3,755 research outputs found
Sharing Non-Anonymous Costs of Multiple Resources Optimally
In cost sharing games, the existence and efficiency of pure Nash equilibria
fundamentally depends on the method that is used to share the resources' costs.
We consider a general class of resource allocation problems in which a set of
resources is used by a heterogeneous set of selfish users. The cost of a
resource is a (non-decreasing) function of the set of its users. Under the
assumption that the costs of the resources are shared by uniform cost sharing
protocols, i.e., protocols that use only local information of the resource's
cost structure and its users to determine the cost shares, we exactly quantify
the inefficiency of the resulting pure Nash equilibria. Specifically, we show
tight bounds on prices of stability and anarchy for games with only submodular
and only supermodular cost functions, respectively, and an asymptotically tight
bound for games with arbitrary set-functions. While all our upper bounds are
attained for the well-known Shapley cost sharing protocol, our lower bounds
hold for arbitrary uniform cost sharing protocols and are even valid for games
with anonymous costs, i.e., games in which the cost of each resource only
depends on the cardinality of the set of its users
The Impact of Marginal Cost Pricing in Resource Allocation Games
A prelimiary version of this paper titled Efficiency and Stability of Nash Equilibria in Resource Allocation Games appeared in the Proceedings of the First International Conference on Game Theory for Networks (GAMENETS), 2009.We study resource allocation games, where users send data along paths and links in the network charge a price equal to marginal cost. When users are price taking, it is known that there exist distributed dynamics that converge towards a fully efficient Nash equilibrium. When users are price anticipating, however, a Nash equilibrium does not maximize total utility in general. In this paper, we explore the inefficiency of Nash equilibria for general networks and semi-convex marginal cost functions. While it is known that for mgeq 2 users, no efficiency guarantee is possible, we prove that an additional differentiability assumption on marginal cost functions implies a bounded efficiency loss of 2/(2m+1). For polynomial marginal cost functions with nonnegative coefficients we precisely characterize the price of anarchy. We also prove that the efficiency of Nash equilibria significantly improves if all users have the same utility function.
We propose a class of distributed dynamics and prove that whenever a game admits a potential function, these dynamics globally converge to a Nash equilibrium. Finally, we show that in general the only} class of marginal cost functions that guarantees the existence of a potential function are affine linear functions
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
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.Comment: 14 pages, 4 figure
Strong Nash Equilibria in Games with the Lexicographical Improvement Property
We introduce a class of finite strategic games with the property that every
deviation of a coalition of players that is profitable to each of its members
strictly decreases the lexicographical order of a certain function defined on
the set of strategy profiles. We call this property the Lexicographical
Improvement Property (LIP) and show that it implies the existence of a
generalized strong ordinal potential function. We use this characterization to
derive existence, efficiency and fairness properties of strong Nash equilibria.
We then study a class of games that generalizes congestion games with
bottleneck objectives that we call bottleneck congestion games. We show that
these games possess the LIP and thus the above mentioned properties. For
bottleneck congestion games in networks, we identify cases in which the
potential function associated with the LIP leads to polynomial time algorithms
computing a strong Nash equilibrium. Finally, we investigate the LIP for
infinite games. We show that the LIP does not imply the existence of a
generalized strong ordinal potential, thus, the existence of SNE does not
follow. Assuming that the function associated with the LIP is continuous,
however, we prove existence of SNE. As a consequence, we prove that bottleneck
congestion games with infinite strategy spaces and continuous cost functions
possess a strong Nash equilibrium
Resource Competition on Integral Polymatroids
We study competitive resource allocation problems in which players distribute
their demands integrally on a set of resources subject to player-specific
submodular capacity constraints. Each player has to pay for each unit of demand
a cost that is a nondecreasing and convex function of the total allocation of
that resource. This general model of resource allocation generalizes both
singleton congestion games with integer-splittable demands and matroid
congestion games with player-specific costs. As our main result, we show that
in such general resource allocation problems a pure Nash equilibrium is
guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure
Nash equilibrium.Comment: 17 page
Tighter Bounds on the Inefficiency Ratio of Stable Equilibria in Load Balancing Games
In this paper we study the inefficiency ratio of stable equilibria in load
balancing games introduced by Asadpour and Saberi [3]. We prove tighter lower
and upper bounds of 7/6 and 4/3, respectively. This improves over the best
known bounds in problem (19/18 and 3/2, respectively). Equivalently, the
results apply to the question of how well the optimum for the -norm can
approximate the -norm (makespan) in identical machines scheduling
Designing cost-sharing methods for Bayesian games
We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria, have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE).We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players
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