Outbidding as Deterrence: Endogenous Demands in the Shadow of Group Competition

Transcription

1 Outbiddig as Deterrece: Edogeous Demads i the Shadow of Group Competitio William Spaiel August 5, 206 Abstract The theory of outbiddig states that terrorist, isurget, ad rebel groups use violece to capture a greater share of their audiece s resources. I argue that oppoets of these groups should aticipate this dyamic, which potetially alters their demads. Although a seemigly obvious implicatio of outbiddig is that violece icreases as the umber of groups (ad thus competitio) icreases, I show that this may or may ot hold if the demad is edogeous. This is because targets, fearig group competitio, limit their aims. The results help explai empirical icosistecies regardig outbiddig. Usig comparative statics from the model, I the discuss the challeges to makig valid empirical ifereces regardig outbiddig. Assistat Professor, Departmet of Political Sciece, Uiversity of Pittsburgh (williamspaiel@gmail.com,

2 Itroductio The theory of outbiddig states that terrorist, isurget, ad rebel groups use violece as a perverse form of marketig to attract ad maitai recruits ad doatios to their orgaizatios (Creshaw 985; Horowitz 985; Oots 989; Bloom 2004; Kydd ad Walter 2006; McCauley ad Moskaleko 2008). Such orgaizatios eed labor ad fudig to operate. However, such resources are scarce, meaig the mapower ad fuds set to oe orgaizatio must also fuel away those assets from other groups. Accordig to outbiddig theory, this competitio forces orgaizatios to commit more attacks, sigal their ability as a superior orgaizatio, ad receive more of these resources (Bloom 2004; Cheoweth 200). Recogizig that terrorism affects coercive demads (Atkiso, Sadler, ad Tschirhart 987; Lapa ad Sadler 988; Bapat 2006; Bapat 204), I explore the broader implicatios of such a mechaism. To wit, cosider the the icetives a target state faces i the shadow of outbiddig. Issuig bold policy declaratios ad expadig territorial demads risks icreasig the supply of terrorists. Give icetives to outbid, that state may the suffer may attacks. Exceptioally violet outbiddig may therefore have a pacifyig effect by covicig the target to moderate its aims. Thus, more groups may lead to greater deterrece ad ultimately less violece. As such, I ivestigate which way the effect actually cuts. To aswer this questio, I develop a formal model of policy demads, violet attacks, ad outbiddig, featurig a target state, multiple competig orgaizatios, ad a pool of citizes. The target state begis by choosig the portio of a policy space it wishes to capture. This policy may represet the extet of Israeli ecroachmet ito Palestiia-claimed territories, America expasio ito the Middle East, or a govermet s restrictios o a miority group. The state prefers capturig a larger share of the policy for itself, but it must also worry that extreme demads radicalize more citizes. Upo observig the state s demad, violece-producig orgaizatios select the level of itesity of attacks to advertise their services to a pool of citizes. Those citizes the choose which orgaizatio to joi, if ay. Throughout the paper, I ofte discuss these groups as terrorists because the literature teds to focus o such orgaizatios. I do ot mea to imply ay ormative judgmets by usig this term. Rather, the model covers ay situatio i which groups use costly violece o a target as a method to icrease recruitmet.

3 Whether more groups implies more violece depeds o a seemigly uimportat detail. Ideed, the shape of the distributio of citize prefereces determies how the umber of groups relates to violece. If citizes grow radicalized at a sufficietly icreasigly rapid rate as the state demads a greater share of the policy, the straightforward effect holds more terrorist groups imply more terrorist attacks. However, if citizes are especially sesitive to iitial ecroachmets, the deterrece effect domiates more terrorist groups ca couterituitively imply fewer terrorist attacks. Why is the shape of the distributio of citize prefereces pivotal? I brief, whe citizes are sesitive to iitial ecroachmets, a target state s demad becomes a allor-othig affair. That is, if it is worth sufferig the great pai to demad the first bit of the policy i dispute, the it is worth demadig all of the good. Thus, the pool of support the groups draw from is either large or o-existet. The outbiddig logic meas that group competitio drives high levels of violece i the former case, ad the violece grows worse as the umber of groups icreases. I tur, if the umber of groups crosses a critical threshold, the target state switches from demadig everythig to demadig othig. The correspodig loss of potetial support reduces violece. The model has importat implicatios for the empirical study of outbiddig. Some scholars have assumed that outbiddig implies a mootoic relatioship betwee the umber of groups ad observed violece (Fidley ad Youg 202; Stato 203; Forta 205). They the fail to ucover this relatioship with large- aalysis, drawig the coclusio that the historical record is ot cosistet with outbiddig o a broad scale. Yet other scholars (Clauset et al 200; Nemeth 204; Jaeger et al 205) recover the relatioship. 2 For the outbiddig literature to progress, scholars ought to address the discrepacy. By brigig itegratig target state ito the strategic discussio, my model offers a explaatio for the icosistet empirical results. I show that assumig a positive, mootoic relatioship betwee the umber of groups ad violece is ujustified without cotrollig for the shape of the market of support. Ideed, certai market shapes lead to egative, o-mootoic, ad zero relatioships betwee violece ad umber of groups. I tur, it is uclear what theoretical purpose simple cotrols for groups serve i regressios. The model reveals two additioal key fidigs. First, violece is icreasig i the 2 See also Lawrece

4 umber of idividuals wishig to joi the orgaizatios, which is i tur icreasig i the target s demads. Thus, violece is correlated with allocatios ufavorable to the resistace orgaizatios. Evidece suggests this is true empirically. Previously, scholars have argued that the coectio idicated that terrorism does ot help groups achieve their policy goals (Abrahms 2006). My model is eutral o the effectiveess of terrorism. It does, however, idicate that the observed coectio betwee terrorism ad lack of policy cocessios may be a artifact of the target believig that such violece is merely the cost of doig busiess. Put differetly, lack of policy cocessios may cause violece, ot the other way aroud. Secod, the model cotributes to a growig literature o limited war aims (Ikeberry 2009; Schultz ad Goemas 204; Coe 205; Spaiel ad Bils 206). Eve after achievig complete military victory, states ofte cocede policy objectives to the vaquished. The causes of such restrait remai uder-aalyzed. If oe iterprets my model as the begiig of post-war policy implemetatio, expectatios of future violece from resistace groups help explai some of the variatio. I particular, the implemetig state is more likely to exercise restrait whe the umber of competig groups is high ad citizes are especially sesitive to iitial ecroachmets. This paper proceeds as follows. I begi by developig the aforemetioed model. Partial equilibrium aalysis the shows that outbiddig occurs edogeously. Afterward, I show how market costraits impact the state s policy decisio, which leads to the ambiguous relatioship betwee the umber of groups ad quatity of attacks. A discussio sectio compares these formal results to existig empirical results, highlightig the ecessity of correct cotrols for proper iferece, ad provides guidelies for future qualitative research o outbiddig. A brief coclusio fiishes the paper. 2 The Model I ow tur to a stylized model to explore the iterestig strategic tradeoffs betwee demadig greater policy cocessios ad provokig more violece. 3 The game has complete iformatio ad cosists of three phases with three groups of players: a state, 3 I that regard, it makes a umber of simplifyig assumptios to keep the substative argumet tractable. For assumptios that appear particularly problematic, I ote how the results differ if they are relaxed. 3

5 {2, 3,...} competig groups, ad a mass of citizes. As a quick preview, the phases are:. The state makes a demad 2. Each group chooses a level of violece 3. Based o the demad, each citize decides whether to support a group, which is i tur iflueced by the level of violece each group chose More thoroughly, the state begis by demadig x [0, ]. portio of the good it cosumes. This represets the Oe might coceptualize this as a govermet s extractive policy over a urepreseted group, America military coverage i the Middle East, or Israeli settlemet expasio. 4 much of the good as possible. All else equal, the state wishes to capture as However, all else is ot equal; larger demads icrease grievaces. 5 A uit mass of citizes exists that are sesitive to those demads. Ideed, citizes receive w i (x) for remaiig civilias ad (without loss of geerality) 0 for joiig a group. I coceptualize joiig a group broadly; citizes may voluteer to become agets, doate to a orgaizatio, or provide material support. The key poit is that a pool of resources becomes available that the orgaizatios wat to compete for. The fuctio w i (x) has may iterpretatios, but four seem particularly saliet. First, it may represet a citize s wage (Bueo de Mesquita 2005; Bueo de Mesquita ad Dickso 2007) ad geeral ejoymet of life; both declie as a target becomes more expasioary. Secod, it captures a idividual s extremist icliatios. Third, it factors i the revege motive (Elster 2005, ; Ricolfi 2005, ), which should be icreasig as the exteral actor etagles itself deeper ito the civilias affairs. Fourth, i the framework of terrorism-as-public-goods, the fuctio represets the distributio of i-group altruistic prefereces ad willigess for self-sacrifice (Azam 2005; Pape 2005, 87-98; Elster 2005; Witrobe 2006). While such idividuals costitute a miority of ay give populatio, a certai subset exhibits these traits (Iaaccoe ad Berma 2006; Berma ad Laiti 2008, 950). 4 Oe may also thik of this as the amout the state wishes to take after havig dispatched traditioal orgaized resistace. 5 This argumet is similar to the provocatio literature (Laqueur 987; Kydd ad Walter 2006, 69-72) except that the state willigly icites violece here as tradeoff for capturig more of the good. 4

6 % of Citizes Joiig Target State s Demad x Figure : Example cumulative probability distributios for λ < (cocave), λ = (liear), ad λ > (covex). The shape of w i (x) helps assist with these iterpretatios. Let w i(x) < 0 be strictly decreasig. I words, the more the state demads, the less attractive civilia life looks. This could be because ecroachmet ito the populatio s domai risks hurtig ecoomic opportuities, killig frieds ad family members, ad geerally radicalizig citizes. 6 Note that some citizes have a particular value for x such that w i (x) = 0. Put differetly, a particular state demad exists that leaves citize i idifferet betwee remaiig a civilia ad joiig a group. 7 Although these idifferece poits might ot appear importat for determiig the relatioship betwee the umber of orgaizatios ad the prevalece of violece, they prove to be critical. Let f(x) represet the probability desity fuctio of these idifferece poits. To permit greater aalysis, suppose the explicit fuctio is f(x) = λx λ, where λ > 0 is a parameter that determies its exact shape. 8 Kow as a power dis- 6 Statistical evidece further idicates that more expasive foreig policy platforms (i.e., greater values of x) lead to more frequet trasatioal terrorism (Savu ad Phillips 2009). 7 The game treats the idifferece poits as complete iformatio, though they are difficult to kow i practice (Kura 99). Nevertheless, the complete iformatio model is a critical first step i uderstadig the strategic tradeoffs betwee capturig more of the good ad facig icreased violece. Furthermore, as the propositios show later, the cocavity of the distributio of these prefereces drives all of the results. Thus, as log as the target state kows the cocavity of the distributio, similar results would hold. 8 Later, I discuss how these results geeralize to other distributio fuctios. 5

7 tributio, this PDF form captures a surprisigly wide rage of distributios. Figure illustrates three such possibilities. Whe λ =, the CDF of the probability distributio is a uiform; each additioal uit the state demads radicalizes the same portio of idividuals. Whe λ <, the fuctio is cocave; each additioal uit the state demads radicalizes decreasigly more citizes. Lastly, whe λ >, the fuctio is covex; each additioal uit the state demads radicalizes icreasigly more citizes. Ex ate, the parameter λ may appear irrelevat to the research questio. To the cotrary, though, the presumed mootoic relatioship betwee violece ad groups requires a specific aligmet of prefereces. 9 Aticipatig that citizes will look for a orgaizatio to joi, the secod phase features the 2 groups compete for their membership; this umber is exogeous, though theoretically idetical results follow if the umber of groups formig were edogeous. 0 Cosistet with outbiddig, each group j simultaeously selects a effort level v j, represetig a amout of violece. I the third phase, the idividuals choose which group to joi. For this fial part, suppose that a ratio cotest success fuctio (Hirshleifer 99) with players captures the probability ay give citize chooses a particular orgaizatio. Thus, for effort v j, orgaizatio j expects to recruit portio of the pool of goods. v j v v The cotest success fuctio models two importat features of competitio for scarce resources. First, the more effort a idividual orgaizatio exerts, the larger the portio of goods it expects to receive. Secod, the more effort other orgaizatios exerts, the smaller the portio of goods the origial orgaizatio expects to receive. 2 differetly, more effort is beeficial for ay give orgaizatio but simultaeously hurts all other orgaizatios. This also avoids modelig recruitmet as a all-pay, wier- 9 The positive support o the distributio fuctio correspods to cases where the some segmet of the populatio will positively support terrorist violece. Per stadard outbiddig theory (Bloom 2004; Brym ad Araj 2008), we would ot expect groups to advertise i this maer otherwise. 0 I tur, oe might coceptualize the game preseted here as a reduced-form iteractio where those etry decisios have cocluded. v That is, j v +...+v is icreasig i v j. 2 v That is, j v +...+v is decreasig i all v j. Put 6

8 take-all auctio that does ot match the empirical record give that small orgaizatios ca persist over time. 3 As for the remaiig payoffs, the state receives x, the share it demads, ad pays a cost for the total amout of violece the orgaizatios commit ad the portio of recruits it radicalizes. Let α > 0 be a scalar measurig how much the state values the good versus the pai it suffers from violece ad F (x) be the cumulative distributio fuctio of f(x). 4 I tur, the formal utility expressio is: [ ] x α v j + F (x) j= For coveiece, I assume that all the orgaizatios are equal i their iitial outlays. 5 They care about the amout of total recruits they receive ad pay for their effort. This cost may arise due to the expeses ecessary to commit a attack ad the risk of retributio from the target. 6 As stadard with cotests success fuctios, assume that this cost equals the amout of violece committed v j. As such, j s overall utility equals: v j F (x) v j v v This utility fuctio clarifies that the orgaizatios value market share, which is the corerstoe of the outbiddig literature (Bloom 2004). Policy prefereces are abset, cosistet with Croi s (20, 40) argumet that forcig policy cocessios are a low priority for a vast majority of terrorist groups. Itegratig a preferece over the policy 3 I this way, groups are perfectly substitutable. Thus, citizes coditio support decisios etirely o violece. See Kapla 205 for a discussio of outbiddig amog groups with differig ideological orietatio. Results would hold if certai groups received a larger share of the ratio per uit of violece created. 4 Recall that the distributio fuctio F (x) mapped ito a portio of idividuals who led support to a orgaizatio. Thus, the scalar α implicitly reflects the overall size of idividuals willig to joi orgaizatios, which depeds o citize sympathy toward the eterprize. Oe might imagie a alterative payoff specificatio i which the state weighs the cost of outbiddig violece ad total group membership differetly. The results below are theoretically idetical, so I use oe scalar for the sake of parsimoy. 5 Relaxig this assumptio would ot alter the theoretical isights I preset here. 6 Although groups may wish to commit violece for violece s sake, oe may alteratively iterpret these costs as the additioal (uecessary) risks groups icur i a rush to compete for resources. Berma (2009, 4) argues that these risks sometimes destroy a etire orgaizatio, helpig explai why oly 40 or so groups exist today despite low ecoomic barriers to etry. 7

9 outcome ito the utility fuctio leads to theoretically aalogous results. 7 Leavig out the policy preferece, however, makes these results all the more iterestig: groups ca credibly iduce policy cocessios despite havig o desire to do so. 3 Partial Equilibrium Aalysis: Outbiddig betwee Groups This is a extesive form game with complete iformatio, so subgame perfect equilibrium is the appropriate solutio cocept. While the overall iteractio ivolves the decisios of the state ad idividuals who choose to joi the orgaizatio, the game featurig the orgaizatios is iterestig i its ow right. Sice the solutio to this portio of the game is ecessary to solve for the overall subgame perfect equilibrium, I start at this poit. Propositio gives the equilibrium strategies of the orgaizatios: Propositio. I all SPE, each orgaizatio j selects violece level v j = F (x)( ) 2. I provide full proofs of all formal claims i the appedix. Here, however, the followig comparative static helps provide ituitio for Propositio : Remark. The quatity of violece is icreasig i the umber of idividuals seekig membership. demads. Thus, the quatity of violece is icreasig i the size of the state s Orgaizatios face a tradeoff betwee capturig larger recruitmet shares ad miimizig their operatig costs. Whe the market for recruits is small (i.e., whe F (x) is close to 0), ay oe orgaizatio has little icetive to ivest heavily i advertisig. I cotrast, whe the market for recruits is eormous (i.e., whe F (x) is close to ), each uit of effort brigs back a larger retur, holdig all other decisios costat. But all 7 I particular, oe might imagie that more violece may probabilistically lead to a greater likelihood that the state withdraws. This effectively lowers the margial cost of violece for state j to βv j, where β (0, ). I tur, equilibrium strategies chage, with the orgaizatios ivestig more i violece (because it is cheaper ), which i tur causes the state to capture less of the policy (because it faces more costs ad greater risk of failure). The key theoretical fidig that the effect of umber of groups o violece depeds o the cocavity of the recruitmet fuctio remais uchaged. 8

10 other orgaizatios face the idetical icetive. Cosequetly, they all icrease their levels of violece, compoudig the effect across all the groups. 8 These market icetives give the state a troublesome tradeoff later the state would like to capture more of the good, but doig so guaratees more attacks o it. Remark also demostrates the challeges of iferrig group success based o the amout of violece observed ad the level of violece the target suffers. Yet Abrahms (2006), for example, otes that foreig orgaizatios achieve their policy objectives a remarkably low percetage of the time. He argues that this fidig suggests that the poor success rate is iheret to the tactic of terrorism itself (Abrahms 2006, 43-44). That is, such violece causes policy outcomes ufavorable to the audieces of terrorist orgaizatios. Whether terrorism is a useful coercive tactic is beyod the scope of my model. 9 However, the model gives a alterative explaatio for such a empirical correlatio: policy outcomes extremely ufavorable to a audiece result i greater levels of violece. That is, the causal relatioship may flow i the opposite directio. If the optimal demad x is high which I show below is sometimes the case forward lookig targets uderstad that oerous policy demads will result i such high levels of violece. They evertheless take a large portio of the good because its margial value exceeds the margial icrease i violece. Terrorist violece may prove expesive, but sometimes states are willig to suffer those costs to achieve policy goals. model: The ext remark shows that the outbiddig mechaism arises edogeously i this Remark 2. (Edogeous Outbiddig) Holdig fixed the size of the market, the quatity of violece is icreasig i the umber of groups. I particular, the appedix shows that the groups i total commit F (x)( ) quatity of violece, which is icreasig i. A simple ituitio makes sese of this result. If a sigle orgaizatio exists, it eed ot commit costly violece to compete for resources. Cosequetly, it ca reap all 8 As a result, public support is partially edogeized i this model. Usually, outbiddig scholars preface their theories by statig that they ought to oly apply whe the public is receptive to attacks (Bloom 2004). But ote that whether the public is receptive of the attacks is itself a fuctio of the iitial demads made by the target state. 9 Others (Dershowitz 2002; Pape 2003) argue for its effectiveess. 9

11 the profits without payig ay advertisig costs. With two groups, the orgaizatios must compete for resources. Nevertheless, some profits still remai, as the margial value of a group s advertisemets is decreasig. Maitaiig these levels of profits with three groups proves impossible, though, as the third group would wat to exert some effort ad capture some of the remaiig surplus. Thus, while each group s idividual effort decreases, the total effort icreases. These priciples hold true as icreases, as Remark 2 claimed. Remark 2 s result is critical for the discussio movig forward. It shows that competitio matters i this model i the maer that the outbiddig literature has previously expressed. That is, it takes outbiddig theory seriously. Yet the results below show that more groups might ot ultimately imply more violece despite the icetives for competitio. 4 Why the Shape of Aggregate Citize Prefereces Matters Now cosider the state s demad. As previewed above, the state wishes to capture as much of the good as it ca, subject to the costs it suffers from attacks ad radicalizatio from the populatio. The decisio for the mass of citizes is trivial those for whom w i (x) > 0 joi ad those for whom w i (x) < 0 do ot. 20 From there, because the game has complete iformatio ad occurs sequetially, the state also aticipates sufferig F (x) ( ) i violece. Coveietly, the state ca calculate this for each possible value x it could choose. Combiig this with the value it receives for capturig the good, the cost of icreased membership, ad a scalar differetiatig the value of the two, the state s objective fuctio is: ( Let x = αλ(2 ) [ ( x α F (x) ) ] + F (x) ) λ. This is sufficiet for the ext propositio: Propositio 2. Suppose λ >. The game has a uique SPE. I it, the state demads 20 Because the citizes are a cotiuum ad f(x) is atomless, the case where w i (x) = 0 is immaterial. 0

12 the miimum of x ad. Here, the ituitio is straightforward. Recall the covex CDF i Figure, which correspods to the case where λ >. Capturig small values of x iduces few idividuals to joi the orgaizatio. Thus, orgaizatios have little icetive to advertise through attacks. Facig oly mior resistace iitially, the state should demad somethig. However, the rate of radicalizatio evetually becomes large, yieldig more recruitmet ad greater advertisemet. This effect decreases the margial value of demadig more. Note that if α is sufficietly small that is, the state cares about violece relatively little compared to the good the margial value for demadig more may ever tur egative. I that case, the state demads everythig. Otherwise, it cuts off its demad at the poit where the margial value turs egative, which is x. Perhaps uexpectedly, these results do ot carry over to situatios where λ < : Propositio 3. Suppose λ <. The game has a uique SPE. If α <, the state 2 demads. If α >, the state demads 0. 2 Put differetly, the state makes the demad a all-or-othig affair whe λ < ; it ever captures a middlig amout as it might whe λ >. Why is there such a stark cotrast betwee Propositios 2 ad 3? Cosider the state s decisio to icrease its demad from oe of the good to a arbitrarily small portio. Doig so iduces more citizes to joi a group ad i tur icreases the outbiddig violece. Recallig back to Figure, the margial differece betwee 0 ad that arbitrarily small portio is also vaishigly small whe λ >. Oly later i the distributio does takig a additioal fixed uit substatially alter the recruitmet patters. Thus, the state keeps demadig more util it reaches the poit where the margial recruitmet ad outbiddig violece costs exceed the margial value of the amout captured. I cotrast, whe λ <, icreasig the state s demad from 0 to a arbitrarily small amout has a disproportioately sigificat effect o recruitmet; demadig each additioal uit of the good results i less radicalizatio tha takig the first portio does. As such, if demadig that first amout proves worthwhile, demadig all of the good must be worthwhile as well. I tur, the state merely eeds to check whether takig othig is better tha demadig everythig ad sufferig the cosequeces. The cutpoit α < 2 determies which is optimal.

13 5 Empirical Implicatios: Deterrece, Violece, ad Group Size Havig solved the model, I ow tur to its comparative statics. To begi, cosider how the state s demad chages as a fuctio of the umber of groups: Remark 3. (Credible Deterrece) The state s demad is weakly decreasig i the umber of competig groups; group surplus is weakly decreasig i the umber of orgaizatios but citize welfare is weakly icreasig. Give that this is a model of competitive orgaizatioal advertisig, it might ot be surprisig that the overall group surplus decreases as the umber of orgaizatios icrease. Similar to competitive Courot markets where additioal firms push equilibrium quatities below the moopoly quatity, Remark says that addig aother group forces the overall violece to icrease. I tur, icreasig the umber of orgaizatios slowly eradicates all of the surplus. However, competitio is oly part of the story. Orgaizatios also suffer due to the state s edogeous respose to potetial outbiddig. Iteralizig additioal attacks with more orgaizatios preset, the state reduces its demads. I tur, fewer idividuals wish to provide resistace, leavig orgaizatios with a smaller pool of potetial recruits ad less fudig. May scholars have oted terrorists desire to provoke a overreactio from their targets, thereby polarizig moderates ad covicig them to led their support (Mishal ad Sela 2000; Zirakzadeh 2000; Rosedorff ad Sadler 2004; Bueo de Mesquita ad Dickso 2007). Give that the deterrece effect would ordiarily lead to less support for terrorism, the model suggests that the icetives to provoke are the strogest whe more groups exist. Although the groups collectively lose out as they become more umerous, citizes who choose ot to joi a group collectively beefit. Recall that coditioal o remaiig a civilia, each strictly prefers the state demad a smaller amout of the good. But havig more groups creates a deterret effect o the target state, leadig it to temper its demads. Thus, all civilias beefit from the credible threat to use violece eve if outbiddig is ultimately costly to the orgaizatios. 2 2 This reveals a deeper credibility issue that the citizes face. Citizes would collectively beefit if they could threate to joi a orgaizatio regardless of whether their demads are met. If the 2

14 It is worth emphasizig that the state s limited aims are ot the result of some bargaiig process. 22 I the stadard bargaiig model of war setup (Fearo 995), states with all of the proposal power ask for less tha their ideal policy outcome because the oppoet ca implemet a less favorable outcome via war. Here, the state faces o such rejectio costrait; if the state wats to capture the etire policy good, it may. But the model illustrates that states sometimes wish to limit their share i the dictator game, recogizig that the cost of violece may exceed the margial value of a additioal portio of the policy good. 23 For example, may Europea coutries chose ot to participate i the Iraq because eve a successful ivasio would icrease terrorist attacks. 24 Observig this exact result from the Iraq War has icreased reticece i egagig elsewhere i the Arab world. This helps explai why most state resposes to attacks oly ivolve the police ad judiciary (Duyvestey 2008; Creshaw 205), with oly aroud % triggerig a military respose from a post-war Wester Europea coutry (Carter 206). Movig o, the above welfare aalysis does ot yet aswer how the levels of violece a importat comparative static ad ofte a key depedet variable i empirical work chages with the umber of groups. The followig two remarks address this questio, separatig the parameter space by λ: Remark 4. If λ is sufficietly high (i.e., λ > max{2, 2 }), equilibrium violece is 3α icreasig i. Although the comparative static here matches the covetioal coectio betwee umber of groups ad violece, the ituitio is ot as straightforward as the literature implies. As λ grows, the recruitmet curve grows icreasigly covex. Substatively, this meas that the umber of idividuals wishig to joi a group icreases precipitously target state believed those threats, it would further recede its demads, as violece is icreasig i the umber of citizes providig support (Remark ). All citizes would beefit. However, such threats are ot iheretly credible the oly citizes that joi are those for which w i (x) < Cuigham (20), for example, fids that states are more likely to grat self-determiatio whe facig divided movemets tha uited frots because cocessios ca stregthe the target s preferred factio. My model highlights a alterative mechaism, amely that icreased competitio disicetivizes larger claims. 23 I practice, this effect may reverberate; Schultz ad Goemas (204) demostrate that these ultimate goals affect crisis bargaiig decisios. Future research could exploit variatio i competig groups to explore these theories empirically. 24 See, for istace, Parmetier This was also a poit of objectio for doves i the Uited States. 3

15 just before the full demad of. Thus, the target state ca sigificatly reduce the violece agaist it by oly slightly backig away from that full demad. Accordigly, the state demads the iterior solutio x. 25 Icreasig the umber of groups decreases this amout oly slightly, agai because the fuctio is sufficietly covex. Cosequetly, the effect of greater competitio domiates, resultig i the additioal violece that the black dots i Figure 2 illustrate. 26 The simple mootoic relatioship disappears whe λ is ot as covex. Because these results deped o a umber of cutpoits, I save specific details for the appedix. I short, because the recruitmet curve does ot icrease as precipitously, a demad of is possible provided that is sufficietly small. Icreasig (but still keepig it below the critical threshold) leads to a mootoic icrease i violece because the state s demad remais costat. However, movig past that threshold causes the state to switch its demad to the iterior solutio. This ca trigger a discotiuous decrease i violece, leadig to a o-mootoic relatioship. Violece ca cotiue to decrease here as icreases further depedig o the specific values of ad λ. Iterestigly, the results grow further complicated whe λ < : Remark 5. For λ <, if the state is sufficietly sesitive to violece, equilibrium violece drops to 0 for sufficietly large. Put differetly, whe the cumulative distributio fuctio of citize turig poits is cocave (like the fuctio of the red lie i Figure ), the covetioal wisdom o outbiddig o loger assuredly holds; violece oly icreases i the umber of groups to a poit, at which it drops off precipitously. The key is uderstadig the two cases from Propositio 3. Recall that whe λ <, the state adopts a go big or go home strategy it either demads everythig or it demads othig. This is because a demad slightly greater tha 0 correspods to a high jump i idividuals voluteerig. However, ay demad slightly greater tha that icreases the umber of voluteers at a smaller rate. Thus, if the state is willig to demad ay positive amout, it ought to go all the way to. The costs icurred through violece determies the state s choice betwee 0 ad. Whe the cost is low, demadig is optimal; whe the cost is high, stickig to 25 As the appedix shows, the formal requiremet to guaratee a demad of x across is λ > 2 3α. 26 The formal requiremet for the effect of greater competitio to domiate is λ > 2. This combied with the above formal requiremet geerates the cutpoit o λ foud i Remark 4. 4

16 Equilibrium Level of Violece 0 Remark 5 Remark Number of Orgaizatios Figure 2: Equilibrium levels of violece as a fuctio of the umber of groups ad the shape of the ideological distributio of the citizes. Note violece is strictly icreasig i for for the parameters of Remark 4. However, a critical threshold exists for Remark 5 whe λ <. Below that threshold, violece icreases i ; after that threshold, equilibrium violece drops precipitously. 5

17 the safe 0 amout is preferable. The umber of orgaizatios determies the extet of violece. I tur, icreasig the umber of orgaizatios covices the state to switch from to 0, leadig to a decrease i violece. Figure 2 illustrates these results. The red dots correspod to the equilibrium violece whe λ < as a fuctio of the umber of orgaizatios. Below the critical threshold, the state demads all of the good. Equilibrium violece is therefore icreasig i the umber of groups, as the total demaded remais costat but the outbiddig icetives magify. However, oce exceeds the threshold, the state switches to demadig oe of the good. The orgaizatios have o icetive to outbid, ad thus equilibrium violece drops off etirely. Before movig o, a couple of otes are i order about the geerality of the results i Figure 2. First, the cocavity of the cumulative distributio fuctio drives the drop off. If a segmet of a cumulative distributio fuctio is cocave ad the state optimally chooses a demad withi that rage, it will select oe of the ed poits. Thus, the drop off is ot a cosequece of this particular family of probability distributios I aalyze. Secod, the reaso levels of violece remai costat at 0 for high values of whe λ < is because F (0) = 0 for the probability distributios I aalyze. Oe may alteratively suppose that a fixed portio of idividuals prefer to joi a orgaizatio regardless of the state s demad. Here, eve if the state optimally chooses 0, the orgaizatios have icetive to outbid oe aother. I tur, the fuctio would maitai the precipitous drop off but would rise oce more immediately afterward. 6 Challeges for Empirical Outbiddig Research A commoly-held implicatio of outbiddig is that icreasig competitio leads to more violece. For example, Fidley ad Youg (202, 708) state that The greater the umber of oppositio groups, the more likely ay terrorist acts will occur durig armed coflict. Stato (203, 04) echoes this, statig that outbiddig argumets predict that terrorism is more likely i coflicts ivolvig multiple rebel groups. Nemeth (204, 345) argues that Groups i competitive ad favorable eviromets will commit more terrorist acts tha groups i ocompetitive ad ofavorable eviromets. Ad Forta (205, 5) posits that The outbiddig argumet suggests that terrorism is more likely whe there are several rebel groups active as part of the same struggle. 6

18 Each of these scholars the operatioalizes group competitio ad tests whether the presumed outbiddig effect holds. 27 Their results are mixed. Nemeth fids evidece to support the hypothesis, though the coectio ituitively higes o whether the audiece will respod favorably to violece. Stato ad Forta fid o results. Fidley ad Youg make a stroger claim, ceterig their argumet o the ull hypothesis, ultimately cocludig that their results clearly suggest that the outbiddig argumet may ot be geeralizable to a wide variety of coutries ad coflicts (79). More tha a decade has passed sice Bloom s work revitalized iterest i outbiddig. For the literature to mature ad for kowledge to accumulate, we must recocile the discrepacies i these empirical results. The model provides a explaatio: the relatioship betwee the umber of groups ad levels of violece ca go either way depedig o the shape of the distributio fuctio determiig recruitmet. I that light, it is usurprisig to see differet empirical results depedig o the research desig. Cosequetly, Fidley ad Youg s claim that their empirical results fail to support the outbiddig hypothesis overlooks how the iitial demads complicate the associatio. If simple couts of competig groups are isufficiet to recover the correct relatioship, the what is? Fidig a solutio is a challegig task. But Fidley ad Youg provide some isight. They ote that It may also be the case that there are heterogeeous dyamics at work i which the umber of groups icreases terrorism i some coutries such as Israel, but decreases it i others (79). The model supports this assertio. I some cases, violece icreases i group size. True to that, they fid that the predicted relatioship holds with attacks o Israel. More specifically, per Remark 4, the model shows that the straightforward relatioship holds for λ sufficietly great. Thus, to properly test the above hypotheses, oe could gather data o the shape of the recruitmet curves. Followig that, a model that subsets the data o high levels of λ would recover the appropriate relatioship betwee the umber of groups ad violece. This may be a directio for future qualitative research i outbiddig theory. Now that the model has revealed that the shape of distributio matters, qualitative scholars may wish to reivestigate cases where the 27 The operatioalizatio varies from paper to paper. Stato ad Forta both use dummy variables. Nemeth uses a firm cocetratio idex from Herfidahl 950 ad Hirschma 945. Fidley ad Youg use a variety of measures, a ecessary task to persuasively argue for a ull hypothesis. 7

19 outbiddig mechaism came ito play to the back out what caused the curve to take a particular shape, which would the allow for this subsettig. Of course, the devil is i the details developig measures of supply curves requires a sigificat effort eve with vast amouts of data. Oe cadidate for a λ proxy worth explorig is foreig itervetio. For example, cosider the origis of discotet i Lebao prior to the 983 Beirut barrack bombigs. Coli Powell, who was as assistat to Defese Secretary Caspar Weiberger at the time, wrote i retrospect that U.S. operatios agaist Shiite targets led that audiece to assume...the America referee had take sides (Powell ad Perscio 996, 29). I effect, by takig oe step ito the fray, Washigto had mobilized a large segmet of the populatio agaist it. This is like havig a recruitmet curve with cocave shape, like red curve from Figure. That is, the iitial icursio leads to a sharp icrease i iitial recruitmet, perhaps due to social solidarity (Witrobe 2006, 08-43). The model predicts a o-mootoic relatioship uder these coditios. Domestic policy grabs could reasoably have the opposite cocavity. Here, iitial icursios are less likely to cross a clear lie, as there are o foreig troops flowig across the border. Ecoomic distortios ought to be less sigificat as well, as the home govermet already regulates commerce to some degree. Combied, these factors mea that the populatio is more likely to overlook mior extractios. I tur, the relatioship betwee violece ad umber of groups may hold a more regular patter i these cases. That said, a complete uderstadig of this pheomeo requires solvig a difficult selectio problem. 28 Whe the deterret effect is at its weakest, the target willigly captures the policy at stake, resultig i violece that the target writes off as a cost of doig busiess. I tur, the historical record shows a large umber of cases where terrorism fails to yield policy cocessios (Abrahms 2006). Whe the deterret effect of terrorism is at its strogest, the potetial targets withdraw from the policy domai before ay violece ca take place. These are relative o-evets that historias ad political scietists have less icetive to study, which distorts the record See Fearo 2002 for a aalogous challege i the exteded deterrece literature. 29 To wit, the amout of research o why the Uited States fought the Iraq War is eormous. The amout of research o why Frace, Germay, ad others did ot joi the coalitio is small, eve though they cited fear of icitig terrorism as a reaso ot to fight. Part of the problem is primary source materials are less abudat i the secod case, as ot egagig meas a lack of meetigs ad reports 8

20 Eve if the state opts agaist complete withdrawal, the selectio problem may persist. The parties ca establish spheres of ifluece that the govermet respects to reduce violece (Stailad 202, ). For istace, Pakista has passed de facto cotrol over portios of cities to the Muttahida Qaumi Movemet (Lieve 20, 36-37) ad larger segmets of the coutry to the Taliba (Rashid 2008, 385). Although the military could have applied pressure to these oppoets, the govermet preferred ot to suffer the costs of terrorist ad isurget violece. Other examples iclude Idia s hadlig of Naga atioalists ad pockets of relative peace betwee Afghaista ad the Mujahidee durig the Soviet itervetio (Stailad 202, 25) ad the Soviet/Russia approach to Tatarsta s push for regioal autoomy (Kodrashov 2000, 42). These agai disicetivize research o the subject, agai distortig the record. I addressig these problems of cotrollig for λ, oe temptig solutio to avoid is usig state fixed effects. Fixed effects are useful whe uobservable characteristics of a uit remai uchagig over time ad correlate with the depedet variable at had. Thus, if the recruitmet supply curves remai costat over time, fixed effects may appear to provide a solutio to test the relatioship betwee the umber of groups ad violece, eve if they caot test whether λ holds i the expected maer. Agai, though, the solutio is ot that simple. Fixed effects oly add (or subtract) a amout to the estimates for all observatios of their respective coutries. Thus, a fixed effect is the portio of the depedet variable attributable to beig a part of that uit which might iclude latet coflict, wealth, opportuity, ad a supply curve. However, the model demostrates that the supply curve matters isofar as it relates to the umber of groups. As such, the appropriate statistical solutio is to subset the data or use a iteractio term. This is ifeasible with fixed effects for two reasos. First, the large quatity of iteractio terms would prevet cross-coutry aalysis, which is a key motivator for the large- empirical studies i the first place (Fidley ad Youg 202). Secod, the iteractio would also absorb ay coutryspecific characteristics that remai costat over time. We would therefore be uable to adequately differetiate whether the estimates are the result of the supply curve s iteractive effect with violece or some other uobserved characteristic. Aother approach Fidley ad Youg take is to subset o istaces with violece. Although Figure 2 appears to support this strategy, o-mootoic effects persist uder o how to deal with the o-existet provoked violece. 9

21 two coditios. First, as described i the comparative statics ad detailed i the appedix, slightly covex recruitmet curves ca lead to o-mootoicities. Secod, recall that the mootoic relatioship coditioal o violece for cocave fuctios oly holds whe o citizes voluteer if the state demads 0. If some would voluteer here, the precipitous declie for cocave fuctios see i Figure 2 still occurs. I tur, the amout of violece begis icreasig agai after the heavy drop. But this meas that the discotiuous drop would occur i data subsetted o observed violece, agai meaig that such a empirical model would be uable to recover the correct relatioship. Nevertheless, the model provides a fial alterative. The relatioship is straightforward ad mootoic after subsettig o istaces where the state demads. The deterrece effect has o bite here because the state willigly demads everythig, ad Remark already revealed that violece is icreasig i the umber of groups for ay give size of the market. Icreasig the umber of groups may evetually lead to a drop off i violece, but this will be because the state reduces its demad ad therefore that observatio would ot appear i the dataset. The oly challege is codig a demad size to subset o. This has bee a sigificat obstacle to empirical tests of the bargaiig model of war. Fortuately, the task here is substatially easier a researcher oly eeds to code for all-ecompassig demads to produce the proper subset ad does ot eed to sort through the miutiae of differetiatig various fractios of a good. Overall, these issues idicate that researchers ought to thik about outbiddig more holistically, as measurig demad curves requires cosiderig both the terrorist attacks that did ad did ot happe. Accordigly, empirical tests eed to take seriously the plausible source of terrorist ager, as outbiddig may have the differetial effects outlied above. 7 Coclusio Does outbiddig deter aggressive demads? This paper ivestigated a target state s demad decisio i the shadow of itergroup competitio for scarce terrorist resources. Fearig that especially large demads will lead to a greater supply of terrorist recruits ad greater competitio for them, target states edogeously limit their aims. Further, they demad less as the umber of groups ad thus the icetives to compete grows. 20

22 The deterrece effect leads to uexpected results regardig the overall relatioship betwee the umber of groups ad violece. Whereas researchers have traditioally assumed that outbiddig implies that more groups yield more violece, the deterrece effect sometimes domiates. Icreasig groups ca therefore lead to a sharp drop off i the quatity of violece. Formal aalysis idicates that the covexity of terrorist supply curves etirely determies whether the expected effect holds. These results help illustrate the utility of formal theory, as it is uclear ex ate why the covexity of that fuctio would matter for empirical implicatios of the outbiddig theory. This model was also the first step i thikig about secod-order effects of outbiddig. The literature o outbiddig is maturig. Future research the ought to slowly move beyod providig microfoudatioal or empirical support ad advace to askig how states, terrorist groups, ad other actors strategically respod to the outbiddig icetives. The cotest model itroduced i the paper which edogeously supports the otio of outbiddig provides a useful baselie to expad o. Such research would yield ew testable hypotheses, which could provide further empirical support outbiddig from aother agle. 8 Appedix This sectio gives full proofs for claims ot previously show. 8. Proof of Propositio The state s move has determied that F (x) of the idividuals will comprise the market. Each orgaizatio j therefore has a objective fuctio of: v j F (x) v j v v I words, group j ears a share of the F (x) umber of idividuals equal to the percetage of all effort v j v +...+v it exerts. It must also pay for its ow effort. If F (x) = 0 (because x = 0), the optimal v j is 0. For ozero F (x), takig the first order coditio of this objective fuctio yields: 2

23 F (x) v v v j (v v ) 2 = 0 Sice the iteractio cotais orgaizatios, there are such first order coditios. Substitutig v j = v j ito the above first order coditio yields: F (x) ( )v j (v j ) 2 = 0 v j = F (x)( ) 2 The secod order coditio is fulfilled because the secod derivative of the objective fuctio is: (v v v j ) 2(v v ) (v v ) 4 Thus, each orgaizatio commits to F (x)( ) 2 quatity of violece. 8.2 Proof of Remark For the first setece of Remark, per Propositio, each group commits to F (x)( ) 2 violece. The derivative of this with respect to x equals f(x)( ). This is strictly 2 positive, so icreasig the umber of citizes supportig the competig groups (i.e., icreasig F (x)) icreases violece. extesio of the first, otig that F (x) is icreasig i x. The secod setece of the remark is a simple 8.3 Proof of Remark 2 Per Propositio, each group commits to F (x)( ) violece. There are such groups. 2 Therefore, the total amout of violece across all groups equals: F (x)( ) 2 ( = F (x) ) The first derivative of this with respect to is positive. Therefore, the equilibrium level of violece is icreasig i holdig fixed the size of the market. 22

24 8.4 Proof of Propositio 2 The derivative of the state s above objective fuctio is: ( f(x)(α) 2 ) Settig this equal to 0, substitutig the fuctioal form of f(x), ad solvig for x yields x. 30 Because x [0, ], the most the state ca possibly take is. Thus, the state demads the miimum of x ad. 8.5 Proof of Propositio 3 The proof follows from the same setup as Propositio 2. The state s objective fuctio remais the same. Thus, the objective fuctio s critical poit is idetical. However, because λ <, the secod derivative is positive, meaig that the critical poit is ow a miimum. I tur, the state s optimal demad must be o a corer. Usig the objective fuctio, demadig is better tha demadig 0 if: [ α () ( )] + > 0 α < 2 This geerates the cutpoits i Propositio Proof of Remark 3 For clarity, I split this proof ito three parts Demads are weakly decreasig i the umber of terrorist orgaizatios. First, cosider the case whe λ <. The state demads either 0 or here, so provig the claim oly requires showig that icreasig caot lead to a switch from demadig 0 to demadig. Recall from Propositio 3 that the state demads 0 if α > This is a maximizer because secod derivative equals αλ(λ ) ( 2 ) x λ 2, which is strictly egative for λ >. 23