informs DOI /moor.xxxx.xxxx c 200x INFORMS

MATHEMATICS OF OPERATIONS RESEARCH Vol. xx, No. x, Xxxxxxx 200x, pp. xxx xxx ISSN 0364-765X EISSN 1526-5471 0x xx0x 0xxx nforms DOI 10.1287/moor.xxxx.xxxx c 200x INFORMS On the Complexty of Pure-Strategy Nash Equlbra n Congeston and Local-Effect Games 1 Julane Dunkel Operatons Research Center, Massachusetts Insttute of Technology, 77 Massachusetts Avenue, Cambrdge, MA 02139 emal: julane@mt.edu Andreas S. Schulz Sloan School of Management, Massachusetts Insttute of Technology, 77 Massachusetts Avenue, Cambrdge, MA 02139 emal: schulz@mt.edu Rosenthal s congeston games consttute one of the few known classes of noncooperatve games possessng purestrategy Nash equlbra. In the network verson, each player wants to route one unt of flow on a sngle path from her orgn to her destnaton at mnmum cost, and the cost of usng an arc depends only on the total number of players usng that arc. A natural extenson s to allow for players controllng dfferent amounts of flow, whch results n so-called weghted congeston games. Whle examples have been exhbted showng that pure-strategy Nash equlbra need not exst anymore, we prove that t s actually strongly NP-hard to determne whether a gven weghted network congeston game has a pure-strategy Nash equlbrum. Ths s true regardless of whether flow s unsplttable or not. In the unsplttable case, the problem remans strongly NP-hard for a fxed number of players. In addton to congeston games, we provde complexty results on the exstence and computablty of pure-strategy Nash equlbra for the closely related famly of bdrectonal local-effect games. Theren, the cost of a player takng a partcular acton depends not only on the number of players choosng the same acton but also on the number of players settlng for (locally) related actons. ey words: Noncooperatve games, Pure-strategy Nash equlbra, Computatonal complexty, Congeston games, Local-effect games MSC2000 Subject Classfcaton: Prmary: 90B10, 90B20, 91A10; Secondary: 90C27, 91A43 OR/MS subject classfcaton: Prmary: networks/graphs: multcommodty, theory; Secondary: games: noncooperatve, mathematcs: combnatorcs Hstory: Receved: May 21, 2007; Revsed: December 20, 2007. 1. Introducton. Game theory n general and the concept of Nash equlbrum n partcular has lately (re)emerged as a hot topc n the operatons research and computer scence lterature. The complexty of computng a mxed Nash equlbrum of a fnte game gven n strategc form s a case n pont. (Goldberg and Papadmtrou 2006) showed that fndng a mxed Nash equlbrum n a game wth a constant number of players can be reduced to solvng a 4-player game. Daskalaks, Goldberg, and Papadmtrou (2006) showed n turn that the latter problem s PPADcomplete,.e., t s as dffcult as computng a Brouwer fx-pont of a contnuous functon from the closed unt ball to tself. Subsequently, Chen and Deng (2005) and Daskalaks and Papadmtrou (2005) proved that computng mxed Nash equlbra n games wth three players s PPAD-complete as well. Eventually, Chen and Deng (2006) establshed the same result for the two-player case. Whle Nash (1951) showed that mxed Nash equlbra do exst n any fnte noncooperatve game, t s well known that pure-strategy Nash equlbra may not, as s demonstrated by classcal games such as matchng pennes (e.g., Shor 2005). It s therefore natural to ask whch games have pure-strategy Nash equlbra and, f applcable, how dffcult t s to fnd one. In ths artcle, we study these questons for certan classes of weghted congeston games and local-effect games. Congeston games were ntroduced by Rosenthal (1973a), who proved that they are guaranteed to possess pure-strategy Nash equlbra. In fact, Monderer and Shapley (1996) showed that every exact potental game s somorphc to a congeston game. In a congeston game, a player s strategy conssts of a subset of resources, and her cost depends only on the number of players choosng the same resources. An 1

2 Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS mportant subclass of congeston games can be represented by means of networks. 2 Every player wants to route one unt of flow from her orgn to her destnaton, on a path of mnmal cost. The network arcs are the resources, and a player s set of pure strateges conssts of the sets of arcs correspondng to paths connectng her orgn-destnaton par. Fabrkant, Papadmtrou, and Talwar (2004) studed the computatonal complexty of fndng pure-strategy Nash equlbra n congeston games. For symmetrc network congeston games, where all players have the same orgn-destnaton par, they presented a polynomaltme algorthm for computng a pure-strategy Nash equlbrum. On the other hand, they proved that ths problem s PLS-complete for symmetrc congeston games as well as for asymmetrc network congeston games. A smpler proof of the latter result was gven by Ackermann, Rögln, and Vöckng (2006a), who also showed that ths result stll holds for affne cost functons. In weghted congeston games, players control dfferent, ntegral amounts of flow. Dependng on whether players are allowed to splt ther flows or not, a player s strategy conssts of a set of paths wth correspondng nteger flow values between her orgn-destnaton par, or a sngle path. Lbman and Orda (2001) constructed a smple nstance of an unsplttable weghted network congeston game that does not possess a pure-strategy Nash equlbrum. A smlar example was presented by Fotaks, ontoganns, and Spraks (2005), who also observed that for the specal case of affne cost functons, a pure-strategy Nash equlbrum s always guaranteed to exst. Awerbuch, Azar, and Epsten (2005) derved a tght bound of ( 5 + 3)/2 on the pure prce of anarchy for ths specal case. The pure prce of anarchy s the rato of the cost of a worst pure-strategy Nash equlbrum to that of a globally optmal soluton. Awerbuch et al. also gave upper bounds for nstances wth polynomal cost functons of degree greater than 1. Tght bounds for ths case were later provded by Aland et al. (2006). Goemans, Mrrokn, and Vetta (2005) showed that a pure-strategy Nash equlbrum need not exst for nstances wth cost functons that are polynomals of degree at most 2. Mlchtach (1996) had earler shown that weghted congeston games wth player-specfc cost functons on networks consstng of parallel arcs only do not always have a pure-strategy Nash equlbrum ether. Ths was elaborated on by Garng, Monen, and Temann (2006) who consdered dfferent cost functons, slght modfcatons n the network topology, and both the weghted and the unweghted case. Mlchtach (2006) characterzed topologcal propertes of networks that guarantee the exstence of purestrategy Nash equlbra n network congeston games f players control dfferent amounts of flow or cost functons are player-specfc. In ths artcle, we prove that the problem of decdng whether a weghted network congeston game wth smple, non-lnear cost functons possesses a pure-strategy Nash equlbrum s strongly NP-hard, regardless of whether one consders splttable or unsplttable flows. In the unsplttable case, we are able to show that the problem remans strongly NP-complete even f the number of players s fxed, or f all players have the same orgn and destnaton. We also establsh strong NP-completeness for weghted congeston games wth affne player-specfc cost functons on networks consstng of parallel arcs only. Leyton-Brown and Tennenholtz (2003) ntroduced local-effect games to model stuatons n whch the use of one resource can affect the cost of usng other resources. Local-effect games are n general not guaranteed to possess pure-strategy Nash equlbra. However, Leyton-Brown and Tennenholtz showed that so-called bdrectonal local-effect games wth lnear local-effect functons belong to the class of exact potental games, and therefore always have pure-strategy Nash equlbra. The queston of whether there exsts a polynomal-tme algorthm for fndng a pure-strategy Nash equlbrum for these games was left open. We prove that computng a pure-strategy Nash equlbrum s, n fact, PLS-complete. Because the proof uses a tght PLS-reducton, our result mples the exstence of nstances of bdrectonal local-effect games wth lnear local-effect functons that have exponentally long shortest mprovement paths. It also mplcates that the problem of computng a pure-strategy Nash equlbrum that s reachable from a gven strategy state va selfsh mprovement steps s PSPACE-hard. In addton, we show that, gven an ntal strategy profle for a bdrectonal local-effect game wth lnear local-effect functons and a postve nteger k (unarly encoded), t s strongly NP-complete to decde whether there s a sequence of at most k selfsh steps that transforms the ntal state nto a pure-strategy Nash equlbrum. We also prove that the problem of decdng whether a bdrectonal local-effect game wth general local-effect functons has a pure-strategy Nash equlbrum, s strongly NP-complete. Before we present the detals of our results on weghted congeston games and local-effect games n 2 Network congeston games are partcularly nterestng from a computatonal pont of vew because players strateges can be encoded compactly.

Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS 3 Sectons 3 and 4, respectvely, we conclude ths ntroducton by brefly dscussng addtonal related work on the computatonal complexty of pure-strategy Nash equlbra. Gottlob, Greco, and Scarcello (2005) consdered restrctons of strategc games ntended to capture certan aspects of bounded ratonalty. Among other results, they proved that even n the settng where each player s payoff functon depends on the actons of at most three other players and where each player s strategy set conssts of at most three actons, the problem of determnng whether a strategc game has a pure-strategy Nash equlbrum s NP-complete. Ths result was strengthened by Fscher, Holzer, and atzenbesser (2006), who showed that the problem remans NP-hard even f each player has only two actons to choose from and her payoff depends on the actons of at most two other players. Àlvarez, Gabarró, and Serna (2005) studed how varous representatons of a strategc game nfluence the computatonal complexty of decdng the exstence of a pure-strategy Nash equlbrum. They showed that ths problem s NP-complete when the number of players s large and the number of strateges for each player s constant, whle the problem s p 2-complete when the number of players s constant and the szes of the strategy sets are exponental (wth respect to the lengths of the strateges). Schoenebeck and Vadhan (2006) analyzed the computatonal complexty of decdng whether pure-strategy Nash equlbra exst n graph games and crcut games. Brandt, Fscher, and Holzer (2007) studed the mpact of varous notons of symmetry n strategc games on the computatonal complexty of fndng pure-strategy Nash equlbra. Expandng on a lne of research started by Ieong et al. (2005), who consdered sngleton congeston games, Ackermann, Rögln, and Vöckng (2006a) proved that the lengths of all best-response sequences are polynomally bounded n the number of players and resources, n congeston games where the strategy space of each player conssts of the bases of a matrod over the set of resources. Ths especally mples that pure-strategy Nash equlbra for congeston games wth ths matrod property can be computed n polynomal tme, even n the case of player-specfc costs (Ackermann, Rögln, and Vöckng 2006b). In the latter paper, Ackermann et al. also showed the exstence of pure-strategy Nash equlbra n weghted congeston games wth the same matrod property. 2. Prelmnares. Noncooperatve Games. A strategc game s defned by a set N of n players, a fnte set of actons S for each player N, and a payoff or utlty functon u for each player mappng S := N S to Q. The set S s called the strategy or acton space of the game, and ts elements are the pure-strategy states. A pure-strategy Nash equlbrum of a strategc game s a state s = (s 1, s 2,..., s n ) S such that for each player N, u (s) u (s 1,..., s 1, s, s +1,..., s n ), for all s S. Thus, no player can beneft from changng her strategy whle the other players retan ther current strateges. Although every game has a Nash equlbrum f players are allowed to randomze over ther set of pure actons (Nash 1951), pure-strategy Nash equlbra are, n general, not guaranteed to exst. A fundamental class of strategc games, whch always have a pure-strategy Nash equlbrum, are potental games. Every game n ths class s characterzed by the exstence of a potental functon that assocates wth each strategy profle a real number such that the change n the potental functon value of two states dfferng only n a sngle player s strategy s postve f and only f the dfference n payoff to ths partcular player s postve. A potental functon s exact f these two values always concde. Congeston Games. An unweghted congeston game s defned by a set of players N = {1, 2,..., n} and a set of resources E. For each player N, her set of avalable strateges s a collecton of subsets of the resources;.e., S 2 E. A nondecreasng cost functon f e : N Q 0 s assocated wth each resource ( e E. Gven a strategy profle s = (s 1, s 2,..., s n ) S, the cost of player s c (s) = u (s) = e s f e ne (s) ), where n e (s) denotes the number of players usng resource e n s. In other words, n a congeston game each player chooses a subset of resources that are avalable to her; and the cost to a player s the sum of the costs of the resources used by her, where the cost of a resource depends only on the total number of players sharng ths resource. A network congeston game s a congeston game n whch the arcs of an underlyng drected network represent the resources. Each player N has an orgn-destnaton par (r, t ), where r and t are nodes of the network, and the set S of pure strateges avalable to player s the set of drected (smple) paths from r to t. In a weghted network congeston game, each player N has a postve nteger weght w, whch consttutes the amount of flow that player wants to shp from r to t. In the case of unsplttable flows, the cost of player adoptng strategy s n a strategy profle s = (s 1, s 2,..., s n ) S s gven

4 Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS by c (s) = e s f e ( θe (s) ), where θ e (s) = :e s w denotes the total flow on arc e n s. In ntegersplttable network congeston games, a player wth weght greater than one can choose a subset of paths on whch to route her flow smultaneously; that s, player s strategy conssts of the specfcaton of the r -t -paths used and the nteger amounts of flow routed on them, whch sum up to w. The correspondng cost s the total cost of the paths that player uses, weghted by the respectve amounts of flow player routes on them. An (un)weghted network congeston game s called symmetrc or a sngle-commodty game f all players have the same orgn-destnaton par. In terms of the nput sze of a weghted network congeston game, we assume that the cost functons are explctly specfed; that s, for each nteger value θ wth 0 θ N w and each arc e, the value f e (θ) s gven n bnary encodng. 3 Local-Effect Games. In a local-effect game wth player set N = {1, 2,..., n}, all players have the same set of avalable actons, A. For each acton a A, there s a nondecreasng cost functon f a : N Q 0 that depends merely on the number of players who play ths acton. Furthermore, for each par of actons a, a A, a a, a functon f a,a : N Q 0 expresses the mpact of acton a on the cost of acton a. Its value depends only on the number of players that choose acton a. The functons f a,a are called local-effect functons, and t s assumed that f a,a(0) = 0. Moreover, local-effect functons are ether strctly ncreasng or dentcal zero. For a gven strategy state s = (s 1, s 2,..., s n ) A n, n a (s) denotes the number of players playng( acton a n s. The cost to a player N for playng acton s n strategy state s s gven by c (s) = f s ns (s) ) + ( a A,a s f a,s na (s) ). If the local-effect functons f a,a are zero for all a a, the local-effect game s equvalent to a symmetrc network congeston game wth parallel arcs. A local-effect game s called a bdrectonal local-effect game f, for all a, a A, a a, and for all x N, f a,a(x) = f a,a (x). PLS. The complexty class PLS was ntroduced by Johnson, Papadmtrou, and Yannakaks (1988) n order to characterze the computatonal complexty of local search problems. A combnatoral optmzaton problem Π conssts of a collecton of nstances (F, c), where F denotes the set of feasble solutons and c : F Q s the objectve functon. A combnatoral optmzaton problem Π together wth a neghborhood functon N : F 2 F belongs to PLS f (a) nstances are recognzable n polynomal tme and a feasble soluton can be computed effcently, (b) the feasblty of a proposed soluton can be checked effcently and ts objectve functon value can be evaluated n polynomal tme, and (c) the neghborhood of a feasble soluton can be searched n polynomal tme to determne a better feasble soluton, f one exsts. The computatonal problem assocated wth a local search problem s to fnd, for a gven nstance (F, c), a locally optmal soluton w.r.t. the neghborhood functon N,.e., an s F such that there s no soluton n the neghborhood of s wth strctly better cost. A local search problem L 2 n PLS s PLS-complete f, for any other problem L 1 n PLS, there are polynomal-tme computable functons φ and ψ such that φ maps nstances x of L 1 to nstances φ(x) of L 2, ψ maps solutons of φ(x) to solutons of x, and f s s a locally optmal soluton for the nstance φ(x) of L 2, then ψ(s, x) s a locally optmal soluton for x. Such a reducton s called tght f for any nstance x of L 1 one can dentfy a subset X of feasble solutons of φ(x) so that (a) X contans all local optma of φ(x), (b) for every soluton f of x one can construct n polynomal tme a soluton s X such that ψ(s, x) = f, and (c) f the transton graph of φ(x) contans a drected path from s X to s X whose nternal nodes are not n X, then ether ψ(s, x) = ψ(s, x) or the transton graph of x contans an arc from ψ(s, x) to ψ(s, x) (Schäffer and Yannakaks 1991). In partcular, the length of a longest path from any soluton to a locally optmal soluton n the transton graph of L 2 s at least as large as that n the transton graph of L 1. 3. Complexty of Weghted Congeston Games. We begn by gvng a hgh-level descrpton of the common dea that forms the bass of our NP-hardness proofs for the varous classes of games. In each case, we take a counterexample,.e., an nstance that does not have a pure-strategy Nash equlbrum, and couple t wth an nstance of the same class n whch the strategy profles correspond to the feasble solutons n a gven nstance of an NP-complete problem. We also ntroduce an addtonal player who can partcpate n ether game. All other players are lmted by cost or structure to partcpate n ther part of the game only. The partcpaton of the extra player n the counterexample turns that game nto one that has a pure-strategy Nash equlbrum. Therefore, the entre game has a pure-strategy 3 Whle more compact encodngs are often possble, ths assumpton leads to stronger hardness results, whch are the man concern of ths paper.

Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS 5 Nash equlbrum f and only f the part of the game correspondng to the NP-complete problem has a pure-strategy Nash equlbrum that prevents the extra player from jonng the game, whch happens f and only f t corresponds to a Yes-nstance. The only devaton from ths proof scheme occurs n the case of weghted network congeston games wth a fxed number of players. Instead of ntroducng another player, one of the two players from the counterexample s gven access to the other part of the game, whch she wll be able to take advantage of f and only f the state n ths part of the game corresponds to a soluton of a Yes-nstance. 3.1 Unsplttable Flows. Lbman and Orda (2001) presented an example of a weghted network congeston game wth general nondecreasng arc-cost functons that does not have a pure-strategy Nash equlbrum. Fotaks, ontoganns, and Spraks (2005) provded a smlar nstance. We smplfy the latter nstance and turn t nto a gadget to derve the followng result. Theorem 3.1 The problem of decdng whether a weghted symmetrc network congeston game wth unsplttable flows possesses a pure-strategy Nash equlbrum s strongly NP-complete. Proof. The proof s by reducton from 3-Partton, whch s strongly NP-complete (Garey and Johnson 1979, SP15). Consder an arbtrary nstance of 3-Partton: a fnte set A = {1, 2,..., 3m} of tems (m 2), a number B N, and a postve nteger weght w for each tem A such that B/4 < w < B/2 and A w = mb. We wll construct a weghted sngle-commodty network congeston game that has a pure-strategy Nash equlbrum f and only f A can be parttoned nto m dsjont sets A 1, A 2,..., A m such that A k w = B for 1 k m. We ntroduce a player p for each tem A; the correspondng weght s w. In addton, there are three players p 3m+1, p 3m+2, and p 3m+3 wth weghts w 3m+1 = B, w 3m+2 = 2B, and w 3m+3 = B/2, respectvely. Here, B := 2mB. All players have the same orgn, r, and destnaton, t. The network s depcted n Fgure 1. It conssts of a contracted verson of the example by Fotaks, ontoganns, and Spraks (2005) and m addtonal arcs e 1, e 2,..., e m, connectng r and t. We denote the r-t-paths n the lower part of the network by P 1 = (e 1 ), P 2 = (e 2, e 3 ), P 3 = (e 2, e 4, e 6 ), and P 4 = (e 5, e 6 ). e m e 2 e 1 r P 1 e 1 e 2 e 3 P 2 t P 4 e 5 e 6 e 4 P 3 Fgure 1: Illustraton of the weghted sngle-commodty network congeston game used n the proof of Theorem 3.1.

6 Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS The nondecreasng arc-cost functons are defned as follows: f ek (x) := { x, f x < (m + 1)B 240 B, otherwse f e1 (x) := f e2 (x) := f e3 (x) := f e4 (x) := 12 B, f x B 120 B, f B < x 2B 228 B, otherwse, 1 B, f x B 2 B, f B < x 2B 8 B, otherwse, 16 B, f x B 18 B, f B < x 2B 20 B, otherwse, 1 B, f x B 40 B, f B < x 2B 79 B, otherwse, f e5 (x) := f e6 (x) := 10 B, f x B 12 B, f B < x 2B 14 B, otherwse, 2 B, f x B 10 B, f B < x 2B 12 B, otherwse. for k = 1,..., m, Suppose there s a partton A 1, A 2,..., A m of A such that A k w = B for 1 k m. Consder the strategy state s n whch player p chooses the arc e k such that tem A k, and s 3m+1 = P 3, s 3m+2 = P 4, and s 3m+3 = P 1. The cost to player p, for A, s B. If such a player would route her flow on one of the other arcs e l, l k, her cost would ncrease to B + w. By choosng a path P k, k {1, 2, 3, 4}, ths player s cost would ncrease to at least 4B. Player p 3m+1 experences a cost of 14B. A change to path P 1, P 2, or P 4 would result n a cost of 120B, 17B, or 26B. For player p 3m+2, the cost n state s s 24B. Swtchng to path P 1, P 2, or P 3 ncreases her cost to 228B, 26B, or 99B. Player p 3m+3 has a cost of 12B. Routng her flow on path P 2, P 3, or P 4 would result n an ncreased cost of 18B, 54B, or 26B. Fnally, every player p 3m+1, p 3m+2, and p 3m+3 would ncrease her cost to 240B by swtchng to some sngle-arc path e k, k {1, 2,..., m}. Thus, no player can decrease her cost by routng flow over another path: s s a pure-strategy Nash equlbrum. For the other drecton, we clam that every pure-strategy Nash equlbrum s of ths game has the followng propertes: (a) Players p 3m+1, p 3m+2, and p 3m+3 play a strategy n {P 1, P 2, P 3, P 4 }. (b) Every player p, {1, 2,..., 3m}, chooses an arc e k, for some 1 k m. Property (a) clearly holds for players p 3m+1 and p 3m+2. For, f one of these players chose an arc e k for some k {1, 2,..., m}, she would experence a cost of 240B. She could decrease her cost to at most 228B by swtchng to some path P k, k {1, 2, 3, 4}. For property (b), suppose there s a player p, {1, 2,..., 3m}, routng her flow on a path P k, k {1, 2, 3, 4}. In ths case, the cost of player p s at least 4B. Gven that we have already establshed property (a) for players p 3m+1 and p 3m+2, the total weght of players usng an arc e k n s s at most 2mB. Therefore, there must exst some k {1, 2,..., m} such that :s =e k w 2B. By swtchng to arc e k, player p can decrease her cost to, at most, f ek (2B + w ) < 3B. In order to show (a) for player p 3m+3, suppose that she uses an arc e k, k {1, 2,..., m}. Then, only players p 3m+1 and p 3m+2 use one of the paths P 1, P 2, P 3, and P 4. However, the congeston game restrcted to players p 3m+1 and p 3m+2 and strateges P 1, P 2, P 3, and P 4 does not have a pure-strategy Nash equlbrum a contradcton. 4 Table 1 lsts the 16 possble combnatons that we need to consder to show that, n each case at least one of the two players can decrease her cost by 4 Ths subgame concdes wth the contracted verson of the nstance by Fotaks, ontoganns, and Spraks (2005) to whch we referred earler.

Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS 7 routng her flow on a dfferent path. Hence, the only way that the entre game can have a pure-strategy Nash equlbrum s for player p 3m+3 to play a strategy n {P 1, P 2, P 3, P 4 }. s 3m+1 s 3m+2 Devator New strategy Current cost/ B Improved cost/ B P 1 P 1 p 3m+1 P 3 c 3m+1 = 228 c 3m+1 = 4 P 1 P 2 p 3m+1 P 3 c 3m+1 = 12 c 3m+1 = 11 P 1 P 3 p 3m+2 P 2 c 3m+2 = 52 c 3m+2 = 20 P 1 P 4 p 3m+2 P 2 c 3m+2 = 22 c 3m+2 = 20 P 2 P 1 p 3m+2 P 2 c 3m+2 = 120 c 3m+2 = 28 P 2 P 2 p 3m+1 P 1 c 3m+1 = 28 c 3m+1 = 12 P 2 P 3 p 3m+1 P 1 c 3m+1 = 24 c 3m+1 = 12 P 2 P 4 p 3m+1 P 1 c 3m+1 = 17 c 3m+1 = 12 P 3 P 1 p 3m+2 P 2 c 3m+2 = 120 c 3m+2 = 26 P 3 P 2 p 3m+2 P 4 c 3m+2 = 26 c 3m+2 = 24 P 3 P 3 p 3m+1 P 1 c 3m+1 = 99 c 3m+1 = 12 P 3 P 4 p 3m+1 P 1 c 3m+1 = 14 c 3m+1 = 12 P 4 P 1 p 3m+2 P 2 c 3m+2 = 120 c 3m+2 = 20 P 4 P 2 p 3m+1 P 3 c 3m+1 = 12 c 3m+1 = 11 P 4 P 3 p 3m+1 P 1 c 3m+1 = 22 c 3m+1 = 12 P 4 P 4 p 3m+1 P 1 c 3m+1 = 26 c 3m+1 = 12 Table 1: Possble defectons n the subgame defned by the subnetwork of the network dsplayed n Fgure 1 that conssts of arcs e 1, e 2,..., e 6 and players p 3m+1 and p 3m+2 only. Gven a pure-strategy Nash equlbrum s, we can now defne a partton of A by settng A k := { A : s = e k }, k = 1, 2,..., m. We clam that these sets defne a soluton to the 3-Partton problem. Suppose ths s not the case. Then, because of (a), there exsts an ndex k {1, 2,..., m} such that :s =e k w < B. The current cost of player p 3m+3 usng a path P k, k {1, 2, 3, 4}, s at least 4B. By swtchng to arc e k, ths player can decrease her cost to, at most, f ek (B 1 + mb) = (m + 1)B 1. Ths contradcts the assumpton of s beng a Nash equlbrum. To complete the proof, we note that the problem of decdng whether a weghted network congeston game wth unsplttable flows has a pure-strategy Nash equlbrum, s n NP. Indeed, one can verfy n polynomal tme that a gven strategy state s a Nash equlbrum by conductng a shortest-path computaton for each player. Whle the NP-hardness of the correspondng decson problem for weghted network congeston games wth player-specfc payoff functons follows mmedately, we can actually strengthen ths result. Theorem 3.2 The problem of decdng whether a weghted network congeston game wth parallel arcs and affne player-specfc cost functons possesses a pure-strategy Nash equlbrum s strongly NPcomplete. Proof. The problem s obvously n NP. To show NP-completeness, we reduce, as before, from 3-Partton. We are gven a set A = {1, 2,..., 3m} of tems, a number B N, and a postve nteger weght w for each tem A such that B/4 < w < B/2 and A w = mb. We wll construct a weghted network congeston game wth parallel arcs only and player-specfc cost functons such that t has a pure-strategy Nash equlbrum f and only f A can be parttoned nto m dsjont sets A 1, A 2,..., A m such that A k w = B, for 1 k m. We ntroduce a player p for each tem A; the correspondng weght s w. There are four addtonal players p 3m+, for = 1, 2, 3, 4, wth w 3m+1 = 1, w 3m+2 = 2, w 3m+3 = 3, and w 3m+4 = 1. All players want to shp flow from r to t n a network of parallel arcs e 1, e 2,..., e m+3 connectng r and t. Let f,k denote the cost functon of player p for arc e k, and let := 3(mB+7)+1. For = 1, 2,..., 3m and k = 1, 2,..., m + 3, we defne { x, f k {1, 2,..., m} f,k (x) :=, otherwse.

8 Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS For the remanng players p 3m+, {1, 2, 3, 4}, we set f 3m+1,k (x) := f 3m+2,k (x) := f 3m+3,k (x) := f 3m+4,k (x) :=, f k {1, 2,..., m + 1} 7, f k = m + 2 2x, f k = m + 3,, f k {1, 2,..., m + 1} 2x, f k = m + 2 5, f k = m + 3,, f k {1, 2,..., m} {m + 2} 3x, f k = m + 1 2x, f k = m + 3, x, f k {1, 2,..., m}, f k {m + 1, m + 2} B + 1, f k = m + 3. Assume that we are gven a Yes-nstance of the partton problem. Then there s a partton A 1, A 2,..., A m of A such that A k w = B for 1 k m. Consder the strategy state s n whch player p chooses the arc e k wth A k, = 1, 2,..., 3m, and s 3m+1 = e m+3, s 3m+2 = e m+2, s 3m+3 = e m+1, and s 3m+4 = e m+3. In s, each player p correspondng to an tem A has a cost of B. Swtchng to a dfferent arc n {e 1, e 2,..., e m } ncreases her cost to B + w, and routng her flow on an arc n the set {e m+1, e m+2, e m+3 } results n a cost of, whch s no mprovement ether. Player p 3m+1 has a cost of 4. The only other arc yeldng a cost less than s e m+2. However, swtchng to ths arc results n a hgher cost of 7. Player p 3m+2 has a cost of 4 n state s. Changng her strategy to e m+3 gves a new cost of 5; all other arcs have cost for ths player. Player p 3m+3 wth current cost 9 can decrease her cost nether by usng arc e m+3, whch would yeld a cost of 10, nor by takng one of the other arcs, whch would result n a cost of. Player p 3m+4 s cost s B + 1 n s. Swtchng to an arc e k, for some 1 k m, results n the same cost, all other arcs would ncrease her cost. Hence, s s a pure-strategy Nash equlbrum. For the other drecton of the proof, we frst observe that any pure-strategy Nash equlbrum of the constructed game has the followng propertes: (a) Each player p, {1, 2,..., 3m}, uses an arc n {e 1, e 2,..., e m }. (b) None of the players p 3m+1, p 3m+2, p 3m+3 plays a strategy n {e 1, e 2,..., e m }. (c) Player p 3m+4 plays strategy e m+3. Propertes (a) and (b) follow mmedately from the fact that for any player there exsts a strategy wth cost strctly smaller than,.e., n any Nash equlbrum each player pays less than. For property (c), we frst observe that n any Nash equlbrum the only possble strateges for player p 3m+4 are e k, 1 k m, and e m+3. Suppose that p 3m+4 does not use arc e m+3. Then, by propertes (a) and (b), only p 3m+1, p 3m+2, and p 3m+3 play strateges n {e m+1, e m+2, e m+3 }. However, the congeston game restrcted to these three players and strateges does not have a pure-strategy Nash equlbrum, yeldng a contradcton. In fact, we only need to consder all possbltes for the three players to choose ther strateges from {e m+1, e m+2, e m+3 }. We can exclude from the start all possbltes that mply a cost of for one of the players. Eght possble combnatons reman to be consdered, whch are lsted n Table 2. In consequence, the only way for the whole game to have a pure-strategy Nash equlbrum s f player p 3m+4 uses arc e m+3. Gven a pure-strategy Nash equlbrum s of the constructed game, we can now assocate a partton of the tem set wth the m groups of players who route ther flows on arcs e k wth 1 k m. We clam that the so defned sets,.e., A k := { A : s = e k }, 1 k m, form a soluton to the 3-Partton problem. Suppose ths s not true. Then there exsts an ndex k {1, 2,..., m} such that the total weght of players routng ther flows on e k s at most B 1. Consder player p 3m+4, who, by property (c), plays strategy e m+3 n any pure-strategy Nash equlbrum. Her current cost s B + 1. By swtchng to arc e k, she can decrease her cost to at most B. However, ths contradcts the assumpton of s beng a Nash equlbrum. Therefore, A k w = B for 1 k m. The followng result shows that decdng the exstence of pure-strategy Nash equlbra n asymmetrc weghted network congeston games wth unsplttable flows remans strongly NP-complete, even f the

Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS 9 s 3m+1 s 3m+2 s 3m+3 Defector New strategy Current cost Improved cost e m+2 e m+2 e m+1 p 3m+1 e m+3 c 3m+1 = 7 c 3m+1 = 2 e m+2 e m+2 e m+3 p 3m+2 e m+3 c 3m+2 = 6 c 3m+2 = 5 e m+2 e m+3 e m+1 p 3m+1 e m+3 c 3m+1 = 7 c 3m+1 = 6 e m+2 e m+3 e m+3 p 3m+3 e m+1 c 3m+3 = 10 c 3m+3 = 9 e m+3 e m+2 e m+1 p 3m+3 e m+3 c 3m+3 = 9 c 3m+3 = 8 e m+3 e m+2 e m+3 p 3m+1 e m+2 c 3m+1 = 8 c 3m+1 = 7 e m+3 e m+3 e m+1 p 3m+2 e m+2 c 3m+2 = 5 c 3m+2 = 4 e m+3 e m+3 e m+3 p 3m+3 e m+1 c 3m+3 = 12 c 3m+3 = 9 Table 2: Possble defectons n the subgame restrcted to players p 3m+1, p 3m+2, p 3m+3 and strateges e m+1, e m+2, e m+3, as dscussed n the proof of Theorem 3.2. Ths game s smlar to nstances descrbed by Mlchtach (1996). number of players s fxed. Note that ths result does not render Theorem 3.1 obsolete, because that theorem dealt wth symmetrc games. Theorem 3.3 The problem of decdng whether a weghted network congeston game wth a fxed number of players has a pure-strategy Nash equlbrum s strongly NP-complete. Proof. We reduce from Arc-Dsjont Paths: Gven a drected graph G = (N, A) and a set of node pars (r 1, t 1 ), (r 2, t 2 ),..., (r k, t k ), does there exst a collecton of arc-dsjont paths P 1, P 2,..., P k, where P s an r -t -path? Ths problem s NP-complete, even n the case of only two termnal pars (Fortune, Hopcroft, and Wylle 1980). Let G = (N, A), (r 1, t 1 ), and (r 2, t 2 ) be an nstance of Arc-Dsjont Paths wth two termnal pars. We wll construct a congeston game whose underlyng network wll consst of two buldng blocks. The frst component s obtaned from the orgnal network G by replacng each arc a A by a path consstng of three arcs e 1, e 2, and e 3. We denote the resultng graph by G. The second buldng block s the network that we already used as a gadget n the proof of Theorem 3.1. It conssts of four dfferent r-t-paths, whch we wll call Q k here, for k = 1, 2, 3, 4. We wll refer to t as the lower part of the new network. See Fgure 2 for an llustraton. e 1 2 e 2 2 e 3 2 r 1 t 1 e 1 r 2 e 1 1 e 2 1 e 3 1 e 1 m e 2 m e 3 m t 2 e 0 e m G r Q 1 e 1 e 2 e 3 Q2 t Q4 e 5 e 6 e 4 Q 3 Fgure 2: Illustraton of the weghted network congeston game created n the proof of Theorem 3.3. Both buldng blocks are connected as follows. Assume that A = {a 1, a 2,..., a m }. We ntroduce an arc e 0 between r and the startng node of the second arc e 2 1, whch was one of three arcs replacng the orgnal arc a 1 A. Furthermore, for 1 m 1, we create an arc e connectng the end node of e 2 wth the start node of e 2 +1. Fnally, there s an arc e m from the end node of e 2 m to the termnal node t. We denote the r-t-path formed by the arcs e k, k = 0, 1,..., m, and e 2, = 1, 2,..., m, by Q 5.

10 Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS The game has four players. Players p 1 and p 2 wsh to route one unt of flow from r 1 to t 1 and from r 2 to t 2, respectvely. Players p 3 and p 4 have weghts w 3 = m and w 4 = 2m, to be sent from r to t. We defne the cost of the arcs that replaced a A as follows: { 0, f 0 x < m f e 1 (x) := f e 3 (x) :=, f x m, f e 2 (x) := { 0, f 0 x < m + 2, f x m + 2, where := 237m + 1. Moreover, for 0 m, f e (x) := { 0, f 0 x m, f x > m. The costs functons f e, for = 1, 2,..., 6, n the lower part of the network are defned n the same way as the correspondng functons n the proof of Theorem 3.1; however, B s replaced by m. Let us frst assume that the gven nstance of Arc-Dsjont Paths s a Yes-nstance,.e., there exst arc-dsjont paths P 1 and P 2 n G connectng r 1 and t 1, and r 2 and t 2, respectvely. We wll abuse notaton and denote the correspondng paths n G also by P 1 and P 2. Consder the strategy state s n whch players p 1 and p 2 choose P 1 and P 2, respectvely, whle player p 3 uses Q 5, and p 4 routes her flow on Q 2. We clam that s s a Nash equlbrum. Snce at most one of the players p 1 and p 2 uses a partcular arc n G, the cost of any arc e 1, e2, e3, 1 m, n s s zero. Therefore, p 1 and p 2 have zero cost and play optmal strateges. Smlarly, the cost of player p 3 s zero, because n addton to the last observaton, the total weght on any arc e, 0 m, s m,.e., the cost for usng each of these arcs s zero. Fnally, player p 4 uses the cheapest path n the lower part of the network. By routng her flow on any path sharng arcs wth G her cost would ncrease to at least. Thus, no player can decrease her cost by routng flow over another path; s s ndeed a pure-strategy Nash equlbrum. Let us now assume that we are gven a pure-strategy Nash equlbrum s of the constructed game. Snce any arc n G, f used by player p 4, nduces a cost of > 237m, ths player s always better of usng a path n {Q 1, Q 2, Q 3, Q 4 }. We further observe that there s no r 1 -t 1 -path or r 2 -t 2 -path that shares any arc wth the lower part of the network,.e., nether player p 1 nor player p 2 wll use such an arc. However, f we restrct the game to players p 3 and p 4 and the lower part of the network, t follows from the same reasonng used n the proof of Theorem 3.1 that ths subgame does not have a pure-strategy Nash equlbrum. Therefore, player p 3 has to choose a path ntersectng G. Arcs e 1 and e 3, 1 m, are very expensve f the load s greater or equal to w 3 = m; hence, the only way for p 3 to route her flow n a Nash equlbrum s to use path Q 5. Snce she cannot decrease her cost by swtchng to a path n the lower part of the network, the cost for usng Q 5 must be smaller than. Ths mples that at most one of the players p 1 and p 2 uses an arc e 2, 1 m (otherwse the total weght on such an arc would be m + 2). Smlarly, nether p 1 nor p 2 can use any arc e k for k {0, 1,..., m}. Ths, n turn, mplcates that both p 1 and p 2 only use G to route ther flows. Furthermore, the correspondng paths of these two players n G have to be dsjont,.e., the Arc-Dsjont Paths nstance s a Yes-nstance. 3.2 Integer-Splttable Flows. Rosenthal (1973b) gave an example of an asymmetrc weghted network congeston game that does not have a pure-strategy Nash equlbrum f players are allowed to splt ther flows (see Fgure 5). Interestngly, the same game possesses a pure-strategy Nash equlbrum f each player has to route her flow on a sngle path. The followng result shows that one cannot effcently decde the exstence of pure-strategy Nash equlbra n network congeston games wth nteger-splttable flows, unless P = NP. Theorem 3.4 The problem of decdng whether a weghted network congeston game wth ntegersplttable flows possesses a pure-strategy Nash equlbrum s strongly NP-hard, even f there s only one player wth weght 2, and all other players have unt weghts. Proof. The reducton s from Monotone 3Sat, whch s known to be NP-complete (Garey and Johnson 1979, LO2). Consder an nstance of Monotone 3Sat wth set of varables X = {x 1, x 2,..., x n } and set of three-varable clauses C = {c 1, c 2,..., c m }. Each clause contans ether only negated varables or only unnegated varables.

Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS 11 We wll create a game that has one player p x for every varable x X wth weght w x = 1, orgn x, and destnaton x. Moreover, each clause c C gves rse to a player p c wth weght w c = 1, orgn c, and destnaton c. There are three more players p 1, p 2, and p 3 wth weghts w 1 = 1, w 2 = 2, and w 3 = 1 and orgn-destnaton pars (r, t 1 ), (r, t 2 ), and (r, t 3 ), respectvely. For every varable x X, the network contans two dsjont paths Px 1 and Px 0 from x to x. Path Px 0 conssts of 2 {c C x c} + 1 arcs, and Px 1 has 2 {c C x c} + 1 arcs wth cost functons as shown n Fgure 3. For each orgn-destnaton par (c, c), we ntroduce two dsjont paths Pc 1 and Pc 0 from c to c. Path Pc 1 conssts of only two arcs. The paths Pc 0 have seven arcs each and are constructed for c = c j n the order j = 1, 2,..., m as follows. For a postve clause c = c j = (x j1 x j2 x j3 ) wth j 1 < j 2 < j 3, path Pc 0 starts wth the arc connectng c to the replacemen frst nner node v 1 on path Px 1 j1 that has only two ncdent arcs so far. The second arc s the unque arc (v 1, v 2 ) of path Px 1 j1 that has v 1 as ts start vertex. The thrd arc connects v 2 to the frst nner node v 3 on path Px 1 j2 that has only two ncdent arcs so far. The fourth arc s the only arc (v 3, v 4 ) on Px 1 j2 wth start vertex v 3. From v 4, there s an arc to the frst nner node v 5 on Px 1 j3 that has only two ncdent arcs so far, followed by (v 5, v 6 ) of Px 1 j3. The last arc of Pc 0 connects v 6 to c. Fgure 3 llustrates ths constructon. For a negatve clause c = c j = (x j1 x j2 x j3 ) we proceed n the same way, except that we choose the nner nodes v from the upper varable paths Px 0 j1, Px 0 j2, and Px 0 j3. P 0 x1 P 0 x2 P 0 x3 x 1 P 1 x 1 x 2 P 1 x 2 x 3 P 1 x 3 x1 x2 x3 v v 3 v 5 1 v6 v 2 v 4 P 0 c1 c 1 c 1 c 2 P 1 c1 1 0/ 1 2 m Fgure 3: Part of the constructed network correspondng to a postve clause c 1 = (x 1 x 2 x 3 ). The notaton a/b defnes a cost per unt flow of value a for load 1 and b for load 2. For any other arc, the cost does not depend on the amount of flow on that arc. Arcs wthout specfed values have zero cost. The strategy set of player p x s {Px 1, P x 0 }. We wll nterpret t as settng the varable x to true (false) f p x sends her unt of flow over Px 1 (Px 0 ). Note that player p c can only choose between the paths Pc 1 and Pc 0. Ths s due to the order n whch the paths P c 0 are constructed. Dependng on whether player p c sends her unt of flow over path Pc 1 or Pc 0, the clause c wll be satsfed or not. The second part of the network conssts of all orgn-destnaton pars and paths for players p 1, p 2, and p 3 (see Fgure 4). Player p 1 can choose between paths U 1 = {(r, t 2 ), (t 2, t 1 )} and L 1 = {(r, t 1 )}. Player p 2 s the only player who can splt her flow; that s, she can route her two unts ether both over path U 2 = {(r, t 2 )}, both over path L 2 = {(r, t 1 ), (t 1, t 2 )}, or one unt on the upper and the other unt on the lower path;.e., her strategy set s S 2 = {L 2, U 2, L 2 U 2 }. Fnally, player p 3 has three possble paths to choose from. The upper path U 3 shares an arc wth each clause path Pc 1 and has some addtonal arcs to connect these. The paths M 3 = {(r, t 2 ), (t 2, t 3 )} and L 3 = {(r, t 1 ), (t 1, t 2 ), (t 2, t 3 )} have only arcs wth the paths of p 1 and p 2 n common. The cost functons are defned n Fgure 4. Gven a satsfyng truth assgnment, we defne a strategy state s of the game by settng the strategy of player p x to be Px 1 for a true varable x, and P x 0 otherwse. Each player p c plays Pc 1. Furthermore, s 1 = L 1, s 2 = U 2, and s 3 = M 3. It s easy to show that no player can decrease her cost by unlaterally swtchng to another strategy;.e., the defned strategy confguraton s a pure-strategy Nash equlbrum. For the other drecton, we frst observe that any pure-strategy Nash equlbrum s has the followng propertes: (a) player p 3 does not use path U 3, (b) the cost of player p 3 s at least 8, and (c) each player p c routes her unt flow over path Pc 1. Property (a) follows from the fact that the subgame shown n Fgure 5 wth players p 1 and p 2 only does not have a pure-strategy Nash equlbrum. 5 Thus, p 3 wll use ether the mddle or the lower path. No matter how many other players use an edge of the lower path, the cost of p 3 usng L 3 s at least 10. The only possblty for p 3 to face a cost strctly less than 8 s f she uses the 5 Ths subgame concdes wth the nstance orgnally conceved by Rosenthal (1973b) to whch we alluded earler.

12 Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS 1 2 7 P 1 c1 P 1 c2 P 1 c3 0/ 1 m 1 2 0/ 1 m c 1 c 1 c 2 c 2 c 3 c 3 c m c m 1 2 0/ 1 m 1 2 P 1 cm 0/ 1 m r 0/6/7/13 1 t 2 t 3 5/7/9/13 4/8/12/13 t 1 Fgure 4: Part of the network used by players p 1, p 2, and p 3. A sngle number a on an arc defnes a constant cost of a per unt flow for ths arc. Unlabeled arcs have cost zero. mddle path and at most one addtonal unt of p 1 or p 2 s routed over arc (r, t 2 ). Let us consder the case s 1 = L 1, s 2 = L 2, and s 3 = M 3 frst. Then p 2 has a cost of 34, and she can decrease her cost by swtchng to strategy U 2 wth a new cost of 14. If s 1 = L 1, s 2 = L 2 U 2, and s 3 = M 3, the cost of player p 2 s 17. Choosng strategy U 2 nstead yelds a lower cost of 14. The last case to consder s s 1 = U 1, s 2 = L 2, and s 3 = M 3. Then, player p 2 has a cost of 30, whch can be decreased by swtchng to strategy L 2 U 2 leadng to a cost of 16. Consequently, property (b) holds for any pure-strategy Nash equlbrum of the game. For property (c), suppose there s a player p c, c C, routng her unt flow on Pc 0. By (a) and (b), we know that p 3 uses ether the lower or the mddle path wth a cost of at least 8. Consder a change of p 3 to the upper path U 3. Her new cost would be at most 7 + (m 1) 1 m < 8, whch would contradct that s s a Nash equlbrum. r 5/7/9/13 0/6/7/13 4/8/12/13 t 2 L U LL LU UU 9,34 7,11 5,12 0,30 6,15 7,14 t 1 Fgure 5: On the left: Subgame wth two players wthout pure-strategy Nash equlbrum. The players weghts are one and two, respectvely. On the rght: Dgraph of all strategy states and mprovng moves. We clam that the truth assgnment that sets a varable x to true f the correspondng player uses Px 1, and x to false otherwse, satsfes all clauses. Suppose that all varables of a postve clause c = (x 1 x 2 x 3 ) are false;.e., s x = Px 0 for = 1, 2, 3. By property (c), player p c uses Pc 1. Because of (a), her current cost s 1/2. Choosng path Pc 0 nstead would decrease the cost to zero, whch contradcts the assumpton of s beng a Nash equlbrum. A smlar argument holds for a negatve clause. Note that we have not clamed that the problem of decdng whether a weghted network congeston game wth nteger-splttable flows possesses a pure-strategy Nash equlbrum s n NP. Whle ths can be easly shown f all cost functons are convex, t follows from a result by Meyers and Schulz (2005) that, n general, the problem of decdng whether a gven strategy profle s a pure-strategy Nash equlbrum s n tself a co-np-complete problem. 4. Bdrectonal Local-Effect Games. Leyton-Brown and Tennenholtz (2003) presented a characterzaton of local-effect games that have an exact potental functon and whch are therefore guaranteed to possess pure-strategy Nash equlbra. One of these subclasses are bdrectonal local-effect games wth lnear local-effect functons. However, wthout lnear local-effect functons, decdng the exstence of pure-strategy Nash equlbra s dffcult. Theorem 4.1 The problem of decdng whether a bdrectonal local-effect game possesses a pure-strategy Nash equlbrum s strongly NP-complete.

Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS 13 Proof. The proof uses a reducton from 3-Partton. Consder an arbtrary nstance of 3- Partton wth fnte set A = {1, 2,..., 3m} of tems, a number B N, and a postve nteger weght w for each tem A such that B/4 < w < B/2 and A w = mb. We may assume that B 12. We wll construct a bdrectonal local-effect game such that t has a pure-strategy Nash equlbrum f and only f A can be parttoned nto m dsjont sets A 1, A 2,..., A m such that A k w = B for 1 k m. The acton set A conssts of 3m 2 + 2m + 3 actons that are avalable to 4m + 3 players. For each tem A, there s a set of m correspondng actons a 1, a2,..., am ; we wll make sure that n any Nash equlbrum of the game exactly one player wll pck one of these actons, for each tem. The remanng actons are denoted by d j and d j, j = 1, 2,..., m, and h 1, h 2, and h 3. We defne the cost functons for actons a {a j : = 1, 2,..., 3m, j = 1, 2,..., m} {d j : j = 1, 2,..., m} as { 0, f x 1 f a (x) :=, otherwse, where := 3(4m + 3) B + 18. For j = 1, 2,..., m, { B, f x 1 f dj (x) :=, otherwse. Furthermore, we have f h1 (x) := f h2 (x) := f h3 (x) := f h1,h 2 (x) := 0, f x = 0 B + 1, f x = 1 2B + 6, f x = 2 3B + 12, f x = 3 B x, otherwse, 0, f x = 0, B + 1, f x = 1, 2B + 2, f x = 2, 3B + 12, f x = 3, B x, otherwse. The local-effect functons between actons h 1, h 2, and h 3 are gven by f h1,h 3 (x) := f h2,h 3 (x) := 0, f x = 0 B + 1, f x = 1 2B + 4, f x = 2 B x, otherwse, 0, f x = 0 B + 3, f x = 1 2B + 4, f x = 2 B x, otherwse, 0, f x = 0 B + 1, f x = 1 2B + 5, f x = 2 B x, otherwse. All other local-effect functons are defned n Fgure 6. Assume that the gven 3-Partton nstance s a Yes-nstance; that s, there s a partton A 1, A 2,..., A m of the tem set A such that A k w = B for 1 k m. Consder the followng strategy state s of the local-effect game. For = 1, 2,..., 3m and j = 1, 2,..., m, let { 1, f Aj n dj (s) = 1, n dj (s) = 0, n h1 (s) = 0, n h2 (s) = 1, n h3 (s) = 2, and n a j (s) = 0, otherwse. We show that no player can decrease her cost by swtchng to another acton. Frst, a player choosng some acton a j pays a cost of w due to a local effect from acton d j. Swtchng to acton a k for some k j does not change the player s cost. Any other acton apart from h 1, h 2, or h 3 would mply a cost of at least. Swtchng to acton h 1, h 2, or h 3 leads to a cost of at least B + 1 > w. Thus, none of these players can decrease her cost. Now consder a player wth acton d j. Her cost s A j w = B.

14 Dunkel and Schulz: On the Complexty of Pure-Strategy Nash Equlbra Mathematcs of Operatons Research xx(x), pp. xxx xxx, c 200x INFORMS a 1 1 a 2 1 a 3 1 a m 1 a 1 2 a 2 2 a 3 2 a m 2 w 1 w 2 w 1 w 2 w 1 w 2 w 1 w 2 a 1 p a 2 p a 3 p a m p w p w p w p w p 1/m 1/m 1/m d 1 d 2 d 3 d m 1/m h 1 h 3 h 2 d 1 d 2 d 3 d m Fgure 6: Illustraton of the local-effect graph of the game constructed n the proof of Theorem 4.1. An arc label α corresponds to the coeffcent of a lnear local-effect functon f a,a (x) = α x. Moreover, p := 3m. Swtchng to some acton a j, d k, or d k wth k j ncreases her cost to at least. Acton d j mples an equal cost of B. Swtchng to h 1, h 2, or h 3 results n a cost of at least B + 1. Hence, such a player has no ncentve to change her strategy ether. The player playng acton h 2 has a cost of 3B + 6. Swtchng to some acton n A \ {h 1, h 2, h 3 } ncreases her cost to at least. A change to acton h 1 or h 3 results n a new cost of 3B + 6 or 3B + 12, respectvely. Fnally, a player wth acton h 3 has cost 3B + 3 n state s. She would also strctly ncrease her cost by swtchng to some acton n A \ {h 1, h 2, h 3 }. A change to acton h 1 would result n a cost of 3B + 6, whle swtchng to h 2 ncreases her cost by four cost unts. Consequently, the defned state s a pure-strategy Nash equlbrum. For the other drecton of the proof, we make the followng observatons. Let s be a pure-strategy Nash equlbrum. Then, for each {1, 2,..., 3m}, at most one player chooses a strategy from the set {a j : j {1, 2,..., m}}. Otherwse, at least one of the players could decrease her current cost of at least to less than 3(4m + 3)B < by swtchng to acton h 3. By the same token, for each j {1, 2,..., m}, at most one player chooses an acton from {d j, d j } n s. It follows that at least three players play an acton from {h 1, h 2, h 3 }. Suppose there were more than three. Then, there exsted ether an {1, 2,..., 3m} such that m j=1 n a j (s) = 0 or some j {1, 2,..., m} wth n dj (s) + n dj (s) = 0. In the frst case, one of the players currently playng h 1, h 2, or h 3 could decrease her cost from at least B + 1 to at most w < B f she made the swtch to any of the actons correspondng to tem a. In the second case, swtchng to d j would decrease the cost of some player to B. Consequently, n h1 (s) + n h2 (s) + n h3 (s) = 3, (1) m n a j (s) = 1 for = 1, 2,..., 3m, (2) j=1 and n dj (s) + n dj (s) = 1 for j = 1, 2,..., m. (3) The key nsght left to show s that n dj (s) = 1 for j = 1, 2,..., m. Consder the subgame defned by actons h 1, h 2, and h 3 and three players. Assume that m j=1 n d j (s) < m. Table 3 shows that ths subgame does not have a pure-strategy Nash equlbrum. Therefore, the only way s can be a pure-strategy Nash equlbrum s that the local effects on h 1 due to actons d j sum up to 1 (see the row next to last n Table 3). Consequently, n dj (s) = 1 for j = 1, 2,..., m.