arXiv subjects

Jason R. Marden

Publications and source records attributed to Jason R. Marden.

At least 55 records · Page 3Linked to original sources

arXiv2022

A General Lotto game with asymmetric budget uncertainty

The General Lotto game is a popular variant of the famous Colonel Blotto game, in which two opposing players allocate limited resources over many battlefields. In this paper, we consider incomplete and asymmetric information formulations regarding the resource budgets of the players. In particular, one of the player's resource budget is common knowledge while the other player's is private. We provide complete equilibrium characterizations in the scenario where the private resource budget is drawn from an arbitrary Bernoulli distribution. We then show that these characterizations can be used to analyze a multi-stage resource assignment problem where a commander must decide how to assign resources to sub-colonels that compete against opponents in separate General Lotto games. While optimal deterministic assignments have been characterized in the literature, we broaden the context by deriving optimal (Bernoulli) randomized assignments, which induce asymmetric information General Lotto games to be played. We demonstrate that randomizing can offer a four-fold improvement in the commander's performance over deterministic assignments.

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arXiv2022

Valid Utility Games with Information Sharing Constraints

The use of game theoretic methods for control in multiagent systems has been an important topic in recent research. Valid utility games in particular have been used to model real-world problems; such games have the convenient property that the value of any decision set which is a Nash equilibrium of the game is guaranteed to be within 1/2 of the value of the optimal decision set. However, an implicit assumption in this guarantee is that each agent is aware of the decisions of all other agents. In this work, we first describe how this guarantee degrades as agents are only aware of a subset of the decisions of other agents. We then show that this loss can be mitigated by restriction to a relevant subclass of games.

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arXiv2022

Strategic investments in multi-stage General Lotto games

In adversarial interactions, one is often required to make strategic decisions over multiple periods of time, wherein decisions made earlier impact a player's competitive standing as well as how choices are made in later stages. In this paper, we study such scenarios in the context of General Lotto games, which models the competitive allocation of resources over multiple battlefields between two players. We propose a two-stage formulation where one of the players has reserved resources that can be strategically pre-allocated across the battlefields in the first stage. The pre-allocation then becomes binding and is revealed to the other player. In the second stage, the players engage by simultaneously allocating their real-time resources against each other. The main contribution in this paper provides complete characterizations of equilibrium payoffs in the two-stage game, revealing the interplay between performance and the amount of resources expended in each stage of the game. We find that real-time resources are at least twice as effective as pre-allocated resources. We then determine the player's optimal investment when there are linear costs associated with purchasing each type of resource before play begins, and there is a limited monetary budget.

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arXiv2022

The Impact of Message Passing in Agent-Based Submodular Maximization

This paper considers a set of sensors, which as a group are tasked with taking measurements of the environment and sending a small subset of the measurements to a centralized data fusion center, where the measurements will be used to estimate the overall state of the environment. The sensors' goal is to send the most informative set of measurements so that the estimate is as accurate as possible. This problem is formulated as a submodular maximization problem, for which there exists a well-studied greedy algorithm, where each sensor sequentially chooses a set of measurements from its own local set, and communicates its decision to the future sensors in the sequence. In this work, sensors can additionally share measurements with one another, in order to augment the decision set of each sensor. We explore how this increase in communication can be exploited to improve the results of the nominal greedy algorithm. Specifically, we show that this measurement passing can improve the quality of the resulting measurement set by up to a factor of $n+1$, where $n$ is the number of sensors.

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arXiv2021

Strategically revealing intentions in General Lotto games

Strategic decision-making in uncertain and adversarial environments is crucial for the security of modern systems and infrastructures. A salient feature of many optimal decision-making policies is a level of unpredictability, or randomness, which helps to keep an adversary uncertain about the system's behavior. This paper seeks to explore decision-making policies on the other end of the spectrum -- namely, whether there are benefits in revealing one's strategic intentions to an opponent before engaging in competition. We study these scenarios in a well-studied model of competitive resource allocation problem known as General Lotto games. In the classic formulation, two competing players simultaneously allocate their assets to a set of battlefields, and the resulting payoffs are derived in a zero-sum fashion. Here, we consider a multi-step extension where one of the players has the option to publicly pre-commit assets in a binding fashion to battlefields before play begins. In response, the opponent decides which of these battlefields to secure (or abandon) by matching the pre-commitment with its own assets. They then engage in a General Lotto game over the remaining set of battlefields. Interestingly, this paper highlights many scenarios where strategically revealing intentions can actually significantly improve one's payoff. This runs contrary to the conventional wisdom that randomness should be a central component of decision-making in adversarial environments.

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arXiv2021

Balancing Asymptotic and Transient Efficiency Guarantees in Set Covering Games

Game theoretic approaches have gained traction as robust methodologies for designing distributed local algorithms that induce a desired overall system configuration in multi-agent settings. However, much of the emphasis in these approaches is on providing asymptotic guarantees on the performance of a network of agents, and there is a gap in the study of efficiency guarantees along transients of these distributed algorithms. Therefore, in this paper, we study the transient efficiency guarantees of a natural game-theoretic algorithm in the class of set covering games, which have been used to model a variety of applications. Our main results characterize the optimal utility design that maximizes the guaranteed efficiency along the transient of the natural dynamics. Furthermore, we characterize the Pareto-optimal frontier with regards to guaranteed efficiency in the transient and the asymptote under a class of game-theoretic designs. Surprisingly, we show that there exists an extreme trade-off between the long-term and short-term guarantees in that an asymptotically optimal game-theoretic design can perform arbitrarily bad in the transient.

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arXiv2021

Mission Level Uncertainty in Multi-Agent Resource Allocation

In recent years, a significant research effort has been devoted to the design of distributed protocols for the control of multi-agent systems, as the scale and limited communication bandwidth characteristic of such systems render centralized control impossible. Given the strict operating conditions, it is unlikely that every agent in a multi-agent system will have local information that is consistent with the true system state. Yet, the majority of works in the literature assume that agents share perfect knowledge of their environment. This paper focuses on understanding the impact that inconsistencies in agents' local information can have on the performance of multi-agent systems. More specifically, we consider the design of multi-agent operations under a game theoretic lens where individual agents are assigned utilities that guide their local decision making. We provide a tractable procedure for designing utilities that optimize the efficiency of the resulting collective behavior (i.e., price of anarchy) for classes of set covering games where the extent of the information inconsistencies is known. In the setting where the extent of the informational inconsistencies is not known, we show -- perhaps surprisingly -- that underestimating the level of uncertainty leads to better price of anarchy than overestimating it.

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arXiv2021

When Smoothness is Not Enough: Toward Exact Quantification and Optimization of the Price of Anarchy

The price of anarchy (PoA) is a popular metric for analyzing the inefficiency of self-interested decision making. Although its study is widespread, characterizing the PoA can be challenging. A commonly employed approach is based on the smoothness framework, which provides tight PoA values under the assumption that the system objective consists in the sum of the agents' individual welfares. Unfortunately, several important classes of problems do not satisfy this requirement (e.g., taxation in congestion games), and our first result demonstrates that the smoothness framework does *not* tightly characterize the PoA for such settings. Motivated by this observation, this work develops a framework that achieves two chief objectives: i) to tightly characterize the PoA for such scenarios, and ii) to do so through a tractable approach. As a direct consequence, the proposed framework recovers and generalizes many existing PoA results, and enables efficient computation of incentives that optimize the PoA. We conclude by highlighting the applicability of our contributions to incentive design in congestion games and utility design in distributed welfare games.

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arXiv2021

The Division of Assets in Multiagent Systems: A Case Study in Team Blotto Games

Multi-agent systems are designed to concurrently accomplish a diverse set of tasks at unprecedented scale. Here, the central problems faced by a system operator are to decide (i) how to divide available resources amongst the agents assigned to tasks and (ii) how to coordinate the behavior of the agents to optimize the efficiency of the resulting collective behavior. The focus of this paper is on problem (i), where we seek to characterize the impact of the division of resources on the best-case efficiency of the resulting collective behavior. Specifically, we focus on a team Colonel Blotto game where there are two sub-colonels competing against a common adversary in a two battlefield environment. Here, each sub-colonel is assigned a given resource budget and is required to allocate these resources independent of the other sub-colonel. However, their success is dependent on the allocation strategy of both sub-colonels. The central focus of this manuscript is on how to divide a common pool of resources among the two sub-colonels to optimize the resulting best-case efficiency guarantees. Intuitively, one would imagine that the more balanced the division of resources, the worse the performance, as such divisions restrict the sub-colonels' ability to employ joint randomized strategies that tend to be necessary for optimizing performance guarantees. However, the main result of this paper demonstrates that this intuition is actually incorrect. A more balanced division of resources can offer better performance guarantees than a more centralized division. Hence, this paper demonstrates that the resource division problem is highly non-trivial in such enmeshed environments and worthy of significant future research efforts.

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arXiv2021

The Effectiveness of Subsidies and Tolls in Congestion Games

Are rewards or penalties more effective in influencing user behavior? This work compares the effectiveness of subsidies and tolls in incentivizing user behavior in congestion games. The predominantly studied method of influencing user behavior in network routing problems is to institute taxes which alter users' observed costs in a manner that causes their self-interested choices to more closely align with a system-level objective. Another conceivable method to accomplish the same goal is to subsidize the users' actions that are preferable from a system-level perspective. We show that, when users behave similarly and predictably, subsidies offer superior performance guarantees to tolls under similar budgetary constraints; however, in the presence of unknown player heterogeneity, subsidies fail to offer the same robustness as tolls.

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arXiv2021

Tractable mechanisms for computing near-optimal utility functions

Large scale multiagent systems must rely on distributed decision making, as centralized coordination is either impractical or impossible. Recent works approach this problem under a game theoretic lens, whereby utility functions are assigned to each of the agents with the hope that their local optimization approximates the centralized optimal solution. Yet, formal guarantees on the resulting performance cannot be obtained for broad classes of problems without compromising on their accuracy. In this work, we address this concern relative to the well-studied problem of resource allocation with nondecreasing concave welfare functions. We show that optimally designed local utilities achieve an approximation ratio (price of anarchy) of 1-c/e, where c is the function's curvature and e is Euler's constant. The upshot of our contributions is the design of approximation algorithms that are distributed and efficient, and whose performance matches that of the best existing polynomial-time (and centralized) schemes.

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arXiv2021

Utility Design for Distributed Resource Allocation -- Part II: Applications to Submodular, Covering, and Supermodular Problems

A fundamental component of the game theoretic approach to distributed control is the design of local utility functions.Relative to resource allocation problems that are additive over the resources, Part I showed how to design local utilities so as to maximize the associated performance guarantees [Paccagnan et al., TAC 2019] which we measure by the price of anarchy. The purpose of the present manuscript is to specialize these results to the case of submodular, covering, and supermodular problems. In all these cases we obtain tight expressions for the price of anarchy that often match or improve the guarantees associated to state-of-the-art approximation algorithms. Two applications and corresponding numerics are presented: the vehicle-target assignment problem and a coverage problem arising in wireless data caching.

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arXiv2020

The Impact of Complex and Informed Adversarial Behavior in Graphical Coordination Games

How does system-level information impact the ability of an adversary to degrade performance in a networked control system? How does the complexity of an adversary's strategy affect its ability to degrade performance? This paper focuses on these questions in the context of graphical coordination games where an adversary can influence a given fraction of the agents in the system, and the agents follow log-linear learning, a well-known distributed learning algorithm. Focusing on a class of homogeneous ring graphs of various connectivity, we begin by demonstrating that minimally connected ring graphs are the most susceptible to adversarial influence. We then proceed to characterize how both (i) the sophistication of the attack strategies (static vs dynamic) and (ii) the informational awareness about the network structure can be leveraged by an adversary to degrade system performance. Focusing on the set of adversarial policies that induce stochastically stable states, our findings demonstrate that the relative importance between sophistication and information changes depending on the the influencing power of the adversary. In particular, sophistication far outweighs informational awareness with regards to degrading system-level damage when the adversary's influence power is relatively weak. However, the opposite is true when an adversary's influence power is more substantial.

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arXiv2020

The Cost of Denied Observation in Multiagent Submodular Optimization

A popular formalism for multiagent control applies tools from game theory, casting a multiagent decision problem as a cooperation-style game in which individual agents make local choices to optimize their own local utility functions in response to the observable choices made by other agents. When the system-level objective is submodular maximization, it is known that if every agent can observe the action choice of all other agents, then all Nash equilibria of a large class of resulting games are within a factor of $2$ of optimal; that is, the price of anarchy is $1/2$. However, little is known if agents cannot observe the action choices of other relevant agents. To study this, we extend the standard game-theoretic model to one in which a subset of agents either become \emph{blind} (unable to observe others' choices) or \emph{isolated} (blind, and also invisible to other agents), and we prove exact expressions for the price of anarchy as a function of the number of compromised agents. When $k$ agents are compromised (in any combination of blind or isolated), we show that the price of anarchy for a large class of utility functions is exactly $1/(2+k)$. We then show that if agents use marginal-cost utility functions and at least $1$ of the compromised agents is blind (rather than isolated), the price of anarchy improves to $1/(1+k)$. We also provide simulation results demonstrating the effects of these observation denials in a dynamic setting.

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arXiv2020

Policy Evaluation and Seeking for Multi-Agent Reinforcement Learning via Best Response

This paper introduces two metrics (cycle-based and memory-based metrics), grounded on a dynamical game-theoretic solution concept called sink equilibrium, for the evaluation, ranking, and computation of policies in multi-agent learning. We adopt strict best response dynamics (SBRD) to model selfish behaviors at a meta-level for multi-agent reinforcement learning. Our approach can deal with dynamical cyclical behaviors (unlike approaches based on Nash equilibria and Elo ratings), and is more compatible with single-agent reinforcement learning than alpha-rank which relies on weakly better responses. We first consider settings where the difference between largest and second largest underlying metric has a known lower bound. With this knowledge we propose a class of perturbed SBRD with the following property: only policies with maximum metric are observed with nonzero probability for a broad class of stochastic games with finite memory. We then consider settings where the lower bound for the difference is unknown. For this setting, we propose a class of perturbed SBRD such that the metrics of the policies observed with nonzero probability differ from the optimal by any given tolerance. The proposed perturbed SBRD addresses the opponent-induced non-stationarity by fixing the strategies of others for the learning agent, and uses empirical game-theoretic analysis to estimate payoffs for each strategy profile obtained due to the perturbation.

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arXiv2020

Multiagent Maximum Coverage Problems: The Trade-off Between Anarchy and Stability

The price of anarchy and price of stability are three well-studied performance metrics that seek to characterize the inefficiency of equilibria in distributed systems. The distinction between these two performance metrics centers on the equilibria that they focus on: the price of anarchy characterizes the quality of the worst-performing equilibria, while the price of stability characterizes the quality of the best-performing equilibria. While much of the literature focuses on these metrics from an analysis perspective, in this work we consider these performance metrics from a design perspective. Specifically, we focus on the setting where a system operator is tasked with designing local utility functions to optimize these performance metrics in a class of games termed covering games. Our main result characterizes a fundamental trade-off between the price of anarchy and price of stability in the form of a fully explicit Pareto frontier. Within this setup, optimizing the price of anarchy comes directly at the expense of the price of stability (and vice versa). Our second results demonstrates how a system-operator could incorporate an additional piece of system-level information into the design of the agents' utility functions to breach these limitations and improve the system's performance. This valuable piece of system-level information pertains to the performance of worst performing agent in the system.

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arXiv2020

When showing your hand pays off: Announcing strategic intentions in Colonel Blotto games

In competitive adversarial environments, it is often advantageous to obfuscate one's strategies or capabilities. However, revealing one's strategic intentions may shift the dynamics of the competition in complex ways. Can it ever be advantageous to reveal strategic intentions to an opponent? In this paper, we consider three-stage Colonel Blotto games in which one player can choose whether or not to pre-commit resources to a single battlefield before play begins. This pre-commitment is public knowledge. In response, the opponent can either secure the battlefield by matching the pre-commitment with its own forces, or withdraw. In a two-player setting, we show that a weaker player never has an incentive to pre-commit any amount of resources to a battlefield regardless of how valuable it is. We then consider a three-player setting in which two players fight against a common adversary on separate fronts. Only one of the two players facing the adversary has the option of pre-committing. We find there are instances where this player benefits from pre-committing. The analysis indicates that under non-cooperative team settings and no possibility of forming alliances, there can be incentives to publicly announce one's strategic intentions to an adversary.

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arXiv2020

When Smoothness is Not Enough: Toward Exact Quantification and Optimization of the Price-of-Anarchy

Today's multiagent systems have grown too complex to rely on centralized controllers, prompting increasing interest in the design of distributed algorithms. In this respect, game theory has emerged as a valuable tool to complement more traditional techniques. The fundamental idea behind this approach is the assignment of agents' local cost functions, such that their selfish minimization attains, or is provably close to, the global objective. Any algorithm capable of computing an equilibrium of the corresponding game inherits an approximation ratio that is, in the worst case, equal to its price-of-anarchy. Therefore, a successful application of the game design approach hinges on the possibility to quantify and optimize the equilibrium performance. Toward this end, we introduce the notion of generalized smoothness, and show that the resulting efficiency bounds are significantly tighter compared to those obtained using the traditional smoothness approach. Leveraging this newly-introduced notion, we quantify the equilibrium performance for the class of local resource allocation games. Finally, we show how the agents' local decision rules can be designed in order to optimize the efficiency of the corresponding equilibria, by means of a tractable linear program.

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