Play2Perfect: Multi-Domain AI Approaches

• Play2Perfect is a multifaceted concept in AI that combines perfect-information guidance during training with execution based on realistic, imperfect observations.
• It spans applications in dexterous robotics, imperfect-information games, and equilibrium refinement, each employing distinct asymmetric learning mechanisms.
• These frameworks demonstrate enhanced performance and sample efficiency by coupling task-agnostic pretraining with precise finetuning or deferred reasoning strategies.

Searching arXiv for the cited Play2Perfect-related papers to ground the article in the current literature. Play2Perfect is used in recent arXiv literature in several related but non-identical senses. In dexterous robotics, it names a two-stage reinforcement-learning framework in which a robot first learns task-agnostic play on diverse objects and goals, and is then finetuned on precise assembly (Lum et al., 24 Jun 2026). In imperfect-information games, the term is also used for schemes that exploit perfect information during training or internal evaluation while constraining the deployed decision rule to the information actually available at execution, most explicitly in PerfectDou’s Perfect-Training–Imperfect-Execution framework for DouDizhu and in determinization-based search variants such as EPIMC (Yang et al., 2022, Arjonilla et al., 2024). In the supplied literature, the label is further used to organize equilibrium-computation pipelines aimed at perfect or refinement-based play in extensive-form games [(Ganzfried, 2022); (Bernasconi et al., 2022); (Hou et al., 20 May 2025); (Etessami, 2014)]. This suggests a family resemblance rather than a single formalism: “play” supplies informative data, structure, or perturbations, and “perfect” names either stronger execution-time behavior or a refined equilibrium target.

1. Scope of the term across the literature

The term appears in at least three technically distinct research programs.

Setting	What “Play2Perfect” denotes	Core mechanism
DouDizhu RL	Perfect-Training–Imperfect-Execution in PerfectDou	Actor uses imperfect information; critic uses perfect information (Yang et al., 2022)
Dexterous robotics	Task-agnostic play pretraining followed by precise assembly finetuning	Goal-conditioned play prior specialized by sparse-reward RL (Lum et al., 24 Jun 2026)
Imperfect-information search and equilibrium computation	Editorial label for perfect-information reasoning or refinement pipelines	Deferred determinization, perturbations, and path-following [(Arjonilla et al., 2024); (Ganzfried, 2022); (Bernasconi et al., 2022); (Hou et al., 20 May 2025); (Etessami, 2014)]

A common misconception is to treat Play2Perfect as a single algorithmic template. The cited work does not support that reading. In PerfectDou, the central asymmetry is between actor and critic; in robotics, it is between pretraining and finetuning; in EPIMC, it is between determinized leaf evaluation and information-set-consistent backup; and in equilibrium-refinement work, it is between ordinary Nash play and perturbation-robust or path-followed refinements (Yang et al., 2022, Lum et al., 24 Jun 2026, Arjonilla et al., 2024, Ganzfried, 2022).

2. Perfect-training–imperfect-execution in DouDizhu

In "PerfectDou: Dominating DouDizhu with Perfect Information Distillation" (Yang et al., 2022), Play2Perfect is instantiated as Perfect-Training–Imperfect-Execution (PTIE), a variant of CTDE specialized to imperfect-information games. The domain is DouDizhu, a three-player shedding game in which one Landlord competes against two Peasants. The game is difficult because it combines imperfect information, a large action space of over 27,000 unique combinations, irreversible combination destruction, and mixed cooperative–competitive credit assignment between the two Peasants and the Landlord (Yang et al., 2022).

The PTIE decomposition is asymmetric. The deployed actor sees only imperfect information at the infoset node $h$, yielding a student policy $\pi_S(a \mid o)$. The critic is trained on the distinguishable node $D(h)$ for the strategically identical perfect-information game and sees full information, including all players’ current hands and state features. Knowledge transfer is implicit rather than explicit: the actor is updated using advantages computed from the teacher critic. The paper states the PTIE policy gradient as

$$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$

PerfectDou does not add an explicit KL distillation loss between a perfect-information policy and an imperfect-information policy, and it does not train a separate student value on partial observations (Yang et al., 2022).

Optimization uses proximal policy optimization with generalized advantage estimation. The clipped PPO objective is standard,

$$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$

but the key PTIE modification is that $V$ and hence $A_t$ are computed from perfect-information inputs $V(D(h))$, while the actor continues to condition only on imperfect observations (Yang et al., 2022).

The system also densifies sparse terminal rewards with an oracle intermediate reward based on the minimum steps to play out all cards for each player, computed by dynamic programming. If $N_t^{\text{role}}$ denotes minimum play-out steps and

$$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$

then the round-level reward is

$\pi_S(a \mid o)$0

with scaling factor $\pi_S(a \mid o)$1. This reward is explicitly designed to encourage Peasant cooperation through the $\pi_S(a \mid o)$2 operator and to align competitive and cooperative incentives (Yang et al., 2022).

The representation stack is highly structured. Card and combination features use a $\pi_S(a \mid o)$3 matrix over 15 ranks. Imperfect-information policy features are a flattened $\pi_S(a \mid o)$4 card-feature tensor plus 6 scalars; perfect-information value features are a flattened $\pi_S(a \mid o)$5 tensor plus 8 scalars. Legal actions are encoded by a $\pi_S(a \mid o)$6 action card matrix plus 7 tag features. To control combinatorial explosion, the system uses a reduced abstract action space of size 621, with kicker selection deferred to a decoding heuristic. The policy network employs an LSTM over state features and a per-action scoring network with “target attention”; the value network combines imperfect and perfect features and predicts a scalar through MLP layers $\pi_S(a \mid o)$7 (Yang et al., 2022).

Training is role-specific and distributed. Three separate policies are maintained, one for the Landlord and one for each Peasant position. Rollout workers generate self-play trajectories in an IMPALA-style architecture; GPU learners update models and synchronize across 8 GPUs with Horovod. Reported hyperparameters include learning rate $\pi_S(a \mid o)$8 with Adam, $\pi_S(a \mid o)$9, $D(h)$0, entropy weight $D(h)$1, LSTM sequence length 15, and total batch size 1024 (Yang et al., 2022).

Empirically, PerfectDou is reported to dominate the published DouDizhu baselines on both WP and ADP over 10k decks with role swaps.

Baseline	WP	ADP
DouZero (Public)	0.543	0.143
DeltaDou	0.584	0.420
RHCP-v2	0.543	0.506
CQN	0.862	2.090
Random	0.994	3.146

The sample-efficiency comparison is equally central. PerfectDou at $D(h)$2 steps defeats DouZero trained for roughly the same scale with WP $D(h)$3 and ADP $D(h)$4, and even remains competitive against DouZero at roughly $D(h)$5 steps. The ablations identify PTIE as crucial: “Vanilla PPO (no PTIE, no oracle)” yields WP $D(h)$6 and ADP $D(h)$7, whereas “ImperfectDou” and “RewardlessDou” remain stronger but below the full system (Yang et al., 2022).

The paper also states clear boundaries. The reported system omits the bidding phase, relies on a domain-specific dynamic program for the oracle, uses no explicit opponent modeling at inference, and performs no explicit KL policy distillation. These are substantive constraints rather than implementation details (Yang et al., 2022).

3. Dexterous play pretraining for precise assembly

In "Play2Perfect: What Matters in Dexterous Play Pretraining for Precise Assembly?" (Lum et al., 24 Jun 2026), Play2Perfect denotes a two-stage framework: task-agnostic goal-conditioned play pretraining in simulation on diverse objects and goals, followed by sparse-reward RL finetuning on CAD-defined precise assembly tasks. The motivation is that precise assembly with multi-fingered robots is contact-rich and sparse-reward, so direct RL from scratch would need to discover grasping, in-hand reorientation, alignment, and contact-rich insertion before receiving any reward (Lum et al., 24 Jun 2026).

The play phase is explicitly goal-conditioned. A single policy controls both arm and hand,

$D(h)$8

where $D(h)$9 are robot proprioception, $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$0 are current and target object poses, and $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$1 encodes object geometry as 3D bounding-box dimensions. Objects are procedurally generated primitives, often with two rigid components, and randomized densities are used to vary center of mass and inertia. Goals are sampled as sequences of 6D objectives: the first requires grasping and lifting, and subsequent goals remain near the previous goal with up to $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$2 translation and up to $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$3 rotation, encouraging repeated in-hand reorientation (Lum et al., 24 Jun 2026).

Pose control is defined through a keypoint-based 6D metric. Four object-frame keypoints are placed at corners of a box with dimensions $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$4. For pose $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$5, the world-frame keypoints are $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$6, and the distance to goal $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$7 is

$$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$8

Goal success is declared when $$\nabla_\theta J = \mathbb{E}_\pi[\nabla_\theta \log \pi_\theta(a \mid h)\, Q^\pi(D(h), a)].$$9, with default $$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$0. The paper emphasizes that smaller $$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$1 increases learned precision (Lum et al., 24 Jun 2026).

The observation vector is 140-dimensional and contains 29 joint positions and velocities, palm pose, fingertip positions, object orientation, current keypoints relative to the palm, relative displacements to goal keypoints, and object primary-component dimensions. Actions are 29 joint-position commands for a 7-DoF arm and 22-DoF hand. Arm actions are delta joint-position targets, hand actions are absolute joint positions, commands are clipped to joint limits, and an EMA with $$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$2 is applied. Physics runs at 120 Hz and the policy at 60 Hz (Lum et al., 24 Jun 2026).

Play pretraining uses a dense reward with smoothness, grasping, lift, goal progress, and sparse success components:

$$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$3

The smoothness term is

$$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$4

with $$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$5 and $$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$6. Goal progress uses

$$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$7

with $$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$8, $$r_t(\theta)=\frac{\pi_\theta(a_t\mid o_t)}{\pi_{\theta_{\mathrm{old}}}(a_t\mid o_t)}, \qquad L_{\mathrm{CLIP}}(\theta)=\mathbb{E}_t\!\left[\min\!\big(r_t(\theta)A_t,\operatorname{clip}(r_t(\theta),1-\epsilon,1+\epsilon)A_t\big)\right],$$9, and $V$0. Finetuning removes grasp, lift, and pose-progress terms, retaining only smoothness and sparse goal bonuses plus a sparse retraction bonus. Intermediate goals are derived from CAD by assembly-by-disassembly; for insertion, an aligned pre-insertion pose is added, and for screwing, successive poses along the thread at $V$1 rotational offsets are used (Lum et al., 24 Jun 2026).

Optimization uses Split and Aggregate Policy Gradients (SAPG), a population-based variant of PPO reported to outperform PPO for dexterous play. The surrogate uses PPO-style clipping,

$V$2

with $V$3, entropy scale $V$4, discount $V$5, and GAE $V$6. The actor is asymmetric with respect to the critic: the actor receives only deployment-available observations, while the critic receives privileged, noise-free signals including undelayed observations, palm and object velocities, reward signals, and progress features (Lum et al., 24 Jun 2026).

The architecture is large-scale but simple: Actor $V$7 LSTM[1024] $V$8 MLP[1024, 1024, 512, 512], Critic $V$9 MLP[1024, 1024, 512, 512]. Domain randomization is extensive in both stages and includes pose noise, observation and action delays, joint-velocity noise, object dimension scale changes, table-height perturbations, and external forces and torques. Contact-critical holes and mating interfaces are modeled with SDFs at resolution 256, while most remaining geometry uses convex decomposition (Lum et al., 24 Jun 2026).

The principal empirical claims are unusually concrete. The paper reports that the prior is 33x more sample-efficient than RL training from scratch, even when the scratch baseline is provided with dense, multi-stage rewards. On a simplified Tight-Insertion variant, Scratch needed more than 100 hours to reach near-perfect success, whereas Play2Perfect reached the same success in about 4 hours. Across four contact-rich tasks, Play2Perfect learned successful policies within 2–5 hours, while scratch training with either sparse or dense rewards produced no successful rollouts after 24 hours. Zero-shot sim-to-real transfer reached 60% success on tight insertions with only 0.5 mm contact clearance, and over 50% success on long-horizon multi-part assembly and screwing (Lum et al., 24 Jun 2026).

The design-study component identifies what matters in play pretraining. Increasing object diversity helps, with diminishing returns between 100 and 1000 procedurally generated objects. Translation-only pretraining transfers poorly, whereas rotation-only transfers substantially better; full 6D pose-reaching is reported as the most consistent and fastest. Online random goal trajectories learn faster and transfer better than fixed banks of 10 or 100 trajectories. Tight goal tolerances matter: $A_t$0 transfers poorly, $A_t$1 can learn but is slower, and $A_t$2 best induces fine in-hand manipulation useful for tight-clearance assembly (Lum et al., 24 Jun 2026).

The limitations are equally specific. Perception can degrade under occlusions and fast motion, especially for symmetric parts; real fixtures taped to foam can move under contact unlike rigid simulated fixtures; the policy does not directly observe fixture geometry beyond goals; and the reported scope is short-horizon skill learning rather than broader sequencing or active-part selection (Lum et al., 24 Jun 2026).

4. Deferred perfect-information reasoning in imperfect-information search

"Perfect Information Monte Carlo with Postponing Reasoning" (Arjonilla et al., 2024) uses “Play2Perfect” in a determinization sense. The baseline is Perfect Information Monte Carlo (PIMC), which samples hidden information, converts the current information set into a perfect-information instantiation, evaluates root actions with a perfect-information leaf evaluator, and aggregates the resulting values. The canonical estimate is

$A_t$3

The difficulty is strategy fusion: a player must use the same strategy at all histories consistent with an information set, but determinization can implicitly prescribe different actions in different determinizations within the same information set (Arjonilla et al., 2024).

The paper defines strategy fusion formally. A policy $A_t$4 creates strategy fusion at infostate $A_t$5 if there exist $A_t$6 such that $A_t$7, and quantifies the phenomenon with

$A_t$8

EPIMC addresses the pathology by postponing perfect-information resolution to depth $A_t$9. Instead of calling the perfect-information evaluator immediately after the root action, it constructs a sampled subgame $V(D(h))$0 in information-set space to depth $V(D(h))$1, averages deferred leaf evaluations, and then solves $V(D(h))$2 with an imperfect-information solver such as ISS or CFR+ (Arjonilla et al., 2024).

The operational contrast with PIMC is central. EPIMC explores one action per sampling iteration and evaluates a single leaf per iteration, keeping leaf evaluations at one per iteration and the number of added nodes near $V(D(h))$3. The build phase therefore costs $V(D(h))$4 with $V(D(h))$5 iterations, while the solve phase costs whatever the chosen imperfect-information subgame solver requires. Memory scales with the size of $V(D(h))$6, i.e. $V(D(h))$7 in the worst case (Arjonilla et al., 2024).

The theoretical contribution is monotone mitigation of strategy fusion with increasing depth. Proposition 1 states that for EPIMC policies at depths $V(D(h))$8 and $V(D(h))$9, the degree of strategy fusion never increases:

$N_t^{\text{role}}$0

Proposition 2 states that if fusion remains at depth $N_t^{\text{role}}$1, then extending the deferral sufficiently strictly reduces it. Proposition 3 states that in finite-horizon games, sufficiently deep deferral eliminates strategy fusion entirely (Arjonilla et al., 2024).

Empirically, EPIMC shows the strongest gains in games with substantial private information. The evaluated domains include Card Game, Battleship, Dark Chess, Phantom Tic-Tac-Toe, and Dark Hex. Gains from increased depth are large in Dark Chess, Dark Hex, and Phantom Tic-Tac-Toe, but minor in Card Game and Battleship, where observations are mostly public. The paper reports that at 100 seconds in Dark Chess, win rates are approximately 80% / 65% / 45% for depths 3 / 2 / 1, respectively, and that at 1000 seconds EPIMC at depth 3 wins close to 100% in both Dark Hex and Dark Chess (Arjonilla et al., 2024).

The paper also emphasizes that reduced fusion is not identical to improved performance. In Dark Hex at 1000 seconds, similar performance was observed at depths 2 and 1 despite reduced fusion, and in some matchups against IS-MCTS, removing a previously beneficial fused behavior can hurt. This matters for interpretation: postponing perfect-information reasoning is not a universal monotone improvement in utility, even though it is a monotone improvement in consistency with the information structure (Arjonilla et al., 2024).

5. Equilibrium refinement, observable mistakes, and path-following to perfect play

Several papers connect the broader Play2Perfect idea to equilibrium refinement in extensive-form games. "Observable Perfect Equilibrium" (Ganzfried, 2022) argues that classical refinements such as trembling-hand perfect equilibrium, quasi-perfect equilibrium, and one-sided quasi-perfect equilibrium may not be the correct concept for developing strong agents in sequential games of imperfect information. It introduces observable perfect equilibrium (OPE), defined by robustness over trembles in publicly observable action probabilities rather than over all action probabilities. In sequence form, after observing public path $N_t^{\text{role}}$2, OPE imposes a lower-bound constraint only on opponent sequences consistent with that path, yielding the saddle-point problem

$N_t^{\text{role}}$3

The paper proves that every OPE is a Nash equilibrium and that OPE always exists. In no-limit clairvoyance with stack $N_t^{\text{role}}$4, OPE prescribes calling a bet of size 1 with probability $N_t^{\text{role}}$5, whereas EFTHPE, QPE, and OSQPE prescribe $N_t^{\text{role}}$6 (Ganzfried, 2022).

"Last-iterate Convergence to Trembling-hand Perfect Equilibria" (Bernasconi et al., 2022) addresses computation of EFPE in two-player zero-sum imperfect-information extensive-form games. It introduces a continuation method through regularized-perturbed games with saddle objective

$N_t^{\text{role}}$7

where the dilated regularizers include tremble offsets. The inner solver is a composite optimistic mirror-descent procedure over sequence-form domains, and the paper proves last-iterate convergence to the unique saddle point of each regularized-perturbed game. When $N_t^{\text{role}}$8 faster than $N_t^{\text{role}}$9, the tracked path converges to an EFPE. The reported implementation computes updates by visiting the game tree, avoids explicit materialization of $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$0, and empirically yields much lower average infoset regret than CFR and related baselines on Kuhn poker, Leduc poker, Goofspiel, deep Rock-Paper-Scissors, and a 3×3 matrix game (Bernasconi et al., 2022).

"A Sequence-Form Characterization and Differentiable Path-Following Computation of Normal-Form Perfect Equilibria in Extensive-Form Games" (Hou et al., 20 May 2025) treats normal-form perfect equilibrium (NFPE) for $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$1-player extensive-form games with perfect recall. The paper develops a sequence-form characterization via perturbed games, then constructs an artificial logarithmic-barrier game and a differentiable path-following method. A smooth equilibrium path starts from an arbitrary positive realization plan and converges to an NFPE as the continuation variable approaches zero. The paper also extends Harsanyi’s linear and logarithmic tracing procedures to sequence form, yielding three methods—LOGB, HLTP, and HLOG—and reports that LOGB shows the best robustness, 0% failure across tested configurations, and the fastest runtime (Hou et al., 20 May 2025).

The complexity boundary for exact refined play in multi-player extensive-form games is given in "The complexity of computing a (quasi-)perfect equilibrium for an n-player extensive form game of perfect recall" (Etessami, 2014). For finite EFGPRs with $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$2, computing a behavior profile within $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$3 of a PE, QPE, NF-PE, or sequential equilibrium is FIXP$$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$4-complete, while computing $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$5-almost $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$6-PE, $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$7-almost $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$8-QPE, and $$\mathrm{Adv}_t = N_t^{\text{Landlord}} - \min(N_t^{\text{Peasant1}}, N_t^{\text{Peasant2}}),$$9-almost SGPE is PPAD-complete. The paper provides explicit fixed-point maps $\pi_S(a \mid o)$00 and $\pi_S(a \mid o)$01 for PE and QPE, respectively, and shows that weak approximate fixed points imply almost-refined equilibria (Etessami, 2014).

Taken together, these papers show that “perfect” in Play2Perfect can refer to multiple targets: robustness to publicly observable mistakes, trembling-hand or quasi-perfect refinements, or NFPE limits under sequence-form continuation. They also show that the computational status of those targets varies sharply by setting: two-player zero-sum admits tree-structured iterative methods, whereas general $\pi_S(a \mid o)$02-player imperfect-information refinement is computationally harder [(Ganzfried, 2022); (Bernasconi et al., 2022); (Hou et al., 20 May 2025); (Etessami, 2014)].

6. Common structure, differences, and boundaries

Across these literatures, the most stable commonality is asymmetry between the information available during optimization and the information available during deployment. In PerfectDou, only the critic sees perfect information, and the actor never does at test time (Yang et al., 2022). In the robotics framework, the actor receives only deployment-available observations while the critic receives privileged, noise-free signals during training (Lum et al., 24 Jun 2026). In EPIMC, perfect-information evaluation is retained, but its influence is deferred and then filtered through an information-set solver, so that consistency is enforced over the early part of the subgame (Arjonilla et al., 2024). In OPE, public observations determine which trembles matter for refinement, rather than unobserved private deviations (Ganzfried, 2022).

The differences are equally important. PerfectDou is a multi-agent RL system for a three-player card game with a role-specific actor-critic architecture, PPO, GAE, oracle shaping, and a reduced action space (Yang et al., 2022). The robotics Play2Perfect framework is a goal-conditioned dexterous-manipulation system optimized with SAPG, domain randomization, CAD-derived sparse goals, and sim-to-real transfer constraints (Lum et al., 24 Jun 2026). EPIMC is an online search method, not a learned policy, and its main object is a sampled subgame solved by ISS or CFR+ (Arjonilla et al., 2024). The equilibrium-refinement papers are neither search nor imitation nor actor-critic RL; they are about perturbation models, sequence form, and continuation to refinement concepts such as OPE, EFPE, and NFPE (Ganzfried, 2022, Bernasconi et al., 2022, Hou et al., 20 May 2025).

Several boundaries recur across the literature. PerfectDou does not model the bidding phase, uses no explicit opponent modeling at inference, and relies on a domain-specific dynamic program for oracle rewards (Yang et al., 2022). The dexterous assembly system remains sensitive to perception failures, contact mismatch, and the absence of explicit scene awareness from fixture geometry (Lum et al., 24 Jun 2026). EPIMC may offer little benefit in domains with mostly public observations, and in some matchups fusion reduction can remove behavior that happened to be advantageous against a specific opponent (Arjonilla et al., 2024). Exact refined equilibrium computation in general $\pi_S(a \mid o)$03-player extensive-form games remains FIXP$\pi_S(a \mid o)$04-complete at the approximation level considered in the complexity paper (Etessami, 2014).

A final misconception is that “perfect” here means omniscient execution. The supplied literature repeatedly rejects that interpretation. PerfectDou’s actor is explicitly constrained to imperfect information at execution; the dexterous assembly policy deploys with ordinary observations plus tracked part pose rather than critic privileges; EPIMC retains hidden information and only uses perfect-information evaluators at deferred leaves; and OPE is based on what is publicly observable rather than globally known (Yang et al., 2022, Lum et al., 24 Jun 2026, Arjonilla et al., 2024, Ganzfried, 2022). The more accurate reading is that Play2Perfect denotes methods that use additional structure during learning, search, or refinement so that execution can be stronger without becoming informationally unrealistic.