Malicious Bayesian Congestion Games
MMalicious Bayesian Congestion Games (cid:63)
Martin Gairing
International Computer Science Institute, Berkeley, CA, [email protected]. In this paper, we introduce malicious Bayesian congestion games as anextension to congestion games where players might act in a malicious way. In sucha game each player has two types. Either the player is a rational player seeking tominimize her own delay, or (cid:21) with a certain probability (cid:21) the player is malicious inwhich case her only goal is to disturb the other players as much as possible.We show that such games do in general not possess a Bayesian Nash equilibriumin pure strategies (i.e. a pure Bayesian Nash equilibrium). Moreover, given a game,we show that it is NP-complete to decide whether it admits a pure Bayesian Nashequilibrium. This result even holds when resource latency functions are linear, eachplayer is malicious with the same probability, and all strategy sets consist of singletonsets of resources. For a slightly more restricted class of malicious Bayesian congestiongames, we provide easy checkable properties that are necessary and su(cid:30)cient for theexistence of a pure Bayesian Nash equilibrium.In the second part of the paper we study the impact of the malicious types on theoverall performance of the system (i.e. the social cost). To measure this impact, weuse the Price of Malice. We provide (tight) bounds on the Price of Malice for aninteresting class of malicious Bayesian congestion games. Moreover, we show that forcertain congestion games the advent of malicious types can also be bene(cid:28)cial to thesystem in the sense that the social cost of the worst case equilibrium decreases. Weprovide a tight bound on the maximum factor by which this happens.
Motivation and Framework. Over the last decade, the study of strategic behavior indistributed systems has improved our understanding of modern computer artifacts such asthe Internet. Normally, the users of such distributed systems are modeled as rational, util-ity optimizing players. However, in many real world scenarios, users do not necessarily actrational, but rather irrational. In this paper, we address one form of irrationality, namely,we allow that players act in a malicious way. In this case, the only goal of a maliciousplayer is to disturb the (non-malicious) players as much as possible. The presence of De-nial of Service attacks in the Internet is an example showing that such systems are quiterealistic. In many such systems with malicious players, the players have only incompleteinformation about the set of malicious players. A standard approach for modeling gameswith incomplete information uses the Harsanyi transformation [14], which converts a gamewith incomplete information to a game where players have di(cid:27)erent types. The type of aplayer represents its private information that is not common knowledge to all players. Inthe resulting Bayesian game, each player’s uncertainty about each other’s type is describedby a probability distribution. (cid:63)
This work was supported by a fellowship within the Postdoc-Programme of the German AcademicExchange Service (DAAD). a r X i v : . [ c s . G T ] J un Martin Gairing
One aspect of Game Theory that was studied extensively in recent years is the Price ofAnarchy as introduced by Koutsoupias and Papadimitriou [16]. The Price of Anarchy is theworst case ratio between the value of the social cost in an equilibrium state of the system andthat of some social optimum. Usually, the equilibrium state is de(cid:28)ned as Nash equilibrium(cid:21) a state in which no player can unilaterally improve her private objective function, alsocoined as private cost. A Nash equilibrium is pure if all players choose a pure strategy andmixed if players choose probability distributions over pure strategies.While the celebrated result of Nash [20] guarantees the existence of a mixed Nash equi-librium for ever (cid:28)nite game, pure Nash equilibria are not guaranteed to exists (see e.g.[9,12,17,18]). An natural question to ask, is whether a given game possesses a pure Nashequilibrium or not. We address this question by asking about the complexity of this decisionproblem.A class of games that always possess pure Nash equilibria is the class of congestion gamesas introduced by Rosenthal [21]. Here, the strategy set of each player is a subset of the powerset of given resources, the latency on each resource is described by a latency function in thenumber of players sharing this resource, and the private cost of each player is the sum ofthe latencies of its chosen resources. Milchtaich [18] considered weighted congestion gamesas an extension to congestion games in which the players have weights and thus di(cid:27)erentin(cid:29)uence on the latency of the resources.To measure the in(cid:29)uence of malicious behavior, Moscibroda et al. [19] introduced thePrice of Byzantine Anarchy as the worst case ratio between the social cost in an equilibriumstate of the system under some assumption on the malicious players and the social cost ofsome social optimum without malicious players. They further de(cid:28)ne the Price of Malice asthe ratio between the Price of Byzantine Anarchy and the Price of Anarchy. We will use asimilar de(cid:28)nition and de(cid:28)ne the equilibrium state as a Bayesian Nash equilibrium.Contribution. In this paper, we introduce malicious Bayesian congestion games as anextension to congestion games where players might act in a malicious way. Following Har-sanyi’s transformation [14], we allow each player to be of two types. Either the player isa rational player seeking to minimize her own delay, or (cid:21) with a certain probability (cid:21) theplayer is malicious in which case her only goal is to disturb the other players as much aspossible. For such games we study the complexity of deciding whether a given game has apure Bayesian Nash equilibrium. Moreover, we study the impact of the malicious types onthe overall performance of the system (i.e. the social cost). To measure this impact, we usethe Price of Malice, which we de(cid:28)ne similarly as Moscibroda et al. [19].We now describe our results in more detail. As our main result, we show that it isNP-complete to decide whether a given malicious Bayesian congestion game admits a pureBayesian Nash equilibrium. This result even holds if resource latency functions are linear,each player is malicious with the same probability, and all strategy sets consist singleton sets(Theorem 1). The same result even holds if we further restrict to the case that each playerhas at most four strategies and at most three players can be assigned to each resource(Theorem 2). For symmetric Bayesian congestion games with identical type probability,identical latency functions and strategy sets that consist only of singletons, we provide easycheckable properties that are necessary and su(cid:30)cient for the existence of a pure BayesianNash equilibrium (Theorem 3).We then shift gears and present results related to the Price of Malice. For general mali-cious Bayesian congestion games with linear latency functions, we show an upper bound onthe the Price of Byzantine Anarchy (Theorem 4). Moreover, we proof a lower bound on thesame ratio that already holds for the case of identical type probabilities (Theorem 5). As a alicious Bayesian Congestion Games 3 corollary, we get an asymptotic tight bound on the Price of Malice (Corollary 2). We closethe paper with a tight lower bound on the maximum factor by which the social cost of aworst case (Bayesian) Nash equilibrium of a congestion game might decrease by introducingmalicious types (Theorem 6).Related Work. Congestion games and variants thereof have long been used to model non-cooperative resource sharing among sel(cid:28)sh players. Rosenthal [21] showed that congestiongames always possess pure Nash equilibria. The complexity of computing such a pure Nashequilibrium has been settled for arbitrary latency functions by Fabrikant et al. [8] andlater for linear latency functions by Ackermann et al. [1]. On the other hand, for weightedcongestion games, Libman and Orda [17], Fotakis et al. [9] and Goemans et al. [12] provideexamples that do not allow for a pure Nash equilibrium. Dunkel and Schulz [7] showed thatit is NP-complete to decide the existence of a pure Nash equilibrium for a given weightedcongestion games.The Price of Anarchy for weighted congestion games has been studied extensively (see e.g.[3,2,5]). In case of linear latency functions, the Price of Anarchy is exactly for unweightedcongestion games [5] and Φ for weighted congestion games [3], where Φ = √ is thegolden ratio. The exact value of the Price of Anarchy is also known for the case of polynomiallatency functions [2]. For bounds on the Price of Anarchy of (weighted) congestion gameswith each strategy set being a singleton set of resources, we refer to [11] and referencestherein.Several recent papers considered games allowing for malicious player behavior [4,15,19].Moscibroda et al. [19] introduced the Price of Malice and gave bounds on the Price ofMalice for a virus inoculation game where some of the players are malicious. In fact, ourde(cid:28)nition of Price of Malice is motivated by the corresponding de(cid:28)nition from this paper.Karakostas et al. [15] and Babaio(cid:27) et al. [4], study malicious player behavior in non-atomiccongestion games. Here, each player from a continuum of in(cid:28)nitely many players controlsonly an in(cid:28)nitesimal small amount of weight and a fraction of those players is malicious. Incontrast to those papers, our games are atomic, and thus have only (cid:28)nitely many players.This yields to di(cid:27)erent results.For general Bayesian games, questions concerning the complexity of deciding the exis-tence of a pure Bayesian Nash equilibrium have been addressed in two recent works [6,13].On the one hand, if the game is given in standard normal form, i.e. the utility functions andthe type probability distribution are represented extensively as tables, then deciding the ex-istence of a pure Bayesian Nash equilibrium is NP-complete [6]. On the other hand, if both(cid:21) the utility functions and the type probability distribution (cid:21) are succinctly encoded, thenthe problem becomes PP-complete [13]. In contrast to [6], malicious Bayesian congestiongames are succinctly represented but they are more structured as the games considered byGottlob et al. [13].A certain class of Bayesian congestion game has been introduced in [10]. Here, players actcompletely rational but they are uncertain about each others weight. Among other results,the authors show that such games always possess pure Bayesian Nash equilibria if latencyfunctions are linear. Martin Gairing Γ is a tuple Γ = ( N , E, ( S u ) u ∈N , ( f e ) e ∈ E ) . Here, N is the set of players and E is the (cid:28)nite set of resources. Throughout, we denote n = |N | and r = | E | and assume n ≥ and r ≥ . For every player u ∈ N , S u ⊆ E is the strategy set of player u . Denote S = S × . . . × S n . For every resource e ∈ E , thelatency function f e : N → R is a non-negative, non-decreasing function that describes thelatency on resource e . For most of our results, we consider a(cid:30)ne latency functions withnon-negative coe(cid:30)cients, that is, for all resources e ∈ E , the latency function is of the form f e ( δ ) = a e · δ + b e with a e , b e ≥ . A(cid:30)ne latency functions are linear if b e = 0 for all e ∈ E .A congestion game is called symmetric, if S u = S (cid:48) u for any pair of players u, u (cid:48) .Strategies and Strategy Pro(cid:28)les. A pure strategy for player u is some speci(cid:28)c strategy s u ∈ S u , while a mixed strategy Q u = ( q ( u, s u )) s u ∈ S u is a a probability distribution over S u ,where q ( u, s u ) denotes the probability that player u chooses the pure strategy s u .A pure strategy pro(cid:28)le is an n -tuple s = ( s , . . . , s n ) whereas a mixed strategy pro(cid:28)le Q = ( Q , . . . , Q n ) is represented by an n -tuple of mixed strategies. For a mixed strategypro(cid:28)le Q , denote by q ( s ) = (cid:89) u ∈N q ( u, s u ) the probability that the players choose the pure strategy pro(cid:28)le s .Load and Private Cost. For a pure strategy pro(cid:28)le s , denote by δ e ( s ) = (cid:80) u ∈N : e ∈ s u the load on resource e ∈ [ m ] , i.e. the number of players assigned to e . In the same way, for apartial strategy pro(cid:28)le s − i , denote δ e ( s − i ) = (cid:80) u ∈N ,u (cid:54) = i : e ∈ s u the load on resource e ∈ [ m ] without player i .Fix a pure strategy pro(cid:28)le s . The private cost PC u ( s ) of player u ∈ N is de(cid:28)ned by thelatency of the chosen resources. ThusPC u ( s ) = (cid:88) e ∈ s u f e ( δ e ( s )) . For a mixed strategy pro(cid:28)le Q , the private cost of player u ∈ N isPC u ( Q ) = (cid:88) s ∈ S q ( s ) · PC u ( s ) . Social Cost. Associated with a congestion game Γ and a pure strategy pro(cid:28)le s is thesocial cost SC ( Γ, s ) as a measure of social welfare. In particular we use the expected averagelatency. That is, SC ( Γ, s ) = 1 n (cid:88) e ∈ E δ e ( s ) · f e ( δ e ( s ))= 1 n (cid:88) s ∈ S q ( s ) (cid:88) u ∈N (cid:88) e ∈ s u f e ( δ e ( s ))= 1 n (cid:88) u ∈N PC u ( Q ) . alicious Bayesian Congestion Games 5 Observe, that this measure di(cid:27)ers from the total latency [22] only by the factor n .The optimum associated with a congestion game Γ is the least possible social cost, overall pure strategy pro(cid:28)les s ∈ S . Thus,OPT ( Γ ) = min s ∈ S SC ( Γ, s ) . Nash Equilibria. Given a congestion game and an associated mixed strategy pro(cid:28)le Q ,player u ∈ N is satis(cid:28)ed if the player cannot improve its private cost by unilaterally changingits strategy. Otherwise, player u is unsatis(cid:28)ed. The mixed strategy pro(cid:28)le Q is a Nashequilibrium if and only if all players u ∈ N are satis(cid:28)ed, that is, PC u ( Q ) ≤ PC u ( Q − u , s u ) for all u ∈ N and s u ∈ S u .Depending on the type of strategy pro(cid:28)le we di(cid:27)er between pure and mixed Nash equi-libria.Price of Anarchy. Let G be a class of congestion games. The Price of Anarchy, also calledcoordination ratio and denoted by PoA, is the supremum, over all instances Γ ∈ G and Nashequilibria Q , of the ratio SC ( Γ, Q ) OPT ( Γ ) . Thus,PoA = sup Γ ∈G , Q SC ( Γ, Q ) OPT ( Γ ) . Ψ is an extension to congestion games,where each player is malicious with a certain probability. Following Harsanyi’s approach,we model such a game with incomplete information as a Bayesian game, where each player u ∈ N can be of two types: Either u is sel(cid:28)sh or malicious. For each type of player u ∈ N we introduce two independent type-agents u s and u m , denoting the sel(cid:28)sh and malicioustype-agent of player u , respectively.Let p u be the probability that player u ∈ N is malicious and call p u the type probabilityof player u . De(cid:28)ne the type probability vector p = ( p , . . . , p n ) in the natural way. Denote p min = min u ∈N p u . In the case of identical type probabilities p u = p for all player u ∈ N .De(cid:28)ne ∆ = (cid:80) u ∈N p u as the expected number of malicious players. Observe, that for identicaltype probabilities ∆ = p · n . Denote by Γ Ψ the congestion game that arises from the maliciousBayesian congestion game Ψ by setting p u = 0 for all player u ∈ N .Summing up, a malicious Bayesian congestion game Ψ is given by a tuple Ψ = ( N , E, ( S u ) u ∈N , ( p u ) u ∈N , ( f e ) e ∈ E ) . Strategies and Strategy Pro(cid:28)les. A pure strategy σ u for player u ∈ N is now a tuple σ u = ( σ ( u s ) , σ ( u m )) ∈ S u , where σ ( u s ) and σ ( u m ) denote the strategy of the sel(cid:28)sh type-agent and malicious type-agent of player u , respectively. Denote σ = ( σ , . . . , σ n ) . A mixedstrategy Q i is now a probability distribution over S i × S i . De(cid:28)ne Q and q ( σ ) as before.Private Cost. For any type probability vector p and pure strategy pro(cid:28)le σ , denote theexpected sel(cid:28)sh load on resource e ∈ E by δ e ( σ ) = (cid:80) u ∈N : e ∈ σ ( u s ) (1 − p u ) and the expectedmalicious load by κ e ( σ ) = (cid:80) u ∈N : e ∈ σ ( u m ) p u . For a partial assignment σ − u de(cid:28)ne δ e ( σ − u ) and κ e ( σ − u ) accordingly, by disregarding player u . Martin Gairing
Fix any type probability vector p and pure strategy pro(cid:28)le σ . The private cost PC u ( p , σ ) of player u ∈ N is de(cid:28)ned byPC u ( p , σ ) = (cid:88) e ∈ σ ( u s ) f e ( δ e ( σ − u ) + κ e ( σ − u ) + 1) . In other words PC u ( p , σ ) is the expected latency that player u experiences if player u issel(cid:28)sh. For each player u ∈ N , type-agent u s aims to minimize PC u ( p , σ ) . Observe, thatPC u ( p , σ ) does not depend on σ ( u m ) . For a mixed strategy pro(cid:28)le Q , de(cid:28)ne PC u ( p , Q ) accordingly.Social Cost. Let Ψ be a malicious Bayesian congestion game with type probability vector p and let Q be a pure strategy pro(cid:28)le for Ψ . We generalize the de(cid:28)nition of social costSC ( Ψ, Q ) to the weighted average latency of the sel(cid:28)sh type-agents. That is,SC ( Ψ, Q ) = (cid:80) u ∈N (1 − p u ) · PC u ( p , Q ) n − ∆ . Bayesian Nash equilibria. A sel(cid:28)sh type-agent is satis(cid:28)ed if she cannot unilaterally de-crease her private cost, that is, PC u ( Q ) ≤ PC u ( Q − u s , σ ( u s )) for all u ∈ N and σ ( u s ) ∈ S u .In contrast to the sel(cid:28)sh type-agents, each malicious type-agent aims to maximize socialcost. So, a malicious type-agent is satis(cid:28)ed if she cannot increase social cost by unilaterallychanging her strategy.For a malicious Bayesian congestion game, a mixed strategy pro(cid:28)le Q is a Bayesian Nashequilibrium if and only if both type-agents of all players u ∈ N are satis(cid:28)ed. Depending onthe type of strategy pro(cid:28)le we again di(cid:27)er between pure and mixed Bayesian Nash equilibria.Price of Byzantine Anarchy and Price of Malice. For a (cid:28)xed expected number ofmalicious players ∆ , let G ( ∆ ) be the class of malicious Bayesian congestion games where (cid:80) u ∈N p u = ∆ . Similarly to [19], we de(cid:28)ne the Price of Byzantine Anarchy, denoted by PoB,as the supremum, over all instances Ψ ∈ G ( ∆ ) and Bayesian Nash equilibria Q , of the ratiobetween the social cost in Q and the optimum social cost of the corresponding congestiongame Γ Ψ . Thus, PoB ( ∆ ) = sup Ψ ∈G ( ∆ ) , Q SC ( Ψ, Q ) OPT ( Γ Ψ ) . Observe that for ∆ = 0 , the Price of Byzantine Anarchy PoB (0) reduces to the Price ofAnarchy PoA as de(cid:28)ned in Section 2.1.Again similarly to [19], we de(cid:28)ne the Price of Malice byPoM ( ∆ ) = PoB ( ∆ ) PoB (0) . In this section, we study the complexity of deciding whether a given malicious Bayesiancongestion game possesses a pure Bayesian Nash equilibrium or not. alicious Bayesian Congestion Games 7
Theorem 1. The problem of deciding whether a malicious Bayesian congestion game withlinear latency functions possesses a pure Bayesian Nash equilibrium is NP-complete, evenif all strategy sets consist of singletons and either of the following properties holds:(a) All players are malicious with the same probability p for any < p < .(b) Only one player is malicious with positive probability p for any < p ≤ .Proof. Our proof uses a reduction from a restricted version of 3-SAT. Here, 3-SAT is re-stricted to instances where each clause is a disjunction of 2 or 3 variables and each variableoccurs at most three times. Tovey [23] showed that it is NP-complete to decide the satis-(cid:28)ability of such instances. Consider an arbitrary instance of 3-SAT with set of variables X = { x , . . . x (cid:96) } and set of clauses C = { c , . . . c k } . Without loss of generality, we mayassume that each variable occurs at most twice unnegated and at most twice negated. u x u x u x (cid:96) e x e x e x e x e x (cid:96) e x (cid:96) e e u u u c u c u c k βM u e e e Fig. 1. Construction for the proof of Theorem 1Part (a): We will construct a malicious Bayesian congestion game with singleton strategysets and identical type probability p . Our construction imposes one player u c for each clause c ∈ C , one player u x and two resources e x , e x for each variable x ∈ X , 3 additional players u , u , u , and 5 additional resources e , e , e , e , e . Our construction is summarized inFigure 1. Resources are depicted as squares and players as circles and an edge (solid ordotted) between a resource e and a player u indicates that { e } is in u ’s strategy set. Anumber α above a resource e de(cid:28)nes the slope of the corresponding linear latency function f e ( δ ) = α · δ . Denote E v = { e x , e x , . . . , e x (cid:96) , e x (cid:96) } . For the proof of part (a), let β = 2 − p .So, all resources e ∈ E v share the latency function f e ( δ ) = (2 − p ) · δ .Player u can only be assigned to e . Both u and e are used to collect the malicioustype-agents of all players except player u . Thus all those players have e in their strategyset and M is chosen su(cid:30)ciently large, such that for all those malicious type-agents e is adominant strategy and no sel(cid:28)sh type other than u s will ever prefer to choose e . Choosing M = (cid:96) +1 su(cid:30)ces. Player u and u are connected to e , e , and e , while u can also choose e and e . For each variable x ∈ X , the corresponding variable player u x is connected to e , e , e x and e x . Assigning the sel(cid:28)sh type-agent u sx to e x (resp. e x ) will be interpreted assetting x to true (resp. false). For each clause c ∈ C , the corresponding clause player u c isconnected to e and to all resources e x ( e x ) with x ∈ X and x appears negated (unnegated) Martin Gairing in c . For the example in Figure 1, c = ( x ∨ x ∨ x (cid:96) ) , c = ( x ∨ x ) , and c k = ( x ∨ x ∨ x (cid:96) ) .Observe that by the structure of our 3-SAT instance, no more than two clause players areconnected to each resource in E v . This (cid:28)nishes the construction of the malicious Bayesiancongestion game.We will (cid:28)rst show that if the 3-SAT instance is satis(cid:28)able then the correspondingBayesian congestion game possesses a pure Bayesian Nash equilibrium. Given a satisfy-ing truth assignment, we de(cid:28)ne a strategy pro(cid:28)le σ of the malicious Bayesian congestiongame as follows:(cid:21) Both type-agents of player u can only be assigned to e .(cid:21) All malicious type-agents except u m are assigned to resource e . By the choice of M ,none of those malicious type-agents can improve.(cid:21) Both type-agents of player u are assigned to e and no type-agent of any player isassigned to e or e . It is easy to see that neither u m nor u s have an incentive to switch.(cid:21) Type agent u s is the only type-agent assigned to e . So, u s cannot improve.(cid:21) For each x ∈ X , the sel(cid:28)sh type-agent u sx of variable player u x is assigned to resource e x if x = true in the satisfying truth assignment, and to e x otherwise. Each of these sel(cid:28)shtype-agents is the only type-agent assigned to her resource. So, they all experience anexpected latency of β = 2 − p and changing to e would yield the same expected latency.Thus, the sel(cid:28)sh type-agents of all variable players are satis(cid:28)ed.(cid:21) Denote by E (cid:48) v the subset of resources from E v to which no sel(cid:28)sh type-agent of a variableplayer is assigned. Since we have a satisfying truth assignment, each clause player isconnected to some resource from E (cid:48) v . For each c ∈ C , the sel(cid:28)sh type-agent u sc isassigned to some resource in E (cid:48) v as follows:Consider the sub-game that consists only of the sel(cid:28)sh type-agents of the clause players u c , c ∈ C and the set of resources E (cid:48) v . Observe that this sub-game is a (non-malicious)congestion game and thus admits a pure Nash equilibrium [21]. Assign the sel(cid:28)sh type-agents of each clause player according to this Nash equilibrium. So, none of these sel(cid:28)shtype-agents can improve by changing to some other resource in E (cid:48) v . Moreover, at mosttwo sel(cid:28)sh type-agents are assigned to each resource in E (cid:48) v and there is exactly onesel(cid:28)sh type-agent of a variable player assigned to each resource in E v \ E (cid:48) v . Thus, thesel(cid:28)sh type-agents of all clause players are satis(cid:28)ed.Since no type-agent can improve be changing her strategy, it follows that σ in a pureBayesian Nash equilibrium.For the other direction observe that any pure Bayesian Nash equilibrium σ ful(cid:28)lls thefollowing structural properties:(I) All malicious type-agents except u m are assigned to resource e and u s is the only sel(cid:28)shtype-agent assigned to e .(II) The sel(cid:28)sh type-agent u s is assigned to e and no other type-agent is assigned to e .Property (I) follows immediately by the choice of M . We will now prove property (II).By way of contradiction assume that u s is assigned to a resource in { e , e , e } in apure Bayesian Nash equilibrium σ . In this case u m will always choose the same resource as u s . However, then there must be an empty resource in { e , e , e } and u s can improve bychoosing this empty resource. This contradicts our assumption that σ is a pure BayesianNash equilibrium. Thus, u s is assigned to e . If some other type-agent is also assigned to e ,then u s experiences an expected latency of at least − p and u s could decrease her expectedlatency to by switching to the empty resource in { e , e , e } . Again a contradiction to σ alicious Bayesian Congestion Games 9 being a pure Bayesian Nash equilibrium. It follows that u s is the only type-agent assignedto e in σ . This completes the proof of property (II).Since u s is the only type-agent assigned to e it follows that for each variable x ∈ X thecorresponding sel(cid:28)sh type-agent u sx is either assigned to e x or to e x . If u sx is not the onlytype-agent on that resource then her expected latency is at least (2 − p ) while changing to e would improve her expected latency to − p , a contradiction to σ being a pure BayesianNash equilibrium. It follows that the sel(cid:28)sh type-agents of all clause players are only assignedto resources in E v to which no sel(cid:28)sh type-agent of a variable player is assigned. This isonly possible if the strategies of the sel(cid:28)sh type-agents u sx , x ∈ X correspond to a satisfyingtruth assignment. This (cid:28)nishes the proof of part (a).Part (b): To see that (b) holds we alter the construction depicted in Figure 1 slightlyby deleting player u and resource e . Furthermore, in the new construction player u isthe only player that is malicious with positive probability p . For the slope of the latencyfunctions of resources in E v , let β = (in fact any < β < would also do). The rest ofthe construction does not change. The proof now follows the same line of arguments as inpart (a) with only minor changes. (cid:117)(cid:116) Theorem 2. The results from Theorem 1 hold, even if | S u | ≤ for all players u ∈ N andfor each resource e ∈ E there are at most three players u ∈ N with { e } ∈ S u .Proof (Sketch). We will slightly alter the construction from Figure 1. First observe that wehave already | S u | ≤ for all players u ∈ N . Furthermore, the only resources that are in thestrategy set of more than three players are e and for part (a) also e . e , e ,x e ,x e ,x β β u u x u x u x Fig. 2. Tree for (cid:96) = 3
To resolve this for e , disconnect all players from e and replace the single resource e with a binary tree of resources with root e , that has (cid:96) leaves e ,x , . . . e ,x (cid:96) , all with depth (cid:100) log( (cid:96) ) (cid:101) . For a resource e at level j the latency function is de(cid:28)ned by f e ( δ ) = β j · δ . So f e , ( δ ) = 1 and f e ,x ( δ ) = β (cid:100) log( (cid:96) ) (cid:101) · δ for all leaves x ∈ X . For each pair of resources fromtwo consecutive levels, we introduce a new player to connect them. Call those players treeplayers. Figure 2 shows the construction for (cid:96) = 3 . Player u is connected to resource e , and each variable player x ∈ X is connected to e ,x . We also change the latency function ofall resources e ∈ E v (cf. Theorem 1) to f e ( δ ) = β (cid:100) log( (cid:96) ) (cid:101) · δ .Moreover, for part (a) we have to resolve that more than three players are connected to e . To do so, we simply copy resource e together with player u multiple times and connectall players (including the tree players) except player u to the new set of resources thatevolve from e . By having su(cid:30)ciently many copies, this can be done, such that no morethan three players are connected to each new resource. Again, M is chosen su(cid:30)ciently large,e.g. M = 2 (cid:100) log( (cid:96) ) (cid:101) +1 . Observe that u s will only sel(cid:28)shly choose e , if all tree players choose the strategy thatis closer to the leaves. The rest of the proof now simply follows the proof of Theorem 1. (cid:117)(cid:116) For the more restricted class of symmetric malicious Bayesian congestion game withsingleton strategy sets, identical type probability p and identical latency functions we caneasily decide whether a pure Bayesian Nash equilibrium exists or not.Theorem 3. A symmetric malicious Bayesian congestion game with singleton strategy sets,identical type probability p and identical (not necessarily linear) latency functions possessesa pure Bayesian Nash equilibrium if and only if either of the following properties holds:(a) p ≤ and r = 2 (b) p ≤ and r | n Proof. Assume that at least one of the properties holds. We will show that in each case thisimplies the existence of a pure Bayesian Nash equilibrium.First assume that (a) holds: For each player u i assign the sel(cid:28)sh type-agent u si alternatelyto the two resources. At each time, we assign the corresponding malicious type-agent to theother resource. It’s not hard to see that the resulting strategy pro(cid:28)le is a pure BayesianNash equilibrium.Now assume that (b) holds: In this case assign nr sel(cid:28)sh type-agents and nr malicioustype-agents to each resource such that the sel(cid:28)sh and malicious type-agent of each (cid:28)xedplayer are not assigned to the same resource. Again it is easy to see that the resultingstrategy pro(cid:28)le is a pure Bayesian Nash equilibrium.For the other direction we will show that if neither (a) nor (b) holds then the maliciousBayesian congestion game does not possess a pure Bayesian Nash equilibrium. By way ofcontradiction assume there exists a malicious Bayesian congestion game Γ satisfying neither(a) nor (b) but Γ admits a pure Bayesian Nash equilibrium σ . We consider 3 sub-cases:Case 1: p > and r = 2 By way of contradiction assume σ assigns more than (cid:100) n (cid:101) sel(cid:28)sh type-agents to some resource e . If this is the case then σ will also assign all malicious type-agents to e . But then allthe sel(cid:28)sh type-agents on resource e can improve by switching to the other resource, acontradiction to σ being a pure Bayesian Nash equilibrium. It follows that at most (cid:100) n (cid:101) sel(cid:28)sh type-agents are assigned to each resource. This again implies that (cid:100) n (cid:101) sel(cid:28)sh type-agents are assigned to one resource (say e ) and (cid:98) n (cid:99) sel(cid:28)sh type-agents are assigned to theother resource (say e ). Denote N = { u ∈ N | σ ( u s ) = e } the set of players with sel(cid:28)shtype-agent assigned to resource e and denote N = N \ N . For each player u ∈ N wehave δ e ( σ − u ) > δ e ( σ − u ) which implies that σ ( u m ) = e for all u ∈ N . Will now showthat σ ( u m ) = e for all u ∈ N . If n is even then this holds immediately by symmetry. Soassume n is odd and ∃ u ∈ N with σ ( u m ) = e . Now consider an arbitrary player u (cid:48) ∈ N with u (cid:48) (cid:54) = u . Since n is odd it follows that n ≥ and thus such a player exits. Furthermore,it follows that (cid:4) n (cid:5) = (cid:6) n (cid:7) − . Then δ e ( σ − u (cid:48) ) + κ e ( σ − u (cid:48) ) ≥ (1 − p ) · ( (cid:108) n (cid:109) −
1) + p · ( (cid:106) n (cid:107) + 1)= (cid:108) n (cid:109) − p alicious Bayesian Congestion Games 11 while δ e ( σ − u (cid:48) ) + κ e ( σ − u (cid:48) ) ≤ (1 − p ) · (cid:106) n (cid:107) + p · ( (cid:108) n (cid:109) − (cid:108) n (cid:109) − − p< δ e ( σ − u (cid:48) ) + κ e ( σ − u (cid:48) ) , a contradiction to σ being a pure Bayesian Nash equilibrium. So, σ ( u m ) = e for all u ∈ N .Summing up, for all u ∈ N we have σ ( u s ) = e and σ ( u m ) = e while for all u ∈ N wehave σ ( u s ) = e and σ ( u m ) = e . In such an assignment each sel(cid:28)sh type-agent on resource e can improve be switching to e . This contradicts our initial assumption that σ is a pureBayesian Nash equilibrium.Case 2: p > and r | n First assume by way of contradiction that there exists a resource e ∈ E to which σ assignsmore than nr sel(cid:28)sh type-agents. It follows that there also exists some other resource e (cid:48) towhich σ assigns less than nr sel(cid:28)sh type-agents. Since σ is a pure Bayesian Nash equilibrium,no malicious type-agent is assigned to e (cid:48) . However, then all sel(cid:28)sh type-agents on e improveby switching to e (cid:48) , a contradiction. It follows that σ assigns exactly nr sel(cid:28)sh type-agents toeach resource. This also implies that σ ( u m ) (cid:54) = σ ( u s ) for all players u ∈ N .If more than nr malicious type-agents are assigned to some resource e ∈ E then all sel(cid:28)shtype-agents on e can improve since p > . So, σ assigns exactly nr malicious type-agentsto each resource. However, in such a pure strategy pro(cid:28)le the sel(cid:28)sh type-agent u s of eachplayer u can improve by switching to σ ( u m ) . This contradicts our initial assumption that σ is a pure Bayesian Nash equilibrium.Case 3: r ≥ and nr (cid:54)∈ N First observe that if n < r and the malicious type-agents are satis(cid:28)ed then there is alwayssome resource e ∈ E to which no type-agent is assigned. Each sel(cid:28)sh type-agent can thenimprove by switching to e , a contradiction to σ being a pure Bayesian Nash equilibrium. Sowe may assume that n > r .Let E + be the set of resources to which σ assigns at least (cid:6) nr (cid:7) sel(cid:28)sh type-agents. Since nr (cid:54)∈ N it follows that ≤ | E + | ≤ r − . If | E + | ≥ then σ assigns all malicious type-agentsto a resource in E + . This implies that there exists a sel(cid:28)sh type-agents assigned to someresource in E + that can improve by switching to some resource in E \ E + . It follows that | E + | = 1 . Without loss of generality assume E + = { e } .Since | E + | = 1 it follows that σ assigns exactly (cid:4) nr (cid:5) = (cid:6) nr (cid:7) − sel(cid:28)sh type-agents toeach resource e ∈ E \ E + . Now, for all u ∈ N , if σ ( u s ) ∈ E \ E + then σ ( u m ) = e . It followsthat σ assigns at least ( r − · (cid:4) nr (cid:5) malicious type-agents to e and (by the pigeon holeprinciple) there exists a resource e (cid:48) ∈ E \ E + to which σ assigns at most (cid:22) (cid:100) nr (cid:101) r − (cid:23) malicioustype-agents. Since r ≥ and n > r it follows that all sel(cid:28)sh type-agents on resource e canimprove by switching to resource e (cid:48) . This contradicts our initial assumption that σ is a pureBayesian Nash equilibrium.In each case we got a contradiction to our assumption that σ is a pure Bayesian Nashequilibrium, proving that Γ does not admit a pure Bayesian Nash equilibrium. This (cid:28)nishesthe proof of the theorem. (cid:117)(cid:116) Observe, that the previous proof is constructive. So, if the requirements for the existenceof a pure Bayesian Nash equilibrium are ful(cid:28)lled, then this equilibrium can also be easilyconstructed in linear time.
We now shift gears and present our results that are related to the Price of Malice. We startwith a general upper bound on the Price of Byzantine Anarchy. The proof of this upperbound uses a technique from [5] adapted to the model of malicious Bayesian congestiongames. Furthermore, it makes use of the following technical lemma, which has an easyproof.Lemma 1. For all x, y ∈ R and c > we have x · y ≤ c · x + c · y Theorem 4. Consider the class of malicious Bayesian congestion games G ( ∆ ) with a(cid:30)nelatency functions. Then,PoB ( ∆ ) ≤ nn − ∆ (1 − p min ) (cid:18) ∆ + 3 + √ ∆ (cid:19) . Proof. Let Ψ be an arbitrary malicious Bayesian congestion game from G ( ∆ ) and let Γ Ψ bethe corresponding (non-malicious) congestion game. Let Q be an arbitrary Bayesian Nashequilibrium for Ψ . Furthermore, let s ∗ be an optimum pure strategy pro(cid:28)le for Γ Ψ . For eachplayer u ∈ N , we havePC u ( p , Q ) = (cid:88) σ ∈S q ( σ ) · PC u ( p , σ ) ≤ PC u ( p , ( Q − u s , s ∗ u ))= (cid:88) σ ∈S q ( σ ) · PC u ( p , ( σ − u s , s ∗ u ))= (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ s ∗ u f e ( δ e ( σ − u ) + κ e ( σ − u ) + 1) ≤ (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ s ∗ u f e ( δ e ( σ ) + ∆ + 1) , where the (cid:28)rst inequality follows since Q is a Bayesian Nash equilibrium and the secondinequality holds, since κ e ( σ − u ) ≤ κ e ( σ ) ≤ ∆ for all e ∈ E . So, we get (cid:88) u ∈N (1 − p u ) · PC u ( p , Q ) ≤ (cid:88) σ ∈S q ( σ ) (cid:88) u ∈N (cid:88) e ∈ σ ∗ ( u ) (1 − p u ) · f e ( δ e ( σ ) + ∆ + 1) ≤ (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ E (1 − p min ) · δ e ( s ∗ ) · f e ( δ e ( σ ) + ∆ + 1) ≤ (1 − p min ) · (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ E δ e ( s ∗ ) · f e ( ∆ + 1)+ (1 − p min ) · (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ E a e · δ e ( σ ) · δ e ( s ∗ ) alicious Bayesian Congestion Games 13 Observe, that δ e ( s ∗ ) ≥ and thus δ e ( s ∗ ) ≤ δ e ( s ∗ ) . Moreover, by applying Lemma 1 with x = δ e ( σ ) and y = δ e ( s ∗ ) , we get (cid:88) u ∈N (1 − p u ) · PC u ( p , Q ) ≤ (1 − p min ) · ( ∆ + 1) · (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ E δ e ( s ∗ ) · f e ( δ e ( s ∗ ))+ (1 − p min ) · c · (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ E a e · δ e ( σ ) + (1 − p min ) · c · (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ E a e · δ e ( s ∗ ) ≤ (1 − p min )( ∆ + 1 + 14 c ) (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ E δ e ( s ∗ ) · f e ( δ e ( s ∗ ))+ c · (cid:88) σ ∈S q ( σ ) (cid:88) e ∈ E δ e ( σ ) f e ( δ e ( σ ))= (1 − p min )( ∆ + 1 + 14 c ) · (cid:88) u ∈N (cid:88) e ∈ s ∗ u f e ( δ e ( s ∗ ))+ c · (cid:88) σ ∈S q ( σ ) (cid:88) u ∈N (cid:88) e ∈ σ ( u s ) (1 − p u ) · f e ( δ e ( σ )) ≤ (1 − p min )( ∆ + 1 + 14 c ) · (cid:88) u ∈N (cid:88) e ∈ s ∗ u f e ( δ e ( s ∗ ))+ c · (cid:88) σ ∈S q ( σ ) (cid:88) u ∈N (cid:88) e ∈ σ ( u s ) (1 − p u ) · f e ( δ e ( σ − u ) + 1) ≤ (1 − p min )( ∆ + 1 + 14 c ) · (cid:88) u ∈N PC u ( s ∗ )+ c · (cid:88) u ∈N (1 − p u ) · PC u ( p , Q ) It follows that SC ( p , Q ) SC ( s ∗ ) = nn − ∆ · (cid:80) u ∈N (1 − p u ) · PC u ( Q ) (cid:80) u ∈N PC u ( s ∗ ) ≤ nn − ∆ (1 − p min ) ∆ + 1 + c − c (1)Now, choosing c = − √ ∆ ∆ +1) yields ∆ + 1 + c − c = ∆ + 1 + ∆ +1 − √ ∆ − − √ ∆ ∆ +1) = 4( ∆ + 1) (1 + − √ ∆ )5 + 4 ∆ − √ ∆ = 4( ∆ + 1) (1 + √ ∆ ∆ )5 + 4 ∆ − √ ∆ = ( ∆ + 1)(5 + 4 ∆ + √ ∆ )5 + 4 ∆ − √ ∆ = ( ∆ + 1)( √ ∆ + 1) √ ∆ −
1= ( ∆ + 1)( √ ∆ + 1) ∆ = 14 (4 ∆ + 6 + 2 √ ∆ )= ∆ + 3 + √ ∆ (2)The theorem follows by combining (1) and (2) and since Q is an arbitrary Bayesian Nashequilibrium. (cid:117)(cid:116) For the case of identical type probabilities we can provide a better upper bound on thePrice of Byzantine Anarchy. Observe that for identical type probabilities, ∆ = p · n and p min = p . As an immediate corollary to Theorem 4, we get:Corollary 1. Consider the class of malicious Bayesian congestion games G ( ∆ ) with a(cid:30)nelatency functions and identical type probability p . Then,PoB ( ∆ ) ≤ ∆ + 3 + √ ∆ . We proceed by introducing a malicious Bayesian congestion game that is parameterizedby a parameter α . In the remainder of the paper, we will make use of this constructiontwice, each time with a di(cid:27)erent parameter α .Example 1. Given some α > , construct a malicious Bayesian congestion game Γ ( α ) withlinear latency functions, n ≥ players and identical type probability p and | E | = 2 n asfollows: Let E = E ∪ E with E = { g , . . . , g n } and E = { h , . . . , h n } . Each player u ∈ { , . . . n } has three strategies in her strategy set. So, S u = { s u , s u , s u } with s u = { g u , h u } , s u = { g u +1 , h u +1 , h u +2 } and s u = E ∪ E , where g j = g j − n and h j = h j − n for j > n .Each resource e ∈ E has a latency function f e ( δ ) = α · δ whereas the resources e ∈ E sharethe identity as their latency function, i.e. f e ( δ ) = δ .The following theorem makes use of Example 1 to show the following lower bound onthe Price of Byzantine Anarchy.Theorem 5. Consider the class of malicious Bayesian congestion games G ( ∆ ) with linearlatency functions and identical type probability p . Then ,PoB ( ∆ ) ≥ ∆ + 2 . Proof. Consider the malicious Bayesian congestion game Ψ = Ψ ( α ) given in Example 1 with α = n − p − p . Observe that ∆ = n · p . alicious Bayesian Congestion Games 15 Obviously , the optimum allocation s ∗ for the corresponding non-malicious game Γ Ψ isfor each player u ∈ N to choose strategy s u . This yields SC ( Γ Ψ , s ∗ ) = 1 + α = n − p − p .On the other hand, if σ ( u m ) = s u and σ ( u s ) = s u for all player u ∈ N , then σ is a(pure) Bayesian Nash equilibrium for Ψ , withSC ( Ψ, σ ) = 2(1 + (1 − p ) + ( n − p ) + (1 + ( n − p ) · α = 2(1 − p )(2 + ( n − p ) + (1 + ( n − p ) − p It follows that SC ( Ψ, σ ) SC ( Γ Ψ , s ∗ ) = 2(1 − p ) + (1 + ( n − p ) n − p = 2(1 − p ) + 1 + ( n − p (2 + ( n − p )2 + ( n − p> − p + n · p = ∆ + 2 − p. The Theorem follows for p → , which implies n → ∞ . (cid:117)(cid:116) Recall that the Price of Anarchy of (non-malicious) congestion games with a(cid:30)ne latencyfunctions is [5]. By combining this with Corollary 1 and Theorem 5 we get:Corollary 2. Consider the class of malicious Bayesian congestion games G ( ∆ ) with a(cid:30)nelatency functions and identical type probability. Then,PoM ( ∆ ) = Θ ( ∆ ) . For certain congestion games, introducing malicious types might also be bene(cid:28)cial to thesystem, in the sense that the social cost of the worst case equilibrium (one that maximizessocial cost) decreases. To capture this, we de(cid:28)ne the Windfall of Malice. The term Windfallof Malice is due to [4]. For a malicious Bayesian congestion game Ψ , denote WoM ( Ψ ) as theratio between the worst case Nash equilibrium of the corresponding congestion game Γ Ψ and the worst case Bayesian Nash equilibrium of Ψ . We show,Theorem 6. For each (cid:15) > there is a malicious Bayesian congestion game Ψ with linearlatency functions and identical type probability, such thatWoM ( Ψ ) ≥ − (cid:15). Proof. Consider the malicious Bayesian congestion game Ψ = Ψ ( α ) given in Example 1 with n = 3 and α = 1 . This game (for n ≥ ) was already used in [5] to proof a lower boundon the Price of Anarchy for the corresponding non-malicious congestion games. For thecongestion game Γ Ψ that corresponds to Ψ , all players u choosing s u is a Nash equilibrium s that maximizes social cost and SC ( Γ Ψ , s ) = 5 .Now, consider the malicious Bayesian congestion game Ψ , where p > . First observethat choosing s u is always a strictly dominant strategy for the malicious type-agent u m forall u ∈ N . Moreover, u s will never choose s u . For i ∈ { , } , let q i be the probability that u si chooses s u i . Then u si chooses s u i with probability (1 − q i ) . We will show that for all p > ,the sel(cid:28)sh type-agent u s experiences a strictly lower expected latency, if she chooses s u andnot s u .On the one hand, if u s chooses s u then her expected latency is: p + (1 − q + 1 − q )(1 − p ) (cid:124) (cid:123)(cid:122) (cid:125) h + 1 + 2 p + (1 − q )(1 − p ) (cid:124) (cid:123)(cid:122) (cid:125) g = 2 + 4 p + (1 − p )(3 − q − q ) On the other hand, if u s chooses s u then her expected latency is: p + ( q + 1 − q )(1 − p ) (cid:124) (cid:123)(cid:122) (cid:125) h + 1 + 2 p + q (1 − p ) (cid:124) (cid:123)(cid:122) (cid:125) g + 1 + 2 p + (1 − q + q )(1 − p ) (cid:124) (cid:123)(cid:122) (cid:125) h = 3 + 6 p + (1 − p )(2 + q ) . However, p + (1 − p )(2 + q ) − (2 + 4 p + (1 − p )(3 − q − q ))= 1 + 2 p + (1 − p )( − q + 2 q ) ≥ p. So u s is always better of by choosing s u .By symmetry it follows that for each p > there is a unique (pure) Bayesian Nashequilibrium σ where σ ( u s ) = s u and σ ( u m ) = s u for all players u ∈ N . For its social costwe get SC ( Ψ, σ ) = 2 + 4 p .So, for each (cid:15) > there is a su(cid:30)ciently small p , such thatWoM ( Ψ ) = SC ( Γ Ψ , s ) SC ( Ψ, σ ) = 52 + 4 p ≥ − (cid:15). This (cid:28)nishes the proof of the theorem. (cid:117)(cid:116)
This is actually a tight result, since for the considered class of malicious Bayesian games theWindfall of Malice cannot be larger than the Price of Anarchy of the corresponding class ofcongestion games which was shown to be in [5]. In this paper, we have introduced and studied a new extension to congestion games, thatwe call malicious Bayesian congestion games. More speci(cid:28)cally, we have studied problemsconcerned with the complexity of deciding the existence of pure Bayesian Nash equilibria.Furthermore, we have presented results on the Price of Malice.Although we were able to derive multiple interesting results, this work also gives rise tomany interesting open problems. We conclude this paper by stating those, that we considerthe most prominent ones.(cid:21) Our NP-completeness result in Theorem 1 holds even for linear latency functions, iden-tical type probabilities, and if all strategy sets are singleton sets of resources. However, alicious Bayesian Congestion Games 17 if such games are further restricted to symmetric games and identical linear latencyfunctions, then deciding the existence of a pure Bayesian Nash equilibrium becomes atrivial task. We believe that this task can also be performed in polynomial time fornon-identical linear latency functions and symmetric strategy sets.(cid:21) Another, interesting problem in this perspective is to reduce the constants in Theorem 2or show that this is not possible.(cid:21) Although the upper bound in Corollary 1 and the corresponding lower bound in The-orem 5 are asymptotically tight, there is still potential to improve. We conjecture thatin this case PoB ( ∆ ) = ∆ + O (1) .(cid:21) We believe that the concept of malicious Bayesian games is very interesting and deservesfurther study also in other scenarios. We hope, that our work will encourage others tostudy such malicious Bayesian games. We are very grateful to Christos Papadimitriou and Andreas Maletti for many fruitfuldiscussions on the topic. Moreover, we thank Florian Schoppmann for his helpful commentson an early version of this paper.