Terrorism and the Optimal Defense of Networks of Targets Dan Kovenock Brian Roberson

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1 Terrorsm and the Optmal efense of Networks of Targets an Kovenock ran Roberson Ths paper has benefted from the helpful comments of Ka Konrad and anel G. Arce as well as partcpants n presentatons at the 2006 Mdwest Economc Theory Meetngs, the 2008 Tournaments, Contests and Relatve Performance Evaluaton Conference held at North Carolna State Unversty, and the 2008 Contests: Theory and Applcatons Conference held at Stockhom School of Economcs. Part of ths work was completed whle Kovenock was Vstng Professor at the Socal Scence Research Center erln (WZ. Roberson gratefully acknowledges fnancal support from the Socal Scence Research Center erln (WZ, the Mam Unversty Commttee on Faculty Research, and the Farmer School of usness. The authors, of course, reman solely responsble for any errors or omssons. an Kovenock Purdue Unversty, epartment of Economcs, Krannert School of Management, 403 West State Street, West Lafayette, IN USA t: , f: , E-mal: kovenock@purdue.edu ran Roberson Mam Unversty, epartment of Economcs, Rchard T. Farmer School of usness, 208 Laws Hall, Oxford, OH USA t: , f: , E-mal: robersba@muoho.edu (Correspondent

2 Abstract Ths paper examnes a game-theoretc model of attack and defense of multple networks of targets n whch there exst ntra-network strategc complementartes among targets. The defender s objectve s to successfully defend all of the networks and the attacker s objectve s to successfully attack at least one network of targets. In ths context, our results hghlght the mportance of modelng asymmetrc attack and defense as a conflct between fully strategc actors wth endogenous entry and force expendture decsons as well as allowng for general correlaton structures for force expendtures wthn and across the networks of targets. 2 JEL Classfcaton: C7, 74 Keywords: Asymmetrc Conflct, Attack and efense, Weakest-Lnk, est-shot

3 3 1 Introducton In the lterature on optmal defense from ntentonal attack there has been growng nterest n not only the attack and defense of solated targets 1 but also networks of targets 2 and even complex supra-networks of targets. 3 Ths move towards ncreasng network complexty emphaszes the role that strategc complementartes among targets play n creatng structural asymmetres between the attack and defense of such combnatons of targets. For example n complex nfrastructure supra-networks such as communcaton systems, electrcal power grds, water and sewage systems, ol ppelne systems, transportaton systems, and cyber securty systems there often exst partcular targets or combnatons of targets whch f destroyed would be suffcent to ether: (a dsable the entre supra-network or (b create a terrorst spectacular. In order to hghlght the mportance of modelng the asymmetrc attack and defense of complex supra-networks as a conflct between fully strategc actors wth endogenous entry and force expendture decsons, we examne a contest-theoretc model of the attack and defense of a complex supra-network and allow for the players to use general correlaton structures for force expendtures wthn and across the networks of targets. The supra-network of targets s made up of an arbtrary combnaton of two smple types of networks whch capture the two extreme endponts of an exposure-redundency spectrum for network types. The maxmal exposure network, whch we label a weakest-lnk network, s successfully defended f and only f the defender successfully defends all targets wthn the network. 4 The maxmal redundancy network, whch we label a best-shot network, s successfully defended f the defender successfully defends at least one target wthn the network. At each target the conflct s modeled as a determnstc contest n whch the player who allocates the hgher level of force wns the target wth probablty one. Gven that the loss of a sngle network may be suffcent to ether dsable the entre supra-network or create a terrorst spectacular, we focus on the case that the attacker s 1 See for example er et al. (2007, Powell (2007a, b, and Rosendorff and Sandler ( See for example er and Abhchandan (2003, er et al. (2005, and Clark and Konrad ( See for example Azaez and er (2007, Hausken (2008, and Levtn and en-ham ( See Hrshlefer (1983 who cons the terms best-shot and weakest-lnk n the context of voluntary provson of publc goods.

4 objectve s to successfully attack a sngle network, and that the defender s objectve s to successfully defend all of the networks. A dstnctve feature of ths envronment s that a mxed strategy s a jont dstrbuton functon n whch the randomzaton n the force allocaton to each target s represented as a separate dmenson. A par of equlbrum jont dstrbuton functons specfes not only each player s randomzaton n force expendtures for each target but also the correlaton structure of the force expendtures wthn and across the networks of targets. For all parameter confguratons, we completely characterze the unque set of Nash equlbrum unvarate margnal dstrbutons for each player as well as the unque equlbrum payoff for each player. Furthermore, n any equlbrum we fnd that the attacker launches an attack on at most one network of targets, and there exst parameter confguratons for whch the attacker optmally launches no attack wth postve probablty. Whle at most one network s attacked, the attacker randomzes over whch network s attacked, and each of the networks s attacked wth postve probablty. In the event that a weakest-lnk network s attacked, the attacker optmally launches an attack on only a sngle target. When a best-shot network s attacked, the attacker optmally attacks every target n that network wth a strctly postve force level. As emphaszed n the Natonal Strategy for Homeland Securty, terrorsts are strategc actors. However, much of the exstng lterature [e.g. Azaez and er (2007, er and Abhchandan (2003, er et al. (2005, er et al. (2007, Levtn and en-ham (2008, Powell (2007a, b, and Rosendorff and Sandler (2004.] assumes that terrorsts (henceforth attackers are not fully strategc n the sense that the number of attacks (whch s usually set to one s exogenously specfed. y endogenzng the attacker s entry and force expendture decsons, we examne not only the condtons under whch the assumpton of one attack s lkely to hold, but also related ssues such as how the defender s actons can decrease the number of terrorst attacks. Furthermore, the few prevous models whch allow for the attacker to endogenously choose the number of targets to attack [e.g. Clark and Konrad (2007 and Hausken (2008] 5 obtan the result 5 Utlzng probablstc contest success functons [Clark and Konrad (2007 utlze the Tullock contest success functon, Hausken (2008 utlzes both the Tullock and dfference-form contest success functons], Clark and Konrad (2007 and Hausken (2008 examne a sngle weakest-lnk network and a supra-network consstng of any arbtrary combnaton of weakest-lnk and best-shot networks [as n ths paper, a successful attack on any one network s suffcent to dsable the entre supra-network], respectvely. 4

5 that even when the attacker s objectve s to dsable a sngle network and the attacker derves no addtonal beneft from successfully dsablng more than one network the attacker optmally chooses to attack every target n every network wth certanty. y showng that n all equlbra of our model the attacker optmally engages n a from of stochastc guerlla warfare n whch they attack at most one network of targets (but wth postve probablty each network s chosen as the one to be attacked, our results also provde a sharp contrast wth exstng models of fully strategc attackers. Secton 2 presents the model of attack and defense wth networks of targets. Secton 3 characterzes a Nash equlbrum and explores propertes of the equlbrum dstrbutons of force. Secton 4 concludes. 5 2 The Model Players The model s formally descrbed as follows. Two players, an attacker, A, and a defender,, smultaneously allocate ther forces across a fnte number, n 2, of heterogeneous targets. The players payoffs depend on the composton of each of the networks of targets n the supra-network. We examne a supra-network consstng of any arbtrary combnaton of two types of smple networks. The targets are parttoned nto a fnte number k 1 of dsjont networks, where network j {1,...,k} conssts of a fnte number n j 1 of targets wth k j=1 n j = n. Let N j denote the set of targets n network j. Let W denote the set of weakest-lnk networks and denote the set of best-shot networks. In a best-shot network the network s successfully defended f the defender allocates at least as hgh a level of force to at least one target wthn the network. Conversely, an attack on a best-shot network s successful f the attacker allocates a hgher level of force to each target n the network. Let x A (x denote the level of force allocated by the attacker (defender to target. efne ι 1 f N j x A j = > x. 0 otherwse Observe that for each target, the player that allocates the hgher level of force wns that target, but n order to wn the network the attacker must wn all of the targets. In a bestshot network, a te arses when player A allocates a level of force to each target n the

6 network that s at least as great as player s allocaton, and there exsts at least one target n the network to whch the players allocate the same level of force. In ths case, the defender wns the network. In the second type of network, whch we label a weakest-lnk network, the network s successfully defended f the defender allocates at least as hgh a level of force to all targets wthn the network. Conversely, an attack on a weakest-lnk network s successful f the attacker allocates a hgher level of force to any target n the network. efne ι W 1 f N j x A j = > x. 0 otherwse Agan, n the case of a te, the defender s assumed to wn the network. The players are rsk neutral and have asymmetrc objectves. The attacker s objectve s to successfully attack at least one network, and the attacker s payoff for the successful attack of at least one network s. The attacker s payoff functon s gven by ( {ι } π A (x A,x = max j j,{ ι W j } j W n x A =1 The defender s objectve s to preserve the entre supra-network, and the defender s payoff for successfully defendng the supra-network s v. The defender s payoff functon s gven by ( {ι } π (x A,x = v (1 max j j,{ ι W j } j W The force allocated to each target must be nonnegatve. It s mportant to note that our formulaton utlzes the all-pay aucton contest success functon. 6 Wthn the all-pay aucton lterature t s well known that the equlbrum of the game n whch the players have dfferng unt costs of resources s equvalent up to a lnear scalng of the equlbrum of the game wth asymmetrc valuatons. Ths result extends drectly to the envronment examned here, and thus, our focus on asymmetrc valuatons also covers the case n whch the players have dfferng unt costs of resources. Also observe that n the formulaton descrbed above the supra-network s a weakestlnk supra-network. That s f the defender loses a sngle network then the entre supranetwork s noperable. y nterchangng the denttes of player A and player, our 6 See aye, Kovenock, and de Vres (1996. n =1 x. 6

7 results on weakest-lnk supra-networks apply drectly to the case of best-shot supranetworks (where a best-shot supra-network s a supra-network whch s successfully defended f the defender successfully defends at least one network. Fgure 1 provdes a representatve supra-network consstng of 5 networks (A,, C,, and E. Networks A, C, and E are weakest-lnk (seres networks wth two targets each. Networks and are best-shot (parallel networks wth fve targets each. In order to preserve the entre supra-network player s objectve s to preserve a path across the entre network. If a sngle target n networks A, C, or E s destroyed then the supra-network s noperable. Conversely, n networks and all of the targets must be destroyed n order to render the supra-network noperable. 7 [Insert Fgure 1 here] Strateges It s clear that there s no pure strategy equlbrum for ths class of games. A mxed strategy, whch we term a dstrbuton of force, for player s an n-varate dstrbuton functon P : R n + [0,1]. The n-tuple of player s allocaton of force across the n targets s a random n-tuple drawn from the n-varate dstrbuton functon P. Model of Attack and efense wth Networks of Targets The model of attack and defense wth networks of targets, whch we label { {Nj } AN j,{ } N j j W,,v }, s the one-shot game n whch players compete by smultaneously announcng dstrbutons of force, each target s won by the player that provdes the hgher allocaton of force for that target, tes are resolved as descrbed above, and players payoffs, π A and π, are specfed above. 3 Optmal strbutons of Force It wll be useful to ntroduce a smple summary statstc whch captures both the asymmetry n the players valuatons and the structural asymmetres arsng n the supranetwork.

8 efnton 1 Let α = v /( [ j W n j + j 1 n j ] denote the normed relatve strength of the defender. Several propertes of ths summary statstc should be noted. Frst, the normed relatve strength of the defender s ncreasng n the relatve valuaton of the defender to the attacker (v /, and s decreasng n the level of exposure arsng n the supra-network ( j W n j + j 1 n j. In partcular, the defender s exposure s ncreasng n the number of weakest-lnk targets ( j W n j, and s decreasng n the number of targets wthn each best-shot network ( j 1 n j. For all parameter ranges, Theorem 1 establshes the unqueness of: ( the players equlbrum expected payoffs and ( the players sets of unvarate margnal dstrbutons. Theorem 1 also provdes a par of equlbrum dstrbutons of force for all parameters ranges. Case (1 of Theorem 1 examnes the parameter confguratons for whch the defender has a normed relatve strength advantage,.e. α 1. Case (2 of Theorem 1 addresses the parameter confguratons for whch the defender has a normed relatve strength dsadvantage,.e. α < 1. It s mportant to note that the stated equlbrum dstrbutons of force (n-varate dstrbutons are not unque. However, n Propostons 1-3 we characterze propertes of optmal attack and defense that hold n all equlbra. Theorem 1 For all feasble parameter fguratons of the game AN{{N j } j,{n j } j W,,v } (.e.,,v > 0 and N j /0 for all j there exsts a unque set of Nash equlbrum unvarate margnal dstrbutons and a unque equlbrum payoff for each player. One such equlbrum s for each player to allocate hs forces accordng to the followng n-varate dstrbuton functons. (1 If α 1, then for player A and x j W [0, ] n j j [0, n j ] n j P A (x = 1 1 α + j W Nj x + j mn Nj {x } v Smlarly for player and x j W [0, ] n j j [0, n j ] n j P (x = mn { { mn Nj x } } j W, { } Nj x The expected payoff for player A s 0, and the expected payoff for player s v (1 1 α. j 8

9 9 (2 If α < 1, then for player A and x j W [0,α ] n j j [0, α n j ] n j P A (x = j W Nj x + j mn Nj {x } v Smlarly for player and x j W [0,α ] n j j [0, α n j ] n j ( {mn { x } } { N j Nj x } P (x = 1 α + mn, The expected payoff for player s 0, and the expected payoff for player A s (1 α. Proof The proof of the unqueness of the players equlbrum expected payoffs and sets of unvarate margnal dstrbutons s gven n the appendx. Ths proof establshes that the par of n-varate dstrbuton functons gven n case (1 consttute an equlbrum wthn the case (1 parameter range. The proof of case (2 s analogous. The appendx (see Lemma 5 establshes that n any n-tuple drawn from any equlbrum n-varate dstrbuton P A player A allocates a strctly postve level of force to at most one network of targets. If the network whch receves the strctly postve level of force s a weakestlnk network, then exactly one target n that network receves a strctly postve level of force. Whle not a necessary condton for equlbrum, the P A descrbed n Theorem 1 also dsplays the property that when the network whch receves the strctly postve level of force s a best-shot network the force allocated to each target n that network s an almost surely ncreasng functon of the force allocated to any of the other targets n that network. The appendx (see Lemma 5 also establshes that n any n-tuple drawn from any equlbrum n-varate dstrbuton P player allocates a strctly postve level of force to at most one target n each best-shot network of targets. We wll now show that for each player each pont n the support of ther equlbrum n-varate dstrbuton functon {P A,P } gven n case (1 of Theorem 1 results n the same expected payoff, and then show that there are no proftable devatons from ths support. We begn wth the case n whch player A attacks a sngle target n a sngle weakestlnk network. The probablty that player A wns target n network j W s gven by the unvarate margnal dstrbuton P (x A,{{} N j x =0} j W,{{ n } A j N } j j. Gven that player s usng the equlbrum strategy P descrbed above, the payoff to player A for any allocaton of force x A R n + whch allocates a strctly postve level of j W j

10 10 force to a sngle target n a weakest-lnk network j W s Smplfyng, π A (x A,P = P (x A x A. ( x π A (x A,P = A x A v = 0. A Thus the expected payoff to player A from allocatng a strctly postve level of force to only one target n any weakest-lnk network s 0 regardless of whch target s attacked. Next, we examne the case n whch player A attacks a sngle best-shot network. The probablty that player A wns every target n network j s gven by the n j -varate margnal dstrbuton P ({x A } N j,{{ } N j } j W,{{ n j } N j } j j j, whch we wll denote as P N j ({x A } N j. Gven that player s usng the equlbrum strategy P descrbed above, the payoff to player A for any allocaton of force x A R n + whch allocates a strctly postve level of force only to the targets n a best-shot network j, and allocates zero forces to every other network s Smplfyng, π A (x A,P = P N j π A (x A,P = ( Nj x A ( {x A } Nj N j x A. N j x A = 0. Thus, the expected payoff to player A from allocatng a strctly postve level of force to only one best-shot network s 0 regardless of whch best-shot network s attacked. For player A, possble devatons from the support nclude allocatng a strctly postve level of force to: (a two or more targets n the same weakest-lnk network, (b two or more targets n dfferent weakest-lnk networks, (c two or more best-shot networks, and (d any combnaton of both weakest-lnk and best-shot networks. egnnng wth (a, the probablty that player A wns both targets and n network j W s gven by the bvarate margnal dstrbuton P (x A,x A,{{} N j, } j W,{{ n } j N } j j, whch we wll denote as P, (x A,x A. The payoff to player A for any allocaton of force x A R n + whch allocates a strctly postve level of force to two targets, n a weakest-lnk network j W s π A (x A,P = P ( x A + va P ( x A P, ( x A,x A x A x A.

11 11 Smplfyng, π A (x A,P = x A + x A { mn x A,x A } x A x A < 0. The case of player A allocatng a strctly postve level of force to more than two targets n a weakest-lnk network follows drectly. Clearly, n any optmal strategy player A never allocates a strctly postve level of force to more than one target wthn a weakestlnk network. The proof for type (b devatons follows along smlar lnes. Thus, n any optmal strategy player A never allocates a strctly postve level of force to more than one target wthn a weakest-lnk network of targets or n dfferent weakest-lnk networks. For type (c devatons, the probablty that player A wns all of the targets n both best-shot networks j, j s gven by the (n j + n j -varate margnal dstrbuton P ({x A } N j N j,{{ } N } j j W,{{ n } j N } j j j j, j, whch we wll denote as P N ( j,n j {x A } N j N j. The payoff to player A for any allocaton of force x A R n + whch allocates a strctly postve level of force to exactly two best-shot networks j, j s π A (x A,P = P N j ( {x A } Nj + va P N j ( {x A } N j P N ( j,n j {x A } N j N j N j N j x A. Smplfyng, π A (x A,P = mn { Nj x A, } N x j A The case of player A allocatng a strctly postve level of force to more than two bestshot networks follows drectly. Clearly, n any optmal strategy player A never allocates a strctly postve level of force to more than one best-shot network. The case of type (d, follows along smlar lnes. Thus, the expected payoff from each pont n the support of the n-varate dstrbuton P A results n the same expected payoff, 0, and there exst no allocatons of force whch have a hgher expected payoff. The case for player follows along smlar lnes.

12 Whle the equlbrum dstrbutons of force stated n Theorem 1 are not unque, 7 t s useful to provde some ntuton regardng the exstence of ths partcular equlbrum before movng on to the characterzaton of propertes of optmal attack and defense that hold n all equlbra (Propostons 1-3. The supports of the equlbrum dstrbutons of force stated n Theorem 1 are gven n Fgure (2. Panels ( and ( of Fgure (2 provde the supports for the attacker and defender, respectvely, n the case that there s one weakest-lnk network wth two targets ( = 1,2. Panels ( and (v of Fgure (2 provde the supports for the attacker and defender, respectvely, n the case that there s one best-shot network wth two targets ( = 1,2 and one weakest-lnk network wth one target ( = [Insert Fgure 2] Across all of the Panels (-(v, f α = 1 then each player randomzes contnuously over ther respectve shaded lne segments. In the event that the defender has a normed relatve strength advantage (α > 1, the defender s strategy stays the same, but the attacker now places a mass pont of sze 1 (1/α at the orgn and randomzes contnuously over the respectve lne segments wth the remanng probablty. Conversely, f the defender does not have a normed relatve strength advantage (α < 1 then t s the defender who places a mass pont (of sze 1 α at the orgn. egnnng wth Panels ( and (, recall that f the attacker successfully attacks a sngle target n a weakest-lnk network the entre network s dsabled. As shown n Panel ( the attacker launches an attack on at most one target. To successfully defend a weakest-lnk network, the defender must wn every target wthn the network. As shown n Panel ( the defender s allocaton of force to target s an almost surely strctly ncreasng functon of the force allocated to target. Note that f the attacker launches an attack on at most one target, then the probablty that any sngle attack s successful depends only on the unvarate margnal dstrbutons of the defender s (n-varate jont 7 For example, n the case (1 parameter range of Theorem 1 another equlbrum strategy for player s to use the dstrbuton of force { Nj x } { Nj x } P (x = mn, j W j.

13 dstrbuton of force. In addton, the defender s expected force expendture depends only on hs set of unvarate margnal dstrbutons, and, for a gven set of unvarate margnal dstrbutons, s nvarant to the correlaton structure. 8 Fnally, note that for the gven correlaton structure n the defender s support [panel (] the probablty that the attacker launches at least one successful attack depends only on the maxmum of hs force allocatons across the two targets. That s, gven the defender s dstrbuton of force, f there exsts any ponts n the support of the attacker s dstrbuton of force n whch x A > x A > 0 wth postve probablty, then the attacker can strctly ncrease hs expected payoff by changng to x A 13 = 0 n all such ponts. In such a devaton, the probablty of at least one successful attack s unaffected, but the attacker s expected force expendture decreases. Thus, at each pont n the support of an optmal dstrbuton of force the attacker launches at most one attack. Panels ( and (v examne a smple supra-network wth one best-shot network and one weakest-lnk network. In Panel (, note that the attacker launches an attack on at most one network. In the event that the best-shot network s attacked, the attacker s allocaton of force to target s an almost surely strctly ncreasng functon of the force allocated to target. In Panel (v, note that the defender allocates a strctly postve level of force to at most one of the targets n the best-shot network, and that level of force allocated to the weakest-lnk network s an almost surely ncreasng functon of the level of force allocated to the best-shot network. Gven these correlaton structures, the ntuton for the attacker launchng an attack on at most one network n the supranetwork follows along the lnes gven above for the weakest-lnk network n whch at most one target was attacked. We now characterze the qualtatve features arsng n all equlbrum dstrbutons of force. Proposton 1 examnes the number of networks that are smultaneously attacked as well as the number of targets wthn each network that are smultaneously attacked and defended. Propostons 2 and 3 examne the lkelhood that the attacker optmally chooses to launch an attack on any gven network, and the lkelhood that the attacker launches no attack or the defender leaves the supra-network undefended. Proposton 1 In any equlbrum {P A,P }: 1. Player A allocates a strctly postve level of force to at most one network. 8 More formally, for a gven set of unvarate margnal dstrbuton functons, the expected force expendture s nvarant to the mappng nto a jont dstrbuton functon,.e. the n-copula.

14 2. If the network to whch player A allocates a strctly postve level of force s a weakest-lnk network, then at most one target n that weakest-lnk network receves a strctly postve level of force. 3. In each best-shot network player allocates a strctly postve level of force to at most one target n the network. The formal proof of Proposton 1 s gven n the appendx (see Lemma 5. The ntuton for Proposton 1 follows from the fact that the lkelhood that player successfully defends all of the networks (and therefore player s expected payoff s weakly decreasng n the number of networks that player A chooses to smultaneously attack. However, player has the ablty to vary the correlaton structure of hs force allocatons whle leavng nvarant: ( hs network specfc multvarate margnal dstrbutons of force, ( hs unvarate margnal dstrbutons of force, and ( hs expected expendture. Furthermore, there exst correlaton structures for whch the lkelhood that player successfully defends all of the networks depends only on player A s force allocaton to the one network whch receves the hghest level of force from player A. Gven that player s usng such a correlaton structure, player A optmally attacks at most one network at a tme. A smlar result extends drectly to (2 the case of weakest-lnk networks and to (3 the case of best-shot networks. Proposton 2 If α 1, then n any equlbrum {P A,P }: 1. The probablty that any weakest-lnk network j s attacked (.e., the probablty that the attacker allocates a strctly postve level of force to weakest-lnk network j s (n j /v, whch s ncreasng n the number of targets n network j and the attacker s valuaton of success and decreasng n the defender s valuaton of successfully defendng the entre supra-network. 2. The probablty that any best-shot network j s attacked s ( /(n j v, whch s ncreasng n the attacker s valuaton of success and s decreasng n both the defender s valuaton and the number of targets n network j. 3. The attacker optmally attacks no network n the supra-network wth probablty 1 (1/α. For the attacker s jont dstrbuton, the appendx characterzes the attacker s mass pont at the orgn (see Lemma 9 as well as hs set of unvarate margnal dstrbutons. Proposton 2 follows drectly. The probablty that a network j s attacked s equal to one mnus the attacker s mass pont at zero n the n j -varate margnal dstrbuton for network j, where the n j -varate margnal dstrbuton for network j s gven by P N j A ({x } Nj. 14

15 The lkelhood that the attacker optmally chooses to launch no attack s ncreasng n the defender s valuaton of success and decreasng n the attacker s valuaton of success. In the case (1 parameter range, the attacker s valuaton s low enough relatve to the defender s valuaton that the optmal strategy ncludes not launchng an attack wth postve probablty. As we move to the case (2 parameter range, the attacker optmally launches an attack wth certanty. In ths case the probablty that any gven network of targets s attacked depends only on the number of targets n the network and the type of network. The proof of Proposton 3 also follows from the characterzaton of the equlbrum jont dstrbutons gven n the appendx. Proposton 3 If α < 1, then n any equlbrum {P A,P }: 1. The probablty that any weakest-lnk network j s attacked (.e., the probablty that the attacker allocates a strctly postve level of force to weakest-lnk network j s n j /([ j W n j + j n 1 ], whch s ncreasng n the number of targets n network j j. 2. The probablty that any best-shot network j s attacked s 1/(n j [ j W n j + j n 1 ], j whch s decreasng n the number of targets n network j. 3. The defender optmally leaves the entre supra-network undefended wth probablty 1 α. In the case (1 parameter range, the defender optmally chooses, wth certanty, to allocate a strctly postve level of defensve force. However, n the case (2 parameter range, the defender optmally chooses to leave the entre supra-network undefended wth postve probablty. Furthermore, the lkelhood that the defender chooses to leave the entre supra-network undefended s ncreasng n the attacker s valuaton of success and decreasng n the defender s valuaton of successfully defendng the entre supranetwork. To summarze, the followng condtons hold n all equlbra. In the case (1 parameter range the attacker optmally chooses not to launch an attack wth postve probablty. In both cases (1 and (2, the attacker optmally launches an attack on at most one network. In the event that a weakest-lnk network s attacked, only one target wthn the network s attacked. The lkelhood that any ndvdual network s attacked depends on the number of targets wthn the network. In each weakest-lnk network the lkelhood of attack s ncreasng n the number of targets. In each best-shot network the lkelhood of attack s decreasng n the number of targets. In the case (2 parameter range, the defender optmally leaves the entre supra-network undefended. Lastly, n both cases (1 15

16 and (2 when the defender chooses to defend the supra-network, wthn each best-shot network, the defender randomly chooses at most one target to defend Concluson Ths paper examnes a game-theoretc model of attack and defense wth multple networks of targets and ntra-network strategc complementartes among targets. In equlbrum we fnd that the correlaton structure of the optmal attack and defense strateges depends crtcally on the composton of the supra-network. In addton, network redundances, as n best-shot networks, strengthen the defender s strategc poston. Conversely, the absence of network redundances, as n weakest-lnk networks, weaken the defender s strategc poston. In the context of networks of targets wth asymmetrc attack and defense, our results hghlght the mportance of allowng for endogenous entry and force expendture decsons ncludng general correlaton structures for force expendtures wthn and across the networks of targets. References 1. Azaez, N., and V. M. er (2007, Optmal resource allocaton for securty n relablty systems, European Journal of Operatonal Research 181: aye, M. R.,. Kovenock, and C. G. de Vres (1996, The all-pay aucton wth complete nformaton, Economc Theory 8: er, V. M., and V. Abhchandan (2003, Optmal allocaton of resources for defense of smple seres and parallel systems from determned adversares, n: Rsk- ased ecson Makng n Water Resources X, Amercan Socety of Cvl Engneers, Reston, VA, pp er, V. M., A. Nagaraj, and V. Abhchandan (2005, Protecton of smple seres and parallel systems wth components of dfferent values, Relablty Engneerng and System Safety 87: er, V. M., S. Olveros, and L. Samuelson (2007, Choosng what to protect: strategc defensve allocaton aganst an unknown attacker, Journal of Publc Economc Theory 9: Clark,. J., and K. A. Konrad (2007, Asymmetrc conflct: weakest lnk aganst best shot, Journal of Conflct Resoluton 51: Coughln, P. J., (1992, Pure strategy equlbra n a class of systems defense games, Internatonal Journal of Game Theory 20:

17 8. Hausken, K., (2008, Strategc defense and attack for seres and parallel relablty systems, European Journal of Operatonal Research 186: Hrshlefer, J., (1983, From weakest-lnk to best-shot: the voluntary provson of publc goods, Publc Choce 41: Levtn, G., and H. en-ham (2007, Importance of protecton aganst ntentonal attacks, Relablty Engneerng and System Safety 93: Nelsen, R.., (1999 An Introducton to Copulas. Sprnger, New York. 12. Powell, R., (2007a, efendng aganst terrorst attacks wth lmted resources, Amercan Poltcal Scence Revew 101: Powell, R., (2007b, Allocatng defensve resources wth prvate nformaton about vulnerablty, Amercan Poltcal Scence Revew 101: Sandler, T., and W. Enders (2004, An economc perspectve on transnatonal terrorsm, European Journal of Poltcal Economy 20: Shubk, M., and R. J. Weber (1981, Systems defense games: Colonel lotto, command and control, Naval Research Logstcs Quarterly 28:

18 18 Appendx Ths appendx characterzes the supports of the equlbrum jont dstrbutons, the unque equlbrum payoffs, and the unque sets of equlbrum unvarate margnal dstrbutons. efore proceedng, observe the followng notatonal conventons whch wll be used throughout the appendx. For ponts n R n, wll use the vector notaton x =(x 1,x 2,...,x n. For a k b k for all k = 1,2,...,n, let [a,b] denote the n-box = [a 1,b 1 ] [a 2,b 2 ]... [a n,b n ], the Cartesan product of n closed ntervals. The vertces of an n-box are the ponts (c 1,c 2,...,c n where c k s equal to a k or b k. Gven that the defender s usng the dstrbuton of force P, let ( ( {ι } Pr max j j,{ ι W } j = 1 P j W,x A (1 denote the probablty that wth a force allocaton of x A the attacker wns at least one network. Thus, the attacker s expected payoff from any pure strategy x A s ( ( {ι } Pr max j j,{ ι W } j = 1 P j W,x A x A. (2 It wll also be useful to note that the attacker s expected payoff from any dstrbuton of force P A s ( ( {ι } E PA [Pr max j j,{ ι W } ] [ ] j = 1 P j W,x A E F x A A (3 where E PA denotes the expectaton wth respect to the jont dstrbuton of force P A and E F A denotes the expectaton wth respect to the unvarate margnal dstrbuton for target, henceforth FA, of the jont dstrbuton of force P A. Smlarly, gven that the attacker s usng the dstrbuton of force P A, let ( ( {ι } Pr max j j,{ ι W } j = 0 P j W A,x denote the probablty that wth a force allocaton of x the defender wns all of the networks n the supra-network. Thus, the defender s expected payoff from any pure strategy x s ( ( {ι } v Pr max j j,{ ι W } j = 0 P j W A,x x. (5 Lastly, the defender s expected payoff from any dstrbuton of force P s ( ( {ι } v E P [Pr max j j,{ ι W } ] [ ] j = 0 P j W A,x E F x (4 (6

19 where E P and E F denote the expectaton wth respect to the jont dstrbuton of force P and the expectaton wth respect to the unvarate margnal dstrbuton for target, F, respectvely. Lemma 1 For each and j such that N j j W, s A = s = s j W and s A = s = 0. For and j such that N j j, s A = s = s j and s A = s = 0. Proof We begn wth the proof that s A = s = 0 for all. y way of contradcton, suppose s A s. Let ŝ max{s A,s }, and let k be the dentty of the player attanng ŝ (that s ŝ = s k and ŝ > s k. If s k > 0, when player k allocates s k to target player k s losng target wth certanty and can strctly ncrease hs payoff by settng s k = 0. It follows drectly, that player k does not randomze over the open nterval (0,ŝ, and thus player k must have a mass pont 0. In the case that s k = 0 (where player k does not randomze over the open nterval (0,ŝ and has a mass pont at 0, we know that ( both players can not have a mass pont at s k and ( player k can strctly ncrease hs payoff by lowerng s k to a neghborhood above 0. Thus, we conclude that s A = s = 0 for all. Lastly, for the proof that for each and j such that N j j W, s A = s = s j W, note that for,k N j j W t follows that f s A = s < sk A = sk then player A would do better by movng mass from s k A to s A. The proof that for and j such that N j j, s A = s = s j follows for the same reasons. Lemma 2 In any equlbrum {P A,P }, for each target j nether player s unvarate margnal dstrbutons place postve mass on any pont except possbly at zero. Proof If x j s such a pont for player j, then player j would ether beneft from movng mass from an ε-neghborhood below x j to zero or to a δ-neghborhood above x j. Lemma 3 In any equlbrum {P A,P }, each player s expected payoff s constant over the support of ther jont dstrbuton except possbly at ponts of dscontnuty of the payoff functon. Proof y Lemma 2, for each target there are no mass ponts n the half-open nterval (0, s ]. Thus for each pont n the support of player j s jont dstrbuton, player j must make hs equlbrum payoff except for possbly at ponts of dscontnuty of the payoff functon. 19

20 Lemma 4 In any equlbrum {P A,P }, for each target each player randomzes contnuously over the nterval (0, s ]. 20 Proof y way of contradcton, suppose that there exsts an equlbrum n whch for some target, player j s unvarate margnal dstrbuton for target, Fj, s constant over the nterval [α,β (0, s ] and strctly ncreasng above β n ts support. For ths to be an equlbrum, t must be the case that F j s also constant over the nterval [α,β. Otherwse, player j could ncrease hs payoff. If F j (α = F j (β, then for any ε > 0 spendng β + ε n target cannot be optmal for player j. Indeed, by dscretely reducng hs expendture from β + ε to α + ε player j s payoff would strctly ncrease. Consequently, f Fj s constant over [α,β t must also be constant over [α, s ], a contradcton to the defnton of s. Lemma 5 In any equlbrum {P A,P }: (a If x A s an n-tuple contaned n the support of P A, then x A allocates a strctly postve level of force to at most one network. (b If the network to whch the n-tuple x A (contaned n the support of P A allocates a strctly postve level of force s a weakest-lnk network, then at most one target n that weakest-lnk network receves a strctly postve level of force. (c If x s an n-tuple contaned n the support of P, then wthn each best-shot network x allocates a strctly postve level of force to at most one target n the network. Proof egnnng wth (a, by way of contradcton suppose that there exsts an equlbrum n whch player A smultaneously allocates a strctly postve level of force to two or more networks. Wthout loss of generalty, we wll also assume that there exsts at least one pont n the support of an equlbrum strategy P A for whch only networks j and j smultaneously receve a strctly postve level of force from player A (henceforth networks j and j are smultaneously attacked. Observe that ths assumpton allows for any number and/or combnaton of networks to be smultaneously attacked as long as at some pont n the support of P A only networks j and j are smultaneously attacked. Furthermore, whle the focus on the case n whch the mnmum number of networks beng smultaneously attacked s equal to two smplfes the expressons that follow, the case n whch the mnmum number of networks whch are smultaneously attacked s greater than two follows drectly.

21 Snce max({ι j } j,{ι W j } j W s equal to ether 0 or 1, the expected payoff for player may be wrtten as ( v v E PA [Pr max ( ] P ι j,ι j = 1,x A s.t. only j and j attacked ( ( {ι } v E PA [Pr max j j,{ ι W } ] j = 1 P j W,x A s.t. not j and j attacked F ( x. (7 The expectaton n the frst lne of (7 s the probablty that player A successfully attacks at least one of the networks j or j gven that player A attacks only networks j and j. The expectaton n the second lne of (7 s the probablty that player A successfully attacks at least one network condtonal on the attack beng on any sngle network or any combnaton of networks other than only j and j. Let x j A denote the restrcton of the vector x A to the set of targets contaned n network j,.e. {x A } N j. Note that ( Pr max ( ( P ι j,ι j = 1,x A s.t. j and j attacked = Pr ι j = 1 P n j,xj A ( + Pr ι j = 1 P n ( j,xj Pr ι j = 1 and ι j = 1 P n j,n j A,x j A,xj A 21. (8 If network j s a best-shot network then Pr(ι j = 1 P n j,xj A = Pn j (x j A. If network j s a weakest-lnk network then the probablty that player A wns at least one target, Pr(ι W j = 1 P n j,xj A, depends crtcally on both Pn j and the number of targets n network j whch are attacked. In both cases, t s clear that player s payoff depends not only on the n j -varate margnal dstrbuton for network j, P n j, and the n j dstrbuton for network j,p n j -varate margnal, but also on the correlaton between these two multvarate margnal dstrbutons. There are 3 possble cases to consder: ( both networks j and j are best-shot networks, ( both networks j and j are weakest-lnk networks, and ( ether network j or j s a best-shot network and the other s a weakest-lnk network. If both networks j and j are best shot networks, then (8 becomes P n j (x j A +Pn j j (x A P n j,n j (x j A,xj A. However, f Pn j,n j (x j A,xj A mn{pn j (x j A,Pn j j (xa} then player could ncrease the frst lne of (7 wthout affectng the unvarate margnals and thus the thrd lne of (7. Furthermore, f P (x = mn j {P n j (x j } then each n j -varate margnal dstrbuton (P n j (x j s preserved, each unvarate margnal dstrbuton (F (x s preserved, and for any set of networks j J the jont margnal dstrbuton for the set J s P J (xj = mn j J {P n j (x j }. Clearly, f networks j and j are the only two networks

22 whch player A smultaneously attacks, then the devaton to such a correlaton strategy strctly ncreases player s payoff. A contradcton to the assumpton that {P A,P } s an equlbrum. Furthermore, f P n j,n j (x j A,xj A = mn{pn j (x j A,Pn j j (xa} then player A could ncrease hs payoff by attackng network j or network j but not both smultaneously; also a contradcton. If networks j and j are not the only networks whch player A smultaneously attacks but all of the networks n the supra-network are best-shot networks, then the second lne of (7 can be broken nto components for each of the sets of networks whch player A smultaneously attacks and each of the networks whch are attacked n solaton. In ths case, the proof follows along the lnes of the proceedng case. That s f the supranetwork s comprsed of only best-shot networks, player A attacks at most one network. For cases ( and ( as well as the remanng case ( network confguratons, note that the result on a supra-network consstng of only best-shot networks can be modfed to show that wthn each weakest-lnk network player A attacks at most one target. That s, wthout loss of generalty, assume that there exsts at least one pont n the support of an equlbrum strategy P A for whch wthn weakest-lnk network j only targets and smultaneously receve a strctly postve level of force from player A. In such a case, Pr(ι W j = 1 P n j,xj A = F (x A + F (x A Pn j (x A,x A,{ s j W } N j, where (x A,x A,{ s j W } N j, denotes the vector formed by replacng each zero n x j A (all targets except and wth s j W. Settng Pn j (x j A = mn j{f (x }, the result follows drectly, as does the result for the case n whch the number of targets whch are smultaneously attacked s greater than two or nvolves any arbtrary combnatons of weakest-lnk targets. Thus, player A attacks at most one target n each weakest-lnk network, and Pr(ι j = 1 P n j,xj A = F (x A. Insertng ths back nto (8, the proof for cases ( and ( as well as the remanng case ( network confguratons follows drectly. The proof for part (c follows from a symmetrc argument. Lemma 6 j, j W, s j W = s j W s W. Proof Followng from Lemmas 2 and 5, n the support of any optmal strategy, when player A allocates s j W to a sngle target n network j the force allocated to each of the remanng targets s 0, player A wns network j wth certanty, and player A s expected payoff s s j W. From Lemma 3, player A s expected payoff s constant across all ponts n the support of P A. Thus, j, j W, s j j W = s W s W. Lemma 7 j, s W = n j s j. 22

23 Proof From Lemma 5 part (a n the support of any optmal strategy player A attacks at most one network. In the case that player A attacks best-shot network j, from Lemma 3 there exsts a k A 0 such that ( Pr ι j = 1 P n j,x A 23 k A + x A (9 whch holds wth equalty for each x A n the support of P A such that player A attacks best-shot network j. From Lemma 5 part (c n the support of any optmal strategy player allocates a strctly postve level of force to at most one target n network j, and thus the support of player s n j -varate margnal dstrbuton for network j, P n j s contaned on the each of the n j axes n R n j. From Lemmas 2 and 4, t follows that equaton (1 holds wth equalty not only for each x A n the support of P A such that player A attacks best-shot network j, but gven that network j s the only network attacked for all n j -tuples x j [0, s j ]n j. That s gven that the support of player s n j -varate margnal dstrbuton for network j s: ( contaned on the each of the n j axes n R n j, ( has no mass ponts except possble at the orgn n R n j, and ( s contnuous on each axs, t follows that for x j A [0, s j ]n j, P n j (x j A = Pr(ι j = 1 P n j,x A = k A va + x A. Thus, f player A chooses the n j -tuple wth s j for each element then from Lemma 2 player A s expected payoff from such an n j -tuple s n j s j. From Lemmas 3 and 6, k A = s W. From Lemma 3, player A s expected payoff s constant across all ponts n the support of P A. Thus, j, s W = n j s j. Lemma 8 s W = mn{,v /[ j W n j + j (1/n j ]}. Proof If player allocates: ( s W to each target n each weakest-lnk network, ( s j to exactly one target n each best-shot network j, and ( 0 to each of the remanng targets n the best-shot networks, then player wns wth certanty and has an expected payoff of v j W n j s W + j ( s W /n j 0. Thus, n such a case t must be that s W v /[ j W n j + j (1/n j ]. Smlarly, player A s expected payoff s s W 0, and thus, s W. Snce player A attacks at most one network, and n the case of a weakest-lnk network only one target, we know that the orgn s contaned n the support of any equlbrum dstrbuton of force for player A, P A. y way of contradcton suppose that there exsts an equlbrum {P A,P } n whch the orgn s not contaned n the support of P. Thus, there exsts an ε > 0 such that for at least two targets, denoted as targets 1 and 2, the ntersecton of the projecton

24 of player s support onto the x 1,x 2 -plane wth the box [0,ε] 2 s empty. There are fve confguratons to consder: ( targets 1 and 2 are n the same weakest-lnk network, ( targets 1 and 2 are n separate weakest-lnk networks, ( targets 1 and 2 are n the same best-shot network, (v targets 1 and 2 are n separate best-shot networks, (v target 1 s n a weakest-lnk network and target 2 s n a best-shot network. From Lemma 5, t s clear that we can rule out case (. In cases ( and (, for any (x 1,x 2 [0,ε] 2 player s bvarate margnal dstrbuton for targets 1 and 2, P 1,2, s equal to zero and player A can strctly ncrease hs payoff by allocatng a level of force less than ε to both targets 1 and 2 a contradcton to Lemma 5 f targets 1 and 2 are n the same weakest-lnk network or f targets 1 and 2 are n separate weakest-lnk networks. Followng along smlar lnes, cases (v and (v lead to a smlar contradcton to Lemma 5. Thus, the orgn s contaned n the support of any equlbrum dstrbuton P for player. Snce only one player can have a mass pont at the orgn, we have that f s W > 0 player A must outbd player wth a probablty that s bounded away from zero. Thus, player places postve mass at the orgn, but f player has a mass pont at the orgn then t must be the case that v j W n j s W + j ( s W /n j = 0. Smlarly, f v j W n j s W + j ( s W /n j > 0 then player must outbd player A wth a probablty that s bounded away from zero. Thus, s W = 0 and player A places postve mass at the orgn. 24 The next two lemmas follow drectly from Lemma 8. Recall that α = v /( [ j W n j + j 1 n j ]. Lemma 9 If α 1, then ( player A places mass 1 (1/α at the orgn, ( player A s expected payoff s 0, ( player does not place postve mass at the orgn, and (v player s expected payoff s v (v /α. Lemma 10 If α < 1, then ( player places mass 1 α at the orgn, ( player s expected payoff s 0, ( player A does not place postve mass at the orgn, and (v player A s expected payoff s α. Lemma 11 There exsts a unque set of equlbrum unvarate margnal dstrbutons {{F j A } j W,{F j } j W }. Proof Ths proof s for the unqueness of player s set of unvarate margnal dstrbutons. The proof for player A s analogous. For each best-shot network j, from Lemma 7 for x j [0, s j ]n j, P n j (x j = s W + x, where s W = mn{,v /[ j W n j +

25 25 j (1/n j ]} and s j = s W n j. Thus, n each best-shot network j player s unque unvarate margnal dstrbutons follow from player s unque n j -varate margnal dstrbuton for network j. From Lemma 5 parts (a and (b, player A attacks at most one target n one weakestlnk network. From Lemmas 2, 3, and 4 t follows that for each target n each weakestlnk network j W, F ( x A x A = s W for x [0, s W ]. Thus, player s unvarate margnal dstrbutons are unquely determned n each weakest-lnk network.

26 26 A A C C E E Fg. 1 Example Supra-Network wth Fve Networks (A,, C,, and E

27 27 One weakest-lnk network wth two targets ( = 1,2 x 2 x 2 ṽ A ṽ A x 1 x 1 ṽ A ṽ A ( Attacker ( efender One best-shot network wth two targets ( = 1,2 and one weakest-lnk network wth one target ( = 3 x 2 x 2 ṽ A2 ṽ A2 x 1 x 1 ṽ A2 ṽ A2 ṽ A x 3 ( Attacker ṽ A x 3 (v efender Fg. 2 Supports of the equlbrum jont dstrbutons stated n Theorem 1 (ṽ A = mn{α, }.