Hierarchical Hypergames: A Recursive Perspective

• Hierarchical hypergames are recursively organized perceptual games that model agents' misaligned beliefs and nested strategic reasoning.
• They provide a formal framework for analyzing dynamic multi-agent interactions, deception, and cyber-physical security through recursive belief hierarchies.
• Recent computational and graph-based models illustrate practical applications in cybersecurity and agent simulation.

Hierarchical hypergames are extensions of hypergame theory in which strategic interaction is modeled not by a single commonly known game, but by recursively organized perceptual games that encode how agents understand the conflict and how they attribute understanding to others. In this literature, the basic departure from classical game theory is that agents may disagree about the players, the available actions, the relevant outcomes, or the preference orderings, and hierarchy enters when those disagreements themselves become objects of strategic reasoning through beliefs about beliefs and higher-order expectations (Trencsenyi et al., 25 Jul 2025).

1. Formal basis in perceptual games

The contemporary hypergame literature defines a hypergame as a structured collection of agent-relative games. A baseline formalization is

$H=(N,\{G_i\}_{i\in N}),$

where $N=\{1,2,\ldots,n\}$ is the set of agents and each $G_i$ is agent $i$'s perceived game. Each perceived game is written as

$G_i=(N_i,A_i,R_i),$

with $N_i\subseteq N$ the set of agents as perceived by $i$,

$A_i=\bigtimes_{j\in N_i}A_{ij},$

$A_{ij}$ agent $i$'s perception of player $N=\{1,2,\ldots,n\}$0's available actions, and $N=\{1,2,\ldots,n\}$1 the preference relations as perceived by $N=\{1,2,\ldots,n\}$2. For outcomes $N=\{1,2,\ldots,n\}$3, the notation $N=\{1,2,\ldots,n\}$4 indicates that $N=\{1,2,\ldots,n\}$5 prefers outcome $N=\{1,2,\ldots,n\}$6 to outcome $N=\{1,2,\ldots,n\}$7 (Trencsenyi et al., 25 Jul 2025).

This formal move replaces the classical assumption of one objective game under common knowledge with a family of subjective strategic objects. The review literature emphasizes that this is motivated by multi-agent systems characterized by uncertainty, misaligned perceptions, and nested beliefs rather than by rationality, complete information, and common knowledge of payoffs. Hierarchical hypergames arise when this subjective structure is itself nested, so that an agent models not only its own perceptual game but also another agent’s perceptual game, and possibly that agent’s beliefs about further agents (Trencsenyi et al., 25 Jul 2025).

Hierarchical hypergames are one of the two major extensions highlighted in the recent systematic review of hypergame theory. That review analyzes 44 selected studies across cybersecurity, robotics, social simulation, communications, and general game-theoretic modeling, and reports the prevalence of hierarchical and graph-based models in deceptive reasoning together with a tendency for practical applications to simplify more extensive theoretical frameworks (Trencsenyi et al., 25 Jul 2025). This suggests that hierarchical hypergames function both as a formal theory of misaligned perception and as a modeling template for applied strategic systems.

2. Recursive hierarchy and levels of perception

A standard recursive presentation introduces explicit levels of perception. In graph-based cyber-defense work, a level-1 hypergame is

$N=\{1,2,\ldots,n\}$8

where each player has its own perceptual game but neither is aware of the other’s perception. A level-2 hypergame is

$N=\{1,2,\ldots,n\}$9

meaning that player 1 perceives the interaction as a level-1 hypergame while player 2 still perceives only $G_i$0. A general schema is also stated: $G_i$1 In that formulation, hierarchy is recursive depth of perception rather than organizational rank or move order (Kulkarni et al., 2020).

A second influential notation, attributed in later work to Wang et al. (1988), makes the indexing of nested perspectives explicit: $G_i$2 with

$G_i$3

Here $G_i$4 is described as player $G_i$5's perceptual game defining player $G_i$6's belief of player $G_i$7's belief of player $G_i$8's perspective of the base game. Recent work on beauty contest games connects this indexing to belief hierarchies through the proposed identification

$G_i$9

while also stating that a rigorous proof of the relationship is beyond the scope of that work (Trencsenyi et al., 11 Feb 2025).

The same paper introduces a perspective-sequence notation

$i$0

where $i$1 is the interpreter or creator of the perceptual game and subsequent $i$2's encode nested perspectives. A player-specific perceptual game is then written as

$i$3

Taken together, these notations show that hierarchical hypergames are not merely “games with asymmetric information.” They are recursively indexed representations of ordered perspectives, in which the object of strategic modeling is a hierarchy of perceived games rather than a single strategic form (Trencsenyi et al., 11 Feb 2025).

3. Rationality and solution concepts

Because players reason inside different perceptual games, solution concepts in hierarchical hypergames are typically defined relative to subjective models rather than a common ground-truth game. One recent computational framework defines a subjective game as

$i$4

and a subjective best response condition as

$i$5

On this basis it defines strong hypergame Nash equilibrium (s-HNE) by

$i$6

and weak hypergame Nash equilibrium (w-HNE) by

$i$7

The same framework explicitly supports $i$8, $i$9, and $G_i=(N_i,A_i,R_i),$0, and states that higher-level hypergames capture recursive reasoning and are specified by the highest order of expectation involved (Trencsenyi, 12 Dec 2025).

Dynamic hypergame work adopts a different vocabulary. In temporal-logic planning with incomplete information, the static hierarchy is

$G_i=(N_i,A_i,R_i),$1

where $G_i=(N_i,A_i,R_i),$2 is the true game induced by player 1’s objective and $G_i=(N_i,A_i,R_i),$3 is player 2’s perceived game under hypothesis $G_i=(N_i,A_i,R_i),$4. The solution concept is subjective rationalizability (SR), refined dynamically into behaviorally subjectively rationalizable (BSR) strategies as the adversary updates its hypothesis from observed play. In this setting, player 2 best-responds in the game she believes she is playing, while player 1 best-responds in a perceptual game that includes both the true task and player 2’s misperception (Li et al., 2020).

The recent review identifies a broader methodological fact: practical applications often simplify extensive theoretical frameworks, there is limited adoption of HNF-based models, and formal hypergame languages are lacking (Trencsenyi et al., 25 Jul 2025). This suggests that hierarchical hypergames do not yet have a single settled equilibrium calculus. Instead, the field contains several solution families—subjective best response, strong and weak hypergame Nash equilibrium, subjectively rationalizable strategy—each tied to a particular representation of nested perception.

4. Dynamic and graph-based operationalizations

A major applied strand embeds hierarchical hypergames in dynamic state-transition systems. In cyber defense on graphs, the underlying attack-defend game is written as

$G_i=(N_i,A_i,R_i),$5

with finite state space $G_i=(N_i,A_i,R_i),$6, deterministic transition map

$G_i=(N_i,A_i,R_i),$7

and defender objective $G_i=(N_i,A_i,R_i),$8 expressed in LTL, often restricted to scLTL. Hierarchy enters through subjective arenas

$G_i=(N_i,A_i,R_i),$9

where the defender knows the true labeling and the attacker acts on a misperceived labeling induced by deception. The resulting model is explicitly a level-2 hypergame,

$N_i\subseteq N$0

which the paper characterizes as “a Hypergame on a Graph with One-sided Misperception of Labeling Function” (Kulkarni et al., 2020).

A closely related dynamic formulation treats deception as control of the opponent’s evolving perception. The physical interaction is a two-player concurrent stochastic game

$N_i\subseteq N$1

and the adversary’s evolving hypothesis is updated by

$N_i\subseteq N$2

The corresponding dynamic hypergame augments state with the physical state, history, automaton state for an scLTL task, and current adversary hypothesis: $N_i\subseteq N$3 In this construction, one action simultaneously changes the world state, the task-progress automaton state, and the game the adversary thinks is being played. The paper’s central claim is that this makes deception operational: strategic planning proceeds by shaping both the physical trajectory and the opponent’s inference dynamics (Li et al., 2020).

Optimization-based cyber-physical security provides another operationalization. There the defender minimizes

$N_i\subseteq N$4

while the attacker perturbs perceived parameters such as

$N_i\subseteq N$5

The paper explicitly states that the basic attack model is a second-level hypergame: the defender is unaware of deception, while the attacker knows the defender is solving a distorted optimization problem. It also studies defender awareness and “double-bluff” variants, which instantiate a deeper belief hierarchy in optimization form (Bakker et al., 2018).

These formulations explain why the review finds hierarchical and graph-based hypergames especially prevalent in deceptive reasoning (Trencsenyi et al., 25 Jul 2025). They are well suited to settings where one player shapes another’s perception of states, actions, labels, or objectives while preserving a dynamic model of interaction.

5. Agent-compatible implementations and computational formalisms

Recent work has moved hierarchical hypergames toward computationally explicit, agent-compatible architectures. In LLM-based strategic simulation, a two-player beauty contest is represented as

$N_i\subseteq N$6

with $N_i\subseteq N$7, $N_i\subseteq N$8, $N_i\subseteq N$9, and

$i$0

An interpretation function

$i$1

constructs player-specific perceptual games; a reasoning function

$i$2

maps natural-language game descriptions to a reasoning trace and an expected opponent move; and a reasoning-analysis function

$i$3

estimates reasoning depth from the number of nested beliefs present in the trace. The paper then assembles an aggregate hierarchical hypergame $i$4 from the set of individual perceptual games, with $i$5 indexed by the maximal reasoning depth present (Trencsenyi et al., 11 Feb 2025).

The representational significance of this work is that hierarchy is encoded simultaneously by nested indices such as $i$6, by hypergame levels $i$7, and by the ordered perspective sequence $i$8. The paper emphasizes that this use of hypergames is both representational and operational: hypergames encode nested beliefs and asymmetric perceptions, and the system also constructs those perceptual games from LLM reasoning as part of the decision pipeline (Trencsenyi et al., 11 Feb 2025).

A different computational advance addresses a gap identified by the systematic review, namely the lack of formal hypergame languages (Trencsenyi et al., 25 Jul 2025). “Hypergame Rationalisability” introduces a declarative, logic-based domain-specific language and an answer-set-programming pipeline. It starts from a base game

$i$9

uses an interpretation function

$A_i=\bigtimes_{j\in N_i}A_{ij},$0

to generate subjective games, and formalizes an umpire as a rationaliser

$A_i=\bigtimes_{j\in N_i}A_{ij},$1

where $A_i=\bigtimes_{j\in N_i}A_{ij},$2 is a set of hypergame structures under which the observed outcome $A_i=\bigtimes_{j\in N_i}A_{ij},$3 is rationalizable. The framework states explicitly that it supports hypergames up to level 2, not an arbitrary recursive hierarchy, and it uses the s-HNE and w-HNE criteria to filter candidate subjective-game structures (Trencsenyi, 12 Dec 2025).

This development is best read as a partial computational answer to the review’s concern about absent formal languages. It does not provide a full general theory of arbitrary-depth hierarchical hypergames, but it does turn level-1 and level-2 hypergame construction into a declarative search problem (Trencsenyi, 12 Dec 2025).

6. Boundaries, adjacent literatures, and open problems

The term hypergame is not unique to the literature on misaligned perceptions. In coalgebraic game theory, “hypergames” denote non-wellfounded Conway-style games, defined as the final coalgebra of

$A_i=\bigtimes_{j\in N_i}A_{ij},$4

with infinite play treated as a draw and strategy theory centered on non-losing strategies. That usage concerns infinite or cyclic game trees rather than divergent subjective perceptions, and it is conceptually distinct from hierarchical hypergames in the perceptual-game sense (Honsell et al., 2011).

A second boundary concerns the many works on hierarchical games that are not hypergames. Hierarchical simple games study ranked player classes and threshold rules for winning coalitions, not mismatched perceptions [(Gvozdeva et al., 2011); (Hameed et al., 2012)]. Structured hierarchical games and Differential Backward Induction concern tree-structured move order and payoff dependence, again without subjective games (Li et al., 2021). Hierarchical pursuit-evasion decompositions, dynamic hierarchical reactive synthesis, and Stackelberg intervention methods all develop multilevel strategic architectures, but their hierarchies are objective, shared, and non-epistemic (Guan et al., 2022, Schmuck et al., 2015, Grontas et al., 2023). The hierarchical public-goods game likewise exhibits cross-level incentive conflict without perceptual divergence (Fujimoto et al., 2016). These are adjacent literatures, not instances of hierarchical hypergame theory.

The systematic review isolates several unresolved issues inside hypergame research proper. It identifies the limited adoption of HNF-based models, the lack of formal hypergame languages, and unexplored opportunities for modeling human-agent and agent-agent misalignment (Trencsenyi et al., 25 Jul 2025). Recent computational papers partially address the language problem and the connection to recursive reasoning, but they typically stop at shallow hierarchies or application-specific semantics (Trencsenyi, 12 Dec 2025, Trencsenyi et al., 11 Feb 2025). This suggests that the central open problem is not only richer recursion, but also the construction of scalable representational standards and solution procedures for nested subjective games in dynamic multi-agent environments.