67,269 research outputs found
Approximate Nash Equilibria Algorithms for Shapley Network Design Games
We consider a weighted Shapley network design game, where selfish players
choose paths in a network to minimize their cost. The cost function of each
edge in the network is affine linear with respect to the sum of weights of the
players who choose the edge. We first show the existence of \alpha-approximate
pure Nash equilibrium by constructing a potential function and establish an
upper bound O(log2(W)) of \alpha, where W is the sum of the weight of all
players. Furthermore, we assume that the coefficients of the cost function
(affine linear function) of the edge all are \phi-smooth random variables on
[0, 1]. In this case, we show that \epsilon-best response dynamics can compute
the (1 + \epsilon)\alpha-approximate pure Nash equilibrium (\epsilon is a
positive constant close to 0) in polynomial time by proving the expected number
of iterations is polynomial in 1/\epsilon, \phi, the number of players and the
number of edges in the network
Designing Network Protocols for Good Equilibria
Designing and deploying a network protocol determines the rules by which end users interact with each other and with the network. We consider the problem of designing a protocol to optimize the equilibrium behavior of a network with selfish users. We consider network cost-sharing games, where the set of Nash equilibria depends fundamentally on the choice of an edge cost-sharing protocol. Previous research focused on the Shapley protocol, in which the cost of each edge is shared equally among its users. We systematically study the design of optimal cost-sharing protocols for undirected and directed graphs, single-sink and multicommodity networks, and different measures of the inefficiency of equilibria. Our primary technical tool is a precise characterization of the cost-sharing protocols that induce only network games with pure-strategy Nash equilibria. We use this characterization to prove, among other results, that the Shapley protocol is optimal in directed graphs and that simple priority protocols are essentially optimal in undirected graphs
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
Resource Buying Games
In resource buying games a set of players jointly buys a subset of a finite
resource set E (e.g., machines, edges, or nodes in a digraph). The cost of a
resource e depends on the number (or load) of players using e, and has to be
paid completely by the players before it becomes available. Each player i needs
at least one set of a predefined family S_i in 2^E to be available. Thus,
resource buying games can be seen as a variant of congestion games in which the
load-dependent costs of the resources can be shared arbitrarily among the
players. A strategy of player i in resource buying games is a tuple consisting
of one of i's desired configurations S_i together with a payment vector p_i in
R^E_+ indicating how much i is willing to contribute towards the purchase of
the chosen resources. In this paper, we study the existence and computational
complexity of pure Nash equilibria (PNE, for short) of resource buying games.
In contrast to classical congestion games for which equilibria are guaranteed
to exist, the existence of equilibria in resource buying games strongly depends
on the underlying structure of the S_i's and the behavior of the cost
functions. We show that for marginally non-increasing cost functions, matroids
are exactly the right structure to consider, and that resource buying games
with marginally non-decreasing cost functions always admit a PNE
Designing Networks with Good Equilibria under Uncertainty
We consider the problem of designing network cost-sharing protocols with good
equilibria under uncertainty. The underlying game is a multicast game in a
rooted undirected graph with nonnegative edge costs. A set of k terminal
vertices or players need to establish connectivity with the root. The social
optimum is the Minimum Steiner Tree. We are interested in situations where the
designer has incomplete information about the input. We propose two different
models, the adversarial and the stochastic. In both models, the designer has
prior knowledge of the underlying metric but the requested subset of the
players is not known and is activated either in an adversarial manner
(adversarial model) or is drawn from a known probability distribution
(stochastic model).
In the adversarial model, the designer's goal is to choose a single,
universal protocol that has low Price of Anarchy (PoA) for all possible
requested subsets of players. The main question we address is: to what extent
can prior knowledge of the underlying metric help in the design? We first
demonstrate that there exist graphs (outerplanar) where knowledge of the
underlying metric can dramatically improve the performance of good network
design. Then, in our main technical result, we show that there exist graph
metrics, for which knowing the underlying metric does not help and any
universal protocol has PoA of , which is tight. We attack this
problem by developing new techniques that employ powerful tools from extremal
combinatorics, and more specifically Ramsey Theory in high dimensional
hypercubes.
Then we switch to the stochastic model, where each player is independently
activated. We show that there exists a randomized ordered protocol that
achieves constant PoA. By using standard derandomization techniques, we produce
a deterministic ordered protocol with constant PoA.Comment: This version has additional results about stochastic inpu
Routing Games with Progressive Filling
Max-min fairness (MMF) is a widely known approach to a fair allocation of
bandwidth to each of the users in a network. This allocation can be computed by
uniformly raising the bandwidths of all users without violating capacity
constraints. We consider an extension of these allocations by raising the
bandwidth with arbitrary and not necessarily uniform time-depending velocities
(allocation rates). These allocations are used in a game-theoretic context for
routing choices, which we formalize in progressive filling games (PFGs).
We present a variety of results for equilibria in PFGs. We show that these
games possess pure Nash and strong equilibria. While computation in general is
NP-hard, there are polynomial-time algorithms for prominent classes of
Max-Min-Fair Games (MMFG), including the case when all users have the same
source-destination pair. We characterize prices of anarchy and stability for
pure Nash and strong equilibria in PFGs and MMFGs when players have different
or the same source-destination pairs. In addition, we show that when a designer
can adjust allocation rates, it is possible to design games with optimal strong
equilibria. Some initial results on polynomial-time algorithms in this
direction are also derived
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
Joint Channel Selection and Power Control in Infrastructureless Wireless Networks: A Multi-Player Multi-Armed Bandit Framework
This paper deals with the problem of efficient resource allocation in dynamic
infrastructureless wireless networks. Assuming a reactive interference-limited
scenario, each transmitter is allowed to select one frequency channel (from a
common pool) together with a power level at each transmission trial; hence, for
all transmitters, not only the fading gain, but also the number of interfering
transmissions and their transmit powers are varying over time. Due to the
absence of a central controller and time-varying network characteristics, it is
highly inefficient for transmitters to acquire global channel and network
knowledge. Therefore a reasonable assumption is that transmitters have no
knowledge of fading gains, interference, and network topology. Each
transmitting node selfishly aims at maximizing its average reward (or
minimizing its average cost), which is a function of the action of that
specific transmitter as well as those of all other transmitters. This scenario
is modeled as a multi-player multi-armed adversarial bandit game, in which
multiple players receive an a priori unknown reward with an arbitrarily
time-varying distribution by sequentially pulling an arm, selected from a known
and finite set of arms. Since players do not know the arm with the highest
average reward in advance, they attempt to minimize their so-called regret,
determined by the set of players' actions, while attempting to achieve
equilibrium in some sense. To this end, we design in this paper two joint power
level and channel selection strategies. We prove that the gap between the
average reward achieved by our approaches and that based on the best fixed
strategy converges to zero asymptotically. Moreover, the empirical joint
frequencies of the game converge to the set of correlated equilibria. We
further characterize this set for two special cases of our designed game
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