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Effectively predicting intent and behavior requires inferring leadership in multi-agent interactions. Dynamic games provide an expressive theoretical framework for modeling these interactions. Employing this framework, we propose a novel method to infer the leader in a two-agent game by observing the agents’ behavior in complex, long-horizon interactions. We make two con- tributions. First, we introduce an iterative algorithm that solves dynamic two-agent Stackelberg games with nonlinear dynamics and nonquadratic costs, and demonstrate that it consistently converges in repeated trials. Second, we propose the Stackelberg Leadership Filter (SLF), an online method for identifying the leading agent in interactive scenarios based on observations of the game interactions. We validate the leadership filter’s efficacy on simulated driving scenarios to demonstrate that the SLF can draw conclusions about leadership that match right-of-way expectations. 

We consider the problem of trajectory planning for optimal relative orbit determination in the cislunar environment. The recent interest in cislunar space has created a need to develop autonomous tracking technologies that can maintain situational awareness of this dynamically complex regime. Optical sensors provide an ideal observation platform because of their low cost and versatility in tracking both cooperative and non-cooperative space objects. The estimation performance of an optical observer can be significantly enhanced through manuevering. This work develops a trajectory planning tool, compatible with low-thrust propulsion, for tracking one or multiple targets operating in proximity to the observer. We formulate an objective function that is a convex combination of the mutual information between target states and measurements, and the low-thrust control effort. The subsequent optimal control problem is solved via direct collocation using the successive convexification algorithm, which, we argue, is well suited for a potential onboard trajectory planning application. We demonstrate the tool for several relevant scenarios with one and multiple targets operating around periodic orbits in the circular restricted three-body problem. A sequential estimator's performance is evaluated using the Cramer-Rao lower bound and, compared to a purely passive observer, we show that optimizing the observer's trajectory can decrease this bound by up to several orders of magnitude within a planning window. This investigation provides an initial proof-of-concept to future onboard planning technologies for relative tracking in the cislunar domain. 

Contingency planning, wherein an agent generates a set of possible plans conditioned on the outcome of an uncertain event, is an increasingly popular way for robots to act under uncertainty. In this work we take a game-theoretic perspective on contingency planning, tailored to multi-agent scenarios in which a robot’s actions impact the decisions of other agents and vice versa. The resulting contingency game allows the robot to efficiently interact with other agents by generating strategic motion plans conditioned on multiple possible intents for other actors in the scene. Contingency games are parameterized via a scalar variable which represents a future time when intent uncertainty will be resolved. By estimating this parameter online, we construct a game-theoretic motion planner that adapts to changing beliefs while anticipating future certainty. We show that existing variants of game-theoretic planning under uncertainty are readily obtained as special cases of contingency games. Through a series of simulated autonomous driving scenarios, we demonstrate that contingency games close the gap between certainty-equivalent games that commit to a single hypothesis and non-contingent multi-hypothesis games that do not account for future uncertainty reduction. 

Markov games model interactions among multiple players in a stochastic, dynamic environment. Each player in a Markov game maximizes its expected total discounted reward, which depends upon the policies of the other players. We formulate a class of Markov games, termed affine Markov games, where an affine reward function couples the players’ actions. We introduce a novel solution concept, the soft-Bellman equilibrium, where each player is boundedly rational and chooses a soft-Bellman policy rather than a purely rational policy as in the well-known Nash equilibrium concept. We provide conditions for the existence and uniqueness of the soft-Bellman equilibrium and propose a nonlinear least-squares algorithm to compute such an equilibrium in the forward problem. We then solve the inverse game problem of inferring the players’ reward parameters from observed state-action trajectories via a projected-gradient algorithm. Experiments in a predator-prey OpenAI Gym environment show that the reward parameters inferred by the proposed algorithm outper- form those inferred by a baseline algorithm: they reduce the Kullback-Leibler divergence between the equilibrium policies and observed policies by at least two orders of magnitude. 

Decision-making in multi-player games can be extremely challenging, particularly under uncertainty. In this work, we propose a new sample-based approximation to a class of stochastic, general-sum, pure Nash games, where each player has an expected-value objective and a set of chance constraints. This new approximation scheme inherits the accuracy of objective approximation from the established sample average approximation (SAA) method and enjoys a feasibility guarantee derived from the scenario optimization literature. We characterize the sample complexity of this new game-theoretic approximation scheme, and observe that high accuracy usually requires a large number of samples, which results in a large number of sampled constraints. To accommodate this, we decompose the approximated game into a set of smaller games with few constraints for each sampled scenario, and propose a decentralized, consensus-based ADMM algorithm to efficiently compute a generalized Nash equilibrium (GNE) of the approximated game. We prove the convergence of our algorithm to a GNE and empirically demonstrate superior performance relative to a recent baseline algorithm based on ADMM and interior point method. 

An atomic routing game is a multiplayer game on a directed graph. Each player in the game chooses a path—a sequence of links that connect its origin node to its destination node—with the lowest cost, where the cost of each link is a function of all players’ choices. We develop a novel numerical method to design the link cost function in atomic routing games such that the players’ choices at the Nash equilibrium minimize a given smooth performance function. This method first approximates the nonsmooth Nash equilibrium conditions with smooth ones, then iteratively improves the link cost function via implicit differentiation. We demonstrate the application of this method to atomic routing games that model noncooperative agents navigating in grid worlds. 

In multi-agent dynamic games, the Nash equilibrium state trajectory of each agent is determined by its cost function and the information pattern of the game. However, the cost and trajectory of each agent may be unavailable to the other agents. Prior work on using partial observations to infer the costs in dynamic games assumes an open-loop information pattern. In this work, we demonstrate that the feedback Nash equilibrium concept is more expressive and encodes more complex behavior. It is desirable to develop specific tools for inferring players’ objectives in feedback games. Therefore, we consider the dynamic game cost inference problem under the feedback information pattern, using only partial state observations and incomplete trajectory data. To this end, we first propose an inverse feedback game loss function, whose minimizer yields a feedback Nash equilibrium state trajectory closest to the observa- tion data. We characterize the landscape and differentiability of the loss function. Given the difficulty of obtaining the exact gradient, our main contribution is an efficient gradient approximator, which enables a novel inverse feedback game solver that minimizes the loss using first-order optimization. In thorough empirical evaluations, we demonstrate that our algorithm converges reliably and has better robustness and generalization performance than the open-loop baseline method when the observation data reflects a group of players acting in a feedback Nash game.