y Max-Planck-Institut fur Informatik

Tight Bounds for Worst-Case EquilibriaArtur Czumaj Department of Computer Science New Jersey Institute of Technology czumaj@Berthold V¨o ckingMax-Planck-Institut f¨u r Informatik Saarbr¨u cken,Germanyvoecking@mpi-sb.mpg.deAbstractThe coordination ratio is a game theoretic measure that aims to reflect the price of selfish routing in a network.We show that the worst-case coordination ratio on parallel links(of possibly different speeds)islogSupported in part by NSF grant CCR-0105701and by SBR grant No. 421090.Supported in part by DFG grant V o889/1-1and the IST program of the EU under contract number IST-1999-14186(ALCOM-FT).In this paper,we address the most basic case of a routing problem.As proposed by Koutsoupias and Papadimitriou [3],we consider a network that consist of parallel linksfrom an origin to a destination,all with possibly different speeds.There are agents, each having an amount of traffic to send from the origin to the destination(we assume throughout the paper that all are non-negative).Each agent,,sends the traffic using a possibly randomized mixed strategy,with denoting the probability that agent sends the entire traffic to a link.We assume the agents are selfish in the sense that each of them tries to minimize its individual cost. Assuming each agent is aware of the strategies of the other agents and behaves in a non-cooperative and selfish way,the system will come to a Nash equilibrium,i.e.,a combination of typically randomized choices(mixed strategies)from which no agent has an incentive to deviate.We notice that the model considered in this paper is a simplification of the problems arising in real networks. However,as pointed out in[3,5,7],this model seems to be appropriated to describe several basic networking problems.We believe that understanding the ratio between worst possible Nash quilibrium and the social optimum in simple situations is necessary for making rigorous analyzes in more complicated networks.Readers interested in more detailed exposition of this model and in its applications are referred to[3,5,7,9].The model.We define now our model formally trying to follow the notation used by Koutsoupias and Papadim-itriou[3].The routing model described above can be formally de-fined as an allocation problem with independent links with speeds and independent tasks with weights .The goal is to allocate the tasks to the links to minimize the maximum load of the links in the system.We use the standard notation to denote set .The set of pure strategies for task is therefore and a mixed strategy is a distribution on this set.Let be a combination of pure strategies,one for each task,its cost for task isGiven tasks of length and links of speed,let denote the social optimum,that is,the minimum cost of a pure strategy:min max. Furthermore,it is easy to see that in any systemmaxFor a task,the expected cost of task on link(or its finish time when its load is allocated to link[3])is equal to1In the original formulation of Koutsoupias and Papadimitriou[3],an additional additive term was used.However,since in all papers we are aware of all analyzes assumed,we skipped that term in our presentation.We want to point out,however,that our bounds are not affected by these additive terms.its strategy.As an example,we observe that in the system considered above in which all links have the same unit speed and all weights are the same,the uniform probabilities.Let denote the maximum expected load over all links, that is,maxNotice that E,and therefore max E.Finally,we define the social cost to be the expected maximum load(instead of maximum expected load),that is,E maxObserve that and possibly.Recall that denotes the social optimum(i.e.,the minimum cost of a pure strategy).In this paper our main focus is on estimating the coordination ratio which is the worst-case ratiomaxlog logand that for2.1.1Previous results.Koutsoupias and Papadimitriou[3] initiated the study of the worst-case coordination ratio and show the following results for networks consisting of parallel links:,from which one can easily derive the re-quired inequality.For two identical links the worst-case coordination ratio is exactlylog logand it is at mostlog log (see also[1]).For(not necessarily identical)links and identical weights in the fully-mixed Nash equilibrium,if, then the worst-case coordination ratio is loglog log has remained unproved priorto our work.1.2New results.Ourfirst result is an upper bound for the worst-case coordination ratio.4T HEOREM1.1.The coordination ratio for parallel links is bounded from above bymin loglog loglog log loglog logloglog loglog logby Theorem1.1.Recently,and independently of our work,also Koutsoupiaset al.[4]obtained the same upper bound.However,in thisspecial case we get a much stronger bound that is actuallytight up to an additive constant.T HEOREM1.2.For identical links the worst-case coor-dination ratio is at mostlogfor every;the classical result of Gonnet[2]implies that in such a system the worst-case coordinationratio islog log logloglogIn particular,the worst-case coordination ratio for paral-lel links islogIn fact,we will show,analogously to the upper bound, that,for every positive real and every,there exists a set of links withlog logloglog loglog logand loglog log.Proof.For,define to be the smallest index in such that or,if no such index exists, .Let us observe that the following properties hold:for every with,all links have load at least,andfor every with,link has load less than.Let.We will show bining this inequality with the obvious constraint will imply an appropriate upper bound on.In order to estimate,we start with estimating. Observe that link does not need to be the link with highest expected load.The following claim,however,shows thatis close to.C LAIM2.1.,and hence.Proof.For the purpose of contradiction,assume. This implies that.Let denote the link with the maximum expected load.Then.We observe that all the tasks that have positive proba-bility on must have weight larger than.Indeed,if one such a task had weight,then it would have ex-pected cost on link to be at mostOn the other hand,since we assumed that and O PT S tr allocates all tasks in to the links,we obtainCombining these inequalities gives.Since the sequence of link speeds is non-increasing,this implies that.This com-pletes the proof of Claim2.2.Finally,we combine the two claims above and obtainBy definition,.Consequently,which implies loglog log.Recall that is a random variable describing the load on link.We have E and E max.Thus,we need to show,for every,that it is unlikely that deviates much from its expectation.For this purpose,we will use a Hoeffding bound.In order to apply this bound,we need to show that the weights of the tasks assigned to link cannot be much larger than.This is shown in the next lemma.L EMMA2.3.For every link and every task with.First,assume thatfor some.Then,on one hand,the expected cost of task on link isbecause and.This proves Lemma2.3.Now we will apply the lemma in order to show that for every,it is unlikely that deviates much from itsexpectation.First,let us focus on a single link.Fix a link, .Let denote the set of tasks with.Furthermore,let and denote random variables that describe the cost on link only counting tasks in and,respectively. Clearly,only tasks with can be allocated to link. Hence,.First,let us consider the tasks in only.Recall that is defined as the weighted sum of independent-random variables,Pr,such that.Hence,we can apply a Hoeffding bound5to obtainPrEfor every.Now,let us consider the tasks in. Since5In this paper we use the following standard version of Hoeffding bound:Let be independent random variables with values in the interval for some,and let,then for any it holds that Pr E.where the last inequality holds for,where is a sufficiently large constant.Finally,if we substitutefor a suitably large constant,we obtainlog log,to conclude thatlog2.3Extension of Lemma1.2for identical links:Proof of Theorem1.2.It is easy to simplify the proof and to improve Lemma1.2when all links are identical,that is,all are the same.In that case,one can assume without loss of generality that and for every.Let,, denote the set of all tasks with.Given that,we can show the following lemma.L EMMA2.4.In the systems with identical links it holds thatfor all.Proof.We use similar arguments as in the proof of Lemma 2.3.The cost of task on link is. Let be any link with.This easily implies Theorem1.2.3Lower Bound:Proof of Theorem1.3This section is devoted to the proof of Theorem1.3,which states that our upper bounds proven in the previous section are essentially tight.Our analysis follows a course similar to the one for the upper bound in the previous section.First,we will describe a mixed strategy in Nash equilibrium with andlogshowing loghaving a Nash equilib-rium satisfying min log,log,and.3.1Lower Bound for Pure Strategies.We start by defin-ing a pure strategy that we will transform afterwards into a mixed strategy.Without loss of generality,let, for,the speed of the links in group is,for,for each link in group,there are exactly tasks of weight each having probability one to be allocated to this link.In our construction can be chosen to be any positive integer that satisfies.Thus,in particular,our analysis can be carried over for all satisfying.L EMMA3.1.Strategy satisfies the following properties:1.the maximum load is,2.the social optimum is,and3.the system is in Nash equilibrium.Proof.1.This property follows from the fact that if a link is ingroup then its load is.3.Let us take any task that is allocated to link ingroup and let be any link,,in group,.In order to prove that the system is in a Nash equilibrium,we must prove only that.Observe that andlog log.We focus our attention on group.Let denote the set of links in this group.contains.Now,we change the pure strategy into a mixed strategy by settingfor every,.We observe the following properties for our new mixed strategy.L EMMA3.2.Strategy satisfies the following properties:1.the maximum load is.2.the social optimum is,3.the system is in Nash equilibrium,and4.the social cost is log,we haveballs uniformly at random into,log logwhich is logloglog log。