategy)	Goofspiel 15	$N = \{1,2\}$12

In all of these cases, LAMIR outperforms RNaD (Kubíček et al., 6 Oct 2025).

The paper also reports tabular K-means experiments on Goofspiel with $N = \{1,2\}$13, described as purely offline, showing that exploitability degrades gracefully as $N = \{1,2\}$14 shrinks and that richer clustering properties $N = \{1,2\}$15 outperform mere legal-actions features (Kubíček et al., 6 Oct 2025). In Leduc Hold’em, where chance nodes are present, chance is approximated by stochastically unrolling the first round; under that approximation, LAMIR with $N = \{1,2\}$16—at least as large as the number of real infosets—matches low exploitability, and even $N = \{1,2\}$17 still beats RNaD (Kubíček et al., 6 Oct 2025).

7. Limitations, scope, and relation to adjacent research

The paper identifies several limitations. First, scalability remains constrained because subgame size grows approximately as $N = \{1,2\}$18, which limits the maximal feasible abstraction size $N = \{1,2\}$19 and search depth $N = \{1,2\}$20. It explicitly notes that future work could integrate GT-CFR or other iterative solvers to ameliorate this issue (Kubíček et al., 6 Oct 2025). Second, chance nodes are not modeled directly in $N = \{1,2\}$21; extending the method to stochastic transitions is left for future work (Kubíček et al., 6 Oct 2025). Third, action abstraction is absent: LAMIR does not reduce $N = \{1,2\}$22, so domains with huge or continuous action spaces would require action-abstraction methods layered on top (Kubíček et al., 6 Oct 2025). Fourth, the imperfect-recall issue leaves convergence guarantees unavailable outside the games satisfying the cited recall condition (Kubíček et al., 6 Oct 2025).

Within the broader landscape, LAMIR is explicitly framed as combining model-based look-ahead “à la MuZero” with learned information-state abstraction to make CFR-style reasoning tractable in large imperfect-information games (Kubíček et al., 6 Oct 2025, Schrittwieser et al., 2019). That positioning is technically specific. MuZero learns a latent dynamics model that supports tree search, but it is designed for perfect-information games and MDP-like settings (Schrittwieser et al., 2019). CFR and CFR+ provide principled equilibrium-seeking procedures in extensive-form games (Tammelin, 2014). LAMIR’s contribution is to interpose a learned abstraction layer between those traditions so that look-ahead is performed not in the original hidden-state game, but in a bounded abstract public-state subgame (Kubíček et al., 6 Oct 2025).

The stated real-world implication is that, by obviating the need for explicit rules or simulators at test time, LAMIR can be applied to proprietary or dynamically changing games, or more generally to partially observable settings in which a small abstract model suffices (Kubíček et al., 6 Oct 2025). This suggests a broader interpretation of LAMIR as a framework for learned subgame construction under partial observability, though the empirical evidence currently reported is concentrated in game benchmarks rather than non-game POMDPs.