1,727 research outputs found
Approximate Pure Nash Equilibria in Weighted Congestion Games
We study the existence of approximate pure Nash equilibria in weighted congestion games and develop techniques to obtain approximate potential functions that prove the existence of alpha-approximate pure Nash equilibria and the convergence of alpha-improvement steps. Specifically, we show how to obtain upper bounds for approximation factor alpha for a given class of cost functions. For example for concave cost functions the factor is at most 3/2, for quadratic cost functions it is at most 4/3, and for polynomial cost functions of maximal degree d it is at at most d + 1. For games with two players we obtain tight bounds which are as small as for example 1.054 in the case of quadratic cost functions
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
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
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
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