Coverage Games: Models & Applications

• Coverage games are a family of models that use game theory and control to maximize the coverage of objectives, resources, or regions in multi-agent systems.
• They encompass multi-objective, resource-selection, spatial, and testing scenarios by evaluating strategic interactions and equilibrium outcomes.
• Key insights include equilibrium robustness, communication constraints, and adaptation to adversarial conditions across domains like robotics, network control, and software testing.

Coverage games are a family of game-theoretic and control-theoretic models in which “coverage” is the central objective, but the term is used in several technically distinct senses. In one line of work, coverage means ensuring that every objective is satisfied by at least one agent in a multi-agent game graph (Kupferman et al., 20 Mar 2026). In another, it means maximizing the total value of covered resources under strategic interaction, as in single-selection coverage games and related welfare-sharing formulations (Seaton et al., 2021, Gopalakrishnan et al., 2014). In robotics and networked control, coverage games model teams of agents that must cover regions, users, or targets while accounting for communication, adversaries, or failures (Fernando et al., 2021, Bhargav et al., 18 Mar 2026, Zheng et al., 8 Jun 2026, Song et al., 2019). In software and game testing, coverage games formalize the maximization of node, code, branch, or behavioral coverage under nondeterminism or reinforcement learning (Wang et al., 2013, Mu et al., 14 Dec 2025). These uses share a common structure: strategic or algorithmic decisions are evaluated by the extent to which a space of targets, resources, objectives, code paths, or regions is covered.

1. Definitions and major interpretations

The term “coverage game” has at least four recurring meanings in the literature. A useful organizing principle is the object being covered and the source of strategic interaction.

Family	Coverage object	Representative papers
Multi-objective games	Objectives across multiple plays	(Kupferman et al., 20 Mar 2026)
Resource-selection games	Valuable resources / welfare	(Seaton et al., 2021, Gopalakrishnan et al., 2014)
Spatial / robotic coverage	Regions, targets, users, or risk fields	(Fernando et al., 2021, Bhargav et al., 18 Mar 2026, Zheng et al., 8 Jun 2026, Song et al., 2019)
Testing and software coverage	Nodes, lines, branches, behavior space	(Wang et al., 2013, Mu et al., 14 Dec 2025, Bernhard et al., 2020)

In the formal framework titled “Coverage Games,” the game is played between a coverer and a disruptor. The coverer operates several agents, the disruptor controls the environment, and the coverer wins if every objective is satisfied by at least one agent; otherwise, the disruptor wins (Kupferman et al., 20 Mar 2026). This explicitly generalizes traditional two-player games with multiple objectives by allowing a decomposition of objectives among different agents.

In single-selection coverage games, a group of agents choose resources, and social welfare is the total value of the covered resources, counted once per resource (Seaton et al., 2021). In welfare-sharing formulations, coverage games arise as a subclass of separable games in which local welfare is attached to resources and agents select subsets of those resources (Gopalakrishnan et al., 2014). A plausible implication is that “coverage” here is best understood as a monotone submodular welfare phenomenon rather than as purely geometric coverage.

In robotics and swarms, coverage games usually mean positioning or action-selection problems in which sensing, communication, or mobility decisions determine coverage quality. “CoCo Games” formulates communication-aware coverage as a cooperative graphical game over UAV actions and routing-induced neighborhoods (Fernando et al., 2021). “Distributed Equilibrium-Seeking in Target Coverage Games via Self-Configurable Networks under Limited Communication” studies a zero-sum game between a sensing team and an attacker over target configurations (Bhargav et al., 18 Mar 2026). “Game-Theoretic Area Coverage Control with Cooperative-Adversarial Multi-Agent Systems” casts area coverage as a two-player zero-sum game in which adversarial agents generate the spatial risk field (Zheng et al., 8 Jun 2026).

In testing, coverage games treat coverage itself as the payoff. “Coverage Games for Testing Nondeterministic Systems” models testing as a two-player turn-based game on a finite graph where the tester maximizes node coverage and the system under test minimizes it (Wang et al., 2013). “Coverage-Aware Game Playtesting with LLM-Guided Reinforcement Learning” makes code and branch coverage of modified game code an explicit component of the reward for an RL playtester (Mu et al., 14 Dec 2025).

2. Formal models and objective functions

A central formalization is the multi-agent coverage game $\mathcal{G} = \langle G, k, \beta \rangle$, where $G$ is a two-player game graph, $k$ is the number of agents controlled by the coverer, and $\beta = \{\alpha_1,\ldots,\alpha_m\}$ is a set of Büchi or co-Büchi objectives (Kupferman et al., 20 Mar 2026). A profile covers $\beta$ iff every objective is satisfied in at least one play. This differs sharply from standard All-$\gamma$ objectives, where a single play must satisfy all objectives simultaneously.

Single-selection coverage games are defined by agents $N = \{1,\dots,n\}$, a finite resource set $\mathcal{R}$, admissible action sets $A_i \subseteq \mathcal{R}$, and a value $v_r \in [0,1]$ for each resource $G$0. Each agent chooses exactly one resource, and the system-level objective is

$G$1

where $G$2 is the set of resources selected by at least one agent (Seaton et al., 2021). The nominal utility design is the marginal contribution utility

$G$3

while compromised agents use $G$4, reflecting communication failure (Seaton et al., 2021).

In welfare-sharing coverage games, total welfare is separable across resources: $G$5 and player $G$6's utility is

$G$7

Coverage games appear as the case where a resource is valuable once it is selected by at least one player (Gopalakrishnan et al., 2014). This formulation places coverage games alongside facility location, routing, and network formation within a unified local-welfare framework.

Graphical and robotic variants replace set coverage by spatial or communication-aware payoff functions. In “CoCo Games,” the payoff of robot $G$8 is

$G$9

combining cooperative RSS coverage of a probabilistic region of interest with connectivity to neighbors (Fernando et al., 2021). In adversarial target coverage, the defender’s expected payoff is $k$0, where $k$1 is a mixed strategy over joint sensing actions and $k$2 is a mixed strategy over target deployments (Bhargav et al., 18 Mar 2026). In cooperative-adversarial area coverage, the payoff is the coverage functional

$k$3

where adversarial agents induce the risk density $k$4 (Zheng et al., 8 Jun 2026).

Testing formulations are similarly explicit. In node coverage games, a play $k$5 induces coverage payoff

$k$6

often with $k$7, so payoff is the number of distinct visited nodes (Wang et al., 2013). In coverage-aware game playtesting, structural anchors are modified lines $k$8 and modified branches $k$9, with $\beta = \{\alpha_1,\ldots,\alpha_m\}$0, and the RL reward combines semantic subgoal completion with one-time rewards for newly covered anchors (Mu et al., 14 Dec 2025).

3. Equilibrium structure, efficiency, and robustness

Coverage games support several equilibrium notions, depending on the formulation. In multi-agent coverage games on graphs, determinacy fails in general: for Büchi and co-Büchi objectives, coverage games need not be determined, and undeterminacy already appears with two agents and three objectives (Kupferman et al., 20 Mar 2026). However, special cases are determined, including $\beta = \{\alpha_1,\ldots,\alpha_m\}$1, $\beta = \{\alpha_1,\ldots,\alpha_m\}$2, and one-player arenas (Kupferman et al., 20 Mar 2026).

In single-selection coverage games without communication failures, marginal contribution utilities yield a valid-utility game over a submodular objective, so every pure Nash equilibrium attains at least half the optimal welfare and an optimal equilibrium exists (Seaton et al., 2021). With compromised agents, prior work gave the bound $\beta = \{\alpha_1,\ldots,\alpha_m\}$3, but the paper’s main result refines this via a distance $\beta = \{\alpha_1,\ldots,\alpha_m\}$4 to games with $\beta = \{\alpha_1,\ldots,\alpha_m\}$5: $\beta = \{\alpha_1,\ldots,\alpha_m\}$6 The point is not merely that failures worsen efficiency, but that the worst-case degradation is structurally fragile (Seaton et al., 2021).

Welfare-sharing coverage games admit a stronger characterization. For arbitrary local welfare functions, all games in the class possess a pure Nash equilibrium if and only if each local distribution rule is equivalent to a generalized weighted Shapley value on some ground welfare function (Gopalakrishnan et al., 2014). If budget balance is also required, the ground welfare must equal the original welfare (Gopalakrishnan et al., 2014). The paper’s explicit conclusion is that potential games are necessary to guarantee pure Nash equilibria in this class, which includes coverage games (Gopalakrishnan et al., 2014).

Potential structure also appears in spatial coverage games. In Edge-weighted Budgeted Maximum Coverage games, the locally altruistic variants are exact potential games with the social coverage objective as potential, so the centralized optimum is a pure Nash equilibrium (Lee et al., 2024). By contrast, the selfish variants may fail to possess a pure Nash equilibrium, although sufficient conditions are given for existence in special cases (Lee et al., 2024). This suggests that small changes in how local coverage benefits are counted can destroy equilibrium structure.

For communication-aware swarm coverage, “CoCo Games” proves that the mean-field variational inference fixed point is a correlated equilibrium, and because the distribution factorizes, specifically a mixed-strategy Nash equilibrium of the stage game (Fernando et al., 2021). In the target-coverage zero-sum setting, the empirical strategies of the defender and attacker converge to an $\beta = \{\alpha_1,\ldots,\alpha_m\}$7-Nash equilibrium, with $\beta = \{\alpha_1,\ldots,\alpha_m\}$8 controlled by submodular curvature, Value of Coordination, and a $\beta = \{\alpha_1,\ldots,\alpha_m\}$9 term (Bhargav et al., 18 Mar 2026).

4. Distributed control, communication, and adversarial spatial coverage

Communication constraints and local interaction graphs are now central in several coverage-game formulations. In “CoCo Games,” robot neighborhoods are defined by hop distance in the routing graph,

$\beta$0

and payoffs depend only on these local neighborhoods, producing a graphical game with scalable local dependencies (Fernando et al., 2021). The associated Markov random field and mean-field variational updates enable real-time stage-game optimization, and the reported optimization time is less than $\beta$1 ms per stage for realistic neighborhood sizes and ROI discretizations (Fernando et al., 2021).

“Distributed Equilibrium-Seeking in Target Coverage Games via Self-Configurable Networks under Limited Communication” makes the communication graph itself adaptive. Each agent chooses both a sensing action and a communication neighborhood under a bandwidth constraint $\beta$2 (Bhargav et al., 18 Mar 2026). The Value of Coordination quantifies how much overlap reduction and coordination benefit a neighborhood induces, and maximizing it through distributed bandit-submodular optimization tightens the defender’s regret and the approximate-Nash bound (Bhargav et al., 18 Mar 2026). A plausible implication is that in communication-constrained coverage games, topology design is part of the equilibrium problem, not merely an implementation detail.

In cooperative-adversarial area coverage, the payoff is the coverage metric $\beta$3, ordinary agents perform gradient descent, and adversarial agents perform gradient ascent (Zheng et al., 8 Jun 2026). The resulting dynamics produce a generalized centroidal Voronoi tessellation for the ordinary agents at equilibrium, while adversarial agents converge to their equilibrium centroids when the equilibrium is stable (Zheng et al., 8 Jun 2026). Analysis of the low-dimensional case identifies a Hopf bifurcation governed by the gain ratio $\beta$4: an adversary-dominated regime yields periodic chase-evasion cycles, whereas an ordinary-agent-dominated regime converges to a fixed configuration (Zheng et al., 8 Jun 2026).

Resilience under failures is treated in CARE, which uses distributed discrete event supervisors to trigger event-driven games when robots fail or become idle (Song et al., 2019). The local game is again a potential game, with utilities designed as marginal contributions to expected task worth. The task worth is adjusted by remaining expected targets and by already assigned robots’ success probabilities, yielding a potential function aligned with overall team performance (Song et al., 2019). This is a distinct but related interpretation of coverage games: not equilibrium over static allocations, but repeated local games embedded in a supervisory control architecture.

Wireless-network coverage adds another variation. In small-cell networks, a repeated coverage game is defined over downlink transmit powers, with utility

$\beta$5

so “coverage” is the integrated rate delivered to users under interference (Treust et al., 2010). The one-shot game is dominance-solvable with the unique Nash equilibrium at full power, but repeated-game strategies can enforce Pareto-optimal outcomes such as the Kalai–Smorodinsky bargaining solution under a 2-connected observation graph (Treust et al., 2010).

5. Testing, software verification, and game playtesting

Testing-oriented coverage games treat nondeterminism or incomplete knowledge as the adversary. In node coverage games for testing nondeterministic systems, the maximal coverage guarantee is

$\beta$6

the largest number of nodes the tester can force from an initial node $\beta$7, regardless of SUT responses (Wang et al., 2013). The decision problem “is $\beta$8?” is NP-complete for both constant and general gain functions, and the restart variant remains NP-complete (Wang et al., 2013). The paper also develops repetitive execution strategies for deterministic test suites applied to nondeterministic systems, showing that game-inspired prioritization improves coverage on industrial benchmarks (Wang et al., 2013).

Offline congestion games approach coverage from a data-identifiability viewpoint. The problem is not covering resources in the environment but covering the data required to recover an approximate Nash equilibrium from logged interaction (Jiang et al., 2022). With facility-level feedback, the relevant assumption is one-unit deviation coverage; with agent-level or game-level feedback, this is insufficient, and increasingly strong covariance-domination assumptions are required (Jiang et al., 2022). This use of “coverage” is statistical rather than operational, but it is structurally parallel: equilibrium recovery depends on whether strategically relevant deviations are sufficiently represented in data.

Robust Stochastic Bayesian Games for behavior space coverage treat the continuous space of other agents’ possible behaviors as the object to be covered (Bernhard et al., 2020). The behavior space $\beta$9 is partitioned into finitely many regions, each defining a hypothesis type, and planning is robust within types: $\gamma$0 The stated motivation is to ensure coverage over all physically feasible behavioral variations while reducing sample complexity relative to standard stochastic Bayesian games (Bernhard et al., 2020).

In game QA, SMART treats coverage explicitly as part of the gameplay objective. Modified lines $\gamma$1 and modified branches $\gamma$2 define structural anchors $\gamma$3, LLMs translate AST diffs into subgoals $\gamma$4 and executable reward functions $\gamma$5, and the RL reward combines semantic and structural terms (Mu et al., 14 Dec 2025). On Overcooked and Minecraft, the reported result is over $\gamma$6 branch coverage of modified code with a $\gamma$7 task completion rate (Mu et al., 14 Dec 2025). This is a literal coverage game: the agent must simultaneously realize gameplay intent and maximize coverage of changed code paths.

Finally, in offline grid-based coverage path planning for guards in games, the problem is exhaustive level exploration in a known 2D polygonal map with holes, using a grid-based world representation (Enezi et al., 2020). The abstract explicitly states that current heuristics often produce poor performance or unnatural behavior and that the proposed offline algorithm is intended as a step toward more efficient coverage path planning for non-player characters (Enezi et al., 2020). Because the detailed methodology was unavailable in the supplied data, only this high-level characterization is warranted.

6. Applications, tensions, and open directions

Coverage games appear in surveillance, patrolling, swarms, network control, software testing, and game QA. The application logic varies, but the recurring technical tension is between complete or high-value coverage and the cost of decentralization, failures, or strategic opposition.

In multi-agent graph coverage games, a priori decomposition of objectives among agents is often impossible even when coverage is achievable; dynamic decomposition through “forks” is essential (Kupferman et al., 20 Mar 2026). In resource-selection games, efficiency depends sensitively on utility design and information structure, and worst-case inefficiency under failures may be fragile rather than representative (Seaton et al., 2021). In robotic coverage, communication structure, limited bandwidth, and adversarial motion alter both the attainable equilibrium and the algorithmic route to it (Fernando et al., 2021, Bhargav et al., 18 Mar 2026, Zheng et al., 8 Jun 2026). In testing, the main issue is whether the tester can force or statistically recover sufficient coverage under nondeterminism or offline data limitations (Wang et al., 2013, Jiang et al., 2022).

Several open directions recur across the literature. Extending coverage games beyond Büchi and co-Büchi objectives to richer $\gamma$8-regular or quantitative objectives is explicitly identified as open in the formal multi-agent setting (Kupferman et al., 20 Mar 2026). Extending distance-based fragility results from single-selection coverage games to more general submodular games is also open (Seaton et al., 2021). Communication-aware robotic work points to partial observability, concurrent dynamics, richer adversary models, and automatic network adaptation as natural extensions (Bhargav et al., 18 Mar 2026, Zheng et al., 8 Jun 2026). Testing-oriented work suggests combining reinforcement learning with symbolic or fuzzing-style exploration to reach defensive or low-probability code paths that semantic reward shaping alone may miss (Mu et al., 14 Dec 2025).

Taken together, the literature suggests that “coverage games” are not a single model but a technically coherent family. The common thread is strategic allocation or motion under a coverage-based objective, coupled with adversarial uncertainty, communication constraints, or decentralized decision-making. The differences lie in what is being covered, who controls the dynamics, and whether equilibrium, control, or verification is the primary analytical lens.