43 research outputs found
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
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
Equilibrium Computation in Resource Allocation Games
We study the equilibrium computation problem for two classical resource
allocation games: atomic splittable congestion games and multimarket Cournot
oligopolies. For atomic splittable congestion games with singleton strategies
and player-specific affine cost functions, we devise the first polynomial time
algorithm computing a pure Nash equilibrium. Our algorithm is combinatorial and
computes the exact equilibrium assuming rational input. The idea is to compute
an equilibrium for an associated integrally-splittable singleton congestion
game in which the players can only split their demands in integral multiples of
a common packet size. While integral games have been considered in the
literature before, no polynomial time algorithm computing an equilibrium was
known. Also for this class, we devise the first polynomial time algorithm and
use it as a building block for our main algorithm.
We then develop a polynomial time computable transformation mapping a
multimarket Cournot competition game with firm-specific affine price functions
and quadratic costs to an associated atomic splittable congestion game as
described above. The transformation preserves equilibria in either games and,
thus, leads -- via our first algorithm -- to a polynomial time algorithm
computing Cournot equilibria. Finally, our analysis for integrally-splittable
games implies new bounds on the difference between real and integral Cournot
equilibria. The bounds can be seen as a generalization of the recent bounds for
single market oligopolies obtained by Todd [2016].Comment: This version contains some typo corrections onl
Bounding the price of anarchy for games with player-specific cost functions
9 pagesWe study the efficiency of equilibria in atomic splittable congestion games on networks. We consider the general case where players are not affected in the same way by the congestion. Extending a result by Cominetti, Correa, and Stier-Moses (The impact of oligopolistic competition in networks, Oper. Res., 57, 1421--1437 (2009)), we prove a general bound on the price of anarchy for games with player-specific cost functions. This bound generalizes some of their results, especially the bound they obtain for the affine case. However our bound still requires some dependence between the cost functions of the players. In the general case, we prove that the price of anarchy is unbounded, by exhibiting an example with affine cost functions and only two players
Exact Price of Anarchy for Weighted Congestion Games with Two Players
This paper gives a complete analysis of worst-case equilibria for various
versions of weighted congestion games with two players and affine cost
functions. The results are exact price of anarchy bounds which are parametric
in the weights of the two players, and establish exactly how the primitives of
the game enter into the quality of equilibria. Interestingly, some of the
worst-cases are attained when the players' weights only differ slightly. Our
findings also show that sequential play improves the price of anarchy in all
cases, however, this effect vanishes with an increasing difference in the
players' weights. Methodologically, we obtain exact price of anarchy bounds by
a duality based proof mechanism, based on a compact linear programming
formulation that computes worst-case instances. This mechanism yields
duality-based optimality certificates which can eventually be turned into
purely algebraic proofs.Comment: 17 pages, 9 figures, 4 table
Altruism in Atomic Congestion Games
This paper studies the effects of introducing altruistic agents into atomic
congestion games. Altruistic behavior is modeled by a trade-off between selfish
and social objectives. In particular, we assume agents optimize a linear
combination of personal delay of a strategy and the resulting increase in
social cost. Our model can be embedded in the framework of congestion games
with player-specific latency functions. Stable states are the Nash equilibria
of these games, and we examine their existence and the convergence of
sequential best-response dynamics. Previous work shows that for symmetric
singleton games with convex delays Nash equilibria are guaranteed to exist. For
concave delay functions we observe that there are games without Nash equilibria
and provide a polynomial time algorithm to decide existence for symmetric
singleton games with arbitrary delay functions. Our algorithm can be extended
to compute best and worst Nash equilibria if they exist. For more general
congestion games existence becomes NP-hard to decide, even for symmetric
network games with quadratic delay functions. Perhaps surprisingly, if all
delay functions are linear, then there is always a Nash equilibrium in any
congestion game with altruists and any better-response dynamics converges. In
addition to these results for uncoordinated dynamics, we consider a scenario in
which a central altruistic institution can motivate agents to act
altruistically. We provide constructive and hardness results for finding the
minimum number of altruists to stabilize an optimal congestion profile and more
general mechanisms to incentivize agents to adopt favorable behavior.Comment: 13 pages, 1 figure, includes some minor adjustment
Project Games
International audienceWe consider a strategic game called project game where each agent has to choose a project among his own list of available projects. The model includes positive weights expressing the capacity of a given agent to contribute to a given project The realization of a project produces some reward that has to be allocated to the agents. The reward of a realized project is fully allocated to its contributors, according to a simple proportional rule. Existence and computational complexity of pure Nash equilibria is addressed and their efficiency is investigated according to both the utilitarian and the egalitarian social function