Game-theoretic Approach for Non-Cooperative Planning

When two or more self-interested agents put their plans to execution in the same environment, conflicts may arise as a consequence, for instance, of a common utilization of resources. In this case, an agent can postpone the execution of a particular action, if this punctually solves the conflict, or it can resort to execute a different plan if the agent’s payoff significantly diminishes due to the action deferral. In this paper, we present a game-theoretic approach to non-cooperative planning that helps predict before execution what plan schedules agents will adopt so that the set of strategies of all agents constitute a Nash equilibrium. We perform some experiments and discuss the solutions obtained with our game-theoretical approach, analyzing how the conflicts between the plans determine the strategic behavior of the agents.

💡 Research Summary

The paper addresses the problem of non‑cooperative multi‑agent planning (MAP) in a shared environment where agents are self‑interested and may experience conflicts over shared resources or mutually exclusive actions. To predict which plan schedules agents will adopt before execution, the authors propose a two‑level game‑theoretic framework.

The first level, called the “general game,” is a normal‑form game in which each agent i chooses one plan π_i from its finite set Π_i. A plan’s intrinsic benefit β_i(π_i) depends on factors such as the number of goals achieved, makespan, and action costs. Because agents execute their chosen plans simultaneously, conflicts (mutex relations) can arise, potentially reducing the realized benefit.

To resolve these conflicts, the second level, the “joint plan schedule game” (internal game), takes a concrete plan profile p = (π_1,…,π_n) as input and searches for a feasible joint schedule s = (σ_1,…,σ_n) where each σ_i is a time‑ordered schedule of the actions of π_i. The internal game is modeled as a perfect‑information extensive‑form game. At each discrete time step t, every agent either executes the next action of its plan or plays the empty action ⊥, which represents a deliberate delay to avoid a mutex conflict. The empty action is allowed only when at least one other agent performs a non‑empty action at the same time, or when the agent has already finished its plan.

The set of all possible schedules for a plan π is denoted Ψ_π; the earliest schedule ψ⁰ finishes at time |π|‑1 and yields utility μ_i(ψ⁰)=β_i(π). Any delayed schedule ψ incurs a utility loss that can be agent‑specific, reflecting different sensitivities to time. The payoff of a terminal node (i.e., a complete, conflict‑free schedule profile) is μ_i(σ_i) for each agent i. The internal game thus returns a Nash equilibrium schedule: no agent can improve its utility by unilaterally changing its own schedule while the others keep theirs.

The payoff function of the general game is defined as ρ_i(p)=μ_i(σ_i), where σ_i is the equilibrium schedule obtained from the internal game for the plan profile p. Consequently, the overall solution of the framework is a pure‑strategy Nash equilibrium of the general game, guaranteeing that the selected plans together with their equilibrium schedules are stable: no single agent has an incentive to deviate either by picking a different plan or by altering its execution timing.

The authors implement the general game using the Gambit software library, which computes Nash equilibria for normal‑form games. The internal game is solved by exhaustive search over the extensive‑form tree, pruning schedules that violate action preconditions or produce mutex conflicts. Experiments are conducted on small synthetic instances (2–4 agents, each with 2–3 short plans of length ≤5). Results show that when conflicts are severe, agents often choose to delay or switch to alternative plans, leading to higher overall utilities compared with naïve simultaneous execution. The framework successfully identifies equilibria that balance individual optimality with conflict avoidance.

Key contributions include: (1) introduction of soft goals to handle situations where achieving all hard goals simultaneously is infeasible; (2) explicit modeling of action conflicts and a penalty mechanism for schedule delays; (3) a concrete two‑level game architecture that can be solved with existing game‑theoretic tools.

The paper acknowledges the exponential blow‑up inherent in both the normal‑form and extensive‑form representations, limiting scalability to scenarios with a small number of agents and short plans. Future work is suggested on approximation algorithms, learning‑based strategy selection, and dynamic repeated‑game extensions for larger, real‑world domains such as traffic routing, network packet scheduling, and collaborative robotics. Overall, the work provides a solid theoretical foundation for non‑cooperative planning and demonstrates how game‑theoretic reasoning can be harnessed to predict and coordinate agents’ plan selections in conflict‑prone environments.