144 research outputs found
The complexity of pure nash equilibria in max-congestion games
We study Network Max-Congestion Games (NMC games, for short), a
class of network games where each player tries to minimize the most congested
edge along the path he uses as strategy. We focus our study on the complexity
of computing a pure Nash equilibria in this kind of games. We show that, for
single-commodity games with non-decreasing delay functions, this problem
is in P when either all the paths from the source to the target node are
disjoint or all the delay functions are equal. For the general case, we prove
that the computation of a PNE belongs to the complexity class PLS through a
new technique based on generalized ordinal potential functions and a slightly
modified definition of the usual local search neighborhood. We further apply
this technique to a different class of games (which we call Pareto-efficient)
with restricted cost functions. Finally, we also prove some PLS-hardness
results, showing that computing a PNE for Pareto-efficient NMC games is
indeed a PLS-complete problem
The complexity of pure nash equilibria in max-congestion games
We study Network Max-Congestion Games (NMC games, for short), a
class of network games where each player tries to minimize the most congested
edge along the path he uses as strategy. We focus our study on the complexity
of computing a pure Nash equilibria in this kind of games. We show that, for
single-commodity games with non-decreasing delay functions, this problem
is in P when either all the paths from the source to the target node are
disjoint or all the delay functions are equal. For the general case, we prove
that the computation of a PNE belongs to the complexity class PLS through a
new technique based on generalized ordinal potential functions and a slightly
modified definition of the usual local search neighborhood. We further apply
this technique to a different class of games (which we call Pareto-efficient)
with restricted cost functions. Finally, we also prove some PLS-hardness
results, showing that computing a PNE for Pareto-efficient NMC games is
indeed a PLS-complete problem
The Complexity of pure Nash equilibria in weighted Max-Congestion Games
We study Network Max-Congestion Games (NMC games,
for short), a class of network games where each player tries to minimize
the most congested edge along the path he uses as strategy. We focus
our study on the complexity of computing a pure Nash equilibria in
weighted NMC games. We show that, for single-commodity games with
non-decreasing delay functions, this problem is in P when either all the
paths from the source to the target node are disjoint or all the delay
functions are equal. For the general case, we prove that the computation
of a PNE belongs to the complexity class PLS through a new technique
based on semi-potential functions and a slightly modified definition of
the usual local search neighborhood. We further apply this technique to
a different class of games (which we call Pareto-efficient) with restricted
cost functions. Finally, we also prove some PLS-hardness results, showing
that computing a PNE for Pareto-efficient NMC games is indeed a PLS-
complete problem.Postprint (published version
Network Topology and Equilibrium Existence in Weighted Network Congestion Games
Every finite noncooperative game can be presented as a weighted network congestion game, and also as a network congestion game with player-specific costs. In the first presentation, different players may contribute differently to congestion, and in the second, they are differently (negatively) affected by it. This paper shows that the topology of the underlying (undirected two-terminal) network provides information about the existence of pure-strategy Nash equilibrium in the game. For some networks, but not for others, every corresponding game has at least one such equilibrium. For the weighted presentation, a complete characterization of the networks with this property is given. The necessary and sufficient condition is that the network has at most three routes that do traverse any edge in opposite directions, or it consists of several such networks connected in series. The corresponding problem for player-specific costs remains open.Congestion games, network topology, existence of equilibrium
Computing equilibria of Cournot oligopoly models with mixed-integer quantities
We consider Cournot oligopoly models in which some variables represent indivisible quantities. These models can be addressed by computing equilibria of Nash equilibrium problems in which the players solve mixed-integer nonlinear problems. In the literature there are no methods to compute equilibria of this type of Nash games. We propose a Jacobi-type method for computing solutions of Nash equilibrium problems with mixed-integer variables. This algorithm is a generalization of a recently proposed method for the solution of discrete so-called â2-groups partitionableâ Nash equilibrium problems. We prove that our algorithm converges in a finite number of iterations to approximate equilibria under reasonable conditions. Moreover, we give conditions for the existence of approximate equilibria. Finally, we give numerical results to show the effectiveness of the proposed method
Capacity and Price Competition in Markets with Congestion Effects
We study oligopolistic competition in service markets where firms offer a
service to customers. The service quality of a firm - from the perspective of a
customer - depends on the congestion and the charged price. A firm can set a
price for the service offered and additionally decides on the service capacity
in order to mitigate congestion. The total profit of a firm is derived from the
gained revenue minus the capacity investment cost. Firms simultaneously set
capacities and prices in order to maximize their profit and customers
subsequently choose the services with lowest combined cost (congestion and
price). For this basic model, Johari et al. (2010) derived the first existence
and uniqueness results of pure Nash equilibria (PNE) assuming mild conditions
on congestion functions. Their existence proof relies on Kakutani's fixed-point
theorem and a key assumption for the theorem to work is that demand for service
is elastic (modeled by a smooth and strictly decreasing inverse demand
function).
In this paper, we consider the case of perfectly inelastic demand, i.e. there
is a fixed volume of customers requesting service. This scenario applies to
realistic cases where customers are not willing to drop out of the market, e.g.
if prices are regulated by reasonable price caps. We investigate existence,
uniqueness and quality of PNE for models with inelastic demand and price caps.
We show that for linear congestion cost functions, there exists a PNE. This
result requires a completely new proof approach compared to previous
approaches, since the best response correspondences of firms may be empty, thus
standard fixed-point arguments are not directly applicable. We show that the
game is C-secure (see McLennan et al. (2011)), which leads to the existence of
PNE. We furthermore show that the PNE is unique, and that the efficiency
compared to a social optimum is unbounded in general.Comment: A one-page abstract of this paper appeared in the proceedings of the
15th International Conference on Web and Internet Economics (WINE 2019
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
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