20 research outputs found
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
Path deviations outperform approximate stability in heterogeneous congestion games
We consider non-atomic network congestion games with heterogeneous players
where the latencies of the paths are subject to some bounded deviations. This
model encompasses several well-studied extensions of the classical Wardrop
model which incorporate, for example, risk-aversion, altruism or travel time
delays. Our main goal is to analyze the worst-case deterioration in social cost
of a perturbed Nash flow (i.e., for the perturbed latencies) with respect to an
original Nash flow. We show that for homogeneous players perturbed Nash flows
coincide with approximate Nash flows and derive tight bounds on their
inefficiency. In contrast, we show that for heterogeneous populations this
equivalence does not hold. We derive tight bounds on the inefficiency of both
perturbed and approximate Nash flows for arbitrary player sensitivity
distributions. Intuitively, our results suggest that the negative impact of
path deviations (e.g., caused by risk-averse behavior or latency perturbations)
is less severe than approximate stability (e.g., caused by limited
responsiveness or bounded rationality). We also obtain a tight bound on the
inefficiency of perturbed Nash flows for matroid congestion games and
homogeneous populations if the path deviations can be decomposed into edge
deviations. In particular, this provides a tight bound on the Price of
Risk-Aversion for matroid congestion games
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
Efficiency of Restricted Tolls in Non-atomic Network Routing Games
An effective means to reduce the inefficiency of Nash flows in non-
atomic network routing games is to impose tolls on the arcs of the network. It is a well-known fact that marginal cost tolls induce a Nash flow that corresponds to a minimum cost flow. However, despite their effectiveness, marginal cost tolls suffer from two major drawbacks, namely (i) that potentially every arc of the network is tolled, and (ii) that the imposed tolls can be arbitrarily large.
In this paper, we study the restricted network toll problem in which tolls can be imposed on the arcs of the network but are restricted to not exceed a predefined threshold for every arc. We show that optimal restricted tolls can be computed efficiently for parallel-arc networks and affine latency functions. This generalizes a previous work on taxing subnetworks to arbitrary restrictions. Our algorithm is quite simple, but relies on solving several convex programs. The key to our approach is a characterization of the flows that are inducible by restricted tolls for single-commodity networks. We also derive bounds on the efficiency of restricted tolls for multi-commodity networks and polynomial latency functions. These bounds are tight even for parallel-arc networks. Our bounds show that restricted tolls can significantly reduce the price of anarchy if the restrictions imposed on arcs with high-degree polynomials are not too severe. Our proof is constructive. We define tolls respecting the given thresholds and show that these tolls lead to a reduced price of anarchy by using a (\lambda,\mu)-smoothness approach
Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses
We present a deterministic polynomial-time algorithm for computing
-approximate (pure) Nash equilibria in weighted congestion games
with polynomial cost functions of degree at most . This is an exponential
improvement of the approximation factor with respect to the previously best
deterministic algorithm. An appealing additional feature of our algorithm is
that it uses only best-improvement steps in the actual game, as opposed to
earlier approaches that first had to transform the game itself. Our algorithm
is an adaptation of the seminal algorithm by Caragiannis et al. [FOCS'11, TEAC
2015], but we utilize an approximate potential function directly on the
original game instead of an exact one on a modified game.
A critical component of our analysis, which is of independent interest, is
the derivation of a novel bound of for the
Price of Anarchy (PoA) of -approximate equilibria in weighted congestion
games, where is the Lambert-W function. More specifically, we
show that this PoA is exactly equal to , where
is the unique positive solution of the equation . Our upper bound is derived via a smoothness-like argument,
and thus holds even for mixed Nash and correlated equilibria, while our lower
bound is simple enough to apply even to singleton congestion games