Stackelberg vs. Nash in Security Games: Interchangeability, Equivalence, and Uniqueness

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1 Stackelberg vs. Nash n Securty Games: Interchangeablty, Equvalence, and Unqueness Zhengyu Yn 1, Dmytro Korzhyk 2, Chrstopher Kekntveld 1, Vncent Contzer 2, and Mlnd Tambe 1 1 Unversty of Southern Calforna, Los Angeles, CA 90089, USA {zhengyuy, kekntv, tambe}@usc.edu 2 Duke Unversty, Durham, NC 27708, USA {dma, contzer}@cs.duke.edu ABSTRACT There has been sgnfcant recent nterest n game theoretc approaches to securty, wth much of the recent research focused on utlzng the leader-follower Stackelberg game model; for example, these games are at the heart of major applcatons such as the ARMOR program deployed for securty at the LAX arport snce 2007 and the IRIS program n use by the US Federal Ar Marshals (FAMS). The foundatonal assumpton for usng Stackelberg games s that securty forces (leaders), actng frst, commt to a randomzed strategy; whle ther adversares (followers) choose ther best response after survellance of ths randomzed strategy. Yet, n many stuatons, the followers may act wthout observaton of the leader s strategy, essentally convertng the game nto a smultaneous-move game model. Prevous work fals to address how a leader should compute her strategy gven ths fundamental uncertanty about the type of game faced. Focusng on the complex games that are drectly nspred by realworld securty applcatons, the paper provdes four contrbutons n the context of a general class of securty games. Frst, explotng the structure of these securty games, the paper shows that the Nash equlbra n securty games are nterchangeable, thus allevatng the equlbrum selecton problem. Second, resolvng the leader s dlemma, t shows that under a natural restrcton on securty games, any Stackelberg strategy s also a Nash equlbrum strategy; and furthermore, the soluton s unque n a class of realworld securty games of whch ARMOR s a key exemplar. Thrd, when faced wth a follower that can attack multple targets, many of these propertes no longer hold. Fourth, our expermental results emphasze postve propertes of games that do not ft our restrctons. Our contrbutons have major mplcatons for the real-world applcatons. Categores and Subject Descrptors I.2.11 [Artfcal Intellgence]: Dstrbuted Artfcal Intellgence General Terms Theory Zhengyu Yn and Dmytro Korzhyk are both frst authors of ths paper. Cte as: Stackelberg vs. Nash n Securty Games: Interchangeablty, Equvalence, and Unqueness, Zhengyu Yn, Dmytro Korzhyk, Chrstopher Kekntveld, Vncent Contzer, and Mlnd Tambe, Proc. of 9th Int. Conf. on Autonomous Agents and Multagent Systems (AAMAS 2010),van der Hoek, Kamnka, Lespérance, Luck and Sen (eds.), May, 10 14, 2010, Toronto, Canada, pp Copyrght c 2010, Internatonal Foundaton for Autonomous Agents and Multagent Systems ( All rghts reserved. Keywords Stackelberg Equlbrum, Nash Equlbrum, Securty 1. INTRODUCTION There has been sgnfcant recent research nterest n game-theoretc approaches to securty at arports, ports, transportaton, shppng and other nfrastructure [12, 3, 4, 7]. Much of ths work has used a Stackelberg game framework to model nteractons between the securty forces and attackers. That s, the defender (.e., the securty forces) acts frst by commttng to a patrollng or nspecton strategy, and the attacker chooses where to attack after observng the defender s choce. The typcal soluton concept appled to these games s Strong Stackelberg Equlbrum (SSE), whch assumes that the defender wll choose an optmal mxed (randomzed) strategy based on the assumpton that the attacker wll observe ths strategy and choose an optmal response. Ths leader-follower paradgm appears to ft many real-world securty stuatons. Indeed, Stackelberg games are at the heart two major decson-support applcatons: the ARMOR program n use at the Los Angeles Internatonal Arport snce 2007 to randomze allocaton of checkponts and canne patrols [12], and the IRIS program n use by the US Federal Ar Marshals to randomze assgnments of ar marshals to flghts [13]. However, there are legtmate concerns about whether the Stackelberg model s approprate n all cases. In some stuatons attackers may choose to act wthout acqurng costly nformaton about the securty strategy, especally f securty measures are dffcult to observe (e.g., undercover offcers) and nsders are unavalable. In such cases, a smultaneous-move game model may be a better reflecton of the real stuaton. The defender faces an unclear choce about whch strategy to adopt: the recommendaton of the Stackelberg model, or of the smultaneous-move model, or somethng else entrely? In general settngs, the equlbrum strategy can n fact dffer between these models. Consder the followng game n normal form: c d a 2,1 4,0 b 1,0 3,1 Table 1: Example game where the Stackelberg Equlbrum s not a Nash Equlbrum If the row player has the ablty to commt, the SSE strategy s to play a wth.5 and b wth.5, so that the best response for the column player s to play d, whch gves the row player an expected utlty 1139

2 of On the other hand, f the players move smultaneously the only Nash Equlbrum (NE) of ths game s for the row player to play a and the column player c. Ths can be seen by notcng that b s strctly domnated for the row player. Prevous work has faled to resolve the defender s dlemma of whch strategy to select when the attacker s observaton capablty s unclear. We conduct theoretcal and expermental analyss of the leader s dlemma, focusng on securty games [7]. These are non-zero-sum games motvated by real-world securty domans, and are at the heart of applcatons such as ARMOR and IRIS [7, 12, 13]. We make four prmary contrbutons. Frst, we show that Nash equlbra are nterchangeable n securty games, avodng equlbrum selecton problems. Second, f the game satsfes the SSAS (Subsets of Schedules Are Schedules) property, the defender s set of SSE strateges s a subset of her NE strateges. In ths case, the defender s always playng a best response by usng an SSE regardless of whether the attacker observes the defender s strategy or not. Thrd, we provde counter-examples to ths (partal) equvalence n two cases: (1) when the SSAS property does not hold for defender schedules, and (2) when the attacker can attack multple targets smultaneously. In these cases, the defender s SSE strategy may not be part of any NE profle. Fnally, our expermental tests show that the fracton of games where the SSE strategy played s not part of any NE profle s vanshngly small. However, when attackers can attack multple targets a relatvely large number of games have dstnct SSE and NE strateges. 2. MOTIVATING DOMAINS We study qute general classes of securty games n ths work, but wth assumptons motvated by two real-world applcatons. The frst s the ARMOR securty system deployed at the Los Angeles Internatonal Arport (LAX) [12]. In ths doman polce are able to set up checkponts on roads leadng to partcular termnals, and assgn canne unts (bomb-snffng dogs) to patrol termnals. Polce resources n ths doman are homogeneous, and do not have sgnfcant schedulng constrants. IRIS s a smlar applcaton deployed by the Federal Ar Marshals Servce (FAMS) [13]. Armed marshals are assgned to commercal flghts to deter and defeat terrorst attacks. Ths doman has more complex constrants. In partcular, marshals are assgned to tours of flghts that return to the same destnaton, and the tours on whch any gven marshal s avalable to fly are lmted by the marshal s current locaton and tmng constrants. The types of schedulng and resource constrants we consder n ths work are motvated by those necessary to represent ths doman. Addtonally, there are many other potental securty applcatons, e.g., the Los Angeles Port doman, where port polce patrol docks to ensure the safety and securty of all passenger, cargo, and vessel operatons. 3. DEFINITIONS AND NOTATION A securty game [7] s a two-player game between a defender and an attacker. The attacker may choose to attack any target from the set T = {t 1,t 2,...,t n}. The defender tres to prevent attacks by coverng targets usng resources from the set R = {r 1,r 2,...,r K}. As shown n Fgure 1,U c d(t ) s the defender s utlty f t s attacked whle t s covered by some defender resource. If t s not covered, the defender gets U u d (t ). The attacker s utlty s denoted smlarly by U c a(t ) and U u a (t ). We use ΔU d (t ) = U c d(t ) 2 In these games t s assumed that f the follower s ndfferent, he breaks the te n the leader s favor (otherwse, the optmal soluton s not well defned). U u d (t ) to denote the dfference between defender s covered and uncovered utltes. Smlarly, ΔU a(t )=U u a (t ) U c a(t ). As a key property of securty games, we assume ΔU d(t ) > 0 and ΔU a(t ) > 0. In words, addng resources to cover a target helps the defender and hurts the attacker. Defender U c d (t ) U u d (t ) ΔU d (t ) > 0 ΔU a (t ) > 0 Not covered Covered Fgure 1: Payoff structure of securty games. Attacker U c a (t ) U u a (t ) Motvated by FAMS and smlar real-world domans, we ntroduce resource and schedulng constrants for the defender. Resources may be assgned to schedules coverng multple targets, s T. For each resource r, there s a subset S of the schedules S that resource r can potentally cover. That s, r can cover any s S. In the FAMS doman, flghts are targets and ar marshals are resources. Schedules capture the dea that ar marshals fly tours, and must return to a partcular startng pont. Heterogeneous resources can express addtonal tmng and locaton constrants that lmt the tours on whch any partcular marshal can be assgned to fly. An mportant subset of the FAMS doman can be modeled usng fxed schedules of sze 2 (.e., a par of departng and returnng flghts). The LAX doman s also a subclass of securty games as defned here, wth schedules of sze 1 and homogeneous resources. A securty game descrbed above can be represented as a normal form game, as follows. The attacker s pure strategy space A s the set of targets. The attacker s mxed strategy a = a s a vector where a represents the probablty of attackng t. The defender s pure strategy s a feasble assgnment of resources to schedules,.e., s K =1 S. Snce coverng a target wth one resource s essentally the same as coverng t wth any postve number of resources, the defender s pure strategy can also be represented by a coverage vector d = d {0, 1} n where d represents whether t s covered or not. For example, {t 1,t 4}, {t 2} can be a possble assgnment, and the correspondng coverage vector s 1, 1, 0, 1. However, not all the coverage vectors are feasble due to resource and schedule constrants. We denote the set of feasble coverage vectors by D {0, 1} n. The defender s mxed strategy C specfes the probabltes of playng each d D, where each ndvdual probablty s denoted by C d. Let c = c be the vector of coverage probabltes correspondng to C, where c = d D dcd s the margnal probablty of coverng t. For example, suppose the defender has two coverage vectors: d 1 = 1, 1, 0 and d 2 = 0, 1, 1. Then C =.5,.5 s one defender s mxed strategy, and the correspondng c =.5, 1,.5. Denote the mappng from C to c by ϕ, so that c = ϕ(c). If strategy profle C, a s played, the defender s utlty s U d (C, a) = whle the attacker s utlty s U a(c, a) = a (c Ud(t c )+(1 c )Ud u (t )), a (c Ua(t c )+(1 c )Ua u (t )). 1140

3 If the players move smultaneously, the standard soluton concept s Nash equlbrum. DEFINITION 1. A par of strateges C, a forms a Nash Equlbrum (NE) f they satsfy the followng: 1. The defender plays a best-response: U d(c, a) U d(c, a) C. 2. The attacker plays a best-response: U a(c, a) U a(c, a ) a. In our Stackelberg model, the defender chooses a mxed strategy frst, and the attacker chooses a strategy after observng the defender s choce. The attacker s response functon s g(c) :C a. In ths case, the standard soluton concept s Strong Stackelberg Equlbrum [8, 16]. DEFINITION 2. A par of strateges C,g forms a Strong Stackelberg Equlbrum (SSE) f they satsfy the followng: 1. The leader (defender) plays a best-response: U d (C,g(C)) U d (C,g(C )), for all C. 2. The follower (attacker) plays a best-response: U a(c,g(c)) U a(c,g (C)), for all C,g. 3. The follower breaks tes optmally for the leader: U d(c,g(c)) U d(c,τ(c)), for all C, where τ(c) s the set of follower best-responses to C. We denote the set of mxed strateges for the defender that are played n some Nash Equlbrum by Ω NE, and the correspondng set for Strong Stackelberg Equlbrum by Ω SSE. 4. EQUILIBRIA IN SECURITY GAMES The challenge for us s to understand the fundamental relatonshps between the SSE and NE strateges n securty games. A specal case s zero-sum securty games, where the defender s utlty s the exact opposte of the attacker s utlty. For fnte two-person zero-sum games, t s known that the dfferent game theoretc soluton concepts of NE, mnmax, maxmn and SSE all gve the same answer. In addton, Nash equlbrum strateges of zero-sum games have a very useful property n that they are nterchangeable: an equlbrum strategy for one player can be pared wth the other player s strategy from any equlbrum profle, and the result s an equlbrum, and the payoffs for both players reman the same. Unfortunately, securty games are not necessarly zero-sum (and are not zero-sum n deployed applcatons). Many propertes of zero-sum games do not hold n securty games. For nstance, a mnmax strategy n a securty game may not be a maxmn strategy. Consder the example n Table 2, n whch there are 3 targets and one defender resource. The defender has three actons; each of defender s actons can only cover one target at a tme, leavng the other targets uncovered. Whle all three targets are equally appealng to the attacker, the defender has varyng utltes of capturng the attacker at dfferent targets. For the defender, the unque mnmax strategy, 1/3, 1/3, 1/3, s dfferent from the unque maxmn strategy, 6/11, 3/11, 2/11. Strategcally zero-sum games [10] are a natural and strct superset of zero-sum games for whch most of the desrable propertes of zero-sum games stll hold. Ths s exactly the class of games for whch no completely mxed Nash equlbrum can be mproved upon. Mouln and Val proved a game (A, B) s strategcally zero-sum f and only f there exst u>0 and v>0 such that t 1 t 2 t 3 C U C U C U Def Att Table 2: Securty game whch s not strategcally zero-sum ua + vb = U + V, where U s a matrx wth dentcal columns and V s a matrx wth dentcal rows [10]. Unfortunately, securty games are not even strategcally zero-sum. The game n Table 2 s a counterexample, because otherwse there must exst u, v > 0 such that, u v = a a a b b b + x y z x y z c c c x y z From these equatons, a + y = a + z = b + x = b + z = c + x = c + y = v, whch mples x = y = z and a = b = c. We also know a + x = u, b + y =2u, c + z =3u. However snce a + x = b + y = c + z, u must be 0, whch contradcts the assumpton u>0. Nevertheless, we show n the rest of ths secton that securty games stll have some mportant propertes. We start by establshng equvalence between the set of defender s mnmax strateges and the set of defender s NE strateges. Second, we show Nash equlbra n securty games are nterchangeable, resolvng the defender s equlbrum strategy selecton problem n smultaneousmove games. Thrd, we show that under a natural restrcton on schedules, any SSE strategy for the defender s also a mnmax strategy and hence an NE strategy. Ths resolves the defender s dlemma about whether to play accordng to SSE or NE when there s uncertanty about attacker s ablty to observe the strategy. Fnally, for a restrcted class of games (ncludng the games from the LAX doman), we fnd that there s a unque SSE/NE defender strategy and a unque attacker NE strategy. 4.1 Equvalence of NE and Mnmax We frst prove that any defender s NE strategy s also a mnmax strategy. Then for every defender s mnmax strategy C we construct a strategy a for the attacker such that C, a s an NE profle. DEFINITION 3. For a defender s mxed strategy C, defne the attacker s best response utlty by E(C) = max n =1 U a(c,t ). Denote the mnmum of the attacker s best response utltes over all defender s strateges by E = mn C E(C). The set of defender s mnmax strateges s defned as: Ω M = {C E(C) =E }. We defne the functon f as follows. If a s an attacker s strategy n whch target t s attacked wth probablty a, then f(a) =ā s an attacker s strategy such that ΔU d (t ) ā = λa ΔU a(t ) where λ>0s a normalzng constant such that n ā =1. The nverse functon f 1 (ā) =a s gven by the followng equaton. a = 1 ΔU a(t ) λ ā (1) ΔU d(t ) 1141

4 LEMMA 4.1. Consder a securty game G. Construct the correspondng zero-sum securty game Ḡ n whch the defender s utltes are re-defned as follows. Ud(t) c = Ua(t) c Ud u (t) = Ua u (t) Then C, a s an NE profle n G f and only f C,f(a) s an NE profle n Ḡ. PROOF. Note that the supports of strateges a and ā are the same, and also that the attacker s utlty functon s the same n games G and Ḡ. Thus a s a best response to C n G f and only f ā s a best response to C n Ḡ. Denote the utlty that the defender gets f profle C, a s played n game G by U G d (C, a). To show that C s a best response to a n game G f and only f C s a best response to ā n Ḡ, t s suffcent to show equvalence of the followng two nequaltes. U G d (C, a) U G d (C, a) 0 U Ḡ d (C, ā) U Ḡ d (C, ā) 0 We wll prove the equvalence by startng from the frst nequalty and transformng t nto the second one. On the one hand, we have, U G d (C, a) U G d (C, a) = a (c c )ΔU d (t ). Smlarly, on the other hand, we have, U Ḡ d (C, ā) U Ḡ d (C, ā) = ā (c c )ΔU a(t ). Gven Equaton (1) and λ>0, wehave, U G d (C, a) U G d (C, a) 0 a (c c )ΔU d (t ) 0 1 ΔU a(t ) λ ā ΔU d (t ) (c c )ΔU d (t ) 0 1 ā (c c )ΔU a(t ) 0 λ 1 ( ) U Ḡ d (C, ā) U Ḡ d (C, ā) 0 λ U Ḡ d (C, ā) U Ḡ d (C, ā) 0 LEMMA 4.2. Suppose C s a defender NE strategy n a securty game. Then E(C) =E,.e., Ω NE Ω M. PROOF. Suppose C, a s an NE profle n the securty game G. Accordng to Lemma 4.1, C,f(a) must be an NE profle n the correspondng zero-sum securty game Ḡ. Snce C s an NE strategy n a zero-sum game, t must also be a mnmax strategy [5]. Thus E(C) =E. LEMMA 4.3. In a securty game G, any defender s strategy C such that E(C) =E s an NE strategy,.e., Ω M Ω NE. PROOF. C s a mnmax strategy n both G and the correspondng zero-sum game Ḡ. Any mnmax strategy s also an NE strategy n a zero-sum game [5]. Then there must exst an NE profle C, ā n Ḡ. By Lemma 4.1, C,f 1 (ā) s an NE profle n G. Thus C s an NE strategy n G. THEOREM 4.4. In a securty game, the set of defender s mnmax strateges s equal to the set of defender s NE strateges,.e., Ω M =Ω NE. PROOF. Lemma 4.2 shows that every defender s NE strategy s a mnmax strategy, and Lemma 4.3 shows that every defender s mnmax strategy s an NE strategy. Thus the sets of defender s NE and mnmax strateges must be equal. 4.2 Interchangeablty of Nash Equlbra We now show that Nash Equlbra n securty games are nterchangeable. THEOREM 4.5. Suppose C, a and C, a are two NE profles n a securty game G. Then C, a and C, a are also NE profles n G. PROOF. Consder the correspondng zero-sum game Ḡ. From Lemma 4.1, both C,f(a) and C,f(a ) must be NE profles n Ḡ. By the nterchange property of NE n zero-sum games [5], C,f(a ) and C,f(a) must also be NE profles n Ḡ. Applyng Lemma 4.1 agan n the other drecton, we get that C, a and C, a must be NE profles n G. By Theorem 4.5, the defender s equlbrum selecton problem n a smultaneous-move securty game s resolved. The reason s that gven the attacker s NE strategy a, the defender must get the same utlty by respondng wth any NE strategy. Next, we gve some nsghts on expected utltes n NE profles. We frst show the attacker s expected utlty s the same n all NE profles, followed by an example demonstratng that the defender may have varyng expected utltes correspondng to dfferent attacker s strateges. THEOREM 4.6. Suppose C, a s an NE profle n a securty game. Then, U a(c, a) =E. PROOF. From Lemma 4.2, C s a mnmax strategy and E(C) = E. On the one hand, U a(c, a) = a U a(c,t ) a E(C) =E. On the other hand, because a s a best response to C, t should be at least as good as the strategy of attackng t arg max t U a(c,t) wth probablty 1, that s, U a(c, a) U a(c,t )=E(C) =E. Therefore we know U a(c, a) =E. Unlke the attacker who gets the same utlty n all NE profles, the defender may get varyng expected utltes dependng on the attacker s strategy selecton. Consder the game shown n Table 3. The defender can choose to cover one of the two targets at a tme. The only defender s NE strategy s to cover t 1 wth 100% probablty, makng the attacker ndfferent between attackng t 1 and t 2. One attacker s NE response s always attackng t 1, whch gves the defender an expected utlty of 1. Another attacker s NE strategy s 2/3, 1/3, gven whch the defender s ndfferent between defendng t 1 and t 2. In ths case, the defender s utlty decreases to 2/3 because she captures the attacker wth a lower probablty. 4.3 SSE and Mnmax / NE We have already shown that the set of defender s NE strateges concdes wth her mnmax strateges. If every defender s SSE strategy s also a mnmax strategy, then SSE strateges must also be NE strateges. The defender can then safely commt to an SSE 1142

5 t 1 t 2 C U C U Def Att Table 3: A securty game where the defender s expected utlty vares n dfferent NE profles strategy; there s no selecton problem for the defender. Unfortunately, f a securty game has arbtrary schedulng constrants, then an SSE strategy may not be part of any NE profle. For example, consder the game n Table 4 wth 4 targets {t 1,...,t 4}, 2 schedules s 1 = {t 1,t 2}, s 2 = {t 3,t 4}, and a sngle defender resource. The defender always prefers that t 1 s attacked, and t 3 and t 4 are never appealng to the attacker. t 1 t 2 t 3 t 4 C U C U C U C U Def Att Table 4: A schedule-constraned securty game where the defender s SSE strategy s not an NE strategy. There s a unque SSE strategy for the defender, whch places as much coverage probablty on s 1 as possble wthout makng t 2 more appealng to the attacker than t 1. The rest of the coverage probablty s placed on s 2. The result s that s 1 and s 2 are both covered wth probablty 0.5. In contrast, n a smultaneous-move game, t 3 and t 4 are domnated for the attacker. Thus, there s no reason for the defender to place resources on targets that are never attacked, so the defender s unque NE strategy covers s 1 wth probablty 1. That s, the defender s SSE strategy s dfferent from the NE strategy. The dfference between the defender s payoffs n these cases can also be arbtrarly large because t 1 s always attacked n an SSE and t 2 s always attacked n a NE. The above example restrcts the defender to protect t 1 and t 2 together, whch makes t mpossble for the defender to put more coverage on t 2 wthout makng t 1 less appealng. If the defender could assgn resources to any subset of a schedule, ths dffculty s resolved. More formally, we assume that for any resource r,any subset of a schedule n S s also a possble schedule n S : 1 K : s s S s S. (2) If a securty game satsfes Equaton (2), we say t has the SSAS property. Ths s natural n many securty domans, snce t s often possble to cover fewer targets than the maxmum number that a resource could possble cover n a schedule. We fnd that ths property s suffcent to ensure that the defender s SSE strategy must also be an NE strategy. LEMMA 4.7. Suppose C s a defender strategy n a securty game whch satsfes the SSAS property and c = ϕ(c) s the correspondng vector of margnal probabltes. Then for any c such that 0 c c for all t T, there must exst a defender strategy C such that ϕ(c )=c. PROOF. The proof s by nducton on the number of t where c c, as denoted by δ(c, c ). As the base case, f there s no such that c c, the exstence trvally holds because ϕ(c) =c. Suppose the exstence holds for all c, c such that δ(c, c )=k, where 0 k n 1. We consder any c, c where δ(c, c )= k +1. Then for some j, c j c j. Snce c j 0 and c j <c j,we have c j > 0. There must be a nonempty set of coverage vectors D j that cover t j and receve postve probablty n C. Because the securty game satsfes the SSAS property, for every d D j, there s a vald d whch covers all targets n d except for t j. From the defender strategy C, by shftng C d(c j c j ) c j probablty from every d D j to the correspondng d, we get a defender strategy C where c = c for j, and c = c for = j. Hence δ(c, c )=k, mplyng there exsts a C such that ϕ(c )=c by the nducton assumpton. By nducton, the exstence holds for any c, c. THEOREM 4.8. Suppose C s a defender SSE strategy n a securty game whch satsfes the SSAS property. Then E(C) =E,.e., Ω SSE Ω M =Ω NE. PROOF. The proof s by contradcton. Suppose C,g s an SSE profle n a securty game whch satsfes the SSAS property, and E(C) > E. Let T a = {t U a(c,t ) = E(C)} be the set of targets that gve the attacker the maxmum utlty gven the defender strategy C. By the defnton of SSE, we have U d (C,g(C)) = max U d (C,t ). t T a Consder a defender mxed strategy C such that E(C )=E. Then for any t T a, U a(c,t ) E. Consder a vector c : c c E U a(c,t )+ɛ, t = Ua u (t ) Ua(t c T a, (3a) ) c, t / T a, (3b) where ɛ s an nfntesmal postve number. Snce E U a(c,t )+ ɛ>0, wehavec <c for all t T a. On the other hand, snce for all t T a, U a(c,t )=E + ɛ<e(c) =U a(c,t ), we have c >c 0. Then for any t T,wehave0 c c. From Lemma 4.7, there exsts a defender strategy C correspondng to c. The attacker s utlty of attackng each target s as follows: { E U a(c + ɛ, t T a, (4a),t )= U a(c,t ) E, t / T a. (4b) Thus, the attacker s best responses to C are stll T a. For all t T a, snce c >c, t must be the case that U d(c,t ) <U d(c,t ). By defnton of attacker s SSE response g, wehave, U d(c,g(c )) = max U d(c,t ) t T a > max U d (C,t )=U d (C,g(C)). t T a It follows that the defender s better off usng C, whch contradcts the assumpton C s an SSE strategy of the defender. Theorem 4.4 and 4.8 together mply the followng corollary. COROLLARY 4.9. In securty games wth the SSAS property, any defender s SSE strategy s also an NE strategy. We can now answer the orgnal queston posed n ths paper: when there s uncertanty over the type of game played, should the defender choose an SSE strategy or a mxed strategy Nash equlbrum or some combnaton of the two? For domans that satsfy the SSAS property, we have proven that any of the defender s SSE strateges s also an NE strategy. 1143

6 Among our motvatng domans, the LAX doman satsfes the SSAS property snce all schedules are of sze 1. Other patrollng domans, such as patrollng a port, also satsfy the SSAS property. In such domans, the defender could thus commt to an SSE strategy, whch s also now known to be an NE strategy. The defender retans the ablty to commt, but s stll playng a best-response to an attacker n a smultaneous-move settng (assumng the attacker plays an equlbrum strategy t does not matter whch one, due to the nterchange property shown above). However, the FAMS doman does not naturally satsfy the SSAS property because marshals must fly complete tours (though n prncple they could fly as cvlans on some legs of a tour). The queston of selectng SSE vs. NE strateges n ths case s addressed expermentally n Secton Unqueness n Restrcted Games The prevous sectons show that SSE strateges are NE strateges n many cases. However, there may stll be multple equlbra to select from (though ths dffculty s allevated by the nterchange property). Here we prove an even stronger unqueness result for an mportant restrcted class of securty domans, whch ncludes the LAX doman. In partcular, we consder securty games where the defender has homogeneous resources that can cover any sngle target. The SSAS property s trvally satsfed, snce all schedules are of sze 1. Any vector of coverage probabltes c = c such that n c K s a feasble strategy for the defender, so we can represent the defender strategy by margnal coverage probabltes. Wth a mnor restrcton on the attacker s payoff matrx, the defender always has a unque mnmax strategy whch s also the unque SSE and NE strategy. Furthermore, the attacker also has a unque NE response to ths strategy. THEOREM In a securty game wth homogeneous resources that can cover any sngle target, f for every target t T, Ua(t c ) E, then the defender has a unque mnmax, NE, and SSE strategy. PROOF. We frst show the defender has a unque mnmax strategy. Let T = {t Ua u (t) E }. Defne c = c as c = Ua u (t ) E Ua u (t ) Ua(t, c ) t T, (5a) 0, t / T. (5b) Note that E cannot be less than any Ua(t c ) otherwse, regardless of the defender s strategy, the attacker could always get at least Ua(t c ) >E by attackng t, whch contradcts the fact that E s the attacker s best response utlty to a defender s mnmax strategy. Snce E Ua(t c ) and we assume E Ua(t c ), 1 c = E Ua(t c ) Ua u (t ) Ua(t > 0 c ) c < 1. Next, we wll prove n c K. For the sake of contradcton, suppose n c <K. Let c = c, where c = c + ɛ. Snce c < 1 and n c <K, we can fnd ɛ>0such that c < 1 and n c <K. Then every target has strctly hgher coverage n c than n c, hence E(c ) <E(c )=E, whch contradcts the fact that E s the mnmum of all E(c). Next, we show that f c s a mnmax strategy, then c = c. By the defnton of a mnmax strategy, E(c) = E. Hence, U a(c,t ) E c c. On the one hand n c K and on the other hand n c n c K. Therefore t must be the case that c = c for any. Hence, c s the unque mnmax strategy of the defender. Furthermore, by Theorem 4.4, we have that c s the unque defender s NE strategy. By Theorem 4.8 and the exstence of SSE [2], we have that c s the unque defender s SSE strategy. THEOREM In a securty game wth homogeneous resources that can cover any one target, f for every target t T, Ua(t c ) E and Ua u (t ) E, then the attacker has a unque NE strategy. PROOF. c and T are the same as n the proof of Theorem Gven the defender s unque NE strategy c, n any attacker s best response, only t T can be attacked wth postve probablty, because, { E U a(c t T (6a),t )= Ua u (t ) <E t / T (6b) Suppose c, a forms an NE profle. We have a =1 (7) t T For any t T, we know from the proof of Theorem 4.10 that c < 1. In addton, because Ua u (t) E,wehavec 0. Thus we have 0 < c < 1 for any t T. For any t,t j T, necessarly a ΔU d (t ) = a jδu d (t j). Otherwse, assume a ΔU d (t ) >a jδu d (t j). Consder another defender s strategy c where c = c + ɛ<1, c j = c j ɛ>0, and c k = c k for any k, j. U d(c, a) U d(c, a) =a ɛδu d(t ) a jɛδu d(t j) > 0 Hence, c s not a best response to a, whch contradcts the assumpton that c, a s an NE profle. Therefore, there exsts β > 0 such that, for any t T, a ΔU d (t )=β. Substtutng a wth β/δu d (t ) n Equaton (7), we have 1 β = 1 ΔU t T d (t ) Then we can explctly wrte down a as β a = ΔU, d(t ) t T, (8a) 0, t / T. (8b) As we can see, a defned by (8a) and (8b) s the unque attacker NE strategy. The mplcaton of Theorem 4.10 and Theorem 4.11 s that n the smultaneous-move game, both the defender and the attacker have a unque NE strategy, whch gves each player a unque expected utlty as a result. 5. MULTIPLE ATTACKER RESOURCES To ths pont we have assumed that the attacker wll attack exactly one target. We now extend our securty game defnton to allow the attacker to use multple resources to attack multple targets smultaneously. To keep the model smple, we assume homogeneous resources (for both players) and schedules of sze 1. The defender has K < n resources whch can be assgned to protect any target, and the attacker has L < n resources whch can be used to attack any target. Attackng the same target wth multple resources s equvalent to attackng wth a sngle resource. The defender s pure strategy s a coverage vector d = d D, where d {0, 1} represents whether t s covered or not. Smlarly, the attacker s pure strategy s an attack vector q = q Q.Wehave n d = K and n q = L. If pure strateges d, q are played, the attacker gets a utlty of U a(d, q) = q (d Ua(t c )+(1 d )Ua u (t )) 1144

7 whle the defender s utlty s gven by U d (d, q) = q (d Ud(t c )+(1 d )Ud u (t )) The defender s mxed strategy s a vector C whch specfes the probablty of playng each d D. Smlarly, the attacker s mxed strategy A s a vector of probabltes correspondng to all q Q. In securty games wth multple attacker resources, the defender s SSE strategy may not be part of any NE profle, even f there are no schedulng constrants. Consder the game shown n Table 5. t 1 t 2 t 3 C U C U C U Def ɛ 0 0 ɛ Att 100 ɛ ɛ 5 Table 5: A securty game wth multple attacker resources where the defender s SSE strategy s not an NE strategy. There are 3 targets t 1,t 2,t 3. The defender has 1 resource, and the attacker has 2 resources. Therefore the defender s pure strategy space s the set of targets to protect: {t 1,t 2,t 3}, whle the attacker s pure strategy space conssts of the pars of targets: { t 1,t 2, t 1,t 3, t 2,t 3 }. If the defender protects t 1 and the attacker attacks t 1,t 2, the defender s utlty s U c d(t 1)+U u d (t 2)= 100 ɛ and the attacker s utlty s U c a(t 1)+U u a (t 2) = 110 ɛ. In ths example, t 1 s very appealng to the attacker no matter f t s covered or not, so t 1 s always attacked. If t 2 s attacked, the defender gets a very low utlty, even f t 2 s defended. So n the SSE, the defender wants to make sure that t 2 s not attacked. The defender s SSE strategy places at least.5 probablty on t 2, so that t 1 and t 3 are attacked nstead of t 2 (recall that the attacker breaks tes n the defender s favor n an SSE). The attacker s SSE response s A = 0, 1, 0,.e., to always attack t 1 and t 3. The other.5 defense probablty wll be placed on t 1 because ΔU d (t 1) > ΔU d (t 3). So, the SSE profle s C, A, where C =.5,.5, 0. Next, we show that there s no NE n whch the defender plays C. Suppose there s an NE profle C, A. Gven C, the attacker s utlty for attackng t 1 s hgher than the utlty for attackng t 2, so t must be that t 1 s always attacked n ths NE. Therefore, the attacker never plays t 2,t 3. However, ths mples that t 1 s the most appealng target for the defender to cover, because U d(t 1, A) >U d(t, A), {2, 3}. So, to be a best response the coverage of t 1 would need to be 1 nstead of 0.5, contradctng the assumpton that C s an equlbrum strategy for the defender. 6. EXPERIMENTAL RESULTS Whle our theoretcal results resolve the leader s dlemma for many nterestng classes of securty games, as we have seen, there are stll some cases where SSE strateges are dstnct from NE strateges for the defender. One case s when securty games do not satsfy the SSAS property, and another s when the attacker has multple resources. We conduct experments to further nvestgate these two cases, offerng evdence about the frequency wth whch SSE strateges dffer from all NE strateges across randomly generated games usng 36 dfferent parameter settngs. For a partcular game nstance we frst compute an SSE strategy C usng the DOBSS mxed-nteger lnear program [12]. We then use the lnear feasblty program below to determne whether or not ths SSE strategy s part of some NE profle by attemptng to fnd an approprate attacker response strategy. A q [0, 1], for all q Q (9) A q =1 (10) q Q A q =0, for all U a(q, C) <E(C) (11) A qu d(d, q) Z, for all d D (12) q Q A qu d (d, q) =Z, for all d D wth C d > 0 (13) q Q Here Q s the set of attacker pure strateges, whch s just the set of targets when there s only 1 attacker resource. The probablty that the attacker plays q s denoted by A q whch must be between 0 and 1 (Constrant (9)). Constrant (10) forces the probabltes to sum to 1. Constrant (11) prevents the attacker from placng postve probabltes on pure strateges whch gve the attacker a utlty less than the best response utlty E(C). In constrants (12) and (13), Z s a varable whch represents the maxmum expected utlty the defender can get among all pure strateges gven the attacker s strategy A, and C d denotes the probablty of playng d n C. These two constrants requre the defender s strategy C to be a best response to the attacker s mxed strategy. Therefore a feasble soluton A s an NE strategy for the attacker. Conversely, f C, A s an NE profle, A must satsfy all of the LP constrants. We frst test sngle-attacker games, fxng the number of targets at 10 and the number of defender resources at 3. We vary the number of schedules, the sze of the schedules, and the number of resource types. Each test set conssts of games, wth payoffs drawn from U[ 100, 0] for U u d (t ) and U c a(t ), and U[0, 100] for U c d(t ) and U u a (t ) Table 6 summarzes our results. A column represents a number of schedules, and a row represents a par of schedule sze and number of resource types. For example, lookng at row 2 and column 2, we see that among games wth 5 schedules of sze 2 and 1 resource type, there are 316 cases where the defender s SSE strategy s not an NE strategy. The number of cases where the defender s SSE strategy s not an NE strategy s never more than 10.5% n any of the 36 settngs we tested. Ths number decreases as we ncrease the number of schedules. Wth 20 avalable schedules, the number s less than 2%. The man mplcaton of these results s that n practce, commttng to an SSE strategy s lkely to be a good approach n almost all cases. Ths s partcularly true n domans lke FAMS where schedule szes are relatvely small (2 n most cases) and the number of possble schedules s large relatve to the number of targets S/1R S/2R S/3R S/1R S/2R S/3R S/1R S/2R S/3R Table 6: Number of nstances out of sngle-attacker securty games where SSE s not NE Table 7 shows the results for varyng numbers of attacker re- 1145

8 sources. Agan, each set has games. As the number of attacker resources ncreases, the number of cases where the defender s SSE strategy s not an NE strategy ncreases. Wth 2, 3, and 4 attacker resources, the numbers are 27%, 54%, and 69% respectvely, whch mples the defender cannot smply play an SSE strategy when there are multple attacker resources. Ths result poses an nterestng drecton for future work, snce t s unclear how a defender should play n these games f the attacker s ablty to observe the mxed strategy s uncertan #SSE NE Table 7: Number of nstances out of multple-attacker securty games where SSE s not NE 7. SUMMARY AND RELATED WORK There has been sgnfcant nterest n understandng the nteracton of observablty and commtment n general Stackelberg games. Bagwell s early work [1] questons the value of commtment to pure strateges gven nosy observatons by followers; but the ensung and on-gong debate llustrated that the leader retans her advantage n case of commtment to mxed strateges [14, 6]. The value of commtment for the leader when observatons are costly s also studed n [9]. In contrast wth ths research, our work focuses on real-world securty games, llustratng subset, equvalence, nterchangeablty, and unqueness propertes that are non-exstent n general Stackelberg games studed prevously. Pta et al. [11] provde expermental results on observablty n Stackelberg games: they test a varety of defender strateges aganst human players (attackers) who choose ther optmal attack when provded wth lmted observatons of defender strategy. Results show the superorty of a defender s strategy computed assumng human anchorng bas n attrbutng probablty dstrbuton over the defender s actons. Ths research complements ours, whch provdes new mathematcal foundatons. Testng the nsghts of our research wth the expermental paradgm of [11], wth expert players s an nterestng topc for future research. Gong back to the foundatons of game theory, Von Neumman and Morgenstern [15] provded a key result on nterchangeablty: for two-player zero-sum games, any combnaton of players maxmn strateges s n equlbrum. However, our securty games are nether zero-sum nor strategcally zero-sum (as seen earler). To summarze, ths paper s focused on a general class of defenderattacker Stackelberg games that are drectly nspred by real-world securty applcatons. The paper confronts fundamental questons of how a defender should compute her mxed strategy. In ths context, ths paper provdes four key contrbutons. Frst, explotng the structure of these securty games, the paper shows that the Nash equlbra n securty games are nterchangeable, thus allevatng the defender s equlbrum selecton problem for smultaneousmove games. Second, resolvng the defender s dlemma, t shows that under the SSAS restrcton on securty games, any Stackelberg strategy s also a Nash equlbrum strategy; and furthermore, ths strategy s unque n a class of real-world securty games of whch ARMOR s a key exemplar. Thrd, when faced wth a follower that can attack multple targets, many of these propertes no longer hold, provdng a key drecton for future research. Fourth, our expermental results emphasze postve propertes of securty games that do not ft the SSAS property. In practcal terms, these contrbutons mply that defenders n applcatons such as ARMOR [12] and IRIS [13] can smply commt to SSE strateges, thus helpng to resolve a major dlemma n real-world securty applcatons. 8. ACKNOWLEDGEMENTS Ths research was supported by the Unted States Department of Homeland Securty through the Center for Rsk and Economc Analyss of Terrorsm Events (CREATE). Korzhyk and Contzer are supported by NSF IIS , ARO CI, and an Alfred P. Sloan Research Fellowshp. However, any opnons, conclusons or recommendatons heren are solely those of the authors and do not necessarly reflect vews of the fundng agences. We thank Ronald Parr for detaled comments and dscussons. 9. REFERENCES [1] K. Bagwell. Commtment and observablty n games. Games and Economc Behavor, 8: , [2] T. Basar and G. J. Olsder. Dynamc Noncooperatve Game Theory. Academc Press, San Dego, CA, 2nd edton, [3] N. Baslco, N. Gatt, and F. Amgon. Leader-follower strateges for robotc patrollng n envronments wth arbtrary topologes. In AAMAS-09, [4] V. Contzer and T. Sandholm. Computng the optmal strategy to commt to. In ACM EC-06, pages 82 90, [5] D. Fudenberg and J. Trole. Game Theory. MIT Press, October [6] S. Huck and W. Müller. Perfect versus mperfect observablty an expermental test of Bagwell s result. Games and Economc Behavor, 31(2): , [7] C. Kekntveld, M. Jan, J. Tsa, J. Pta, M. Tambe, and F. Ordonez. Computng optmal randomzed resource allocatons for massve securty games. In AAMAS-09, [8] G. Letmann. On generalzed Stackelberg strateges. Optmzaton Theory and Applcatons, 26(4): , [9] J. Morgan and F. Vardy. The value of commtment n contests and tournaments when observaton s costly. Games and Economc Behavor, 60(2): , August [10] H. Mouln and J. P. Val. Strategcally zero-sum games: The class of games whose completely mxed equlbra cannot be mproved upon. Internatonal Journal of Game Theory, 7(3-4): , September [11] J. Pta, M. Jan, F. Ordóñez, M. Tambe, S. Kraus, and R. Magor-Cohen. Effectve solutons for real-world Stackelberg games: When agents must deal wth human uncertantes. In AAMAS-09, [12] J. Pta, M. Jan, C. Western, C. Portway, M. Tambe, F. Ordonez, S. Kraus, and P. Parachur. Deployed ARMOR protecton: The applcaton of a game-theoretc model for securty at the Los Angeles Internatonal Arport. In AAMAS-08 (Industry Track), [13] J. Tsa, S. Rath, C. Kekntveld, F. Ordóñez, and M. Tambe. IRIS a tool for strategc securty allocaton n transportaton networks. In AAMAS-09 (Industry Track), [14] E. van Damme and S. Hurkens. Games wth mperfectly observable commtment. Games and Economc Behavor, 21(1-2): , [15] J. von Neumann and O. Morgenstern. Theory of Games and Economc Behavor. Prnceton Unversty Press, May [16] B. von Stengel and S. Zamr. Leadershp wth commtment to mxed strateges. Techncal Report LSE-CDAM , CDAM Research Report,