Zhengyuan Zhou

A general, open-loop formulation for reach-avoid games

A reach-avoid game is one in which an agent attempts to reach a predefined goal, while avoiding some adversarial circumstance induced by an opposing agent or disturbance. Their analysis plays an important role in problems such as safe motion planning and obstacle avoidance, yet computing solutions is often computationally expensive due to the need to consider adversarial inputs. In this work, we present an open-loop formulation of a two-player reach-avoid game whereby the players define their control inputs prior to the start of the game. We define two open-loop games, each of which is conservative towards one player, show how the solutions to these games are related to the optimal feedback strategy for the closed-loop game, and demonstrate a modified Fast Marching Method to efficiently compute those solutions.

Multiplayer reach-avoid games via low dimensional solutions and maximum matching

We consider a multiplayer reach-avoid game with an equal number of attackers and defenders moving with simple dynamics on a two-dimensional domain possibly with obstacles. The attacking team attempts to send as many attackers to a certain target location as possible quickly while the defenders aim to capture the attackers to prevent the attacking team from reaching its goal. The analysis of problems like this plays an important role in collision avoidance, motion planning, and aircraft control, among other applications. Computing optimal solutions for such multiplayer games is intractable due to numerical intractibility. This paper provides a first attempt to address such computational intractability by combining maximum matching in graph theory with the classical Hamilton-Jacobi-Isaacs approach. In addition, our solution provides an initial step to take cooperation into account by computing maximum matching in real time.

A general model for resource allocation in utility computing

A utility computing problem is one in which a server (service provider) provides computing resources to clients (service receivers) whose jobs require the resources for processing. In this paper, we propose a general, decentralized and auction-based model for the server-to-clients resource allocation problem. This model combines a general class of queueing processes with a general class of “incentivecompatible” bidding mechanisms. Crucial to this model is the interplay between the nature of queueing costs and the nature of good bidding mechanisms. Insights on how such interplay contributes to the stability of the system can be helpful in guiding specific implementations in real-world applications. This decentralized and auction-based resource allocation model naturally induces a multi-player game, which is the principal object we study in this paper. We show the existence and uniqueness of Nash equilibrium under this general setting. We also present distributed update dynamics that converge to this unique Nash equilibrium. The distributed dynamics given here share the features of being secure and requiring little communications, thus, providing a practical scheme through which Nash equilibrium is reached.

Wireless communications games in fixed and random environments

Most of the work on power control in wireless communications has focused on the fixed deterministic (thermal) noise paradigm, which results in elegant distributed power control schemes with desired convergence properties. In this paper, we lift the deterministic noise assumption. Instead, we consider a generalized stochastic noise framework, akin to a random environment, which not only incorporates the intrinsic thermal noise at the receivers but also the effect of potential interferers that are extraneous to the communication links under consideration. Given this random environment framework, we develop a game-theoretic formulation where N links play a non-cooperative game, inducing a distributed power control scheme. We first examine the deterministic game and show the existence and uniqueness of the Nash equilibrium. We then study the stochastic behavior of the equilibrium under the random environment. The results indicate that the long-run behavior of the corresponding power control is “stable” (explained technically below), demonstrating its robustness and, hence, applicability.

Target-rate driven resource sharing in queueing systems

We consider the problem of multiple packet streams being fed into queues with finite buffers and competing for processing bandwidth. Due to finite buffer size, the packets may be dropped (when buffers become full); therefore the effective throughput of each stream that flows through a server is impacted by the rates at which other streams come into the server. In this paper, we study the case where each source needs to maintain a target throughput in order to provide satisfactory user experience at the destination. We give a simple and efficient distributed update scheme for source rate control: the source adjusts its rate of sending packets based on the observed throughput rate at the destination in each round. We provide the conditions under which the scheme converges to the target throughput in both the single-queue case and the network case.

Dynamics on Linear Influence Network Games Under Stochastic Environments

A linear influence network is a broadly applicable conceptual framework in risk management. It has important applications in computer and network security. Prior work on linear influence networks targeting those risk management applications have been focused on equilibrium analysis in a static, one-shot setting. Furthermore, the underlying network environment is also assumed to be deterministic. In this paper, we lift those two assumptions and consider a formulation where the network environment is stochastic and time-varying. In particular, we study the stochastic behavior of the well-known best response dynamics. Specifically, we give interpretable and easily verifiable sufficient conditions under which we establish the existence and uniqueness of as well as convergence with exponential convergence rate to a stationary distribution of the corresponding Markov chains.

A path defense approach to the multiplayer reach-avoid game

We consider a multiplayer reach-avoid game played between N attackers and N defenders moving with simple dynamics on a general two-dimensional domain. The attackers attempt to win the game by sending at least M of them (1 ≤ M ≤ N) to a target location while the defenders try to prevent the attackers from doing so by capturing them. The analysis of this game plays an important role in collision avoidance, motion planning, and aircraft control, among other applications involving cooperative agents. The high dimensionality of the game makes computing an optimal solution for either side intractable when N > 1. The solution is difficult even when N = 1. To address this issue, we present an efficient, approximate solution to the 1 vs. 1 problem. We call the approximate solution the “path defense solution”, which is conservative towards the defenders. This serves as a building block for an approximate solution of the multiplayer game. Compared to the classical Hamilton-Jacobi-Isaacs approach for solving the 1 vs. 1 game, our new method is orders of magnitude faster, and scales much better with the number of players.

Semigroups of sl3(C) tensor product invariants

Abstract We find presentations for a family of semigroup algebras related to the problem of decomposing sl 3 ( C ) tensor products. Along the way we find new toric degenerations of the Grassmannian variety Gr 3 ( C n ) which are T-invariant for T ⊂ GL n ( C ) the diagonal torus.

Learning in games with continuous action sets and unknown payoff functions

This paper examines the convergence of no-regret learning in games with continuous action sets. For concreteness, we focus on learning via “dual averaging”, a widely used class of no-regret learning schemes where players take small steps along their individual payoff gradients and then “mirror” the output back to their action sets. In terms of feedback, we assume that players can only estimate their payoff gradients up to a zero-mean error with bounded variance. To study the convergence of the induced sequence of play, we introduce the notion of variational stability, and we show that stable equilibria are locally attracting with high probability whereas globally stable equilibria are globally attracting with probability 1. We also discuss some applications to mixed-strategy learning in finite games, and we provide explicit estimates of the method’s convergence speed.

An infinite dimensional model for a many server priority queue

We consider a Markovian many server queueing system in which customers are preemptively scheduled according to exogenously assigned priority levels. The priority levels are randomly assigned from a continuous probability measure rather than a discrete one and hence, the queue is modeled by an infinite dimensional stochastic process. We analyze the equilibrium behavior of the system and provide several results. We derive the Radon-Nikodym derivative (with respect to Lebesgue measure) of the measure that describes the average distribution of customer priority levels in the system; we provide a formula for the expected sojourn time of a customer as a function of his priority level; and we provide a formula for the expected waiting time of a customer as a function of his priority level. We verify our theoretical analysis with discrete-event simulations. We discuss how each of our results generalizes previous work on infinite dimensional models for single server priority queues.