8,625 research outputs found
Algorithms for generalized potential games with mixed-integer variables
We consider generalized potential games, that constitute a fundamental subclass of generalized Nash equilibrium problems. We propose different methods to compute solutions of generalized potential games with mixed-integer variables, i.e., games in which some variables are continuous while the others are discrete. We investigate which types of equilibria of the game can be computed by minimizing a potential function over the common feasible set. In particular, for a wide class of generalized potential games, we characterize those equilibria that can be computed by minimizing potential functions as Pareto solutions of a particular multi-objective problem, and we show how different potential functions can be used to select equilibria. We propose a new Gauss–Southwell algorithm to compute approximate equilibria of any generalized potential game with mixed-integer variables. We show that this method converges in a finite number of steps and we also give an upper bound on this number of steps. Moreover, we make a thorough analysis on the behaviour of approximate equilibria with respect to exact ones. Finally, we make many numerical experiments to show the viability of the proposed approaches
Tolling, Capacity Selection and Equilibrium Problems with Equilibrium Constraints
An Equilibrium problem with an equilibrium constraint is a mathematical construct that can be applied to private competition in highway networks. In this paper we consider the problem of finding a Nash Equilibrium regarding competition in toll pricing on a network utilising 2 alternative algorithms. In the first algorithm, we utilise a Gauss Siedel fixed point approach based on the cutting constraint algorithm for toll pricing. In the second algorithm, we extend an existing sequential linear complementarity approach for finding Nash equilibrium subject to Wardrop Equilibrium constraints. Finally we consider how the equilibrium may change between the Nash competitive equilibrium and a collusive equilibrium where the two players co-operate to form the equivalent of a monopoly operation
Self-organized Relay Selection for Cooperative Transmission in Vehicular Ad-hoc Networks
Cooperation is a promising paradigm to improve spatial diversity in vehicular ad-hoc networks. In this paper, we pose a fundamental question: how the greediness and selfishness of individual nodes impact cooperation dynamics in vehicular ad-hoc networks. We map the self-interest-driven relay selection decision-making problems as an automata game formulation and present a non-cooperative game-theoretic analysis. We show that the relay selection game is an ordinal potential game. A decentralized self-organized relay selection algorithm is proposed based on a stochastic learning approach where each player evolves toward a strategic equilibrium state in the sense of Nash. Furthermore, we study the exact outage behavior of the multi-relay decode-and-forward cooperative communication network. Closed-form solutions are derived for the actual outage probability of this multi-relay system in both independent and identically distributed channels and generalized channels, which need not assume an asymptotic or high signal-to-noise ratio. Two tight approximations with low computational complexity are also developed for the lower bound of the outage probability. With the exact closed-form outage probability, we further develop an optimization model to determine optimal power allocations in the cooperative network, which can be combined with the decentralized learning-based relay selection. The analysis of the exact and approximative outage behaviors and the convergence properties of the proposed algorithm toward a Nash equilibrium state are verified theoretically and numerically. Simulation results are also given to demonstrate that the resulting cooperative network induced by the proposed algorithm achieves high energy efficiency, transmission reliability, and network-wide fairness performance
Computing all solutions of Nash equilibrium problems with discrete strategy sets
The Nash equilibrium problem is a widely used tool to model non-cooperative
games. Many solution methods have been proposed in the literature to compute
solutions of Nash equilibrium problems with continuous strategy sets, but,
besides some specific methods for some particular applications, there are no
general algorithms to compute solutions of Nash equilibrium problems in which
the strategy set of each player is assumed to be discrete. We define a
branching method to compute the whole solution set of Nash equilibrium problems
with discrete strategy sets. This method is equipped with a procedure that, by
fixing variables, effectively prunes the branches of the search tree.
Furthermore, we propose a preliminary procedure that by shrinking the feasible
set improves the performances of the branching method when tackling a
particular class of problems. Moreover, we prove existence of equilibria and we
propose an extremely fast Jacobi-type method which leads to one equilibrium for
a new class of Nash equilibrium problems with discrete strategy sets. Our
numerical results show that all proposed algorithms work very well in practice
Differential evolution based bi-level programming algorithm for computing normalized nash equilibrium
The Generalised Nash Equilibrium Problem (GNEP) is a Nash game with the distinct feature that the feasible strategy set of a player depends on the strategies chosen by all her opponents in the game. This characteristic distinguishes the GNEP from a conventional Nash Game. These shared constraints on each player’s decision space, being dependent on decisions of others in the game, increases its computational difficulty. A special solution of the GNEP is the Nash Normalized Equilibrium which can be obtained by transforming the GNEP into a bi-level program with an optimal value of zero in the upper level. In this paper, we propose a Differential Evolution based Bi-Level Programming algorithm embodying Stochastic Ranking to handle constraints (DEBLP-SR) to solve the resulting bi-level programming formulation. Numerical examples of GNEPs drawn from the literature are used to illustrate the performance of the proposed algorithm
Evolution, dynamics, and fixed points
Sign-compatible dynamics describe changes in the composition of a population driven by differences in fitness. A saturated equilibrium is a fixed point for sign-compatible dynamics where each subgroup with positive population share has highest fitness. An evolutionary stable equilibrium is a saturated equilibrium attracting all trajectories nearby, such that the Euclidean distance to it decreases monotonically. We address existence, multiplicity, and dynamical stability of fixed points of sign-compatible dynamics. A saturated equilibrium may be approximated by using a variable dimension restart algorithm for solving the nonlinear complementarity problem. Journal of Economic Literature Classification Numbers: C62, C68, C72, C73. Keywords: Sign-compatible population dynamics, saturated equilibrium, evolutionary stable equilibrium, dynamic stability, nonlinear complementarity problem.mathematical economics and econometrics
Satisfaction Equilibrium: A General Framework for QoS Provisioning in Self-Configuring Networks
This paper is concerned with the concept of equilibrium and quality of
service (QoS) provisioning in self-configuring wireless networks with
non-cooperative radio devices (RD). In contrast with the Nash equilibrium (NE),
where RDs are interested in selfishly maximizing its QoS, we present a concept
of equilibrium, named satisfaction equilibrium (SE), where RDs are interested
only in guaranteing a minimum QoS. We provide the conditions for the existence
and the uniqueness of the SE. Later, in order to provide an equilibrium
selection framework for the SE, we introduce the concept of effort or cost of
satisfaction, for instance, in terms of transmit power levels, constellation
sizes, etc. Using the idea of effort, the set of efficient SE (ESE) is defined.
At the ESE, transmitters satisfy their minimum QoS incurring in the lowest
effort. We prove that contrary to the (generalized) NE, at least one ESE always
exists whenever the network is able to simultaneously support the individual
QoS requests. Finally, we provide a fully decentralized algorithm to allow
self-configuring networks to converge to one of the SE relying only on local
information.Comment: Accepted for publication in Globecom 201
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