exact free energy functionals for non simply connected lattices we consider the nearest neighbor ising model in thermal equilibrium on a network with no required regularity or symmetry properties both coupling strengths and external fields are site dependent the objective is to describe this system in terms of a free energy magnetization functional whose conjugate variables are the external fields for simply connected networks this inverse problem has a local structure on generalizing to loops the local structure remains if the description is expanded in an overcomplete fashion to include a collective amplitude with respect to which the free energy is stationary for more complex connectivity a superbond representation is developed in terms of which the system can be described by a combined auxiliary set of branch and node collective variables

network formation games with regular objectives classical network formation games are played on a directed graph players have reachability objectives each player has to select a path from his source to target vertices each edge has a cost shared evenly by the players using it we introduce and study network formation games with regular objectives in our setting the edges are labeled by alphabet letters and the objective of each player is a regular language over the alphabet of labels unlike the case of reachability objectives here the paths selected by the players need not be simple thus a player may traverse some edges several times edge costs are shared by the players with the share being proportional to the number of times the edge is traversed we study the existence of a pure nash equilibrium ne the inefficiency of a ne compared to a social optimum solution and computational complexity problems in this setting

multiparameter local bifurcation on considere l equation l 0 u l 1 n u f 1 n u 0 sur un espace de banach reel u on etudie des solutions non triviales bifurcant a l origine

the price of anarchy of affine congestion games with similar strategies abstract affine congestion games are a well studied model for selfish behavior in distributed systems such as transportation and communication networks seminal influential papers in algorithmic game theory have bounded the worst case inefficiency of nash equilibria termed as price of anarchy in several variants of these games in this work we investigate to what extent these bounds depend on the similarities among the players strategies our notion of similarity is modeled by assuming that given a parameter 1 the costs of any two strategies available to a same player when evaluated in absence of congestion are within a factor one from the other it turns out that for the non atomic case better bounds can always be obtained for any finite value of for the atomic case instead 3 2 and 2 are necessary and sufficient conditions to obtain better bounds in games played on general graph topologies and on parallel link graphs respectively it is worth noticing that small values of model the behavioral attitude of players who are partially oblivious to congestion and are not willing to significantly deviate from what is their best strategy in absence of congestion

matroids are immune to braess paradox the famous braess paradox describes the counterintuitive phenomenon in which in certain settings an increase of resources such as a new road built within a congested network may in fact lead to larger costs for the players in an equilibrium in this paper we consider general nonatomic congestion games and give a characterization of the combinatorial property of strategy spaces for which the braess paradox does not occur in short matroid bases are precisely the required structure we prove this characterization by two novel sensitivity results for convex separable optimization problems over polymatroid base polyhedra which may be of independent interest

mixed nash equilibria in selfish routing problems with dynamic constraints we study the problem of routing traffic through a congested network consisting of m parallel links each having a certain speed moreover we are given n selfish non cooperative agents each of them willing to route her own piece of traffic on exactly one link agents are selfish in that they only pick a link which minimize the delay of their own piece of traffic in this context much effort has been lavished in the framework of mixed nash equilibria where the agent s routing choices are regulated by probability distributions one for each agent which let the system thus enter a steady state from which no agent is willing to unilaterally deviate in this work we consider situations in which some agents have constraints on the routing choice in a sense they are forbidden to route their traffic on some links we show that at most one nash equilibrium may exist and in some cases with equal speed links and where each agent is forbidden to route on at most one link we give necessary and sufficient conditions on its existence these conditions correlate the traffic load of the agents we consider also a dynamic behaviour of the network when the constraints may vary in particular when a constraint is removed we establish under which conditions the network is still in equilibrium these conditions are all effective in the sense that given a set of yes no routing constraints on each link for each agent we provide the probability distributions corresponding to the unique nash equilibrium associated to the constraints if it exists moreover these conditions and the possible nash equilibrium are computed in time o mn

the structure and complexity of nash equilibria for a selfish routing game in this work we study the combinatorial structure and the computational complexity of nash equilibria for a certain game that models selfish routing over a network consisting of m parallel links we assume a collection of n users each employing a mixed strategy which is a probability distribution over links to control the routing of her own traffic in a nash equilibrium each user selfishly routes her traffic on those links that minimize her expected latency cost given the network congestion caused by the other users the social cost of a nash equilibrium is the expectation over all random choices of the users of the maximum over all links latency through a link we embark on a systematic study of several algorithmic problems related to the computation of nash equilibria for the selfish routing game we consider in a nutshell these problems relate to deciding the existence of a pure nash equilibrium constructing a nash equilibrium constructing the pure nash equilibria of minimum and maximum social cost and computing the social cost of a given mixed nash equilibrium our work provides a comprehensive collection of efficient algorithms hardness results and structural results for these algorithmic problems our results span and contrast a wide range of assumptions on the syntax of the nash equilibria and on the parameters of the system

integer linear programs and local search for max cut the paper deals with the complexity of the local search a topic introduced by johnson papadimitriou and yannakakis one consequence of their work and a recent paper by schaffer and yannakakis is that the local search does not provide a polynomial time algorithm to find locally optimum solutions for several hard combinatorial optimization problems this motivates us to seek easier instances for which the local search is polynomial in particular it has been proved recently by schaffer and yannakakis that the max cut problem with the flip neighborhood is pls complete and hence belongs among the most difficult problems in the pls class polynomial time local search the flip neighborhood of a 2 partition is defined by moving a single vertex to the opposite class we prove that when restricted to cubic graphs the flip local search becomes easy and finds a local max cut in o n 2 steps to prove the result we introduce a class of integer linear programs associated with cubic graphs and provide a combinatorial characterization of their feasibility

the price of anarchy for polynomial social cost in this work we consider an interesting variant of the well studied kp model for selfish routing on parallel links which reflects some influence from the much older wardrop model j g wardrop some theoretical aspects of road traffic research proc inst of civil eng part ii 1 1956 325 378 in the new model user traffics are still unsplittable and links are identical social cost is now the expectation of the sum over all links of latency costs each latency cost is modeled as a certain polynomial latency cost function evaluated at the latency incurred by all users choosing the link the resulting social cost is called polynomial social cost or monomial social cost when the latency cost function is a monomial all considered polynomials are of degree d where d 2 and have non negative coefficients we are interested in evaluating nash equilibria in this model and we use the monomial price of anarchy mpoa and the polynomial price of anarchy ppoa as our evaluation measures through establishing some remarkable relations of these costs and measures to some classical combinatorial numbers such as the stirling numbers of the second kind and the bell numbers we obtain a multitude of results for the special case of identical users the fully mixed nash equilibrium where all probabilities are strictly positive maximizes polynomial social cost the mpoa is no more than bd the bell number of order d this immediately implies that the ppoa is no more than 1 t dbt for the special case of two links the mpoa is no more than 2d 2 1 1 n d 1 and this bound is tight for n 2 the mpoa is exactly 2d 1 d d 1 2d 2 d 1 d 1 d d for pure nash equilibria this immediately implies that the ppoa is no more than 2 t d 2t 1 t t 1 2t 2 t 1 t 1 t t

intrinsic robustness of the price of anarchy the price of anarchy defined as the ratio of the worst case objective function value of a nash equilibrium of a game and that of an optimal outcome quantifies the inefficiency of selfish behavior remarkably good bounds on this measure are known for a wide range of application domains however such bounds are meaningful only if a game s participants successfully reach a nash equilibrium this drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria such as mixed nash equilibria and correlated equilibria and to sequences of outcomes generated by natural experimentation strategies such as successive best responses and simultaneous regret minimization we establish a general and fundamental connection between the price of anarchy and its seemingly more general relatives first we identify a canonical sufficient condition for an upper bound on the price of anarchy of pure nash equilibria which we call a smoothness argument second we prove an extension theorem every bound on the price of anarchy that is derived via a smoothness argument extends automatically with no quantitative degradation in the bound to mixed nash equilibria correlated equilibria and the average objective function value of every outcome sequence generated by no regret learners smoothness arguments also have automatic implications for the inefficiency of approximate equilibria for bicriteria bounds and under additional assumptions for polynomial length best response sequences third we prove that in congestion games smoothness arguments are complete in a proof theoretic sense despite their automatic generality they are guaranteed to produce optimal worst case upper bounds on the price of anarchy

simple local search problems that are hard to solve many algorithms for np hard optimization problems find solutions that are locally optimal in the sense that the solutions cannot be improved by a polynomially computable perturbation very little is known about the complexity of finding locally optimal solutions either by local search algorithms or using other indirect methods johnson papadimitriou and yannakakis j comput system sci 37 1988 pp 79 100 studied this question by defining a complexity class pls that captures local search problems it was proved that finding a partition of a graph that is locally optimal into equal parts with respect to the acclaimed kernighan lin algorithm is pls complete it is shown here that several natural simple local search problems are pls complete and thus just as hard two examples are finding a partition that cannot be improved by a single swap of two vertices and finding a stable configuration for an undirected connectionist network when edges or other objects are unweighted then a local optimum

the price of anarchy under nonlinear and asymmetric costs in this paper we characterize the price of anarchy i e the inefficiency between user and system optimal solutions when costs are nonseparable asymmetric and nonlinear generalizing earlier work that has addressed the price of anarchy under separable costs the results in this paper apply primarily to nonatomic games such as the traffic equilibrium problem but also in competitive multiperiod pricing and competitive supply chain settings the bounds established in this paper are tight and explicitly account for the degree of asymmetry and nonlinearity of the cost function we first provide a proof method for problems with a positive definite jacobian matrix subsequently we use ideas from semidefinite optimization in order to account for problems with a positive semidefinite jacobian matrix where the first approach does not apply this latter connection also provides a different application of semidefinite optimization

a simple adaptive procedure leading to correlated equilibrium we propose a new and simple adaptive procedure for playing a game regret matching in this procedure players may depart from their current play with probabilities that are proportional to measures of regret for not having used other strategies in the past it is shown that our adaptive procedure guarantees that with probability one the empirical distributions of play converge to the set of correlated equilibria of the game

hopf bifurcation at a degenerate stationary pitchfork soit e un espace de hilbert reel on considere l existence de petites solutions periodiques pour une equation d evolution sur e donnee par du dt g u u e r avec g 0 0 r

bifurcation at multiple eigenvalues and stability of bifurcating solutions if k is a bounded linear operator from the real banach space u into the real banach space v and u r v has the value zero at 0 0 the existence and linear stability of the equilibrium solutions of the dynamical system k dudt u which are close to the origin in u r are studied it is assumed that u 0 0 u v is a freholm operator of index zero the only restriction on the dimension of the null space of u 0 0 and the order of vanishing at 0 0 of restricted to the null space of d 0 0 u r v is that they both be finite positive integers the main result gives conditions under which the equation which determines the equilibrium solutions in a neighborhood of the origin also determines the stability of these equilibrium solutions

convergence and approximation in potential games we study the speed of convergence to approximately optimal states in two classes of potential games we provide bounds in terms of the number of rounds where a round consists of a sequence of movements with each player appearing at least once in each round we model the sequential interaction between players by a best response walk in the state graph where every transition in the walk corresponds to a best response of a player our goal is to bound the social value of the states at the end of such walks in this paper we focus on two classes of potential games selfish routing games and cut games or party affiliation games fabrikant et al 2004 12 other than bounding the price of anarchy of selfish routing games roughgarden and tardos 2002 25 awerbuch et al 2005 2 christodoulou and koutsoupias 2005 9 there are many interesting problems about game dynamics in these games it is known that exponentially long best response walks may exist to pure nash equilibria fabrikant et al 2004 12 and random best response walks converge to solutions with good approximation guarantees after polynomially many best responses goemans et al 2005 17 in this paper we study the speed of convergence on deterministic best response walks in these games and prove that starting from an arbitrary configuration after one round of best responses of players the resulting configuration is a q n approximate solution furthermore we show that starting from an empty configuration the solution after any round of best responses is a constant factor approximation we also provide a lower bound for the multi round case where we show that for any constant number of rounds t the approximation guarantee cannot be better than n e t for some e t 0 we also study cut games that provide an illustrative example of potential games the convergence of potential games to locally optimum solutions has been studied in the context of local search algorithms johnson et al 1988 19 schaffer and yannakakis 1991 28 in these games we consider two social functions the cut defined as the weight of the edges in the cut and the total happiness defined as the weight of the edges in the cut minus the weight of the remaining edges for the cut social function we prove that the expected social value after one round of a random best response walk is at least a constant factor approximation to the optimal social value we also exhibit exponentially long best response walks with poor social value for the unweighted version of this cut game we prove w n and o n lower and upper bounds on the number of rounds of best responses to converge to a constant factor cut in addition we suggest a way to modify the game to enforce a fast convergence in any fair best response walk for the total happiness social function we show that for unweighted graphs of sufficiently large girth starting from a random configuration greedy behavior of players in a random order converges to an approximate solution after one round

atomic routing games on maximum congestion we study atomic routing games on networks in which players choose a path with the objective of minimizing the maximum congestion along the edges of their path the social cost is the global maximum congestion over all edges in the network we show that the price of stability is 1 the price of anarchy poa is determined by topological properties of the network in particular poa o logn where is the length of the longest path in the player strategy sets and n is the size of the network further 1 poa c 2 log2n where is the length of the longest cycle in the network and c is a constant

hopf bifurcation in the presence of symmetry on etablit une generalisation du theoreme de bifurcation de hopf pour des equations differentielles a symetrie

reduced stability of parameter dependent matrices abstract the stability properties of parameter dependent linear systems z a e z with a 0 block diag 0 f f a stable matrix and 0 of order n n are analyzed near e 0 by a reduction principle which amounts to considering the schur complement g e of a e some sufficient conditions are given for a e and g e to have the same stability properties the so called principle of reduced stability in terms of the asymptotic expansion of g e explicit necessary and sufficient conditions are given in the case f i n 2 in terms of the location of the spectrum of g e allowing for geometrical interpretation

selfish routing in capacitated networks according to wardrop s first principle agents in a congested network choose their routes selfishly a behavior that is captured by the nash equilibrium of the underlying noncooperative game a nash equilibrium does not optimize any global criterion per se and so there is no apparent reason why it should be close to a solution of minimal total travel time i e the system optimum in this paper we offer positive results on the efficiency of nash equilibria in traffic networks in contrast to prior work we present results for networks with capacities and for latency functions that are nonconvex nondifferentiable and even discontinuous the inclusion of upper bounds on arc flows has early been recognized as an important means to provide a more accurate description of traffic flows in this more general model multiple nash equilibria may exist and an arbitrary equilibrium does not need to be nearly efficient nonetheless our main result shows that the best equilibrium is as efficient as in the model without capacities moreover this holds true for broader classes of travel cost functions than considered hitherto

price of anarchy and an approximation algorithm for the binary preference capacitated selfish replication game abstract we consider the capacitated selfish replication csr game with binary preferences over general undirected networks we study the price of anarchy of such games and show that it is bounded above by 3 we develop a quasi polynomial algorithm o n 2 ln d where n is the number of players and d is the diameter of the network which can find in a distributed manner an allocation profile that is within a constant factor of the optimal allocation and hence of any pure strategy nash equilibrium ne of the game proof of this result uses a novel potential function we further show that when the underlying network has a tree structure every globally optimal allocation is an ne which can be reached in only linear time we formulate the optimal solutions and ne points of the csr game using integer linear programs finally we introduce the lcsr game as a localized version of the csr game wherein the actions of the players are restricted to only their close neighborhoods

a survey on the continuous nonlinear resource allocation problem our problem of interest consists of minimizing a separable convex and differentiable function over a convex set defined by bounds on the variables and an explicit constraint described by a separable convex function applications are abundant and vary from equilibrium problems in the engineering and economic sciences through resource allocation and balancing problems in manufacturing statistics military operations research and production and financial economics to subproblems in algorithms for a variety of more complex optimization models this paper surveys the history and applications of the problem as well as algorithmic approaches to its solution the most common techniques are based on finding the optimal value of the lagrange multiplier for the explicit constraint most often through the use of a type of line search procedure we analyze the most relevant references especially regarding their originality and numerical findings summarizing with remarks on possible extensions and future research 2006 elsevier b v all rights reserved

design of price mechanisms for network resource allocation via price of anarchy we study the design of price mechanisms for communication network problems in which a user s utility depends on the amount of flow she sends through the network and the congestion on each link depends on the total traffic flows over it the price mechanisms are characterized by a set of axioms that have been adopted in the cost sharing games and we search for the price mechanisms that provide the minimum price of anarchy we show that given the non decreasing and concave utilities of users and the convex quadratic congestion costs in each link if the price mechanism cannot depend on utility functions the best achievable price of anarchy is 4 3 2 sqrt 2 approx 31 4 thus the popular marginal cost pricing with price of anarchy less than 1 3 33 3 is nearly optimal we also investigate the scenario in which the price mechanisms can be made contingent on the users preference profile while such information is available

pricing with markups in industries with increasing marginal costs we study a game that models a market in which heterogeneous producers of perfect substitutes make pricing decisions in a first stage followed by consumers that select a producer that sells at lowest price as opposed to cournot or bertrand competition producers select prices using a supply function that maps prices to production levels solutions of this type of models are normally referred to as supply function equilibria we consider a market where producers convex costs functions are proportional to each other depending on the efficiency of each particular producer we provide necessary and sufficient conditions for the existence of an equilibrium that uses simple supply functions that replicate the cost structure we then specialize the model to monomial cost functions with exponent q 0 which allows us to reinterpret the simple supply functions as a markup applied to the production cost we prove that an equilibrium for the markups exists if and only if the number of producers in the market is strictly larger than 1 q and if an equilibrium exists it is unique the main result for monomials is that the equilibrium nearly minimizes the total production cost when the market is competitive the result holds because when there is enough competition markups are bounded thus preventing prices to be significantly distorted from costs focusing on the case of linear unit cost functions on the production quantities we characterize the equilibrium accurately and refine the previous result to establish an almost tight bound on the worst case inefficiency of equilibria finally we derive explicitly the producers best response for series parallel networks with linear unit cost functions extending our previous result to more general topologies we prove that a unique equilibrium exists if and only if the network that captures the market structure is 3 edge connected for non series parallel markets we provide an example that does not admit an equilibrium on markups

structure and complexity of extreme nash equilibria we study extreme nash equilibria in the context of a selfish routing game specifically we assume a collection of n users each employing a mixed strategy which is a probability distribution over m parallel identical links to control the routing of its own assigned traffic in a nash equilibrium each user selfishly routes its traffic on those links that minimize its expected latency cost the social cost of a nash equilibrium is the expectation over all random choices of the users of the maximum over all links latency through a link we provide substantial evidence for the fully mixed nash equilibrium conjecture which states that the worst nash equilibrium is the fully mixed nash equilibrium where each user chooses each link with positive probability specifically we prove that the fully mixed nash equilibrium conjecture is valid for pure nash equilibria furthermore we show that under a certain condition the social cost of any nash equilibrium is within a factor of 2h 1 e of that of the fully mixed nash equilibrium where h is the factor by which the largest user traffic deviates from the average user traffic considering pure nash equilibria we provide a ptas to approximate the best social cost we give an upper bound on the worst social cost and we show that it is n p hard to approximate the worst social cost within a multiplicative factor better than 2 2 m 1

on the complexity of nash equilibria and other fixed points we reexamine what it means to compute nash equilibria and more generally what it means to compute a fixed point of a given brouwer function and we investigate the complexity of the associated problems specifically we study the complexity of the following problem given a finite game gamma with 3 or more players and given epsilon 0 compute an approximation within epsilon of some actual nash equilibrium we show that approximation of an actual nash equilibrium even to within any nontrivial constant additive factor epsilon 1 2 in just one desired coordinate is at least as hard as the long standing square root sum problem as well as a more general arithmetic circuit decision problem that characterizes p time in a unit cost model of computation with arbitrary precision rational arithmetic thus placing the approximation problem in p or even np would resolve major open problems in the complexity of numerical computation we show similar results for market equilibria it is hard to estimate with any nontrivial accuracy the equilibrium prices in an exchange economy with a unique equilibrium where the economy is given by explicit algebraic formulas for the excess demand functions we define a class fixp which captures search problems that can be cast as fixed point computation problems for functions represented by algebraic circuits straight line programs over basis max min with rational constants we show that the exact or approximate computation of nash equilibria for 3 or more players is complete for fixp the price equilibrium problem for exchange economies with algebraic demand functions is another fixp complete problem we show that the piecewise linear fragment of fixp equals ppad many other problems in game theory economics and probability theory can be cast as fixed point problems for such algebraic functions we discuss several important such problems computing the value of shapley s stochastic games and the simpler games of condon extinction probabilities of branching processes probabilities of stochastic context free grammars and termination probabilities of recursive markov chains we show that for some of them the approximation or even exact computation problem can be placed in ppad while for others they are at least as hard as the square root sum and arithmetic circuit decision problems

nonuniform classical fluid mixture in one dimensional space with next neighbor interactions an overcomplete description is used to represent thermodynamic potentials for a one dimensional classical fluid mixture with next neighbor interaction in compact closed form in descriptions of this class a thermodynamic potential depends not only on minimally sufficient control variables but on others as well with respect to which it is stationary here this is done first in the direct or fugacity controlled format with the grand potential as the relevant generating function it is then transcribed to an indirect relative density functional format with overcompleteness restricted to a set of grand potential densities poly dispersity requires a separate treatment extensions outside of the range of strict one dimensionality are discussed as are several approximation methods

stability of solutions bifurcating from multiple eigenvalues let x and y be real banach spaces and g x r be a twice continuously differentiate function which is not necessarily linear suppose g u0 0 0 and the dimension of the null space of gu u0 0 is m where 1 m usually s u g u 0 in a neighborhood of u0 0 consists of a finite number of curves emanating from u0 0 we will determine the stability of points u in s i e the maximum of the real parts of the spectrum of gu u for each u s using a general perturbation theorem of kato our results contain as a special case the stability theorems of crandall and rabinowitz for the case m 1 we will also tie our stability theorems together with some bifurcation results of decker and keller finally we apply our results to systems of reaction diffusion equations

non atomic one round walks in congestion games abstract in this paper we study the approximation ratio of the solutions achieved after an approximate one round walk in non atomic congestion games prior to this work the solution concept of one round walks had been studied for atomic congestion games with linear latency functions only christodoulou et al 1 bilo et al 2 we give an explicit formula to determine the approximation ratio for non atomic congestion games having general latency functions in particular we focus on polynomial latency functions and we prove that the approximation ratio is exactly 1 p 1 p 1 for every polynomial of degree p then we show that by resorting to static resp dynamic resource taxation the approximation ratio can be lowered to 1 p 1 p 1 p resp 1 p 1 p 1

overcomplete free energy functional for d 1 particle systems with next neighbor interactions we deduce an overcomplete free energy functional for d 1 particle systems with next neighbor interactions where the set of redundant variables are the local block densities i of i interacting particles the idea is to analyze the decomposition of a given pure system into blocks of i interacting particles by means of a mapping onto a hard rod mixture this mapping uses the local activity of component i of the mixture to control the local association of i particles of the pure system thus it identifies the local particle density of component i of the mixture with the local block density i of the given system consequently our overcomplete free energy functional takes on the hard rod mixture form with the set of block densities i representing the sequence of partition functions of the local aggregates of particle numbers i the system of equations for the local particle density of the original system is closed via a subsidiary condition for the block densities in terms of analoguous to the uniform isothermal isobaric technique all our results are expressible in terms of effective pressures we illustrate the theory with two standard examples the adhesive interaction and the square well potential for the uniform case our proof of such an overcomplete format is based on the exponential boundedness of the number of partitions of a positive integer hardy ramanujan formula and on varadhan s theorem on the asymptotics of a class of integrals

computation and efficiency of potential function minimizers of combinatorial congestion games we study the computation and efficiency of pure nash equilibria in combinatorial congestion games where the strategies of each player i are given by the binary vectors of a polytope p i our main goal is to understand which structural properties of such polytopal congestion games enable us to derive an efficient equilibrium selection procedure to compute pure nash equilibria with attractive social cost approximation guarantees to this aim we identify two general properties of the underlying aggregation polytope p n sum i p i which are sufficient for our results to go through namely the integer decomposition property idp and the box totally dual integrality property box tdi our main results for polytopal congestion games satisfying idp and box tdi are as follows i we show that pure nash equilibria can be computed in polynomial time in fact we obtain this result through a general framework for separable convex function minimization which might be of independent interest ii we bound the inefficiency of these equilibria and show that this provides a tight bound on the price of stability iii we also prove that these results extend to strong equilibria for the bottleneck variant of polytopal congestion games examples of polytopal congestion games satisfying idp and box tdi include common source network congestion games symmetric totally unimodular congestion games non symmetric matroid congestion games and symmetric matroid intersection congestion games in particular r arborescences and strongly base orderable matroids

a note on a parameter relating traffic equilibria and system optimal routing in this note we extend results from roughgarden et al t roughgarden e tardos how bad is selfish routing j acm 49 2 2002 236 259 and correa et al j r correa a schulz n e stier moses selfish routing in capacitated networks math oper res 29 4 2004 961 976 concerning the total travel time in traffic equilibria we focus on convex latency functions and derive bounds for a basic parameter b introduced in correa et al 2004 using simple geometrical ideas the upper bound is expressed in terms of the elasticity of the latency function for the special case of piecewise linear convex latency functions we give an efficient algorithm for calculating this parameter exactly

single homopolymers in the relative density representation a single symmetric homopolymer is studied in the grand ensemble of its monomeric units for an arbitrary external field but next neighbor interactions alone such a system is represented as a functional of the local density to local fugacity ratio conjugate to the fugacity with the excess grand potential as thermodynamic generating function a sample case in a spherical enclosure is solved where a complementary exclusion from a spherical volume requires an extension to a grand ensemble of polymers a preliminary extension is also made in mean field form to non neighbor interactions as well as to nonsymmetric interactions

the classical fluid of associating hard rods in an external field two models of one dimensional fluids of associating hard rods in an arbitrary external field are investigated in the first model particles can only form dimers while in the second model which has been solved previously by percus aggregates of any size coexist in both cases the grand canonical potential and the external potential are found exactly as functionals of the density it is shown that wertheim s thermodynamic perturbation theory of polymerization provides a straightforward route to the exact solution by expanding the functional space to include more density parameters this suggests that wertheim s theory should be used also for studying the structure and not only the thermo dynamics of real associating fluids

a new model for selfish routing in this work we introduce and study a new potentially rich model for selfish routing over non cooperative networks as an interesting hybridization of the two prevailing such models namely the kpmodel e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 and the wmodel j g wardrop some theoretical aspects of road traffic research proceedings of the of the institute of civil engineers 1 pt ii 1952 325 378 in the hybrid model each of nusers is using a mixed strategy to ship its unsplittable traffic over a network consisting of m parallel links in a nash equilibrium no user can unilaterally improve its expected individual cost to evaluate nash equilibria we introduce quadratic social cost as the sum of the expectations of the latencies incurred by the squares of the accumulated traffic this modeling is unlike the kp model where social cost e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 is the expectation of the maximum latency incurred by the accumulated traffic but it is like the w model since the quadratic social cost can be expressed as a weighted sum of expected individual costs we use the quadratic social cost to define quadratic coordination ratio here are our main findings quadratic social cost can be computed in polynomial time this is unlike the p completeness d fotakis s kontogiannis e koutsoupias m mavronicolas p spirakis the structure and complexity of nash equilibria for a selfish routing game in p widmayer f triguero r morales m hennessy s eidenbenz r conejo eds proceedings of the 29th international colloquium on automata languages and programming in lecture notes in computer science vol 2380 springer verlag 2002 pp 123 134 of computing social cost for the kp model for the case of identical users and identical links the fully mixed nash equilibrium m mavronicolas p spirakis the price of selfish routing algorithmica 48 1 2007 91 126 where each user assigns positive probability to every link maximizes quadratic social cost as our main result we present a comprehensive collection of tight constant that is independent of m and n strictly less than 2 lower and upper bounds on the quadratic coordination ratio for several interesting special cases some of the bounds stand in contrast to corresponding super constant bounds on the coordination ratio previously shown in a czumaj b vocking tight bounds for worst case equilibria acm transactions on algorithms 3 1 2007 e koutsoupias m mavronicolas p spirakis approximate equilibria and ball fusion theory of computing systems 36 6 2003 683 693 e koutsoupias c h papadimitriou worst case equilibria in g meinel s tison eds proceedings of the 16th annual symposium on theoretical aspects of computer science in lecture notes in computer science vol 1563 springer verlag 1999 pp 404 413 m mavronicolas p spirakis the price of selfish routing algorithmica 48 1 2007 91 126 for the kp model

hopf bifurcation at multiple eigenvalues with zero eigenvalue on donne des resultats locaux relies a la bifurcation de hopf pour des equations paraboliques la partie lineaire autour du point d equilibre peut avoir des valeurs propres nulles

optimal cost sharing for resource selection games joint use of resources with usage dependent cost raises the question who pays how much we study cost sharing in resource selection games where the strategy spaces are either singletons or bases of a matroid defined on the ground set of resources our goal is to design cost sharing protocols so as to minimize the resulting price of anarchy and price of stability we investigate three classes of protocols basic protocols guarantee the existence of at least one pure nash equilibrium separable protocols additionally require that the resulting cost shares only depend on the set of players on a resource uniform protocols are separable and require that the cost shares on a resource may not depend on the instance that is they remain the same even if new resources are added to or removed from the instance we find optimal basic and separable protocols that guarantee the price of stability and price of anarchy to grow logarithmically in the number of players except for the case of matroid games induced by separable protocols where the price of anarchy grows linearly with the number of players for uniform protocols we show that the price of anarchy is unbounded even for singleton games

abstract forced symmetry breaking and forced frequency locking of modulated waves we consider abstract forced symmetry breaking problems of the typef x y it is supposed that for all the mapsf are equivariant with respect to the action of a compact lie group thatf x0 0 0 and hence thatf x 0 0 for all elementsxof the group orbit o x0 ofx0 we look for solutionsxwhich bifurcate from the solution family o x0 as andymove away from 0and zero respectively especially we describe the number of different solutionsx for fixed control parameters andy their dynamic stability and their asymptotic behavior forytending to zero further generalizations are given to problems of the typef x y x finally our results are applied to a forced frequency locking problem of the typex t f x t y t here it is supposed that the vector fieldsf ares1 equivariant that the unperturbed equationx f x 0 has an orbitally stable modulated wave solution and that the forcingy t is a modulated wave

exact price of anarchy for polynomial congestion games we show exact values for the worst case price of anarchy in weighted and unweighted atomic unsplittable congestion games provided that all cost functions are bounded degree polynomials with nonnegative coefficients the given values also hold for weighted and unweighted network congestion games

homopolymers with intrachain interactions the system under consideration is a classical homopolymeric chain under an arbitrary external field and in the grand canonical ensemble of its monomeric units the ideal case of only symmetric next neighbor interactions is first analyzed in the relative density format arbitrary monomer monomer interactions are introduced in a graphical perturbation series and the leading order is expressed as a relative density functional with the aid of a sequence of redundant fields under whose variation the thermodynamic potential here the excess grand potential is stationary various reduction methods are suggested to compress the set of redundant fields of this overcomplete description

on the principle of reduced stability abstract the principle of reduced stability says that the stability of bifurcating stationary or periodic solutions is given by the finite dimensional bifurcation equation obtained by the method of lyapunov schmidt to be more precise the linearized stability is governed by the linearization of the bifurcation equation about the bifurcating branch of solutions and in particular by the signs of the real parts of the perturbation of the eigenvalues along this branch this principle is true for simple eigenvalue bifurcation whereas it may be false for higher dimensional bifurcation equations a condition for the validity of that principle is given a counterexample shows that it cannot be dropped in general

on the impact of combinatorial structure on congestion games we study the impact of combinatorial structure in congestion games on the complexity of computing pure nash equilibria and the convergence time of best response sequences in particular we investigate which properties of the strategy spaces of individual players ensure a polynomial convergence time we show that if the strategy space of each player consists of the bases of a matroid over the set of resources then the lengths of all best response sequences are polynomially bounded in the number of players and resources we also prove that this result is tight that is the matroid property is a necessary and sufficient condition on the players strategy spaces for guaranteeing polynomial time convergence to a nash equilibrium in addition we present an approach that enables us to devise hardness proofs for various kinds of combinatorial games including first results about the hardness of market sharing games and congestion games for overlay network design our approach also yields a short proof for the pls completeness of network congestion games in particular we show that network congestion games are pls complete for directed and undirected networks even in case of linear latency functions

the complexity of computing a nash equilibrium in 1951 john f nash proved that every game has a nash equilibrium ann of math 2 54 1951 pp 286 295 his proof is nonconstructive relying on brouwer s fixed point theorem thus leaving open the questions is there a polynomial time algorithm for computing nash equilibria and is this reliance on brouwer inherent many algorithms have since been proposed for finding nash equilibria but none known to run in polynomial time in 1991 the complexity class ppad polynomial parity arguments on directed graphs for which brouwer s problem is complete was introduced c papadimitriou j comput system sci 48 1994 pp 489 532 motivated largely by the classification problem for nash equilibria but whether the nash problem is complete for this class remained open in this paper we resolve these questions we show that finding a nash equilibrium in three player games is indeed ppad complete and we do so by a reduction from brouwer s problem thus establishing that the two problems are computationally equivalent our reduction simulates a stylized brouwer function by a graphical game m kearns m littman and s singh graphical model for game theory in 17th conference in uncertainty in artificial intelligence uai 2001 relying on gadgets graphical games performing various arithmetic and logical operations we then show how to simulate this graphical game by a three player game where each of the three players is essentially a color class in a coloring of the underlying graph subsequent work x chen and x deng setting the complexity of 2 player nash equilibrium in 47th annual ieee symposium on foundations of computer science focs 2006 established by improving our construction that even two player games are ppad complete here we show that this result follows easily from our proof

interacting squares in arbitrary external field a model of a many body system composed of squares with contact pair interactions in an arbitrary external field is presented the formulation uses a mapping of the system onto polydisperse hard core mixtures on the polydisperse level then a simplified hamiltonian function is specified this assumption together with a further one about the global free energy functional for the pure hard core part of the idealized mixture make the model solvable it is expected to hold for high temperatures low densities or low temperatures the validity of the method of construction in the latter case is illustrated by a further application to a corresponding lattice system for which exact results to compare with are readily available when the temperature is sufficiently low

a principle of reduced stability for reaction diffusion equations a semilinear elliptic boundary value problem au f x u 0 withfu x u bounded below can be shown to be equivalent to a finite dimensional problem b c 0 rd c rd in the sense that their solution sets which are not necessarily singletons are in a one to one correspondence c u u c the functionb c is called the bifurcation function it is shown that for any solutionu c the number of negative resp zero eigenvalues of the matrixbc c is identical to the number of negative resp zero eigenvalues of the linearized elliptic operatorav fu x u c v this results in a version of the principle of reduced stability for the problemut au f x u 0 and its reductionc b c 0

the price of routing unsplittable flow in this paper we study the price of anarchy for the general class of weighted and unweighted atomic congestion games with the sum of players costs as the objective function we show that for linear resource cost functions the price of anarchy is exactly frac 3 sqrt 5 2 approx 2 618 for weighted congestion games and exactly 2 5 for unweighted congestion games we show that for resource cost functions that are polynomials of degree d the price of anarchy is d theta d our results also hold for mixed strategies in particular these results apply to atomic routing games where the traffic demand from a source to a destination must be satisfied by choosing a single path between source and destination

a bifurcation theorem for potential operators abstract a sufficient condition for bifurcation of real solutions of g u 0 where g r d e d e is given in terms of the crossing number of the eigenvalue perturbation of the linearization along the trivial solutions 0 this number describes how many eigenvalues leave the negative real axis wen the parameter varies from negative to positive values whereas in general only an odd crossing number entails bifurcation for potential operators in a hilbert space e a nonzero crossing number is sufficient for bifurcation the proof uses a lyapunov schmidt reduction and conley s index theory

competitive routing in networks with polynomial costs we study a class of noncooperative general topology networks shared by n users each user has a given flow which it has to ship from a source to a destination we consider a class of polynomial link cost functions adopted originally in the context of road traffic modeling and show that these costs have appealing properties that lead to predictable and efficient network flows in particular we show that the nash equilibrium is unique and is moreover efficient these properties make the polynomial cost structure attractive for traffic regulation and link pricing in telecommunication networks we finally discuss the computation of the equilibrium in the special case of the affine cost structure for a topology of parallel links

fast fair and efficient flows in networks we study the problem of minimizing the maximum latency of flows in networks with congestion we show that this problem is np hard even when all arc latency functions are linear and there is a single source and sink still an optimal flow and an equilibrium flow share a desirable property in this situation all flow carrying paths have the same length i e these solutions are fair which is in general not true for optimal flows in networks with nonlinear latency functions in addition the maximum latency of the nash equilibrium which can be computed efficiently is within a constant factor of that of an optimal solution that is the so called price of anarchy is bounded in contrast we present a family of instances with multiple sources and a single sink for which the price of anarchy is unbounded even in networks with linear latencies furthermore we show that an s t flow that is optimal with respect to the average latency objective is near optimal for the maximum latency objective and it is close to being fair conversely the average latency of a flow minimizing the maximum latency is also within a constant factor of that of a flow minimizing the average latency

efficiency of scalar parameterized mechanisms we consider the problem of allocating a fixed amount of an infinitely divisible resource among multiple competing fully rational users we study the efficiency guarantees that are possible when we restrict to mechanisms that satisfy certain scalability constraints motivated by large scale communication networks in particular we restrict attention to mechanisms where users are restricted to one dimensional strategy spaces we first study the efficiency guarantees possible when the mechanism is not allowed to price differentiate we study the worst case efficiency loss ratio of the utility associated with a nash equilibrium to the maximum possible utility and show that kelly s proportional allocation mechanism minimizes the efficiency loss when users are price anticipating we then turn our attention to mechanisms where price differentiation is permitted using an adaptation of the vickrey clarke groves class of mechanisms we construct a class of mechanisms with one dimensional strategy spaces where nash equilibria are fully efficient these mechanisms are shown to be fully efficient even in general convex environments under reasonable assumptions our results highlight a fundamental insight in mechanism design when the pricing flexibility available to the mechanism designer is limited restricting the strategic flexibility of bidders may actually improve the efficiency guarantee

efficient methods for selfish network design intuitively braess s paradox states that destroying a part of a network may improve the common latency of selfish flows at nash equilibrium such a paradox is a pervasive phenomenon in real world networks any administrator who wants to improve equilibrium delays in selfish networks is facing some basic questions is the network paradox ridden how can we delete some edges to optimize equilibrium flow delays how can we modify edge latencies to optimize equilibrium flow delays unfortunately such questions lead to np hard problems in general in this work we impose some natural restrictions on our networks e g we assume strictly increasing linear latencies our target is to formulate efficient algorithms for the three questions above we manage to provide a polynomial time algorithm that decides if a network is paradox ridden when latencies are linear and strictly increasing a reduction of the problem of deciding if a network with arbitrary linear latencies is paradox ridden to the problem of generating all optimal basic feasible solutions of a linear program that describes the optimal traffic allocations to the edges with constant latency an algorithm for finding a subnetwork that is almost optimal wrt equilibrium latency our algorithm is subexponential when the number of paths is polynomial and each path is of polylogarithmic length a polynomial time algorithm for the problem of finding the best subnetwork which outperforms any known approximation for the case of strictly increasing linear latencies a polynomial time method that turns the optimal flow into a nash flow by deleting the edges not used by the optimal flow and performing minimal modifications on the latencies of the remaining ones our results provide a deeper understanding of the computational complexity of recognizing the most severe manifestations of braess s paradox and our techniques show novel ways of using the probabilistic method and of exploiting convex separable quadratic programs

interaction of periodic and stationary bifurcation from multiple eigenvalues on considere des equations d evolution non lineaires du dt g u o dependant d un parametre reel dans un espace de hilbert reel e

approximating congestion dilation in networks via quality of routing games a classic optimization problem in network routing is to minimize c d where c is the maximum edge congestion and d is the maximum path length also known as dilation the problem of computing the optimal c d is np complete even when either c or d is a small constant we study routing games in general networks where each player i selfishly selects a path that minimizes ci di the sum of congestion and dilation of the player s path we first show that there are instances of this game without nash equilibria we then turn to the related quality of routing qor games which always have nash equilibria qor games represent networks with a small number of service classes where paths in different classes do not interfere with each other with frequency or time division multiplexing qor games have o log4 n price of anarchy when either c or d is a constant thus nash equilibria of qor games give poly log approximations to hard optimization problems

applications of the blowing up construction and algebraic geometry to bifurcation problems a generalization of the morse lemma to vector valued functions is proved by a blowing up argument this is combined with a theorem from algebraic geometry on the number of real solutions of a system of homogeneous equations of even degree to yield a new bifurcation theorem bifurcation in a one or multi parameter problem is guaranteed if the leading term is of even degree it is often two and satisfies a regularity condition applications are given to nonlinear eigenvalue problems and to the hopf bifurcation

competition and efficiency in congested markets we study the efficiency of oligopoly equilibria oe in congested markets the motivating examples are the allocation of network flows in a communication network or of traffic in a transportation network we show that increasing competition among oligopolists can reduce efficiency measured as the difference between users willingness to pay and delay costs we characterize a tight bound of 5 6 on efficiency in pure strategy equilibria when there is zero latency at zero flow and a tight bound of 2 2 2 with positive latency at zero flow these bounds are tight even when the numbers of routes and oligopolists are arbitrarily large

stackelberg routing in arbitrary networks we investigate the impact of stackelberg routing to reduce the price of anarchy in network routing games in this setting an fraction of the entire demand is first routed centrally according to a predefined stackelberg strategy and the remaining demand is then routed selfishly by nonatomic players although several advances have been made recently in proving that stackelberg routing can in fact significantly reduce the price of anarchy for certain network topologies the central question of whether this holds true in general is still open we answer this question negatively by constructing a family of single commodity instances such that every stackelberg strategy induces a price of anarchy that grows linearly with the size of the network moreover we prove upper bounds on the price of anarchy of the largest latency first llf strategy that only depend on the size of the network besides other implications this rules out the possibility to construct constant size networks to prove an unbounded price of anarchy in light of this negative result we consider bicriteria bounds we develop an efficiently computable stackelberg strategy that induces a flow whose cost is at most the cost of an optimal flow with respect to demands scaled by a factor of 1 1 finally we analyze the effectiveness of an easy to implement stackelberg strategy called scale we prove bounds for a general class of latency functions that includes polynomial latency functions as a special case our analysis is based on an approach that is simple yet powerful enough to obtain almost tight bounds for scale in general networks

a strategic flow model of traffic assignment in static capacitated networks this work pleads for the use of the concept of strategies and their network theoretic representation as hyperpaths for modeling network assignment problems while this concept describes adequately the behavior of users in transit systems we show that it can apply as well to networks where arc capacities are rigid this opens up a whole new field of research and raises several questions from both the theoretical and computational points of view these are investigated in the paper

settling the complexity of computing two player nash equilibria we prove that bimatrix the problem of finding a nash equilibrium in a two player game is complete for the complexity class ppad polynomial parity argument directed version introduced by papadimitriou in 1991 our result building upon the work of daskalakis et al 2006a on the complexity of four player nash equilibria settles a long standing open problem in algorithmic game theory it also serves as a starting point for a series of results concerning the complexity of two player nash equilibria in particular we prove the following theorems bimatrix does not have a fully polynomial time approximation scheme unless every problem in ppad is solvable in polynomial time the smoothed complexity of the classic lemke howson algorithm and in fact of any algorithm for bimatrix is not polynomial unless every problem in ppad is solvable in randomized polynomial time our results also have a complexity implication in mathematical economics arrow debreu market equilibria are ppad hard to compute