18 research outputs found
Graphical potential games
We study the class of potential games that are also graphical games with
respect to a given graph of connections between the players. We show that,
up to strategic equivalence, this class of games can be identified with the set
of Markov random fields on .
From this characterization, and from the Hammersley-Clifford theorem, it
follows that the potentials of such games can be decomposed to local
potentials. We use this decomposition to strongly bound the number of strategy
changes of a single player along a better response path. This result extends to
generalized graphical potential games, which are played on infinite graphs.Comment: Accepted to the Journal of Economic Theor
On the Impact of Fair Best Response Dynamics
In this work we completely characterize how the frequency with which each
player participates in the game dynamics affects the possibility of reaching
efficient states, i.e., states with an approximation ratio within a constant
factor from the price of anarchy, within a polynomially bounded number of best
responses. We focus on the well known class of congestion games and we show
that, if each player is allowed to play at least once and at most times
any best responses, states with approximation ratio times the
price of anarchy are reached after best
responses, and that such a bound is essentially tight also after exponentially
many ones. One important consequence of our result is that the fairness among
players is a necessary and sufficient condition for guaranteeing a fast
convergence to efficient states. This answers the important question of the
maximum order of needed to fast obtain efficient states, left open by
[9,10] and [3], in which fast convergence for constant and very slow
convergence for have been shown, respectively. Finally, we show
that the structure of the game implicitly affects its performances. In
particular, we show that in the symmetric setting, in which all players share
the same set of strategies, the game always converges to an efficient state
after a polynomial number of best responses, regardless of the frequency each
player moves with
Efficient computation of approximate pure Nash equilibria in congestion games
Congestion games constitute an important class of games in which computing an
exact or even approximate pure Nash equilibrium is in general {\sf
PLS}-complete. We present a surprisingly simple polynomial-time algorithm that
computes O(1)-approximate Nash equilibria in these games. In particular, for
congestion games with linear latency functions, our algorithm computes
-approximate pure Nash equilibria in time polynomial in the
number of players, the number of resources and . It also applies to
games with polynomial latency functions with constant maximum degree ;
there, the approximation guarantee is . The algorithm essentially
identifies a polynomially long sequence of best-response moves that lead to an
approximate equilibrium; the existence of such short sequences is interesting
in itself. These are the first positive algorithmic results for approximate
equilibria in non-symmetric congestion games. We strengthen them further by
proving that, for congestion games that deviate from our mild assumptions,
computing -approximate equilibria is {\sf PLS}-complete for any
polynomial-time computable
Approximate Pure Nash Equilibria in Weighted Congestion Games: Existence, Efficient Computation, and Structure
We consider structural and algorithmic questions related to the Nash dynamics
of weighted congestion games. In weighted congestion games with linear latency
functions, the existence of (pure Nash) equilibria is guaranteed by potential
function arguments. Unfortunately, this proof of existence is inefficient and
computing equilibria is such games is a {\sf PLS}-hard problem. The situation
gets worse when superlinear latency functions come into play; in this case, the
Nash dynamics of the game may contain cycles and equilibria may not even exist.
Given these obstacles, we consider approximate equilibria as alternative
solution concepts. Do such equilibria exist? And if so, can we compute them
efficiently?
We provide positive answers to both questions for weighted congestion games
with polynomial latency functions by exploiting an "approximation" of such
games by a new class of potential games that we call -games. This allows
us to show that these games have -approximate equilibria, where is the
maximum degree of the latency functions. Our main technical contribution is an
efficient algorithm for computing O(1)-approximate equilibria when is a
constant. For games with linear latency functions, the approximation guarantee
is for arbitrarily small ; for
latency functions with maximum degree , it is . The
running time is polynomial in the number of bits in the representation of the
game and . As a byproduct of our techniques, we also show the
following structural statement for weighted congestion games with polynomial
latency functions of maximum degree : polynomially-long sequences of
best-response moves from any initial state to a -approximate
equilibrium exist and can be efficiently identified in such games as long as
is constant.Comment: 31 page
Approximating Generalized Network Design under (Dis)economies of Scale with Applications to Energy Efficiency
In a generalized network design (GND) problem, a set of resources are
assigned to multiple communication requests. Each request contributes its
weight to the resources it uses and the total load on a resource is then
translated to the cost it incurs via a resource specific cost function. For
example, a request may be to establish a virtual circuit, thus contributing to
the load on each edge in the circuit. Motivated by energy efficiency
applications, recently, there is a growing interest in GND using cost functions
that exhibit (dis)economies of scale ((D)oS), namely, cost functions that
appear subadditive for small loads and superadditive for larger loads.
The current paper advances the existing literature on approximation
algorithms for GND problems with (D)oS cost functions in various aspects: (1)
we present a generic approximation framework that yields approximation results
for a much wider family of requests in both directed and undirected graphs; (2)
our framework allows for unrelated weights, thus providing the first
non-trivial approximation for the problem of scheduling unrelated parallel
machines with (D)oS cost functions; (3) our framework is fully combinatorial
and runs in strongly polynomial time; (4) the family of (D)oS cost functions
considered in the current paper is more general than the one considered in the
existing literature, providing a more accurate abstraction for practical energy
conservation scenarios; and (5) we obtain the first approximation ratio for GND
with (D)oS cost functions that depends only on the parameters of the resources'
technology and does not grow with the number of resources, the number of
requests, or their weights. The design of our framework relies heavily on
Roughgarden's smoothness toolbox (JACM 2015), thus demonstrating the possible
usefulness of this toolbox in the area of approximation algorithms.Comment: 39 pages, 1 figure. An extended abstract of this paper is to appear
in the 50th Annual ACM Symposium on the Theory of Computing (STOC 2018