arXiv subjects

Forrest Laine

Publications and source records attributed to Forrest Laine.

At least 19 recordsLinked to original sources

arXiv2025

Online Competitive Information Gathering for Partially Observable Trajectory Games

Game-theoretic agents must make plans that optimally gather information about their opponents. These problems are modeled by partially observable stochastic games (POSGs), but planning in fully continuous POSGs is intractable without heavy offline computation or assumptions on the order of belief maintained by each player. We formulate a finite history/horizon refinement of POSGs which admits competitive information gathering behavior in trajectory space, and through a series of approximations, we present an online method for computing rational trajectory plans in these games which leverages particle-based estimations of the joint state space and performs stochastic gradient play. We also provide the necessary adjustments required to deploy this method on individual agents. The method is tested in continuous pursuit-evasion and warehouse-pickup scenarios (alongside extensions to $N > 2$ players and to more complex environments with visual and physical obstacles), demonstrating evidence of active information gathering and outperforming passive competitors.

cs.GT↗

arXiv2025

Locally Optimal Solutions for Integer Programming Games

Integer programming games (IPGs) are n-person games with integer strategy spaces. These games are used to model non-cooperative combinatorial decision-making and are used in domains such as cybersecurity and transportation. The prevalent solution concept for IPGs, Nash equilibrium, is difficult to compute and even showing whether such an equilibrium exists is known to be Sp2-complete. In this work, we introduce a class of relaxed solution concepts for IPGs called locally optimal integer solutions (LOIS) that are simpler to obtain than pure Nash equilibria. We demonstrate that LOIS are not only faster and more readily scalable in large-scale games but also support desirable features such as equilibrium enumeration and selection. We also show that these solutions can model a broader class of problems including Stackelberg, Stackelberg-Nash, and generalized IPGs. Finally, we provide initial comparative results in a cybersecurity game called the Critical Node game, showing the performance gains of LOIS in comparison to the existing Nash equilibrium solution concept.

cs.GT↗

arXiv2025

Polyhedral Collision Detection via Vertex Enumeration

Collision detection is a critical functionality for robotics. The degree to which objects collide cannot be represented as a continuously differentiable function for any shapes other than spheres. This paper proposes a framework for handling collision detection between polyhedral shapes. We frame the signed distance between two polyhedral bodies as the optimal value of a convex optimization, and consider constraining the signed distance in a bilevel optimization problem. To avoid relying on specialized bilevel solvers, our method exploits the fact that the signed distance is the minimal point of a convex region related to the two bodies. Our method enumerates the values obtained at all extreme points of this region and lists them as constraints in the higher-level problem. We compare our formulation to existing methods in terms of reliability and speed when solved using the same mixed complementarity problem solver. We demonstrate that our approach more reliably solves difficult collision detection problems with multiple obstacles than other methods, and is faster than existing methods in some cases.

cs.CG↗

arXiv2024

Algorithmic collusion in a two-sided market: A rideshare example

With dynamic pricing on the rise, firms are using sophisticated algorithms for price determination. These algorithms are often non-interpretable and there has been a recent interest in their seemingly emergent ability to tacitly collude with each other without any prior communication whatsoever. Most of the previous works investigate algorithmic collusion on simple reinforcement learning (RL) based algorithms operating on a basic market model. Instead, we explore the collusive tendencies of Proximal Policy Optimization (PPO), a state-of-the-art continuous state/action space RL algorithm, on a complex double-sided hierarchical market model of rideshare. For this purpose, we extend a mathematical program network (MPN) based rideshare model to a temporal multi origin-destination setting and use PPO to solve for a repeated duopoly game. Our results indicate that PPO can either converge to a competitive or a collusive equilibrium depending upon the underlying market characteristics, even when the hyper-parameters are held constant.

cs.GT↗

arXiv2024

Decreasing Wages in Gig Economy: A Game Theoretic Explanation Using Mathematical Program Networks

Gig economy consists of two market groups connected via an intermediary. Popular examples are rideshares where passengers and drivers are mediated via platforms such as Uber and Lyft. In a duopoly market, the platforms must compete to attract not only the passengers by providing a lower rate but also the drivers by providing better wages. While this should indicate better driver payout, as platforms compete to attract the driver pool, real world statistics does not indicate such. This goes completely against the intuition that the worker side of a gig economy, given their importance, should always earn better. We attempt to answer the low wages of drivers in the gig economy by modeling the ridesharing game between duopoly platforms, drivers, and passengers using Mathematical Program Networks. Our model is parsimonious, expressive, models the same-side and cross-side externalities of the economy, and has interpretations under both single-homing and multi-homing regimes. We derive the conditions for the existence of a profitable duopoly and show that it can only happen if the platforms collude together to pay the bare minimum to the drivers. This not only answers why drivers are paid less but also provides strong managerial insights to any interested policy maker.

cs.GT↗

arXiv2024

Mathematical Program Networks

Mathematical Program Networks (MPNs) are introduced in this work. An MPN is a collection of interdependent Mathematical Programs (MPs) which are to be solved simultaneously, while respecting the connectivity pattern of the network defining their relationships. The network structure of an MPN impacts which decision variables each constituent mathematical program can influence, either directly or indirectly via solution graph constraints representing optimal decisions for their decedents. Many existing problem formulations can be formulated as MPNs, including Nash Equilibrium problems, multilevel optimization problems, and Equilibrium Programs with Equilibrium Constraints (EPECs), among others. The equilibrium points of an MPN correspond with the equilibrium points or solutions of these other problems. By thinking of a collection of decision problems as an MPN, a common definition of equilibrium can be used regardless of relationship between problems, and the same algorithms can be used to compute solutions. The presented framework facilitates modeling flexibility and analysis of various equilibrium points in problems involving multiple mathematical programs.

math.OC↗

arXiv2024

Mixed Strategy Constraints in Continuous Games

When modeling robot interactions as Nash equilibrium problems, it is desirable to place coupled constraints which restrict these interactions to be safe and acceptable (for instance, to avoid collisions). Such games are continuous with potential mixed strategy equilibria, and this combination of characteristics means special care must be given to setting coupled constraints in a way that respects mixed strategies while remaining compatible with continuous game solution methods. Here, we investigate the problem of constraint-setting in this context, primarily focusing on a chance-based method. We first motivate these chance constraints in a discrete setting, placing them on n-player matrix games as a justifiable approach to handling the probabilistic nature of mixing. Then, we describe a numerical solution method for these chance constrained, continuous games with simultaneous pure strategy optimization. Finally, using a modified pursuit-evasion game as a motivating example, we demonstrate the actual behavior of this solution method in terms of its fidelity, parameter sensitivity, and efficiency

cs.GT↗

arXiv2024

Does bilevel optimization result in more competitive racing behavior?

Two-vehicle racing is natural example of a competitive dynamic game. As with most dynamic games, there are many ways in which the underlying solution concept can be structured, resulting in different equilibrium concepts. The assumed solution concept influences the behaviors of two interacting players in racing. For example, blocking behavior emerges naturally in leader-follower play, but to achieve this in Nash play the costs would have to be chosen specifically to trigger this behavior. In this work, we develop a novel model for competitive two-player vehicle racing, represented as an equilibrium problem, complete with simplified aerodynamic drag and drafting effects, as well as position-dependent collision-avoidance responsibility. We use our model to explore how different solution concepts affect competitiveness. We develop a solution for bilevel optimization problems, enabling a large-scale empirical study comparing bilevel strategies (either as leader or follower), Nash equilibrium strategy and a single-player constant velocity baseline. We find the choice of strategies significantly affects competitive performance and safety.

cs.GT↗

arXiv2023

Monte Carlo Optimization for Solving Multilevel Stackelberg Games

Stackelberg games originate where there are market leaders and followers, and the actions of leaders influence the behavior of the followers. Mathematical modelling of such games results in what's called a Bilevel Optimization problem. There is an entire area of research dedicated to analyzing and solving Bilevel Optimization problems which are often complex, and finding solutions for such problems is known to be NP-Hard. A generalization of Stackelberg games is a Multilevel Stackelberg game where we may have nested leaders and followers, such that a follower is, in turn, a leader for all lower-level players. These problems are much more difficult to solve, and existing solution approaches typically require extensive cooperation between the players (which generally can't be assumed) or make restrictive assumptions about the structure of the problem. In this paper, we present a stochastic algorithm to approximate the local equilibrium solutions for these Multilevel games. We then construct a few examples of such Multilevel problems, including: a) a nested toll-setting problem; and b) an adversarial initial condition determination problem for Robust Trajectory Optimization. We test our algorithm on our constructed problems as well as some trilevel problems from the literature, and show that it is able to approximate the optimum solutions for these problems within a reasonable error margin. We also provide an asymptotic proof for the convergence of the algorithm and empirically analyze its accuracy and convergence speed for different parameters. Lastly, we compare it with existing solution strategies from the literature and demonstrate that it outperforms them.

cs.GT↗

arXiv2023

Compelling ReLU Networks to Exhibit Exponentially Many Linear Regions at Initialization and During Training

In a neural network with ReLU activations, the number of piecewise linear regions in the output can grow exponentially with depth. However, this is highly unlikely to happen when the initial parameters are sampled randomly, which therefore often leads to the use of networks that are unnecessarily large. To address this problem, we introduce a novel parameterization of the network that restricts its weights so that a depth $d$ network produces exactly $2^d$ linear regions at initialization and maintains those regions throughout training under the parameterization. This approach allows us to learn approximations of convex, one dimensional functions that are several orders of magnitude more accurate than their randomly initialized counterparts. We further demonstrate a preliminary extension of our construction to multidimensional and non-convex functions, allowing the technique to replace traditional dense layers in various architectures.

cs.LG↗

arXiv2023

QTOS: An Open-Source Quadruped Trajectory Optimization Stack

We introduce a new open-source framework, Quadruped Trajectory Optimization Stack (QTOS), which integrates a global planner, local planner, simulator, controller, and robot interface into a single package. QTOS serves as a full-stack interface, simplifying continuous motion planning on an open-source quadruped platform by bridging the gap between middleware and gait planning. It empowers users to effortlessly translate high-level navigation objectives into low-level robot commands. Furthermore, QTOS enhances the stability and adaptability of long-distance gait planning across challenging terrain.

cs.RO↗

arXiv2023

Contingency Games for Multi-Agent Interaction

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.

cs.RO↗

arXiv2022

Learning Mixed Strategies in Trajectory Games

In multi-agent settings, game theory is a natural framework for describing the strategic interactions of agents whose objectives depend upon one another's behavior. Trajectory games capture these complex effects by design. In competitive settings, this makes them a more faithful interaction model than traditional "predict then plan" approaches. However, current game-theoretic planning methods have important limitations. In this work, we propose two main contributions. First, we introduce an offline training phase which reduces the online computational burden of solving trajectory games. Second, we formulate a lifted game which allows players to optimize multiple candidate trajectories in unison and thereby construct more competitive "mixed" strategies. We validate our approach on a number of experiments using the pursuit-evasion game "tag."

cs.GT↗

arXiv2021

Incorporating Data Uncertainty in Object Tracking Algorithms

Methodologies for incorporating the uncertainties characteristic of data-driven object detectors into object tracking algorithms are explored. Object tracking methods rely on measurement error models, typically in the form of measurement noise, false positive rates, and missed detection rates. Each of these quantities, in general, can be dependent on object or measurement location. However, for detections generated from neural-network processed camera inputs, these measurement error statistics are not sufficient to represent the primary source of errors, namely a dissimilarity between run-time sensor input and the training data upon which the detector was trained. To this end, we investigate incorporating data uncertainty into object tracking methods such as to improve the ability to track objects, and particularly those which out-of-distribution w.r.t. training data. The proposed methodologies are validated on an object tracking benchmark as well on experiments with a real autonomous aircraft.

eess.SY↗

arXiv2021

The Computation of Approximate Generalized Feedback Nash Equilibria

We present the concept of a Generalized Feedback Nash Equilibrium (GFNE) in dynamic games, extending the Feedback Nash Equilibrium concept to games in which players are subject to state and input constraints. We formalize necessary and sufficient conditions for (local) GFNE solutions at the trajectory level, which enable the development of efficient numerical methods for their computation. Specifically, we propose a Newton-style method for finding game trajectories which satisfy necessary conditions for an equilibrium, which can then be checked against sufficiency conditions. We show that the evaluation of the necessary conditions in general requires computing a series of nested, implicitly-defined derivatives, which quickly becomes intractable. To this end, we introduce an approximation to the necessary conditions which is amenable to efficient evaluation, and in turn, computation of solutions. We term the solutions to the approximate necessary conditions Generalized Feedback Quasi-Nash Equilibria (GFQNE), and we introduce numerical methods for their computation. In particular, we develop a Sequential Linear-Quadratic Game approach, in which a LQ local approximation of the game is solved at each iteration. The development of this method relies on the ability to compute a GFNE to inequality- and equality-constrained LQ games, and therefore specific methods for the solution of these special cases are developed in detail. We demonstrate the effectiveness of the proposed solution approach on a dynamic game arising in an autonomous driving application.

math.OC↗

arXiv2020

Testing for Typicality with Respect to an Ensemble of Learned Distributions

Methods of performing anomaly detection on high-dimensional data sets are needed, since algorithms which are trained on data are only expected to perform well on data that is similar to the training data. There are theoretical results on the ability to detect if a population of data is likely to come from a known base distribution, which is known as the goodness-of-fit problem. One-sample approaches to this problem offer significant computational advantages for online testing, but require knowing a model of the base distribution. The ability to correctly reject anomalous data in this setting hinges on the accuracy of the model of the base distribution. For high dimensional data, learning an accurate-enough model of the base distribution such that anomaly detection works reliably is very challenging, as many researchers have noted in recent years. Existing methods for the one-sample goodness-of-fit problem do not account for the fact that a model of the base distribution is learned. To address that gap, we offer a theoretically motivated approach to account for the density learning procedure. In particular, we propose training an ensemble of density models, considering data to be anomalous if the data is anomalous with respect to any member of the ensemble. We provide a theoretical justification for this approach, proving first that a test on typicality is a valid approach to the goodness-of-fit problem, and then proving that for a correctly constructed ensemble of models, the intersection of typical sets of the models lies in the interior of the typical set of the base distribution. We present our method in the context of an example on synthetic data in which the effects we consider can easily be seen.

cs.LG↗

arXiv2020

Multi-Hypothesis Interactions in Game-Theoretic Motion Planning

We present a novel method for handling uncertainty about the intentions of non-ego players in dynamic games, with application to motion planning for autonomous vehicles. Equilibria in these games explicitly account for interaction among other agents in the environment, such as drivers and pedestrians. Our method models the uncertainty about the intention of other agents by constructing multiple hypotheses about the objectives and constraints of other agents in the scene. For each candidate hypothesis, we associate a Bernoulli random variable representing the probability of that hypothesis, which may or may not be independent of the probability of other hypotheses. We leverage constraint asymmetries and feedback information patterns to incorporate the probabilities of hypotheses in a natural way. Specifically, increasing the probability associated with a given hypothesis from $0$ to $1$ shifts the responsibility of collision avoidance from the hypothesized agent to the ego agent. This method allows the generation of interactive trajectories for the ego agent, where the level of assertiveness or caution that the ego exhibits is directly related to the easy-to-model uncertainty it maintains about the scene.

cs.RO↗

arXiv2020

Eyes-Closed Safety Kernels: Safety for Autonomous Systems Under Loss of Observability

A framework is presented for handling a potential loss of observability of a dynamical system in a provably-safe way. Inspired by the fragility of data-driven perception systems used by autonomous vehicles, we formulate the problem that arises when a sensing modality fails or is found to be untrustworthy during autonomous operation. We cast this problem as a differential game played between the dynamical system being controlled and the external system factor(s) for which observations are lost. The game is a zero-sum Stackelberg game in which the controlled system (leader) is trying to find a trajectory which maximizes a function representing the safety of the system, and the unobserved factor (follower) is trying to minimize the same function. The set of winning initial configurations of this game for the controlled system represent the set of all states in which safety can be maintained with respect to the external factor, even if observability of that factor is lost. This is the set we refer to as the Eyes-Closed Safety Kernel. In practical use, the policy defined by the winning strategy of the controlled system is only needed to be executed whenever observability of the external system is lost or the system deviates from the Eyes-Closed Safety Kernel due to other, non-safety oriented control schemes. We present a means for solving this game offline, such that the resulting winning strategy can be used for computationally efficient, provably-safe, online control when needed. The solution approach presented is based on representing the game using the solutions of two Hamilton-Jacobi partial differential equations. We illustrate the applicability of our framework by working through a realistic example in which an autonomous car must avoid a dynamic obstacle despite potentially losing observability.

eess.SY↗