Provocation and the Strategy of Terrorist and Guerilla s: Online Theory Appendix Overview of Appendix The appendix to the theory section of Provocation and the Strategy of Terrorist and Guerilla s includes the results of two alternative specifications of the model. Specifically, I demonstrate that: the key theoretical results are similar if we allow domestic pressure to act to influence government behavior, the key theoretical results are similar if I assume that not all damage caused by attacks is beneficial to the group, i.e., groups can be inaccurate in targeting,
Alternative Specification of Theoretical Model: Domestic Pressure The game has the structure depicted in figure. The sequence of play is the same as in the model presented in the main text. The only difference between the model in figure and that in the main text is in how the government s utilities are modeled. Specifically, I address here the idea that the government faces public pressure to respond observably and forcefully to an attack (Richardson, 2006; Bueno de Mesquita, 2007). I assume that public pressure to respond to an attack increases in the severity of the attack (in terms of damage to civilians or government forces). Thus, I introduce one additional parameter to the model, γ 0, which measures how significant this cost is. Given that most of the payoffs in this version of the model are not different than those in the main text, I focus only on the differences. The government s payoffs for not responding to either tactic with force are the only payoffs that differ. If the group attacks and employs terrorist tactics, and the government does not respond forcefully, the group receives a payoff of c t k t, which is the same as in the main text. However, the state receives a payoff of (+γ)c t. The parameter c t again represents the damage imposed on the state, of which the most prominent damage involves civilian deaths and injuries. The costs to the government for not responding forcefully to an attack is captured by γ. I assume that γ is a function of the damage imposed by the attack, c t, so that the costs to not responding in an observable way increase as the severity of the attack increases. The group receives positive payoff from more damaging attacks, while the state pays higher costs for an attack as it imposes more damage. As γ 0, the costs from the public to not responding forcefully diminish and approach 0; however, as γ increases, the costs to not responding also increase. This works The cost of not responding forcefully is assumed to be linear in the severity of the attack to simplify expressions. None of the key results change if we make the more general assumption that γ is a monotonic function of c i. 2
in the same way in the government s payoff to not responding to a guerilla attack. 2 [Figure about here.] I use the subgame perfect equilibrium (SPE) refinement to analyze the game. As play is sequential and the players have complete and perfect information, there is a unique equilibrium in pure strategies for any distribution of the model s parameters (Mas-Colell, Whinston and Green, 995, 276). Table summarizes the equilibrium conditions for the state and group in the game shown in figure and is formatted identically to table X in the main text. The table lays out all possible equilibrium paths of play for the state and group along with the corresponding equilibrium conditions that must hold for each path of play to be optimal. The table is constructed to reflect the logic of backwards induction, with each row representing a possible equilibrium path of play for the government and group. Thus, the column on the far left indicates the three possible equilibrium paths of play for the state when it has experienced an attack. The equilibrium condition that must hold for this path of play to be optimal for the state is stated in the next column to the right. The column to the right of the government s equilibrium conditions states the possible paths of play for the group given that the government plays the strategy listed to the left in the same row. The final column states the equilibrium condition for the group that must hold for the path of play to be optimal. The main conclusions regarding the government s decision to employ a forceful response to an attack using tactic i discussed in the main text are unchanged. In short, the government s choice still depends on how precise it can be in targeting group members, α i. Furthermore, the relationship between accuracy (α i ) and provocation is no different here. The key difference is that the level of precision the government must have in its response decreases below 50% as the costs from not responding with force, i.e., γc i, increase. Accordingly, when the costs to not responding are positive, i.e., γ > 0, the government will 2 It is possible and perhaps plausible to make γ conditional on the tactic. We do not do so here as it does not significantly alter our key conclusions. 3
forcefully respond across a wider range of α i. Figure 4(a) shows this, as the threshold for accuracy (α i ) decreases as the severity of the group s attack increases. In contrast, figure 4(b) reproduces the figure from the main text, which depicts a government that always has α i > as its threshold for employing forceful 2 response.3 All common parameters are set to identical values to produce both figures. The group s behavior relative to government accuracy are no different here than in the main text. [Table about here.] [Figure 2 about here.] Alternative Specification of Theoretical Model: Casualty Distinctions for Group The structure of the game, depicted in figure 3, is the same is in the main text. The difference between this specification of the game and that described in the main text is that I allow the group to make distinctions among casualties here. Specifically, I assume that there are two types of civilians: the kind that the group benefits from targeting, the enemy population, and the kind that the group is harmed from targeting, the group s constituency. The key difference here is in how the group s payoffs are specified. In the main text, I model all civilian casualties caused by a terrorist attack as beneficial to the group. However, it is quite plausible that the group does not benefit from all casualties, as some civilians injured or killed may be members of a group s constituency (or potential constituency). Thus, I introduce a parameter to the model, σ t, which measures the group s accuracy in targeting the correct civilians. To address the possibility that the group also makes distinctions among government forces, e.g., from different ethnic groups, I also include an accuracy parameter specific to guerilla attacks, σ g. 3 If γ = 0, then government equilibrium behavior here is no different than it is in the main text. 4
In this modified model, if the group employs terrorist tactics and the government does not respond forcefully, the group receives a payoff of σc t ( σ)c t k t, while the state receives a payoff of c t. The parameter c t still represents the damage imposed by a terrorist attack. The σ [0, ] parameter captures the group s accuracy in targeting the correct civilians. Accordingly, as σ, the group becomes increasingly accurate and also benefits increasingly from more damaging attacks. On the other hand, as σ 0, the group becomes less and less accurate and terrorist attacks become less beneficial. The group s payoff following a terrorist attack that the government responds forcefully to is specified similarly: σc t ( σ)c t k t α t π + ( α t )π. Thus, the σ parameter is analogous to the α i parameter that accounts for government accuracy in targeting group members (rather than civilians). The parameter σ g affects the group s utilities for guerilla attacks in the same way. All other parameters and utility specifications are the same as in the main text. [Figure 3 about here.] As in the main text and in the political pressure model (i.e., figure ), I use the sub-game perfect equilibrium (SPE) refinement to analyze the game. Table 2 is formatted identically to table and again lists all possible equilibrium paths of play for the state and group along with the corresponding equilibrium conditions that must hold for each path of play to be optimal. [Table 2 about here.] The government s equilibrium conditions are unchanged from what is presented in the main text. A comparison of table 2 to the analogous table presented in the main text makes clear that the group s equilibrium behavior is also quite similar. The group still employs the provocative tactic i if a large enough proportion of the damage from the government s response afflicts the civilian population rather than its own members. However, the baseline changes slightly here, as c i ( σ i )c i k i = 0 implies that at least 50% of the damage from 5
the government s response must afflict the civilian population, rather than c i k i = 0. Thus, the basic empirical expectations with regards to government accuracy and provocation are unchanged. The only difference here is that as a group is less accurate, meaning that σ i decreases, the right-hand side of the equilibrium conditions when provocation is possible are increasingly restrictive. Figure 4 shows graphically how group behavior changes when we relax the assumption that its attacks target the correct people. Figures 4(a) and 4(b) are produced setting all common parameters to identical values. The only difference is that the group is 80% accurate in the model that produces 4(b). Comparison of the two figures shows that paying a cost for this inaccuracy (of 20%) decreases the range of parameters for which the group provokes the government. However, the basic relationship between government accuracy and provocation discussed in the main text is substantively the same. [Figure 4 about here.] 6
References Bueno de Mesquita, Ethan. 2007. Politics and the Suboptimal Provision of Counterterror. International Organization 6():9 36. Mas-Colell, Andreu, Michael Whinston and Jerry Green. 995. Microeconomic Theory. New York City, NY: Oxford University Press, USA. Richardson, Louise. 2006. What Terrorists Want: Understanding the Enemy, Containing the Threat. New York, NY: Random House. 7
Figure : The Strategic s Game 0 No G S Target Civilians Forceful Forceful Response Response c t k t c t k t α tπ + ( α t)π ( + γ)c t c t + α tπ ( α t)π Target Government S Forceful Response c g k g ( + γ)c g Forceful Response c g k g α gπ + ( α g)π c g + α gπ ( α g)π 8
Table : Summary of Results State Action Group Action Given Given State Response Always Employ αi, αj > γc i = with Tactic i if 2 2π αi < min { + c i ki, α 2 2π j + (c i ki) (cj kj) 2π Employ αi > γc i > α 2 2π j = to Provoke if αi < min Conditional on Tactic i { + c i ki, + (c i ki) (cj kj) 2 2π 2 2π Never Employ γc i > α 2 2π i, αj = with Tactic i if ci ki > max{0, cj kj} } } 9
Damage Caused by Group (c) -2.0 -.5 -.0-0.5 0.0 0.5.0 No Provocation Damage Caused by Group (c) -2.0 -.5 -.0-0.5 0.0 0.5.0 No Provocation 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 Government Accuracy State Accuracy (a) Public Pressure Model (b) Original Model Figure 2: Provocation and Government-Caused Damage 0
Figure 3: The Strategic s Game 0 0 No G S Terrorist Guerilla S Forceful Forceful Response Response σ tc t ( σ t)c t k t σ tc t ( σ t)c t k t α tπ + ( α t)π c t c t + α tπ ( α t)π Forceful Response σ gc g ( σ g)c g k g c g Forceful Response σ gc g ( σ g)c g k g α gπ + ( α g)π c g + α gπ ( α g)π
Table 2: Summary of Results State Action Group Action Given Given State Response Always Employ αi, αj > 2 = with Tactic i if αi < min{ 2 + σci ( σi)ci ki 2π, αj + (σici ( σi)ci ki) (cj ( σj)cj kj) 2π Employ αi > 2 > α j = to Provoke if αi < min{ 2 + σci ( σi)ci ki 2π, 2 + (σici ( σi)ci ki) (cj ( σj)cj kj) 2π Conditional on Tactic i Never Employ 2 > α i, αj = with Tactic i if ci ( σi)ci ki > max{0, cj ( σj)cj kj} } } 2
Damage Caused by Group (c) -2.0 -.5 -.0-0.5 0.0 0.5.0 No Provocation Damage Caused by Group (c) -2.0 -.5 -.0-0.5 0.0 0.5.0 No Provocation 0.3 0.4 0.5 0.6 0.7 0.3 0.4 0.5 0.6 0.7 Government Accuracy State Accuracy (a) Group Accuracy Model (b) Original Model Figure 4: Provocation and Government-Caused Damage 3