Utility Design for Distributed Resource Allocation -- Part I: Characterizing and Optimizing the Exact Price of Anarchy

Game theory has emerged as a fruitful paradigm for the design of networked multiagent systems. A fundamental component of this approach is the design of agents' utility functions so that their self-interested maximization results in a desirable collective behavior. In this work we focus on a well-studied class of distributed resource allocation problems where each agent is requested to select a subset of resources with the goal of optimizing a given system-level objective. Our core contribution is the development of a novel framework to tightly characterize the worst case performance of any resulting Nash equilibrium (price of anarchy) as a function of the chosen agents' utility functions. Leveraging this result, we identify how to design such utilities so as to optimize the price of anarchy through a tractable linear program. This provides us with a priori performance certificates applicable to any existing learning algorithm capable of driving the system to an equilibrium. Part II of this work specializes these results to submodular and supermodular objectives, discusses the complexity of computing Nash equilibria, and provides multiple illustrations of the theoretical findings.Comment: 15 pages, 5 figure

A risk-security tradeoff in graphical coordination games

A system relying on the collective behavior of decision-makers can be vulnerable to a variety of adversarial attacks. How well can a system operator protect performance in the face of these risks? We frame this question in the context of graphical coordination games, where the agents in a network choose among two conventions and derive benefits from coordinating neighbors, and system performance is measured in terms of the agents' welfare. In this paper, we assess an operator's ability to mitigate two types of adversarial attacks - 1) broad attacks, where the adversary incentivizes all agents in the network and 2) focused attacks, where the adversary can force a selected subset of the agents to commit to a prescribed convention. As a mitigation strategy, the system operator can implement a class of distributed algorithms that govern the agents' decision-making process. Our main contribution characterizes the operator's fundamental trade-off between security against worst-case broad attacks and vulnerability from focused attacks. We show that this tradeoff significantly improves when the operator selects a decision-making process at random. Our work highlights the design challenges a system operator faces in maintaining resilience of networked distributed systems.Comment: 13 pages, double column, 4 figures. Submitted for journal publicatio

Joint strategy fictitious play with inertia for potential games

We consider multi-player repeated games involving a large number of players with large strategy spaces and enmeshed utility structures. In these ldquolarge-scalerdquo games, players are inherently faced with limitations in both their observational and computational capabilities. Accordingly, players in large-scale games need to make their decisions using algorithms that accommodate limitations in information gathering and processing. This disqualifies some of the well known decision making models such as ldquoFictitious Playrdquo (FP), in which each player must monitor the individual actions of every other player and must optimize over a high dimensional probability space. We will show that Joint Strategy Fictitious Play (JSFP), a close variant of FP, alleviates both the informational and computational burden of FP. Furthermore, we introduce JSFP with inertia, i.e., a probabilistic reluctance to change strategies, and establish the convergence to a pure Nash equilibrium in all generalized ordinal potential games in both cases of averaged or exponentially discounted historical data. We illustrate JSFP with inertia on the specific class of congestion games, a subset of generalized ordinal potential games. In particular, we illustrate the main results on a distributed traffic routing problem and derive tolling procedures that can lead to optimized total traffic congestion

Cooperative Control and Potential Games

We present a view of cooperative control using the language of learning in games. We review the game-theoretic concepts of potential and weakly acyclic games, and demonstrate how several cooperative control problems, such as consensus and dynamic sensor coverage, can be formulated in these settings. Motivated by this connection, we build upon game-theoretic concepts to better accommodate a broader class of cooperative control problems. In particular, we extend existing learning algorithms to accommodate restricted action sets caused by the limitations of agent capabilities and group based decision making. Furthermore, we also introduce a new class of games called sometimes weakly acyclic games for time-varying objective functions and action sets, and provide distributed algorithms for convergence to an equilibrium

A Ten-Year Evaluation of Structural and Compositional Development In Unmanaged Central Appalachian Clearcuts

The varied growing conditions, fractured topography, and diversity of woody species in the Central Appalachians ensure that the harvest response of forest stands in the region will never be easily predictable. Clearcutting exposes the seedbed and freshly cut stumps of a harvested forest to full sunlight, encouraging the regeneration of the greatest number of species possible for that region. Gathering information on stand response to a complete removal in the mixed mesophytic forests of West Virginia will add valuable information to the silvicultural database for the most complex hardwood ecosystem in North America.;Beginning in 1998, U.S. Forest Service personnel installed permanent plots in the Cheat Ranger District of the Monongahela National Forest and began recording observations of early growth response in over 90 even age stands. Utilizing a subset of that initial data, and data from subsequent inventories in 2004 and 2009, this project evaluated the 10 year development of 15 of those clearcuts. This study answers the question of how these even age forests changed over time, without stand management.;Results demonstrated the diverse regeneration of over 40 woody species following clearcutting. Yellow-poplar and oaks were the dominant species in the overstory. Yellow-poplar was more important on mesic and southerly aspects, and oak dominated the xeric aspects. Significant increases in basal area/acre, coinciding with significant decreases in tree density, were sustained in stands of up to 30 years of age. Best basal area growth occurred on southerly stands, followed by summit and westerly stands. Poorest basal area development was exhibited by mesic sites, where wild grapevines had an adverse effect. Grapevines were established on 11 of 15 stands in 2009. Only statistically non-significant changes in grapevine levels were exhibited over the study period in all stands combined. This indicated that vine establishment occurs early in stand development, between harvest and canopy closure. Grapevine damage was more severe on mesic aspects than on southerly and westerly aspects, and summits

Characterizing the interplay between information and strength in Blotto games

In this paper, we investigate informational asymmetries in the Colonel Blotto game, a game-theoretic model of competitive resource allocation between two players over a set of battlefields. The battlefield valuations are subject to randomness. One of the two players knows the valuations with certainty. The other knows only a distribution on the battlefield realizations. However, the informed player has fewer resources to allocate. We characterize unique equilibrium payoffs in a two battlefield setup of the Colonel Blotto game. We then focus on a three battlefield setup in the General Lotto game, a popular variant of the Colonel Blotto game. We characterize the unique equilibrium payoffs and mixed equilibrium strategies. We quantify the value of information - the difference in equilibrium payoff between the asymmetric information game and complete information game. We find information strictly improves the informed player's performance guarantee. However, the magnitude of improvement varies with the informed player's strength as well as the game parameters. Our analysis highlights the interplay between strength and information in adversarial environments.Comment: 8 pages, 2 figures. Accepted for presentation at 58th Conference on Decision and Control (CDC), 201