5,163 research outputs found
Dynamic Network Congestion Games
Congestion games are a classical type of games studied in game theory, in which n players choose a resource, and their individual cost increases with the number of other players choosing the same resource. In network congestion games (NCGs), the resources correspond to simple paths in a graph, e.g. representing routing options from a source to a target. In this paper, we introduce a variant of NCGs, referred to as dynamic NCGs: in this setting, players take transitions synchronously, they select their next transitions dynamically, and they are charged a cost that depends on the number of players simultaneously using the same transition.
We study, from a complexity perspective, standard concepts of game theory in dynamic NCGs: social optima, Nash equilibria, and subgame perfect equilibria. Our contributions are the following: the existence of a strategy profile with social cost bounded by a constant is in PSPACE and NP-hard. (Pure) Nash equilibria always exist in dynamic NCGs; the existence of a Nash equilibrium with bounded cost can be decided in EXPSPACE, and computing a witnessing strategy profile can be done in doubly-exponential time. The existence of a subgame perfect equilibrium with bounded cost can be decided in 2EXPSPACE, and a witnessing strategy profile can be computed in triply-exponential time
Resource Competition on Integral Polymatroids
We study competitive resource allocation problems in which players distribute
their demands integrally on a set of resources subject to player-specific
submodular capacity constraints. Each player has to pay for each unit of demand
a cost that is a nondecreasing and convex function of the total allocation of
that resource. This general model of resource allocation generalizes both
singleton congestion games with integer-splittable demands and matroid
congestion games with player-specific costs. As our main result, we show that
in such general resource allocation problems a pure Nash equilibrium is
guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure
Nash equilibrium.Comment: 17 page
On the Convergence Time of the Best Response Dynamics in Player-specific Congestion Games
We study the convergence time of the best response dynamics in
player-specific singleton congestion games. It is well known that this dynamics
can cycle, although from every state a short sequence of best responses to a
Nash equilibrium exists. Thus, the random best response dynamics, which selects
the next player to play a best response uniformly at random, terminates in a
Nash equilibrium with probability one. In this paper, we are interested in the
expected number of best responses until the random best response dynamics
terminates.
As a first step towards this goal, we consider games in which each player can
choose between only two resources. These games have a natural representation as
(multi-)graphs by identifying nodes with resources and edges with players. For
the class of games that can be represented as trees, we show that the
best-response dynamics cannot cycle and that it terminates after O(n^2) steps
where n denotes the number of resources. For the class of games represented as
cycles, we show that the best response dynamics can cycle. However, we also
show that the random best response dynamics terminates after O(n^2) steps in
expectation.
Additionally, we conjecture that in general player-specific singleton
congestion games there exists no polynomial upper bound on the expected number
of steps until the random best response dynamics terminates. We support our
conjecture by presenting a family of games for which simulations indicate a
super-polynomial convergence time
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
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
Strong Nash Equilibria in Games with the Lexicographical Improvement Property
We introduce a class of finite strategic games with the property that every
deviation of a coalition of players that is profitable to each of its members
strictly decreases the lexicographical order of a certain function defined on
the set of strategy profiles. We call this property the Lexicographical
Improvement Property (LIP) and show that it implies the existence of a
generalized strong ordinal potential function. We use this characterization to
derive existence, efficiency and fairness properties of strong Nash equilibria.
We then study a class of games that generalizes congestion games with
bottleneck objectives that we call bottleneck congestion games. We show that
these games possess the LIP and thus the above mentioned properties. For
bottleneck congestion games in networks, we identify cases in which the
potential function associated with the LIP leads to polynomial time algorithms
computing a strong Nash equilibrium. Finally, we investigate the LIP for
infinite games. We show that the LIP does not imply the existence of a
generalized strong ordinal potential, thus, the existence of SNE does not
follow. Assuming that the function associated with the LIP is continuous,
however, we prove existence of SNE. As a consequence, we prove that bottleneck
congestion games with infinite strategy spaces and continuous cost functions
possess a strong Nash equilibrium
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
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