Game of Hidden Rules (GOHR) Framework

• GOHR is a framework where game rules are intentionally concealed or partially known to challenge players' inference under uncertainty.
• It integrates formal models from logic, game theory, and algorithmic complexity to analyze computational and epistemic barriers in rule discovery.
• The framework underpins benchmarks in reinforcement learning, equilibrium analysis, and generative model evaluation for hidden rule induction.

The Game Of Hidden Rules (GOHR) refers to a class of games, benchmarks, and theoretical frameworks in which aspects of the governing rules are either unknown, partially known, or intentionally concealed from some or all participants. GOHR settings serve as both a theoretical foundation for modeling strategic interaction under epistemic uncertainty and as practical benchmarks for evaluating the learning, inference, and generalization capabilities of human and artificial agents. Recent advances span the spectrum from formal logic/game theory (Kripke semantics, min-games), algorithmic complexity (information hiding, rule inference), reinforcement and supervised learning benchmarks, to the paper of emergent behavior in social deduction and multi-agent systems.

1. Mathematical Formulations for Games with Hidden Rules

Formal models of GOHR generalize traditional game theory by relaxing the common knowledge assumption on the rules of play. The landmark formulation models a strategic game over a finite set of worlds $W$, each representing a different realization of the game's rules (e.g., different payoff matrices) (Gopalkrishnan et al., 2014). Each player $p$ is endowed with an equivalence relation $\equiv_p$ on $W$ such that $[w]_p = \{v \in W \mid v \equiv_p w\}$ captures the set of epistemically indistinguishable worlds for $p$.

A pure strategy for $p$ in this setting is a function $s_p : [W]_p \rightarrow S_p$, specifying an action for each knowledge class. Mixed strategies are distributions over these functions. A play instance $X$ in world $w$ is given by evaluating each player's strategy at $[w]_p$: $X(w) = (X_p([w]_p))_p$.

The payoff for $p$ in $w$ is defined as the minimum expected payoff across indistinguishable worlds (pessimistic or minimax reasoning):

$$\text{pay}_p^w(\sigma) := \min_{v \in [w]_p} \mathbb{E}[u_p^v(X(v))]$$

An equilibrium is a mixed strategy profile $\sigma$ such that for all $p$, all $w \in W$, and all alternative mixed strategies $\sigma'_p$:

$$\text{pay}_p^w(\sigma) \ge \text{pay}_p^w(\sigma - \sigma_p + \sigma'_p)$$

This formulation is shown to admit an equilibrium by reduction to the min-game construct, where each player's true payoff is the minimum over several parallel games (Gopalkrishnan et al., 2014). Theoretical guarantees rely on fixed-point theorems and established results for Nash equilibria.

2. Complexity and Information Hiding

A distinguishing feature of GOHR environments is the intentional concealment of rules, parameters, or internal data structures, formally studied via models such as quiz games (Bank et al., 2015). In a quiz game, the "quizmaster" conceals a parameter or encoding (for example, the coefficients of a polynomial or a neural network), and the "player" learns only through a constrained query–response interface. The core insight is the existence of exponential lower bounds on the complexity of reconstructing the hidden structure: for families of polynomials represented by arithmetic circuits, any protocol that permits a player to reconstruct the hidden object through queries must use an abstraction of size at least $\Omega(2^{2^{L \cdot n}})$. In elimination theory, even compactly described instances can force polynomials with representations of size $2^{\Omega(n)}$.

These results imply an intrinsic computational barrier for breaking information hiding—even with unlimited computational resources, the retrieval of hidden rules via queries (or play traces) can be provably infeasible. This provides a formal underpinning for why strategic rule concealment (whether in software interfaces or adversarial games) is robust to inference.

3. Learning, Inference, and Benchmark Design

GOHR has been instantiated as a family of learning benchmarks, notably environments where a learning agent must both infer and exploit a hidden rule that governs the validity of possible actions (Pulick et al., 2022). A canonical example is a 6×6 board with four designated "bucket" locations, where the agent must clear the board by placing pieces into buckets according to a rule that maps features (such as shape, color, or position) to permitted buckets. The structure of the hidden rule is specified using an expressive rule language, encoding stationary (feature-based) and non-stationary (history-dependent, sequential) patterns; for example, rules requiring alternation between buckets or priority-based constraints (e.g., "clockwise" or "one-free" placement).

The environment enables fine-grained modification: each component (rule atom, allowed set, move count, sequencing) can be varied independently, supporting controlled experiment design and enabling systematic attribution of observed task difficulty to rule features. Difficulty is quantified using cumulative error curves and summary statistics such as Terminal Cumulated Error (TCE). The evaluation infrastructure supports a public leader-board for cross-comparison of human and artificial learners (Pulick et al., 2022).

Experiments comparing reinforcement learning (RL, DQN/REINFORCE) to human performance (Pulick et al., 2023) show that increased rule generality (i.e., more permissible actions) facilitates RL learners exploiting the greater policy space, but can impede human deduction—humans find high-generality rules more ambiguous and harder to infer, as reflected in later emergence of correct action "streaks." This divergence highlights fundamental differences in inductive learning versus deductive reasoning.

4. Algorithms and Agent Strategies in GOHR

Practical success in GOHR depends on joint inference and control: the agent must simultaneously hypothesize about the governing rule (from partial observations) and optimize its action policy. State representation critically affects learning efficiency. Two principal schemes are studied (Mathew et al., 7 Sep 2025):

• Feature-Centric (FC): The board is encoded as a stack of spatial feature maps (channels), each indicating the presence/absence of shapes and colors at each position; feature associations are learned implicitly via alignment.
• Object-Centric (OC): Each piece/object is encoded as a vector of attributes, giving explicit access to shape, color, and position per object; suitable for rules dependent on object features rather than positional relationships.

Both representations are processed with Transformer-based A2C agents, with FC excelling in spatially explicit or position-dependent rules, and OC exhibiting greater stability in learning and generalization across rule variations.

Training leverages partial observability: at each timestep the agent receives the current state, history of successful actions, and masked action spaces reflecting legal/illegal moves—with reward and status code feedback. Experimental setups probe both isolated rule learning and transfer effects, demonstrating that pre-training on component rules accelerates compound rule acquisition.

5. Rule Induction and Equilibria in Social Deduction and Imperfect Information Games

GOHR extends beyond single-agent learning benchmarks to multi-agent game-theoretic settings, especially in social deduction games (e.g., Mafia, The Resistance/Avalon) (Carminati et al., 2023). In hidden-role games, players are assigned to teams (with some roles concealed), with the additional challenge of communication protocol design (public, private). A robust solution concept is the team max-min equilibrium (TME) in the split-personality form: the uninformed team commits to a joint strategy to maximize their minimum guaranteed utility against an adaptive adversary.

Efficient computation of equilibrium is feasible under assumptions such as constant number of adversaries, private communication, and mediation—the problem reduces to solving (via LP or counterfactual regret minimization) a two-player zero-sum game over an exponentially reduced game tree. If private communication is disallowed or the adversary is uncoordinated, computing optimal equilibrium becomes NP- or coNP-hard even for small player counts, which underscores the computational complexity induced by hidden-rule structure.

Further, learning in imperfect information games is enhanced by progressive hiding algorithms (Heymann et al., 5 Sep 2024). By gradually transitioning from complete information (where all hidden variables are visible) to true play (with increasingly restrictive information partitions), agents may first master mechanics under full observability before adapting to uncertainty. This method leverages a proximal penalty to enforce information constraints while allowing efficient optimization with regret minimization (CFR), even in non–perfect recall settings.

6. Limitations in Rule Induction within High-Dimensional Generative Models

The challenge of learning hidden inter–feature rules is not restricted to classic or board-like environments but extends to deep generative models. Diffusion models (DMs), while effective at capturing coarse statistical properties, are empirically and theoretically limited in learning fine–grained or mathematically exact inter–feature relationships (e.g., strict geometric or proportional dependencies between image attributes) (Han et al., 7 Feb 2025). Under the denoising score matching (DSM) objective, DMs exhibit a non-vanishing lower bound for error in rule conformity.

Guided diffusion—a post-hoc coupling with classifiers trained to distinguish rule-conforming samples—offers limited improvement but is inherently constrained by classifier weaknesses in discriminating fine-grained signals within a high-dimensional, largely unlabelled sample space. This illustrates that the challenge of hidden rule discovery is fundamental, spanning symbolic, logical, and deep statistical learning frameworks.

7. Implications, Experimental Methodologies, and Open Questions

The GOHR paradigm is central to metrological approaches in artificial intelligence, serving as a benchmark on which the inference capabilities of learning agents can be rigorously compared, analyzed, and quantified (Mathew et al., 7 Sep 2025). Through modular, controllable environments with transparent rule specification and public leader-boards (Pulick et al., 2022), GOHR enables the isolation of learning bottlenecks, the paper of transfer, and the controlled assessment of how specific aspects of rule complexity (generality, sequentiality, feature dependency) affect both human and machine learning.

A common implication is that information hiding introduces intrinsic computational and epistemic barriers, often resulting in exponential overhead for reconstruction/inference (Bank et al., 2015), and that equilibrium concepts must be substantially generalized beyond classical Nash assumptions (Gopalkrishnan et al., 2014, Carminati et al., 2023). Practical advances in agent architecture, training regimes (progressive hiding), and hybrid learning paradigms are necessary to approach human-level proficiency in such environments.

A plausible implication is that future work on GOHR should prioritize algorithms able to self-discover, generalize, and reliably infer hidden structure amid partial feedback, leveraging cross-disciplinary insights from logic, optimization, reinforcement learning, and statistical theory. The open question of scalable, general-purpose rule induction in the face of sophisticated information hiding remains a central challenge in both artificial intelligence and strategic decision-making.

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Table: GOHR Modeling Constructs and Solution Approaches

Construct	Formal Principle	Key Reference
Kripke Semantics	Epistemic games, min-games	(Gopalkrishnan et al., 2014)
Quiz Game Model	Information hiding, lower bounds	(Bank et al., 2015)
Expressive Rule Language	Controlled benchmark tasks	(Pulick et al., 2022)
Split-Personality Equilibria	Team max-min equilibrium	(Carminati et al., 2023)
Progressive Hiding	Gradual information relaxation	(Heymann et al., 5 Sep 2024)
Feature- vs Object-Centric Encoding	Representation learning	(Mathew et al., 7 Sep 2025)
Classifier-Guided Diffusion	Rule-guided generative modeling	(Han et al., 7 Feb 2025)

All claims, empirical results, and frameworks are drawn verbatim or as direct logical consequences from the primary cited works.