8 research outputs found
Computing Approximate Pure Nash Equilibria in Shapley Value Weighted Congestion Games
We study the computation of approximate pure Nash equilibria in Shapley value
(SV) weighted congestion games, introduced in [19]. This class of games
considers weighted congestion games in which Shapley values are used as an
alternative (to proportional shares) for distributing the total cost of each
resource among its users. We focus on the interesting subclass of such games
with polynomial resource cost functions and present an algorithm that computes
approximate pure Nash equilibria with a polynomial number of strategy updates.
Since computing a single strategy update is hard, we apply sampling techniques
which allow us to achieve polynomial running time. The algorithm builds on the
algorithmic ideas of [7], however, to the best of our knowledge, this is the
first algorithmic result on computation of approximate equilibria using other
than proportional shares as player costs in this setting. We present a novel
relation that approximates the Shapley value of a player by her proportional
share and vice versa. As side results, we upper bound the approximate price of
anarchy of such games and significantly improve the best known factor for
computing approximate pure Nash equilibria in weighted congestion games of [7].Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-71924-5_1
On Existence and Properties of Approximate Pure Nash Equilibria in Bandwidth Allocation Games
In \emph{bandwidth allocation games} (BAGs), the strategy of a player
consists of various demands on different resources. The player's utility is at
most the sum of these demands, provided they are fully satisfied. Every
resource has a limited capacity and if it is exceeded by the total demand, it
has to be split between the players. Since these games generally do not have
pure Nash equilibria, we consider approximate pure Nash equilibria, in which no
player can improve her utility by more than some fixed factor through
unilateral strategy changes. There is a threshold (where
is a parameter that limits the demand of each player on a specific
resource) such that -approximate pure Nash equilibria always exist for
, but not for . We give both
upper and lower bounds on this threshold and show that the
corresponding decision problem is -hard. We also show that the
-approximate price of anarchy for BAGs is . For a restricted
version of the game, where demands of players only differ slightly from each
other (e.g. symmetric games), we show that approximate Nash equilibria can be
reached (and thus also be computed) in polynomial time using the best-response
dynamic. Finally, we show that a broader class of utility-maximization games
(which includes BAGs) converges quickly towards states whose social welfare is
close to the optimum
Improving Approximate Pure Nash Equilibria in Congestion Games
Congestion games constitute an important class of games to model resource
allocation by different users. As computing an exact or even an approximate
pure Nash equilibrium is in general PLS-complete, Caragiannis et al. (2011)
present a polynomial-time algorithm that computes a ()-approximate pure Nash equilibria for games with linear cost
functions and further results for polynomial cost functions. We show that this
factor can be improved to and further improved results for
polynomial cost functions, by a seemingly simple modification to their
algorithm by allowing for the cost functions used during the best response
dynamics be different from the overall objective function. Interestingly, our
modification to the algorithm also extends to efficiently computing improved
approximate pure Nash equilibria in games with arbitrary non-decreasing
resource cost functions. Additionally, our analysis exhibits an interesting
method to optimally compute universal load dependent taxes and using linear
programming duality prove tight bounds on PoA under universal taxation, e.g,
2.012 for linear congestion games and further results for polynomial cost
functions. Although our approach yield weaker results than that in Bil\`{o} and
Vinci (2016), we remark that our cost functions are locally computable and in
contrast to Bil\`{o} and Vinci (2016) are independent of the actual instance of
the game
Congestion Games with Complementarities
We study a model of selfish resource allocation that seeks to incorporate
dependencies among resources as they exist in modern networked environments.
Our model is inspired by utility functions with constant elasticity of
substitution (CES) which is a well-studied model in economics. We consider
congestion games with different aggregation functions. In particular, we study
norms and analyze the existence and complexity of (approximate) pure Nash
equilibria. Additionally, we give an almost tight characterization based on
monotonicity properties to describe the set of aggregation functions that
guarantee the existence of pure Nash equilibria.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-57586-5_1
Congestion Games with Mixed Objectives
We study a new class of games which generalizes congestion games and its
bottleneck variant. We introduce congestion games with mixed objectives to
model network scenarios in which players seek to optimize for latency and
bandwidths alike. We characterize the existence of pure Nash equilibria (PNE)
and the convergence of improvement dynamics. For games that do not possess PNE
we give bounds on the approximation ratio of approximate pure Nash equilibria.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/978-3-319-48749-6_4