ND-iBR: Decentralized Game Dynamics

• ND-iBR is a decentralized game-theoretic method where agents update strategies using local information, enabling efficient approximation of Nash equilibria.
• It employs an iterative best-response mechanism localized to dynamic neighborhoods to drastically reduce computational complexity and enhance scalability.
• Empirical evaluations in multi-agent motion planning and networked evolutionary games demonstrate ND-iBR’s robust performance, fast convergence, and improved safety.

Neighborhood-Dominated iterative Best Response (ND-iBR) denotes a class of decentralized game-theoretic dynamics in which agents update their strategies based on local information from neighboring agents, leading to scalable, efficient computation of (approximate) Nash equilibria in multi-agent settings. ND-iBR extends classical iterative best-response schemes by explicitly exploiting problem structure that limits agent interactions to local neighborhoods, yielding both theoretical convergence guarantees and practical computational gains, particularly in dynamic potential games and networked evolutionary games (Mai et al., 15 Nov 2025, Chellig et al., 2020).

1. Game-Theoretic Foundations

ND-iBR operates in environments composed of agents $\mathcal{N} = \{1, \ldots, N\}$, each with its own state $x_i \in \mathcal{X}_i$ and control or strategy $u_i \in \mathcal{U}_i$, evolving via potentially nonlinear, discrete-time dynamics $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$ (Mai et al., 15 Nov 2025). Agents optimize finite-horizon cost functionals of the form

$$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$

where the instantaneous and terminal terms can be decomposed into “global” potential terms, coupling all agents, and agent-independent or “remainder” components: \begin{align*} L_i(x^{(t)}, u_i^{(t)}) &= P(x^{(t)}, u^{(t)}) + \Theta_i(x_{-i}^{(t)}, u_{-i}^{(t)}) \ L_i^F(x^{(T)}) &= R(x^{(T)}) + \Xi_i(x_{-i}^{(T)}). \end{align*} Such structure admits formulation as a dynamic potential game, in which simultaneous minimization of the global potential $$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$ aligns with finding Nash equilibria (Mai et al., 15 Nov 2025).

2. The iε-BR Process and the Emergence of ND-iBR

The classical iterative $\varepsilon$-best response (i$\varepsilon$-BR) provides a constructive path to approximate Nash equilibria. At each iteration, an agent $i$ with possible strategy improvement $$J_i(x^{(0)}, (u_i^k, u_{-i}^k)) - \inf_{v_i} J_i(x^{(0)}, (v_i, u_{-i}^k)) \geq \varepsilon$$ solves the local optimization subproblem

$x_i \in \mathcal{X}_i$0

with other agents’ strategies unchanged (Mai et al., 15 Nov 2025). Each such update strictly decreases the global potential by at least $x_i \in \mathcal{X}_i$1, ensuring finite-step convergence in compact, continuous games.

ND-iBR modifies this process by localizing the coupling in each agent’s cost to a dynamic neighborhood $x_i \in \mathcal{X}_i$2, yielding a local cost functional

$x_i \in \mathcal{X}_i$3

where interaction terms are restricted to current neighbors. ND-iBR then replaces $x_i \in \mathcal{X}_i$4 with $x_i \in \mathcal{X}_i$5 in the i$x_i \in \mathcal{X}_i$6-BR routine, driving decentralized updates (Mai et al., 15 Nov 2025).

3. Distributed Algorithm and Termination Properties

The essential distributed ND-iBR scheme proceeds as follows (Mai et al., 15 Nov 2025):

1. Neighbor Trajectory Exchange: Each agent $x_i \in \mathcal{X}_i$7 at iteration $x_i \in \mathcal{X}_i$8 collects nominal trajectories $x_i \in \mathcal{X}_i$9 of current neighbors.
2. Local MPC-style Update: Agent $u_i \in \mathcal{U}_i$0 solves

$u_i \in \mathcal{U}_i$1

typically via a general-purpose nonlinear solver (e.g., IPOPT).

1. Improvement Assessment: Compute $u_i \in \mathcal{U}_i$2.
2. Termination: If $u_i \in \mathcal{U}_i$3, declare $u_i \in \mathcal{U}_i$4 an $u_i \in \mathcal{U}_i$5-Nash equilibrium; else select the maximizing $u_i \in \mathcal{U}_i$6, set $u_i \in \mathcal{U}_i$7, and repeat.

Termination within at most $u_i \in \mathcal{U}_i$8 iterations is guaranteed, provided that each strategy set $u_i \in \mathcal{U}_i$9 is compact and cost functionals satisfy the DPG decomposition. The final profile $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$0 satisfies the $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$1-Nash condition: $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$2

4. ND-iBR in Networked Evolutionary Games

For $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$3 symmetric games on random graphs $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$4, ND-iBR specializes to a local update wherein each vertex $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$5 selects the strategy maximizing its payoff against its neighbors’ current choices. The update reduces to generalized majority or minority dynamics, parameterized by payoff-skew $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$6,

$x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$7

dictated by the payoff matrix $x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$8 (Chellig et al., 2020). In the majority regime, vertices synchronize to the local majority strategy, and, for sufficiently dense graphs ($x_i^{(t+1)} = f_i(x_i^{(t)}, u_i^{(t)})$9), the system attains unanimity in at most four synchronous rounds. In the skewed ($$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$0) case, critical connectivity thresholds $$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$1 precisely characterize when the largest component achieves consensus; below threshold, discordant “blocking stars” persistently impede full synchronization.

5. Computational Complexity and Scalability

ND-iBR leverages locality to realize substantial reductions in per-iteration computational burden. With all-to-all coupling, each agent’s subproblem has $$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$2 constraints, yielding total coupling $$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$3. Restricting to local neighborhoods, the coupling reduces to $$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$4, where empirical studies find $$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$5 and growing sublinearly as agent population increases (e.g., $$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$6 as $$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$7) (Mai et al., 15 Nov 2025). This yields 2–3$$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$8 reductions in effective coupling and commensurate improvements in solver runtime.

6. Empirical Performance in Multi-Agent Motion Planning

Monte Carlo experiments for multi-agent quadrotor navigation under stochastic disturbances demonstrate that ND-iBR within the Reachability-Enhanced Dynamic Potential Game (RE-DPG) framework maintains stable tracking cost across agents ($$J_i(x^{(0)}, (u_i, u_{-i})) = \sum_{t=0}^{T-1} L_i(x^{(t)}, u_i^{(t)}) + L_i^F(x^{(T)}),$$9) and noise ($$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$0), outperforming probabilistic iLQR and DRL baselines in time efficiency and collision avoidance (Mai et al., 15 Nov 2025). Results indicate

• Consistently low tracking costs across varying $$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$1 and $$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$2
• Near-zero residual distance to goals within horizon, matching or improving upon baselines
• Zero collision incidence in all tested scenarios (with MA-FRS enabled); ablation of MA-FRS yields collisions at higher $$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$3 or $$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$4
• Average per-agent MA-FRS propagation $$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$5 ms, ND-iBR per step $$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$6 s, validating real-time feasibility for receding-horizon planning

A plausible implication is that ND-iBR, especially when combined with explicit uncertainty propagation (via MA-FRS), achieves robust, scalable, and safe coordination in dynamic, highly interactive environments.

7. Theoretical Limits and Network Effects

In dense interaction graphs, ND-iBR rapidly stabilizes collective strategies under broad classes of payoff matrices. However, below critical network connectivity, persistent local dissent arises, fully characterized via "blocking" substructures such as $$\Phi(u) = \sum_{t=0}^{T-1} P(x^{(t)}, u^{(t)}) + R(x^{(T)})$$7-blocking stars. The frequency and persistence of such structures are quantified in terms of random graph parameters and Poisson statistics at threshold regimes (Chellig et al., 2020). This demonstrates the sensitivity of ND-iBR convergence both to local coupling rules and the underlying interaction topology.