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Strategically Robust Game-Theoretic Multi-Agent Trajectory Optimization
Victor L. Qin, Nicolas Lanzetti, Saverio Bolognani, Hamsa Balakrishnan
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper proposes a strategically robust game-theoretic framework for multi-agent trajectory optimization in Advanced Air Mobility (AAM). It addresses the limitation of standard Nash equilibrium approaches, which assume perfect adherence to planned trajectories, by introducing a fictitious adversary that perturbs other agents' control inputs within a bounded budget to minimize distance. The authors prove that this strategically robust formulation preserves the exact dynamic potential game structure, allowing for efficient computation via a single optimization problem. They derive a quasi-closed-form solution for the adversarial component, enabling real-time application. Experiments with up to eight agents demonstrate that the method selects more robust trajectories in high-risk scenarios with only a modest increase in computational runtime compared to nominal methods.
Entities (9)
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Saverio Bolognani → affiliatedwith → ETH Zurich
confidence 95% · Saverio Bolognani 3 ... 3 Automatic Control Laboratory, ETH Zurich.
Victor L. Qin → affiliatedwith → MIT
confidence 95% · Victor L. Qin 1∗ ... 1 Department of Aeronautics and Astronautics, MIT.
Nicolas Lanzetti → affiliatedwith → CalTech
confidence 95% · Nicolas Lanzetti 2 ... 2 Department of Computing and Mathematical Sciences, Caltech.
Strategically Robust Game → preservesstructureof → Dynamic Potential Game
confidence 95% · We show that, under reasonable assumptions on agents' distance cost and robustness levels, the strategically robust game remains an exact dynamic potential game
Fictitious Adversary → perturbs → Control Inputs
confidence 93% · each agent protects against a fictitious adversary that, for each timestep, perturbs other agents' control inputs within a bounded budget
Strategically Robust Game → improvesrobustnessof → Trajectory Optimization
confidence 92% · strategic robustness selects more robust trajectories in high-collision-risk configurations
Logarithmic Distance Cost → usedin → Experiments
confidence 90% · Experiments with up to eight agents using logarithmic distance costs show that strategic robustness selects more robust trajectories
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Abstract
Abstract:Aviation authorities worldwide expect Advanced Air Mobility (AAM) traffic management to be decentralized among service providers, requiring AAM flights to autonomously plan trajectories by predicting other flights' control inputs rather than relying on centralized coordination. Game-theoretic approaches that formulate multi-agent collision avoidance as an exact dynamic potential game can efficiently find open-loop equilibria, but they assume that agents exactly follow their equilibrium trajectories---an unrealistic assumption given uncertainties in actuation, perception, and computation. We propose a strategically robust formulation where each agent protects against a fictitious adversary that, for each timestep, perturbs other agents' control inputs within a bounded budget to minimize distance at that timestep. We show that, under reasonable assumptions on agents' distance cost and robustness levels, the strategically robust game remains an exact dynamic potential game and admits a quasi-closed-form solution to the inner adversarial problem for linear dynamics, which limits computational overhead. Experiments with up to eight agents using logarithmic distance costs show that strategic robustness selects more robust trajectories in high-collision-risk configurations while leaving low-risk trajectories nearly unchanged, with only a modest increase in runtime.
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- Source: https://arxiv.org/abs/2609.35142v1
- Canonical: https://arxiv.org/abs/2609.35142v1
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Strategically Robust Game-Theoretic Multi-Agent Trajectory Optimization Victor L. Qin 1∗ , Nicolas Lanzetti 2 , Saverio Bolognani 3 , and Hamsa Balakrishnan 1 Abstract— Aviation authorities worldwide expect Advanced Air Mobility (AAM) traffic management to be decentralized among service providers, requiring AAM flights to autonomously plan trajectories by predicting other flights’ control inputs rather than relying on centralized coordination. Game-theoretic approaches that formulate multi-agent collision avoidance as an exact dynamic potential game can efficiently find open- loop equilibria, but they assume that agents exactly follow their equilibrium trajectories—an unrealistic assumption given uncertainties in actuation, perception, and computation. We propose a strategically robust formulation where each agent protects against a fictitious adversary that, for each timestep, perturbs other agents’ control inputs within a bounded budget to minimize distance at that timestep. We show that, under reasonable assumptions on agents’ distance cost and robustness levels, the strategically robust game remains an exact dynamic potential game and admits a quasi-closed-form solution to the inner adversarial problem for linear dynamics, which limits computational overhead. Experiments with up to eight agents using logarithmic distance costs show that strategic robustness selects more robust trajectories in high-collision- risk configurations while leaving low-risk trajectories nearly unchanged, with only a modest increase in runtime. I. INTRODUCTION Advanced Air Mobility (AAM) vehicles, such as unmanned aircraft systems (UASs) and electric vertical take-off and landing aircraft (eVTOLs), could transform urban and regional transportation [1,2]. However, AAM operations pose a chal- lenge for current air traffic management systems, as they are expected to operate autonomously on demand between many urban destinations. National aviation authorities worldwide envision AAM traffic management services being provided by several service providers in a region [3, 4]. With the decentralization of traffic management responsi- bilities, autonomous AAM flights cannot rely on a centralized authority to guarantee collision avoidance. Instead, they must solve a game-theoretic trajectory optimization problem, where each flight (each agent) finds the most efficient trajectory V. Qin would like to thank Geoffrey Ding for helpful conversations. V. Qin is supported by the National Science Foundation Graduate Research Fellowship Program under Grant #2141064, by NCCR Automation, grant agreement 51NF40225155 from the Swiss National Science Foundation, and in part by NASA Grant #80NSSC23M0220. N. Lanzetti is supported by the PIMCO Fellows Program, by the National Science Foundation under Grant #CCF-2326609, and by the Resnick Sustainability Institute. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation or of the National Aeronautics and Space Administration. 1 Department of Aeronautics and Astronautics, MIT. 2 Department of Computing and Mathematical Sciences, Caltech. 3 Automatic Control Laboratory, ETH Zurich. ∗ Corresponding author, victorqi@mit.edu Fig. 1: Illustration of the strategically robust approach for a two- agent scenario. The ego agent (blue trajectory) plans against a fictitious adversary that, for each timestep1≤ H ≤ T, perturbs the other agent’s control inputs away from its nominal trajectory (gray) to minimize the distance between agents at that timestep (orange). The adversary’s worst-case trajectory for each timestep (red) is subject to a budget constraint on the total control input deviation ( P H−1 k=0 ∥u j k − ˆu j,H k ∥ 2 ≤ ε j,H 2 , gray shaded region). See (5). while avoiding collisions with other agents. Standard game- theoretic methods that solve for the Nash equilibrium assume that each agent strictly adheres to its Nash equilibrium trajectory—a difficult assumption given the uncertainties inherent in aviation, even if agents are not adversarial (e.g., perception at night, miscalibrated sensors or systems, pilot errors)—and provide no guarantees when agents deviate. Clearly, misspecified control inputs from other AAM flights, due to factors like limited computation or partial information, could lead to catastrophic collisions. It is thus natural for flights to seek protection against such deviations, i.e., against uncertainty in the control inputs of other flights. Such strategic uncertainty—in contrast to the standard uncertainty in robust control or optimization—is not exogenous (e.g., against environmental noise as in robust control) but endogenous to the flights, where the control inputs of a flight affect those of the other flights against which they seek protection. Placing the uncertainty on control inputs rather than directly on states also guarantees that every deviation that is being protected against is dynamically feasible for the deviating flight. To achieve this goal, we adopt a strategically robust approach, introduced for static games [5], and propose that each agent minimizes its control cost against a fictitious adversary. The key idea is illustrated in Fig. 1. Each agent arXiv:2609.35142v1 [eess.SY] 28 Sep 2026 plans its own trajectory while anticipating that other agents may not exactly follow their nominal trajectories, by assuming that a fictitious adversary, at each step, perturbs the nominal control inputs of each of the other agents so as to minimize the distances to the ego agent—that is, to create the worst- case collision scenario. Crucially, the adversary is budget- constrained: for each of the other agents, the total deviation from its nominal control inputs cannot exceed a prescribed budget. By optimizing against this worst-case scenario at every timestep, each agent obtains a trajectory that is robust to bounded perturbations in the other agents’ control inputs. Agents can tune their desired level of robustness through this budget. If the budget is reduced to zero, the fictitious adversary is forced to replicate the control inputs of the other flights, recovering the standard Nash equilibrium in dynamic games. As the budget is increased, players assign more power to their fictitious adversary, thereby increasing robustness to perturbations in the strategies of the other players. The budget can therefore be directly interpreted as the level of robustness. While attractive in spirit, integrating strategic robustness into multi-agent trajectory optimization poses significant challenges. Even in the absence of strategic robustness, finding equilibria in dynamic games involves solving coupled nonlin- ear optimization problems and is computationally expensive. Recent work [6,7] identified that, under mild assumptions, multi-agent trajectory optimization can be structured as a dynamic potential game [8,9,10], which effectively allows us to find multi-agent equilibrium trajectories by solving a single constrained optimization problem. Whether strategic robustness preserves this structure without altering its favorable computational properties is the main challenge in deploying it for multi-agent trajectory optimization. In this paper, we answer this question affirmatively. In particular, our contribution is threefold: 1) We show that, under natural assumptions on the distance cost, using strategic robustness in exact dynamic po- tential games leads to another dynamic potential game with an appropriately modified potential function. 2)While the modified potential function requires solving a worst-case optimization problem, we obtain a quasi- closed-form solution to this adversarial maximization that can be quickly computed online. 3) Through various numerical examples, we show that our strategically robust method does not significantly add to the runtime of the trajectory optimization. A. Related work Game-theoretic planners: ALGAMES [11] solves for generalized Nash equilibria using Newton’s method on KKT conditions, while dynamic potential games [9] enable faster convergence by reducing the game to a single optimization problem [6, 7, 12, 13]. This optimization problem is solved online using iterative trajectory optimization methods based on linear-quadratic approximations, such as iLQR [14] and ALTRO [15]. While dynamic potential games are more restrictive than generalized Nash equilibrium games, multi- agent trajectory problems naturally fit in the dynamic potential game structure [6,9]. However, these planners assume perfect knowledge of other agents’ cost functions and dynamics. Exogenous uncertainty: In decision theory, there are var- ious approaches that are robust against exogenous uncertainty, including distributionally robust optimization [16] and risk measures [17]. Risk measures have been applied to risk-aware robotics [18,19]. Risk-sensitive iLQR games [20] incorporate noise in dynamics via entropic risk. Yet these approaches are primarily robust to exogenous uncertainty, and not to strategic uncertainty which is instead endogenous. Strategically robust game theory: To protect against strategic uncertainty (i.e., uncertainty about the other players), we adopt the strategically robust game-theoretic approach [5]. In strategically robust game theory, agents make decisions against a fictitious agent that aims to inflict maximum damage but is constrained to lie within a prescribed distance of the mixed strategy of all other players. This way, strategically robust equilibria interpolate between Nash and security equilibria. The open-loop strategically robust equilibria we use for this work are precisely inspired by this philosophy and can be interpreted as pure strategically robust equilibria in open-loop dynamic games. Under natural assumptions on the interagent cost structure, we show that our strategically robust game retains the exact dynamic potential game property [9], thereby preserving the computational advantages of solving a single optimization problem while adding robustness. More broadly, our work subscribes to a growing body of recent literature that leverages strategic robustness and risk aversion, sometimes combined with bounded rationality, in multi-agent settings to improve robustness, tractability, and sometimes even collaboration [21, 22, 23, 24, 25]. I. STRATEGICALLY ROBUST TRAJECTORY OPTIMIZATION We consider a multi-agent trajectory optimization problem, where agentsi ∈ 1, 2,...,N : = Noptimize their trajectories over a time horizon0≤ k ≤ T. Letx i k ∈R n and u i k ∈R m be the state and control input of agentiat timek; x k andu k are the concatenations of the state and input of all agents at timek;x i andu i are the concatenations of the state and input of agentifor all timesteps; andx =x k 0≤k≤T andu =u k 0≤k≤T−1 . The agent dynamics are linear and identical across all agents: Assumption 1 (Dynamics). The agent dynamics are x i k+1 = Ax i k + Bu i k ∀i∈N. We define the set of feasible trajectories for an agent given some fixed initial statex i 0 to beC i , and the set of trajectories for all agents to be C : = Q i∈N C i : C i =(x i 0 ,...,x i T ,u i 0 ,...,u i T−1 )| x i k+1 = Ax i k + Bu i k ∀k ∈ [0,T − 1]. Given other agents’ control inputsu −i and fixed initial state x i 0 , each agent seeks to minimize the control cost min u i J i (x 0 ,u) : = L i T (x T ) + T−1 X k=0 L i k (x k ,u k ) s.t. x l k+1 = Ax l k + Bu l k ∀l∈N, k ∈ [0,T − 1], (1) where we assume all functions are continuously differentiable. As in [7], we decomposeL i k (x k ,u k )into a private component L i k , which depends on the agent state and input, and an interagent componentL ij k forj ∈ N \i, which captures effects such as collision avoidance and depends on the states x i k and x j k of agents i and j: L i k (x k ,u k ) = L i k (x i k ,u i k ) + X j∈N\i L ij k (x i k ,x j k ) L i T (x T ) = L i T (x i T ) + X j∈N\i L ij T (x i T ,x j T ). (2) For example, the private cost can be a standard quadratic cost on the distance to a goalx i f , i.e.,L i k (x i k ,u i k ) = (x i k − x i f ) ⊤ Q i k (x i k − x i f ) + u i k ⊤ R i k u i k , while the interagent costL ij k (x i k ,x j k ) =−c ln(∥x i k −x j k ∥ 2 +η)pushes agents apart to avoid collisions, given parameters c > 0, 1≫ η > 0. 1 A. Game-theoretic solution concept We consider a finite horizon open-loop trajectory optimiza- tion problem with horizonT, where agentidecides on a strategyγ i (x 0 ) =u i k k∈[0,T−1] ∈ Γ i , a sequence of control inputs that generates a trajectoryx i =x i k k∈[0,T] such that (x i ,γ i (x 0 ))∈ C i , with the initial statex i 0 given byx 0 . The joint strategy of all players isγ(x 0 ) = (γ 1 (x 0 ),...,γ N (x 0 )), whereΓ = Q i∈N Γ i is the space of all joint strategies; we will writeγ i (x 0 ,k) = u i k . We write the cost to an agent asJ i (x 0 ,γ i (x 0 ),γ −i (x 0 )), where the trajectoryxis generated fromx 0 by the strategies in the second argument. Definition 1 (Nash equilibrium). An open-loop Nash equi- librium for a gameG = (N,J i i∈N , Γ)given initial conditionsx 0 is a set of strategiesγ ⋆ = (γ 1 ⋆ ,...,γ N ⋆ )∈ Γ with(x,γ ⋆ (x 0 ))∈ Csuch that for all agentsi∈N,γ i ∈ Γ i : J i (x 0 ,γ i ⋆ (x 0 ),γ −i ⋆ (x 0 ))≤ J i (x 0 ,γ i (x 0 ),γ −i ⋆ (x 0 )). The computation of open-loop Nash equilibria involves solving nonlinear equations coupled between players and is therefore computationally challenging. However, if the inter- agent costs satisfy symmetry properties—i.e.,L ij k (x i k ,x j k ) = L ji k (x j k ,x i k ) —then the game admits a potential function; see [6, 7, 9] and the proof of our Theorem 1 below for details. Definition 2 (Dynamic potential game). A gameG = (N,J i i∈N , Γ)is an exact dynamic potential game if there exists a potential function Φ(x 0 ,γ(x 0 )) such that J i (x 0 ,γ i (x 0 ),γ −i (x 0 ))−J i (x 0 ,γ i ′ (x 0 ),γ −i (x 0 )) =Φ(x 0 ,γ i (x 0 ),γ −i (x 0 ))−Φ(x 0 ,γ i ′ (x 0 ),γ −i (x 0 )). (3) If a game is an exact dynamic potential game, every minimizer of the potential function is an open-loop Nash equilibrium. Thus, whenever the potential attains its minimum (e.g. when it is coercive), an equilibrium exists and can be found by solving a single optimization problem [6,7]. Dynamic potential games can also be defined as games where 1 The regularizerηsimply keepsL ij k (x i k ,x j k ) finite so that the cost remains well-defined. we can representJ i as the sum of a potential function and a term that is independent of the agent’s own strategy; see [9, Lemma 3] and [26, Theorem 2.1] for more details. Lemma 1 (adapted from [26, Theorem 2.1]). The game G = (N,J i i∈N , Γ)is an exact dynamic potential game with potential functionΦif and only if there exist functions Θ i , i∈N, such that J i (x 0 ,γ(x 0 )) = Φ(x 0 ,γ(x 0 )) + Θ i (x 0 ,γ −i (x 0 ))(4) for all i∈N and all γ ∈ Γ. B. Strategically robust equilibria The Nash equilibrium is a natural solution concept for such a multi-agent decision problem, but in practice agents might fear misbehavior from others, i.e., deviations in their control inputs. We seek protection against such deviations by modifying the distance cost between ego agentiand agentj at time H ∈ [1,T ] as follows: 2 ̃ L ij H (x i H ,x j H ) = max ˆγ j,H L ij H (x i H , ˆx j,H H ) s.t. ˆx j,H k+1 = Aˆx j,H k + B ˆu j,H k ∀k ∈ [0,H − 1] ˆx j,H 0 = x j 0 H−1 X k=0 ∥u j k − ˆu j,H k ∥ 2 ≤ ε j,H 2 . (5) Each agenti ∈ Nevaluates the distance cost against a fictitious adversary. This adversary selects, for each agent j ∈ N \iand timestepH ∈ [1,T ], an open-loop policy ˆγ j,H with the goal of maximizing the distance cost for agent iat each timestepH(equivalently, minimizing the distance between agentsiandjat timeH), but cannot deviate in total by more thanε j,H from the nominal control inputsu j k ,k ∈ [0,H− 1] . 3 As described in Section I,ε j,H is the robustness level: atε j,H = 0the adversary is constrained to the nominal inputsu j k and we recover the standard Nash equilibrium, while larger values protect against larger perturbations. Given this robust distance cost, the strategically robust control cost of each agent i is ̃ J i (x 0 ,γ i (x 0 ),γ −i (x 0 )) =L i T (x i T ) + T−1 X k=0 L i k (x i k ,u i k ) + X j∈N\i T X H=0 ̃ L ij H (x i H ,x j H ). (6) With this, we can define strategically robust equilibria in our context as follows: Definition 3 (Open-loop strategically robust equilibrium). An open-loop strategically robust equilibrium for a game ̃ G = (N, ̃ J i i∈N , Γ) given initial conditionsx 0 is a set of 2 AtH = 0,ˆx j,0 0 = x j 0 , so ̃ L ij 0 = L ij 0 and we recover the nominal cost. 3 The budget constrains deviations fromu j k H−1 k=0 , so the adversary’s reachable set at timeHis fixed, convex, and centered atx j H . Equivalently, under Assumption 2,(8)below depends on agentsiandjonly through the relative positionz H = x i H −x j H . Hence ̃ L ij H is well defined as written and continuously differentiable (since squared distance to a convex set is). strategiesγ ⋆ = (γ 1 ⋆ ,...,γ N ⋆ ) ∈ Γwith(x,γ ⋆ (x 0 )) ∈ C such that for all agents i∈N,γ i ∈ Γ i we have ̃ J i (x 0 ,γ i ⋆ (x 0 ),γ −i ⋆ (x 0 ))≤ ̃ J i (x 0 ,γ i (x 0 ),γ −i ⋆ (x 0 )). Note that our fictitious adversary is solving a target intercept problem with a separate adversarial trajectoryˆγ j,H given deviation budgetε j,H for each timestepH. This is deliberate: a collision between two aircraft at any one timestep is catastrophic. Our fictitious adversary is at least as powerful as one committing to a single trajectory maximizing the more standard formulation of a summed distance cost over the entire horizon [0,T ], given the same budget. I. COMPUTATION OF STRATEGICALLY ROBUST EQUILIBRIA We now show that the strategically robust game preserves the potential structure of the nominal game, which makes its equilibria efficiently computable. A. Strategic robustness in exact dynamic potential games We make two further assumptions. First, we assume the interagent cost is a monotone function of the distance between agents. 4 Assumption 2 (Distance cost). The distance cost is L ij k (x i k ,x j k ) = −μ(∥x i k − x j k ∥ 2 ), whereμ :R ≥0 →Ris a monotonically increasing function. This assumption is natural for collision avoidance, where the cost should grow as agents approach one another. The logarithmic distance penalty from Section I, L ij k (x i k ,x j k ) = −c ln(∥x i k − x j k ∥ 2 + η), satisfies this assumption. Second, we assume players have the same robustness level. Assumption 3 (Symmetric robustness). For all timesH ∈ [1,T ] , agents have the same robustness level, i.e.,ε i,H = ε j,H for all i,j ∈N. Our main theoretical result follows. Theorem 1. LetG = (N,J i i∈N , Γ)be a dynamic game withJ i given by(1)and(2), and let Assumptions 1 and 2 hold so thatGis an exact dynamic potential game. Suppose Assumption 3 holds. Then the strategically robust game ̃ G = (N, ̃ J i i∈N , Γ)is also an exact dynamic potential game with potential function ̃ Φ(x 0 ,γ(x 0 )) = X i∈N " L i T (x i T ) + T−1 X k=0 L i k (x i k ,u i k ) # + X i,j∈N,i<j T X H=0 ̃ L ij H (x i H ,x j H ). (7) We prove Theorem 1 in Appendix A. Because the strate- gically robust costs ̃ J i form an exact dynamic potential game, every minimizer of the potential function(7)is a strategically robust equilibrium. Finally, two comments on the symmetry. First, Assumption 2 impliesL ij H (x i H ,x j H ) = 4 Only the collision-relevant components of the state enter the distance cost; formally,∥D(x i H − x j H )∥ 2 for a fixed selection matrixD. We take D = I to lighten notation. Algorithm 1 Strategically Robust Potential Game Require:GameG, dynamics(A,B), deviation bounds ε j,H 1: Precompute eigendecompositions Q H , Σ σ H 2: repeat 3:for pairs i < j, timesteps H = 1,...,T do 4:Setλ H = 0or find the positive root of(14)via Newton’s method (Step 3) 5:Compute δz j,H H in closed form (Step 2) 6:end for 7:Take minimization step on ̃ Φ(x 0 ,γ(x 0 )) 8: until convergence L ji H (x j H ,x i H ). Relaxing this assumption in certain ways (e.g., allowing agent-dependent weights) would still yield a weighted dynamic potential game [7]. Second, we also hypothesize that relaxing Assumption 3 such thatε i,H ̸= ε j,H under certain conditions could yield an ordinal potential game. Ordinal and weighted potential games, with appropriate assumptions, have convergence guarantees [8]. B. Fast computation of the strategically robust distance cost While attractive, minimizing the potential still entails a computational challenge: the mere evaluation of ̃ L ij H requires solving an optimization problem. We now derive a quasi- closed-form and computationally efficient solution to this worst-case problem in three steps: Step 1 (Monotonicity) By monotonicity ofμ, we can maximize the distance cost by minimizing the squared distance in(5)for a givenj,H. Define the relative position z H = x i H − x j H , and the deviationsδz j,H k = x j k − ˆx j,H k and δu j,H k = u j k − ˆu j,H k . Plugging into (5) gives max ˆγ j,H −μ(∥x i H − ˆx j,H H ∥ 2 ) = max δu j,H k −μ(∥z H + δz j,H H ∥ 2 ) =− μ min δu j,H k ∥z H + δz j,H H ∥ 2 ! , where we used the monotonicity ofμ. Thus, we can focus on solving the inner minimization. The inner minimization satisfies Slater’s condition forε j,H > 0, so strong duality holds. For fixed dual multiplierλ H ≥ 0, we therefore write its Lagrangian relaxation, with δz j,H 0 = 0: min δu j,H k ∥z H + δz j,H H ∥ 2 + λ H H−1 X k=0 ∥δu j,H k ∥ 2 − ε j,H 2 ! s.t. δz j,H k+1 = Aδz j,H k + Bδu j,H k ∀k ∈ [0,H − 1]. (8) Step 2 (Optimal Control) For fixed dual multiplierλ H ≥ 0, (8)is a quadratic optimization problem that gives usδu j,H k and δz j,H H in closed form, whereM H = [A H−1 B,...,AB,B] (see Appendix B): δz j,H H = M H δu j,H =−M H M H ⊤ M H M H ⊤ + λ H I † z H . Here † denotes the Moore–Penrose pseudoinverse, which equals the ordinary inverse whenλ H > 0. We then decompose (a) Head-on scenario.(b) Parallel scenario.(c) 4-agent scenario.(d) 8-agent scenario. Fig. 2: Comparison of trajectory optimization methods. Solid lines show nominal trajectories; different dashes show other methods. All trajectories are open-loop. (a) Head-on: agents start at(0, 0.95)and(2, 1.05)and swap positions. (b) Parallel: agents start at(0, 0)and (2, 0)and travel in they-direction to(0, 2)and(2, 2). (c) 4 agents crossing a region. (d) 8 agents swapping antipodal positions on a circle. TABLE I: Two-agent computation time ×nominal (s)Head-onParallel Nominal1.00× (0.045s) 1.00× (0.012s) Wider0.89× (0.040s) 1.17× (0.014s) Strat. Robust 1.41× (0.064s) 2.01× (0.024s) TABLE I: Deviation from nominal Head-on Parallel Nominal0.000 0.000 Wider0.516 0.449 Strat. Robust 0.252 0.109 TABLE I: Multi-agent computation time ×nominal (s)4 agents8 agents Nominal1.00× (0.095s) 1.00× (1.356s) Wider1.04× (0.099s) 1.57× (2.125s) Strat. Robust 1.99× (0.190s) 1.45× (1.973s) M H M H ⊤ into its eigendecompositionQ H Σ σ H Q H ⊤ , which can be precomputed for all H . Step 3 (Updatingλ H ) We can efficiently compute the dual multipliersλ H by first testing whetherλ H = 0is optimal; otherwise, we take the eigendecomposition ofM H M H ⊤ and solve the first-order condition of the dual using Newton iterations, as shown in Appendix B. The final algorithm is summarized in Algorithm 1. IV. RESULTS We now illustrate strategic robustness on four multi-agent trajectory optimization scenarios, with two, four, and eight agents. We compare three different methods, withη = 10 −8 : •Nominal: We compute the open-loop Nash equilibria by solving the exact dynamic potential game formed byJ i in (1), with L ij k (x i k ,x j k ) =−2 ln(∥x i k − x j k ∥ 2 + η). • Strategically robust: We compute the open-loop strate- gically robust equilibria by solving the exact dynamic potential game formed by ̃ J i using Algorithm 1, with L ij k (x i k ,x j k ) =−2 ln(∥x i k −x j k ∥ 2 +η)and the robustness parameter ε j,H = 2 for all players and times. •Wider: We compute the open-loop Nash equilibria with increased distance costL ij k (x i k ,x j k ) = −5 ln(∥x i k − x j k ∥ 2 + η). This adds robustness by increasing the distance between agents in all scenarios compared to the “nominal” method. We used a direct single shooting method (cf. [27]) to optimize the problem, solved usingscipy’s SLSQP solver [28], which we provide with the function and its gradient (the Hessian is instead approximated numerically). Every example uses single integrator dynamics inR 2 , withA = I,B = 0.1I, and T = 20; positions, times, and budgets are in normalized units. The private cost isL i k = (x i k − x i f ) ⊤ Q k (x i k −x i f )+u i k ⊤ Ru i k , withQ k = Ifor allk ∈ 0,...,T − 1,Q T = 150I, and R = I. The strategically robust method is initialized by first solving the nominal potential game; its reported computational time includes that initialization. 5 A. Two-player system We consider the two scenarios in Figs. 2a and 2b, where two agents are traveling either head-on or in parallel. To start, we observe that the strategically robust trajectories are wider compared to the nominal trajectories. This is a direct consequence of strategic robustness: since agents protect against misbehavior by other agents, they commit to wider trajectories to reduce the collision risk. Increasing the collision parameter: It is natural to ask if strategic robustness can be reproduced by a simple increase in the collision parameter, as in the “wider” method. We argue here that strategic robustness offers protection in a targeted way. Indeed, when comparing the “strategically robust” and “wider” methods, we observe that the effect of strategic robustness differs between Figs. 2a and 2b. In Fig. 2a, the collision risk is concrete and strategic robustness leads to significantly wider trajectories. In Fig. 2b, strategic robustness instead leaves the trajectories nearly unchanged, as it reasons that a significantly larger deviation in control inputs is needed for a collision to occur. Merely increasing the weight of the distance cost instead leads to approximately the same effect in both settings. This qualitative observation can be made quantitative by inspecting the deviation of the trajectories from the nominal Nash equilibrium trajectory in Table I, calculated as the trapezoidal approximation of the area between trajectories. Strategic robustness deviates2.3×more 5 The code can be found athttps://github.com/victor-qin/s trategically_robust_potential_trajopt from the nominal trajectory in the head-on scenario than in the parallel one, whereas merely increasing the collision parameter deviates by a comparable amount in both scenarios. Runtime: We measure the runtime of different methods in Table I, using 100 runs for each scenario, reported as [multiple of nominal time] (median time in seconds). The strategically robust method only modestly increases the runtime compared to simply solving for the nominal trajectory. B. A multi-agent system Finally, we test our strategically robust trajectory planner for four and eight agents. We plot only the nominal and strate- gically robust methods for clarity. The resulting trajectories are shown in Figs. 2c and 2d, and the computational times are listed in Table I. Strategic robustness takes between 1.41×and2.01×the time of solving the nominal trajectory across two to eight agents, and the factor does not grow with the number of agents N . V. CONCLUDING REMARKS We believe our approach of reframing collision avoidance through strategic robustness [5] offers interesting directions for building fast trajectory optimization methods with collision avoidance guarantees. We mention a few. First, we have assumed that agents have identical dynamics, interagent costs, and robustness levels. As discussed in Section I- A, we expect these assumptions can be relaxed to form weighted or ordinal dynamic potential games [8]. 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Cournapeau, E. Burovski, P. Peterson, W. Weckesser, J. Bright, S. J. van der Walt, M. Brett, J. Wilson, K. J. Millman, N. Mayorov, A. R. J. Nelson, E. Jones, R. Kern, E. Larson, C. J. Carey, ̇ I. Polat, Y. Feng, E. W. Moore, J. VanderPlas, D. Laxalde, J. Perktold, R. Cimrman, I. Henriksen, E. A. Quintero, C. R. Harris, A. M. Archibald, A. H. Ribeiro, F. Pedregosa, P. van Mulbregt, and SciPy 1.0 Contributors, “SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python,” Nature Methods, vol. 17, p. 261–272, 2020. APPENDIX A. Proof of Theorem 1 It suffices to show that ̃ L ij H (x i H ,x j H ) = ̃ L ji H (x j H ,x i H )for alli < j ∈N, H ∈ [0,T ], because then the terms of ̃ Φfrom (7) involving agent i are exactly ̃ J i . Define Θ i : = ̃ J i − ̃ Φ: Θ i (x −i 0 ,γ −i (x 0 )) =− X l∈N\i " L l T (x l T ) + T−1 X k=0 L l k (x l k ,u l k ) # − X p,l∈N,p,l̸=i,p<l T X H=0 ̃ L pl H (x p H ,x l H ). Because dynamics are decoupled,Θ i is a dummy function in- dependent ofγ i (x 0 ). Thus ̃ J i i∈N forms an exact dynamic potential game by Lemma 1. Next, we show ̃ L ij H (x i H ,x j H ) = ̃ L ji H (x j H ,x i H ) using Assumptions 2 and 3. Recall ̃ L ij H (x i H ,x j H ) given by (5): max ˆγ j,H L ij H (x i H , ˆx j,H H ) s.t. ˆx j,H k+1 = Aˆx j,H k + B ˆu j,H k ∀k ∈ [0,H − 1] ˆx j,H 0 = x j 0 H−1 X k=0 ∥u j k − ˆu j,H k ∥ 2 ≤ ε j,H 2 . (9) Defineδz j,H k = x j k − ˆx j,H k andδu j,H k = u j k − ˆu j,H k to rewrite the dynamics as ˆx j,H k+1 − x j k+1 = A(ˆx j,H k − x j k ) + B(ˆu j,H k − u j k ) δz j,H k+1 = Aδz j,H k + Bδu j,H k , with the initial conditionδz j,H 0 = 0asˆx j,H 0 = x j 0 . Since z H = x i H − x j H is constant with respect to the adversarial optimization of (9), we can rewrite it as max δu j,H k − μ(∥z H + δz j,H H ∥ 2 ) s.t. δz j,H k+1 = Aδz j,H k + Bδu j,H k ∀k ∈ [0,H − 1] δz j,H 0 = 0 H−1 X k=0 ∥δu j,H k ∥ 2 ≤ ε j,H 2 . (10) Second, we consider ̃ L ji H (x j H ,x i H ), defined similarly to(9). Following similar steps, defineδz i,H k = ˆx i,H k − x i k and δu i,H k = ˆu i,H k − u i k , so thatx j H − ˆx i,H H =−z H − δz i,H H (the sign flip ensures a match with the definition ofz H above). Then, we can rewrite ̃ L ji H (x j H ,x i H ) as max δu i,H k − μ(∥z H + δz i,H H ∥ 2 ) s.t. δz i,H k+1 = Aδz i,H k + Bδu i,H k ∀k ∈ [0,H − 1] δz i,H 0 = 0 H−1 X k=0 ∥δu i,H k ∥ 2 ≤ ε i,H 2 . (11) The caseH = 0is trivial byˆx j,0 0 = x j 0 . Given that ε i,H = ε j,H by Assumption 3 and(A,B)are identical by Assumption 1, the problems(10)and(11)are the same, which implies that ̃ L ij H (x i H ,x j H ) = ̃ L ji H (x j H ,x i H )for all H ∈ [0,T ]. B. Details on Section I-B Step 2 (Optimal Control)Consider(8)for timesteps H ∈ [1,T ], assumingε j,H > 0with fixedλ H ≥ 0and δz j,H 0 = 0. LetM H = [A H−1 B,...,AB,B]andδu j,H be the vertical concatenation ofδu j,H 0 ,...,δu j,H H−1 , so that δz j,H H = M H δu j,H . We can rewrite (8) as min δu j,H ∥z H + M H δu j,H ∥ 2 + λ H δu j,H ⊤ δu j,H − ε j,H 2 . (12) Forλ H > 0, the objective is strictly convex, and setting its gradient to zero gives the unique input minimizer: δu j,H =− M H ⊤ M H + λ H I −1 M H ⊤ z H =−M H ⊤ M H M H ⊤ + λ H I −1 z H . Substituting into δz j,H H gives δz j,H H =−M H M H ⊤ M H M H ⊤ + λ H I −1 z H . The matrixM H M H ⊤ is the positive semidefinite controlla- bility Gramian. We can compute its eigendecomposition of- fline asQ H Σ σ H Q H ⊤ , whereΣ σ H = diag(σ 1,H ,...,σ n,H ), σ l,H ≥ 0, andQ H is an orthogonal matrix. Forλ H > 0, we can then rewrite δz j,H H as δz j,H H =−Q H Σ σ H λ H +σ H Q H ⊤ z H , where Σ σ H λ H +σ H is quickly formed online. Ifλ H = 0, the problem reduces to least squares. We select the minimum-norm inputδu j,H = −M † H z H , giving δz j,H H = −M H M † H z H , whereM † H is the Moore–Penrose pseudoinverse. We can also derive this using the limit as λ H → 0 + and following the singular value decomposition. Step 3 (Updatingλ H ) We maximize the dual function q(λ H )overλ H ≥ 0. Substitutingδu j,H from Step 2 into the Lagrangian and simplifying gives, for λ H > 0, q(λ H ) = z H ⊤ Q H Σ λ H λ H +σ H Q H ⊤ z H − λ H ε j,H 2 .(13) Letw = Q H ⊤ z H . Asλ H → 0 + , the ratioλ H /(λ H + σ l,H ) becomes one forσ l,H = 0and zero otherwise, soq(0) = P l:σ l,H =0 w 2 l , the squared residual in unreachable directions. The dual functionq(λ H )is concave onλ H ≥ 0. An interior maximizer satisfies ∂q ∂λ H = w ⊤ Σ σ H (λ H +σ H ) 2 w− ε j,H 2 = n X l=1 w l 2 σ l,H (λ H + σ l,H ) 2 − ε j,H 2 = 0. (14) Thisuniquepositiverootexistsifandonlyif P l:σ l,H >0 w 2 l /σ l,H > ε j,H 2 . Then we can compute the root using Newton’s method, initialized atλ H > 0with q ′ (λ H ) > 0. Otherwise,λ H = 0is optimal, and the adversary exactly intercepts the agent if and only ifz H ∈ range(M H ).