9,965 research outputs found
Game Efficiency Through Linear Programming Duality
The efficiency of a game is typically quantified by the price of anarchy (PoA), defined as the worst ratio of the value of an equilibrium - solution of the game - and that of an optimal outcome. Given the tremendous impact of tools from mathematical programming in the design of algorithms and the similarity of the price of anarchy and different measures such as the approximation and competitive ratios, it is intriguing to develop a duality-based method to characterize the efficiency of games.
In the paper, we present an approach based on linear programming duality to study the efficiency of games. We show that the approach provides a general recipe to analyze the efficiency of games and also to derive concepts leading to improvements. The approach is particularly appropriate to bound the PoA. Specifically, in our approach the dual programs naturally lead to competitive PoA bounds that are (almost) optimal for several classes of games. The approach indeed captures the smoothness framework and also some current non-smooth techniques/concepts. We show the applicability to the wide variety of games and environments, from congestion games to Bayesian welfare, from full-information settings to incomplete-information ones
Bayes correlated equilibria and no-regret dynamics
This paper explores equilibrium concepts for Bayesian games, which are
fundamental models of games with incomplete information. We aim at three
desirable properties of equilibria. First, equilibria can be naturally realized
by introducing a mediator into games. Second, an equilibrium can be computed
efficiently in a distributed fashion. Third, any equilibrium in that class
approximately maximizes social welfare, as measured by the price of anarchy,
for a broad class of games. These three properties allow players to compute an
equilibrium and realize it via a mediator, thereby settling into a stable state
with approximately optimal social welfare. Our main result is the existence of
an equilibrium concept that satisfies these three properties.
Toward this goal, we characterize various (non-equivalent) extensions of
correlated equilibria, collectively known as Bayes correlated equilibria. In
particular, we focus on communication equilibria (also known as coordination
mechanisms), which can be realized by a mediator who gathers each player's
private information and then sends correlated recommendations to the players.
We show that if each player minimizes a variant of regret called untruthful
swap regret in repeated play of Bayesian games, the empirical distribution of
these dynamics converges to a communication equilibrium. We present an
efficient algorithm for minimizing untruthful swap regret with a sublinear
upper bound, which we prove to be tight up to a multiplicative constant. As a
result, by simulating the dynamics with our algorithm, we can efficiently
compute an approximate communication equilibrium. Furthermore, we extend
existing lower bounds on the price of anarchy based on the smoothness arguments
from Bayes Nash equilibria to equilibria obtained by the proposed dynamics
Bargaining Mechanisms for One-Way Games
We introduce one-way games, a framework motivated by applications in
large-scale power restoration, humanitarian logistics, and integrated
supply-chains. The distinguishable feature of the games is that the payoff of
some player is determined only by her own strategy and does not depend on
actions taken by other players. We show that the equilibrium outcome in one-way
games without payments and the social cost of any ex-post efficient mechanism,
can be far from the optimum. We also show that it is impossible to design a
Bayes-Nash incentive-compatible mechanism for one-way games that is
budget-balanced, individually rational, and efficient. To address this negative
result, we propose a privacy-preserving mechanism that is incentive-compatible
and budget-balanced, satisfies ex-post individual rationality conditions, and
produces an outcome which is more efficient than the equilibrium without
payments. The mechanism is based on a single-offer bargaining and we show that
a randomized multi-offer extension brings no additional benefit.Comment: An earlier, shorter version of this paper appeared in Proceedings of
the Twenty-Fourth International joint conference on Artificial Intelligence
(IJCAI) 201
Network uncertainty in selfish routing
We study the problem of selfish routing in the presence of incomplete network information. Our model consists of a number of users who wish to route their traffic on a network of m parallel links with the objective of minimizing their latency. However, in doing so, they face the challenge of lack of precise information on the capacity of the network links. This uncertainty is modelled via a set of probability distributions over all the possibilities, one for each user. The resulting model is an amalgamation of the KP-model of [13] and the congestion games with user-specific functions of [17]. We embark on a study of Nash equilibria and the price of anarchy in this new model. In particular, we propose polynomial-time algorithms for computing some special cases of pure Nash equilibria and we show that negative results of [17], for the non-existence of pure Nash equilibria in the case of three users, do not apply to our model. Consequently, we propose an interesting open problem in this area, that of the existence of pure Nash equilibria in the general case of our model. Furthermore, we consider appropriate notions for the social cost and the price of anarchy and obtain upper bounds for the latter. With respect to fully mixed Nash equilibria, we propose a method to compute them and show that when they exist they are unique. Finally we prove that the fully mixed Nash equilibrium maximizes the social welfare. 1
Price of Anarchy in Bernoulli Congestion Games with Affine Costs
We consider an atomic congestion game in which each player participates in
the game with an exogenous and known probability , independently
of everybody else, or stays out and incurs no cost. We first prove that the
resulting game is potential. Then, we compute the parameterized price of
anarchy to characterize the impact of demand uncertainty on the efficiency of
selfish behavior. It turns out that the price of anarchy as a function of the
maximum participation probability is a nondecreasing
function. The worst case is attained when players have the same participation
probabilities . For the case of affine costs, we provide an
analytic expression for the parameterized price of anarchy as a function of
. This function is continuous on , is equal to for , and increases towards when . Our work can be interpreted as
providing a continuous transition between the price of anarchy of nonatomic and
atomic games, which are the extremes of the price of anarchy function we
characterize. We show that these bounds are tight and are attained on routing
games -- as opposed to general congestion games -- with purely linear costs
(i.e., with no constant terms).Comment: 29 pages, 6 figure
Welfare guarantees for proportional allocations
According to the proportional allocation mechanism from the network
optimization literature, users compete for a divisible resource -- such as
bandwidth -- by submitting bids. The mechanism allocates to each user a
fraction of the resource that is proportional to her bid and collects an amount
equal to her bid as payment. Since users act as utility-maximizers, this
naturally defines a proportional allocation game. Recently, Syrgkanis and
Tardos (STOC 2013) quantified the inefficiency of equilibria in this game with
respect to the social welfare and presented a lower bound of 26.8% on the price
of anarchy over coarse-correlated and Bayes-Nash equilibria in the full and
incomplete information settings, respectively. In this paper, we improve this
bound to 50% over both equilibrium concepts. Our analysis is simpler and,
furthermore, we argue that it cannot be improved by arguments that do not take
the equilibrium structure into account. We also extend it to settings with
budget constraints where we show the first constant bound (between 36% and 50%)
on the price of anarchy of the corresponding game with respect to an effective
welfare benchmark that takes budgets into account.Comment: 15 page
The Evolutionary Price of Anarchy: Locally Bounded Agents in a Dynamic Virus Game
The Price of Anarchy (PoA) is a well-established game-theoretic concept to shed light on coordination issues arising in open distributed systems. Leaving agents to selfishly optimize comes with the risk of ending up in sub-optimal states (in terms of performance and/or costs), compared to a centralized system design. However, the PoA relies on strong assumptions about agents\u27 rationality (e.g., resources and information) and interactions, whereas in many distributed systems agents interact locally with bounded resources. They do so repeatedly over time (in contrast to "one-shot games"), and their strategies may evolve.
Using a more realistic evolutionary game model, this paper introduces a realized evolutionary Price of Anarchy (ePoA). The ePoA allows an exploration of equilibrium selection in dynamic distributed systems with multiple equilibria, based on local interactions of simple memoryless agents.
Considering a fundamental game related to virus propagation on networks, we present analytical bounds on the ePoA in basic network topologies and for different strategy update dynamics. In particular, deriving stationary distributions of the stochastic evolutionary process, we find that the Nash equilibria are not always the most abundant states, and that different processes can feature significant off-equilibrium behavior, leading to a significantly higher ePoA compared to the PoA studied traditionally in the literature
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