Model

Hamas (H) and Fatah (F) compete over a infinite number of periods indexed by t ∈ ℕ. In our data, a period corresponds to a calendar month. Period t's interaction explicitly depends on a publicly observed state variable $s^t\in {\cal S}$ that represents the popularity of Fatah relative to Hamas among Palestinians.Footnote ²³ The set of states ${\cal S} = { s_1, \;s_2, \;\ldots , \;s_K} \subseteq {\rm {\mathbb R}}$ comprises K ≥ 3 equally spaced popularity levels, where $k > {k}^{\prime}$ if and only if $s_k > s_{{k}^{\prime}}$. We say Fatah is relatively more popular in state s than in state ${s}^{\prime}$ if $s > {s}^{\prime}$, and vice versa for Hamas. In other words, smaller (larger) states represent periods when Hamas (Fatah) is more relatively popular.

In each period t, Hamas and Fatah choose whether to commit a terrorist attack ($a_i^t = 1$) or not ($a_i^t = 0$), where i = H, F indexes the group.Footnote ²⁴ Given an action profile $a^t = ( a_{\rm H}^t , \;a_{\rm F}^t ) $, per-period payoffs are $u_i( a_i^t , \;s^t; \;\theta ) + \varepsilon _i^t ( a_i^t ) $. The term $\varepsilon _i^t \in {\mathbb R}^2$ is a vector of action-specific payoff shocks that are private information to group i, where $\varepsilon _i^t ( a_i^t ) $ refers to the $( a_i^t + 1) $th element of the vector $\varepsilon _i^t $. The shock $\varepsilon _i^t ( a_i) $ is an independent and identically distributed (IID) draw from a standard type-one extreme value (T1EV) distribution.Footnote ²⁵ The shocks account for unobserved factors temporarily affecting the costs and benefits of terrorism and ensure that choices within each period are stochastic.

The term $u_i( a_i^t , \;s^t; \;\theta ) $ is the systematic component of group i's per-period payoff and consists of popularity benefits and attack costs:

(1)$$u_i( a_i^t , \;s^t; \;\theta ) = \underbrace{{\beta _is^t}}_{{{\rm popularity\ benefit}}} + \underbrace{{\kappa _ia_i^t }}_{{{\rm attack\ cost}}}, \;$$

where θ = (β _H, β _F, κ _H, κ _F). Because β _is ^t captures i's benefit from relative popularity level s ^t, we expect β _H < 0 and β _F > 0—that is, groups want more favorable support. This is one incentive for groups to compete. Likewise, κ _i denotes i's cost of attacking, which is another competitive incentive, and we expect κ _i < 0. Note that these inequalities are theoretical expectations from the outbidding literature. We do not impose them as a priori restrictions, but we explicitly test them after estimating the unobserved competitive incentives.

In contrast, outbidding theories do not offer explicit expectations about the relative magnitudes of β _i and κ _i across actors. It could be that Hamas cares more about relative popularity than Fatah (|β _H| > |β _F|) because Fatah has outside support from Israel and the United States, so it might care less about local Palestinian support. A similar argument suggests the opposite, however, because Hamas has outside support from Iran, Syria, and Qatar during our time frame. While intuition suggests that Hamas has lower attack costs given the differences in the groups’ use of violence, outbidding theories do not have explicit predictions about relative attack costs. The model accommodates all possibilities and allows us to quantify the differences post-estimation.

The sequence of the game in period t is as follows.

1. 1. Group i observes s ^t and $\varepsilon _i^t $.

2. 2. Groups simultaneously choose whether to attack: $a_i^t \in { 0, \;1} $.Footnote ²⁶

3. 3. Payoffs are accrued.

4. 4. Transition to period t + 1.

As the game transitions from period t to t + 1, popularity evolves according to an AR-1 process with a mean that depends on the chosen actions and state. Given today's support and attack decisions (a ^t, s ^t), we define the mean of tomorrow's support s ^t+1 as

(2)$$\mu [ a^t, \;s^t; \;\gamma ] = \gamma _0 + \gamma _1s^t + \sum\limits_i {( \gamma _{i, 1} + \gamma _{i, 2}s^t) } a_i^t .$$

The term (γ _i,1 + γ _i,2s ^t) represents group i's ability to use terrorist attacks to increase its support—what we call i's effectiveness of attacks, which is the third competitive incentive in the model. Outbidding theories expect γ _H,1 < 0 and γ _F,1 > 0; that is, attacks by group i pull popular support in i's preferred direction. These inequalities are theoretical expectations but are not imposed in estimation. As with the payoff parameters, outbidding does not have explicit expectations about the relative magnitudes of γ _F,1 and γ _H,1 (that is, about which group is more effective at using terrorism), but the model accommodates either possibility.

Note that Equation 2 allows the effects of i's attacks (γ _i,1 + γ _i,2s ^t) to depend on the current popularity level, s ^t. A priori, it is not clear whether group i's attacks should be more or less effective as its popularity increases. On the one hand, higher popularity could make its attacks more effective due to support from the local population, implying that γ _i,2 > 0; but on the other hand there is less of the population to be won over, implying that γ _i,2 > 0.

In period t + 1, the probability that $s^{t + 1} = {s}^{\prime}$ given action profile a ^t and state s ^t is $f( {s}^{\prime}; \;a^t, \;s^t, \;\gamma ) $. We specify f using a discretized normal distribution:

(3)$$\eqalign{f( {s}^{\prime}; \;a^t, \;s^t, \;\gamma ) = \cr \left\{{\matrix{ {\Phi \left({\displaystyle{{{s}^{\prime} + d-\mu [ a^t, \;s^t; \;\gamma ] } \over \sigma }} \right)-\Phi \left({\displaystyle{{{s}^{\prime}-d-\mu [ a^t, \;s^t; \;\gamma ] } \over \sigma }} \right)} \hfill & {{s}^{\prime}\in { s_2, \;\ldots , \;s_{K-1}} } \hfill \cr {\Phi \left({\displaystyle{{s_1 + d-\mu [ a^t, \;s^t; \;\gamma ] } \over \sigma }} \right)} \hfill & {{s}^{\prime} = s_1} \hfill \cr {1-\Phi \left({\displaystyle{{s_K-d-\mu [ a^t, \;s^t; \;\gamma ] } \over \sigma }} \right)} \hfill & {{s}^{\prime} = s_K} \hfill \cr } } \right.}$$

where Φ is the standard normal cumulative distribution function, σ is the standard deviation parameter, and 2d = s ₂ − s ₁ is the distance between the relative popularity levels. The parameters γ = (γ ₀, γ ₁, γ _H,1, γ _H,2, γ _F,1, γ _F,2, σ) describe the transitions of the game, and we estimate them later. We choose this specification because γ can be estimated using standard techniques for continuous AR-1 models even though popularity levels are discrete.Footnote ²⁷

Equilibria

Given a sequence of states, actions, and payoff shocks ${ s^t, \;a_i^t , \;\varepsilon _i^t } _{t = 1}^\infty $, group i's total payoffs are $\sum _{t = 1}^\infty \delta ^{t-1}[ {u_i( a_i^t , \;s^t) + \varepsilon_i^t ( a_i^t ) } ] $, where δ ∈ [0, 1) is a fixed, common discount factor. Discount factors are difficult to identify in dynamic discrete-choice models.Footnote ²⁸ Following Rust and others,Footnote ²⁹ we estimate the model at several discount factors and fix the discount factor at δ = 0.999, which gives the highest log-likelihood.Footnote ³⁰ This matches anecdotal descriptions of the groups that highlight their long time horizons.Footnote ³¹

Markov equilibria in discrete dynamic games with per-period private-information payoff shocks have a straightforward characterization.Footnote ³² Dropping references to time, let v _i(a _i, s) denote i's net-of-shock expected utility from choosing action a _i in state s and continuing to play the game. The vector $v_i = ( {v_i( a_i, \;s) } ) _{( a_i, s) \in { 0, 1} \times {\cal S}}$ collects these values for each (a _i, s) pair. Given a vector of expected utility values v _i and a vector of random shocks $\varepsilon _i = ( \varepsilon _i( 0) , \;\varepsilon _i( 1) ) $, group i chooses action a _i in state s if and only if

$$a_i = \mathop {\rm argmax}\limits_{_{a_i\in \{ 0, 1\} } } \;\{ v_i( a_i, \;s) + \varepsilon _i( a_i) \} .$$

Thus, v _i implicitly specifies a cutoff strategy for i, where i chooses to attack in state s if and only if $v_i( 1, \;s) -v_i( 0, \;s) > \varepsilon _i( 0) -\varepsilon _i( 1) $, where we sidestep the zero-probability event that i is indifferent. Because $\varepsilon _i( 0) $ and $\varepsilon _i( 1) $ are IID draws from a standard T1EV distribution, i chooses a _i in state s with probability

(4)$$P( a_i, \;s; \;v_i) = \displaystyle{{\exp { v_i( a_i, \;s) } } \over {\exp { v_i( 0, \;s) } + \exp { v_i( 1, \;s) } }}{\mkern 1mu} .$$

Let g denote the joint density of the action-specific payoff shocks, $\varepsilon _i$. Group i's continuation value for state s is

$$\displaylines{V_i( s, \;v_i) =\; {\int} \,\,{\mathop {\max }\limits_{_{a_i\in \{ 0, 1\} } } } { \{v_i( a_i, \;s) + \varepsilon_i( a_i) \} } g( \varepsilon _i) d\varepsilon _i \cr \hskip6pc = \log ( {\exp { v_i( 0, \;s) } + \exp { v_i( 1, \;s) } } ) + C, \;}$$

where C is Euler's constant. The second equality in Equation 5 follows from McFadden because g is the joint density of two IID standard T1EV random variables.Footnote ³³ Consider a profile v = (v _i, v _j) of action-state expected utility values. Group i's iterative net-of-shock expected utility of action a _i in state s, denoted ${\cal V}_i( a_i, \;s, \;v; \;\theta , \;\gamma ) $, is

(6)$${{\cal V}_i( a_i, \;s, \;v; \;\theta , \;\gamma ) = \underbrace{{u_i( a_i, \;s; \;\theta ) }}_{{\scriptstyle i{\rm ^{\prime}{\rm s}\ \rm payoff\ } \atop \scriptstyle {\rm today} } } + \delta \underbrace{{\boldsymbol[ \sum\limits_{a_j} P ( a_j, \;s; \;v_j) \underbrace{{\sum\limits_{{s}^{\prime}\in {\cal S}} f ( {s}^{\prime}; \;a_i, \;a_j, \;s, \;\gamma ) V_i( {{s}^{\prime}, \;v_i} ) }}_{{\scriptstyle i^{\prime}{{\rm s}\ expected\ continuation} \atop \scriptstyle {\rm value\ given\ }a_j } }\boldsymbol] }}_{{{\rm iterated\ expectation\ over\ }j{\rm ^{\prime}{\rm s}\ action}}}}.$$

An equilibrium is a profile v that satisfies the fixed-point condition

(7)$$v = {\cal V}( v; \;\theta , \;\gamma ) \equiv{\times} _i \times _{( a_i, s) }{\cal V}_i( a_i, \;s, \;v; \;\theta , \;\gamma ) .$$

Equations 4 to 7 characterize equilibria as a system of 4K equations, where K is the number of relative popularity levels. In words, starting with i's net-of-shock, action-specific expected utilities, Equations 4 and 5 return i's choice probabilities and continuation values, respectively. Then Equation 6 updates i's net-of-shock action-specific expected utilities, holding fixed i's continuation values and j's choice probabilities. An equilibrium is a fixed point in Equation 7. In Appendix B, we consider a symmetric example, use Equation 7 to compute equilibria, and then study their substantive properties and comparative statics.

Remarks

Three remarks on the model. First, because this is a model of outbidding, it explains variation in violence via intergroup competition. Such a spartan approach is critical for our argument: outbidding produces heterogeneous relationships between competition and violence, and one such relationship is the discouragement effect. Furthermore, this discouragement effect appears in the canonical case of outbidding, and other forces are not necessary to generate discouragement effects. Adding more moving parts to the analysis—while potentially interesting in future work—only obfuscates this central result. Thus, the model does not include other motives for terrorism considered by Kydd and Walter.Footnote ³⁴

We therefore prioritize matching the model to outbidding theories and abstract away from many other features that appear in the broader literature. Overwhelmingly, outbidding theories focus on terrorist groups’ use of violence.Footnote ³⁵ Conrad and Spaniel also use a contest to build a theory of outbidding.Footnote ³⁶ Groups may use other tools, like public goods provision, to boost their support, but studies considering how groups provide public goods focus on their competition with the government, not rival groups.Footnote ³⁷ Likewise, a key assumption in most outbidding theories is that the government is tangential.Footnote ³⁸ Nonetheless, in Appendix D we discuss whether and how unobserved government actions can affect our parameter estimates and explicitly control for government actions when considering the robustness of our approach. Likewise, because this is a model of outbidding, it abstracts away from some of the specifics of the Fatah-Hamas rivalry, such as principal–agent problems within the groups.

The model is not inconsistent with other features affecting terrorism and public support. It includes exogenous shocks to the costs of attacking, and relative popularity evolves stochastically, capturing outside forces. We put assumptions on these features, but they mimic standard assumptions in reduced-form work (for example, $\varepsilon _i^t $ is drawn IID). As discussed by Canen and Ramsay, all quantitative empirical work requires models and assumptions to make causal claims, and this paper is no exception.Footnote ³⁹ In subsequent sections, we examine model fit and the robustness of our inferences to different modeling assumptions. For the former, we test outbidding's theoretical expectations concerning groups’ competitive incentives and explicitly compare the outbidding model to competing models. For the latter, numerous robustness checks show that our estimates and predictions are insensitive to excluding or including specific time frames where there are known shifts in the conflict. Our estimates of the groups’ payoffs are unaffected by uncertainty in the estimated transition function f and by small changes in the discount factor. As mentioned, a key feature of our estimates is that Fatah is more effective than Hamas at using violence to increase public support. Appendix D is dedicated to demonstrating the robustness of this result to different terrorism measures, control variables, and (unobserved) omitted variables.

Second, we do not explicitly model the decisions of individuals who choose a group to support. This is a simplifying assumption that also appears in Conrad and Spaniel and structural models of dynamic elections.Footnote ⁴⁰ Instead, individuals and their choices are captured by Equations 2 and 3, which describe how relative support evolves given the attack decisions of the two groups and their current popularity. Rather than microfounding this behavior, we calibrate it to data by estimating the relevant parameters of interest, γ. This allows us to sidestep additional assumptions detailing the preferences and decisions of local individuals, which could be quite complicated.Footnote ⁴¹ We do not assume that they want violence or that terrorism must increase support; indeed, the model accommodates the possibility that terrorism decreases support. When we estimate the γ _i,1 and γ _i,2 parameters, however, we find that violence does increase relative support for the two groups (that is, γ _i,1 + γ _i,2s ^t is estimated to be positive for Fatah and negative for Hamas at the observed relative popularity levels s ^t). This result is consistent with previous empirical work,Footnote ⁴² Bloom's case study of the conflict,Footnote ⁴³ and the theoretical mechanism underlying outbidding.Footnote ⁴⁴

Third, the model treats competitive incentives as exogenous actor-specific parameters. This matches other contest models, and we explicitly borrow their terminology of “value,” “cost,” and “effectiveness.” On the one hand, this approach allows us to flexibly accommodate time-invariant, group-level heterogeneity. That is, these parameters can be composed of time-invariant, group-specific features, and we avoid additional functional-form assumptions by treating these as fixed effects. This is a strength because we anticipate systematic differences between Hamas and Fatah, such as different corruption levels, relationships with Israel, and capacities for violence.Footnote ⁴⁵

On the other hand, several interesting questions arise about the origins of these incentives—for example, why does one group have lower costs than another? We choose to prioritize a flexible model with actor-specific parameters rather than endogenizing the parameters because the contest literature anticipates that the discouragement effect appears when groups have asymmetric incentives.Footnote ⁴⁶ Also, our data cover only two groups, so we cannot leverage cross-sectional variation to determine the covariates of the competitive incentives. In Appendices D and H, we explore whether these incentives change across time periods, but we find little variation. If future work fits the model to several different conflict environments, then we can compare how these incentives vary in a post-estimation exercise—in a manner similar to how Crisman-Cox and Gibilisco study the correlates of their estimated audience cost parameters.Footnote ⁴⁷

Data Sources and Measurement

Terrorism data are from the Global Terrorism Database (GTD), where we record terrorist attacks by Fatah/PLO and Hamas from January 1994 to December 2018.Footnote ⁴⁸ The GTD records both suicide bombings, which are the focus of Bloom and Findley and Young, and other types of terrorism, such as rocket attacks, which are a greater part of violence against Israelis in recent years.Footnote ⁴⁹ Hamas engages in an average of roughly 1.5 attacks per month, while Fatah engages in an average of less than one per month (see Figure A.1).Footnote ⁵⁰ To measure group i's attack decision in month t, we use a dummy variable for whether the group committed any terrorist attacks in that month.Footnote ⁵¹

The model's state variable represents the relative popularity of the two groups among Palestinians. To measure it, we treat relative popularity as a dynamic latent variable and use observed public-opinion variables as its indicators. To assemble the set of indicators, we use surveys from the Jerusalem Media and Communication Centre (JMCC) and the Palestinian Center for Policy and Survey Research (PCPSR).Footnote ⁵² The JMCC publishes two to six surveys per year using random samples of Palestinian adults. It conducts face-to-face interviews in randomly selected households from randomly selected neighborhoods throughout the West Bank and Gaza Strip; within each home, the subjects inside are selected using Kish tables. Each survey typically occurs over a few days but less than a week. The average sample size is 1,205, with a range between 815 and 1,920.Footnote ⁵³ On average, 63 percent of respondents are from the West Bank. Given their rich data on Palestinian attitudes, these surveys also appear in other studies.Footnote ⁵⁴

The PCPSR runs two to nine surveys per year. It generally uses a multi-step selection process, randomly sampling from locations in proportion to the population from a list of all cities, towns, villages, and refugee camps in the West Bank and Gaza Strip. Once locations are selected, it samples individual blocks and then individual households. Each survey typically occurs over several days but less than a week. Sample sizes vary between 1,076 and 2,006, with a mean of 1,312.Footnote ⁵⁵ West Bank respondents tend to make up 60 to 67 percent (mean, 63 percent) of the overall sample, with Gaza respondents making up the rest.

We search through every survey published by these centers between 1994 and 2018 to track Palestinian public opinion on both actors using three questions. The first (from the JMCC) asks which political or religious group the respondent trusts most. The second (from the PCPSR) asks which political party they support. The third (from the JMCC) asks which party they intend to vote for in the legislative elections. For each of these three questions we track the proportion of respondents who answer Hamas or Fatah.Footnote ⁵⁶ All three questions are open-ended. Appendix C gives details on question wording and variation by geography.

Figure 1 graphs responses to these six survey questions over time. Generally, public attitudes toward the two groups are inversely related. During the 1990s and early 2000s, support for Fatah falls and support for Hamas rises. These trends level out a bit in later years, with Fatah regaining some support at the expense of Hamas. The surveys mostly correlate with each other in the expected directions (Table C.2), which suggests that they can be collapsed onto one dimension. To do this, we use a dynamic factor model that transforms these polling questions into a continuous representation $\tilde{s}^t$ of the theoretical state variable s ^t. See Appendix C for details.

FIGURE 1. Survey responses over time

Having constructed the continuous state variable $\tilde{s}^t$, we assess its validity. All indicators load onto the factor in the expected directions (Table C.3). Figure 2 shows how the state variable evolves from 1994 to 2018. Fatah is favored in earlier periods; its relative popularity peaks during the 1996 Oslo II process (about 12.5 in January 1996). Hamas is at its most popular relative to Fatah in 2006, in the aftermath of the general election in which Hamas took control of Gaza (about –10.9 in August 2006). The mean of this variable is −0.87; the median is about −3, and the standard deviation is 6.59 (with an interquartile range of −6.02 to 4.13). The continuous state variable is easily mapped back onto the original surveys, such that, on average, a one-unit increase in $\tilde{s}^t$ roughly corresponds to increases of 0.9, 1.5, and 2 percentage points in net trust, support, and intention to vote for Fatah over Hamas, respectively.

FIGURE 2. Popularity of Fatah relative to Hamas over time

Several important events are flagged in Figure 2, providing context and face validity to the idea that this variable captures the relative ups and downs of the two groups. The late 1990s are typically regarded as an important inflection point for these two groups, and that is clearly reflected here. Fatah sees its popular support erode as the peace process unravels. Furthermore, our measure has rich variation, with substantial ups and downs that went undetected in earlier measures of these groups’ popularity.Footnote ⁵⁷ In Appendix C we demonstrate that our latent measure of relative popularity is robust to different model specifications. The estimated state variables are closely correlated (0.87–0.99) across specifications.

Estimation and Identification

We use a two-step procedure, where we first estimate how relative support evolves (γ) and then estimate the groups’ payoff parameters (β, κ).Footnote ⁵⁸ To do this, first rewrite the AR-1 model in Equation 2 in terms of the continuous state variable $\tilde{s}^t$:

(8)$$\tilde{s}^t = \gamma _0 + \gamma _1\tilde{s}^{t-1} + \gamma _{{\rm H}, 1}a_{\rm H}^{t-1} + \gamma _{{\rm H}, 2}( \tilde{s}^{t-1}a_{\rm H}^{t-1} ) + \gamma _{{\rm F}, 1}a_{\rm F}^{t-1} + \gamma _{{\rm F}, 2}( \tilde{s}^{t-1}a_{\rm F}^{t-1} ) + \nu ^t, \;$$

where and $a_{\rm H}^{t-1} $ are binary indicators for whether Fatah and Hamas attack, respectively, and ν ^t ~ N(0, σ ²).Footnote ⁵⁹

The first-step estimates are then used to construct the Markov transition probabilities, f. To discretize the continuous state $\tilde{s}^t$, we define the lowest and highest (most Hamas-friendly and most Fatah-friendly) states as the bottom and top 2.5th percentiles of $\tilde{s}^t$. Discrete states between these extremes are defined at equally spaced intervals with distance 2d = 0.05. In the baseline model, K = 440. We then map the continuous measure $\tilde{s}^t$ into the discrete measure s ^t by finding the closest discrete state.Footnote ⁶⁰ Let $\mu [ a, \;s; \;\hat{\gamma }] $ be the fitted values from the first model (reported in Table 1) for all possible combinations of action profiles with the discrete states. Plugging these fitted values and the estimated standard deviation $\hat{\sigma }$ into Equation 3 produces the transition probabilities.

TABLE 1. Regressing relative popularity (state variable) on terrorist attacks

Notes: Newey-West standard errors in parentheses. No standard errors are reported for the AR(1) model due to unit root. ECM refers to the Engle-Granger error correction model.

We use a constrained maximum likelihood estimator (CMLE) to estimate the payoff parameters θ = (β, κ).Footnote ⁶¹ Specifically, let $Y = ( {s^t, \;a_{\rm H}^t , \;a_{\rm F}^t } ) _{t = 1}^T $ denote the time series of observed data (relative popularity levels and attacks). We fix the transition probabilities using the first-step estimates, $\hat{\gamma }$, and the definition of f in Equation 3. The CMLE estimates $( \hat{\theta }, \;\hat{v}) $ maximize the log-likelihood

$$L( v\vert Y) = \sum\limits_{t = 1}^T {[ {\log P( a_{\rm H}^t ; \;s^t, \;v_{\rm H}) + \log P( a_{\rm F}^t ; \;s^t, \;v_{\rm F}) } ] } $$

subject to the equilibrium constraint equations $v = {\cal V}( v; \;\theta , \;\hat{\gamma }) $. For standard errors, we follow Silvey in using the bordered Hessian to compute the variance-covariance matrix and use a two-step correction described in Appendix F.Footnote ⁶²

The game can have multiple equilibria. The CMLE allows for this multiplicity, with its main identification assumption being that the data Y are generated from only one of these equilibria.Footnote ⁶³ By treating the endogenous equilibrium quantities, v, as auxiliary parameters, the CMLE selects the values of v that best describe the data while still being an equilibrium of the model. In other words, the CMLE imposes an empirical selection rule: choose the equilibrium associated with the highest log-likelihood. This process is a computationally feasible alternative to an approach that computes all equilibria at every optimization step and then always chooses the equilibrium that maximizes the log-likelihood at that optimization step.Footnote ⁶⁴ The CMLE imposes this same empirical selection rule, but without the infeasible requirement of repeatedly enumerating all equilibria.

Along with the assumption that one equilibrium is generating the data, three empirical moments pin down our parameters of interest. We estimate γ through observed variation in the state variable over time. We know that each action profile has a positive probability of being played at each relative popularity level given the distributional assumptions on $\varepsilon _i^t $, and that the probability of transitioning from level s to level ${s}^{\prime}$ is positive for all s and . Thus f can be estimated nonparametrically from frequency estimators with a sufficiently long time frame, because the equilibrium path will eventually visit all states, and all action profiles will be played in every state. When the transition probabilities are known, the payoff parameters are identified by their relationship to the equilibrium constraint ${\cal V}$ in Equation 6. A group's attack cost is identified through its baseline propensity to attack regardless of the state, and its value of public support is identified by the variation in its propensity to attack across states.

Formal identification of the payoff parameters θ follows from Propositions 2 and 3 in Pesendorfer and Schmidt-Dengler.Footnote ⁶⁵ The former is a necessary condition stating that up to K payoff parameters per actor can be identified. We seek to estimate two parameters per group using K = 440 states, which satisfies the necessary condition. The connection between K and identification raises a question about how sensitive the estimates are to discretizing relative popularity; in Appendix J we show that our estimates are robust to both small and major changes in this process. The latter is a more involved sufficient condition for identifying θ that depends on the equilibrium choice probabilities, which we can verify given our estimated equilibrium (Appendix E).

Although the groups’ competitive incentives can be identified given data generated from an equilibrium of the game, another reasonable concern is how sensitive the estimated incentives are to forces outside the model, in particular to interventions from Israel. Here, we anticipate that the importance of Israeli actions depends on whether the incentive is group effectiveness or directly enters into the groups’ payoff functions. For the former, we can compare our baseline estimates of γ _i,1 to those in robustness exercises where we control for Israeli actions or their proxies (such as the number of Palestinian fatalities or the time since the last Israeli election) or instrument group attacks with rainfall. Our estimates of groups’ attack effectiveness are stable across specifications (Appendix D).

For the latter, the analysis is murkier because we are unable to conduct such robustness exercises. If we wanted to include Israeli interventions when estimating the groups’ value of support and cost of attacking, then we would need monthly data on Israeli actions against the individual groups during our time frame. Furthermore, we would need to either estimate how these actions evolve according to relative popularity levels and attack decisions, or explicitly model the Israeli government as a third strategic actor. Given the scarcity of high-frequency data on how Israel responds to individual groups, and given that outbidding theories generally treat governments as tangential, we think that the appropriate first step is to structurally estimate an outbidding model without government interventions. Nonetheless, we anticipate that the groups’ costs of attacking include both upfront costs (such as explosives) and strategic backlash from the Israeli government (such as border walls and air strikes). Also, if Israeli interventions are aimed at reducing the likelihood of attacks, then these interventions should target groups precisely when the tug-of-war predicts high attack probabilities. Thus, the observed probability of attacks would appear flatter as a function of relative popularity than in a world without interventions. This would attenuate our estimates of the groups’ values of support, because these are identified by variation in changes in attack probabilities as a function of relative popularity.

Parameter Estimates

Table 1 shows the first-stage estimates. Attacks by Fatah and Hamas move the state space in the expected direction. Recall that estimates of γ _i,1 reflect each group's effectiveness at using terrorism to shift public support toward itself and away from its rival. In months when Hamas attacks, its relative popularity improves by an average of about 0.11 to 0.28 in the following month. When Fatah attacks, it can expect its relative popularity to improve by about 0.87 to 1.4 on average. As mentioned, the scale of $\tilde{s}^t$ can be roughly compared with the net level of trust in Fatah over Hamas, so on average, these magnitudes roughly reflect shifts in net levels of trust for Fatah over Hamas.Footnote ⁶⁶ Both of these effects are statistically significant in the ECM model. These results provide evidence that groups are capable of outbidding and that acts of terrorism carry popularity benefits for the group committing them.Footnote ⁶⁷

We also find that Fatah's use of terrorism increases support for Fatah more effectively than Hamas's use of terrorism increases support for Hamas. Specifically, we reject the hypothesis that the groups are equally effective at moving public opinion (H ₀:γ _H,1 + γ _H,2 ⋅ s + γ _F,1 + γ _H,2 ⋅ s = 0) at every level of relative popularity s using the estimates and Newey-West variance matrix from the ECM model.

One possible explanation is that attacks by Fatah provide more information to the public. This could be for a variety of reasons. For example, it is often seen as the more pro-peace actorFootnote ⁶⁸ or, alternatively, as the more corrupt or possibly inept actor.Footnote ⁶⁹ Attacks by Hamas are expected, so they do little to adjust public opinion. Attacks by Fatah, being more surprising, are more likely to shift the public's beliefs about how committed Fatah is to the Palestinian cause. Thus, even though attacks demonstrate the resolve of both groups, Fatah's attacks give it a larger boost in public opinion. This explanation is consistent with our parameter estimates, but it is of course a conjecture because it involves assumptions about the Palestinian population that we deliberately did not microfound. Future studies should consider the population side of the outbidding process to better explain these asymmetries in the response to terrorism.

In Appendix D, we show that these relationships are not driven by omitted economic or political factors, such as unemployment, Palestinian attitudes toward violence, the Second Intifada, Israeli election timing, or Palestinian fatalities from Israeli forces (which is one proxy for government actions). We also find no evidence that the groups are becoming more or less effective during our time frame (Table D.3). Overall, the relationships between attacks and shifts in public support are largely unchanged in either direction or magnitude across model specifications.

We also consider alternative measures of attacks. Whether we use attack counts, fatalities, or fatalities per attack, we find Fatah is more effective than Hamas (Table D.4). We also study plausibly exogenous variation in attacks driven by extreme rainfall shocks in the Gaza Strip and the West Bank.Footnote ⁷⁰ Our baseline estimates of γ _F,1 and γ _H,1 (in Table 1) are similar in direction and magnitude to those from an instrumental-variables analysis, although we do not want to over-interpret these results (see Appendix D.3 for details).

We also conduct formal sensitivity analyses for our estimates $\hat{\gamma }_{{\rm F}, 1}$ and $\hat{\gamma }_{{\rm H}, 1}$ (Appendix D.4). Here, we ask how strong omitted variables would have to be to make the asymmetry between these estimates disappear or to make the estimates null. We find that these unobserved effects would have to be implausibly large to explain away our findings.

Table 2 presents estimates for the two groups’ value of popularity and cost of attack. The sign on each estimate is in the direction expected by outbidding theory and is statistically significant at conventional levels (one-sided tests). Both actors like being more popular than their opponent. It may be concerning that the β _i estimates are quite close to 0, but we reject the null hypothesis that both are 0. Furthermore, the estimates of β _i have strong impacts on the equilibrium attack probabilities despite their seemingly small magnitudes (see Appendix K). Interestingly, Hamas values its support more than Fatah, with $\vert \hat{\beta }_{\rm H}\vert $ being an order of magnitude larger than $\vert \hat{\beta }_{\rm F}\vert $. One possible explanation for this is that Fatah has more support from outside actors to consider than Hamas. While that would be consistent with our parameter estimates, our analysis cannot rule out other explanations.

TABLE 2. Payoff estimates

Note: Bordered-Hessian (BH) standard errors and two-step corrected standard errors.

Intuitively, we find that terrorism is less costly for Hamas than for Fatah, a finding which itself has several potential explanations. First, it could reflect different preferences between the two groups regarding violence. Second, Hamas has made a concerted effort to build its capacity for violence by developing infrastructure to acquire weapons and better train its members. Hence, the group would find it less costly to engage in violence than Fatah, which has devoted more resources to governance and engagement with the Israeli and US governments. Both explanations fit with the historical record, which typically depicts Hamas as a more extreme actor while Fatah is a more practical political entity.Footnote ⁷¹

Beyond the face validity of the point estimates, we consider the robustness of the estimates in Table 2. In Appendix F, we conduct a sensitivity analysis to demonstrate that they are stable across a range of plausible first-stage estimates. In Appendix H, we consider several shorter time frames that represent potential starts, stops, or change points in the Hamas-Fatah rivalry—for example, ending the analysis with the 2011 coalition agreement, or starting in 1997.Footnote ⁷² In Appendix J, we demonstrate robustness to how we discretize our measure of relative popularity; our results are stable even with a small number of states (15 ≤ K ≤ 22).

Figure 3 graphs the groups’ estimated attack probabilities over time—that is, $P( a_i = 1; \;s^t, \;\hat{v}_i) $. We also graph the relative popularity level s ^t over time on the second horizontal axis for reference. Notice that Hamas has a higher probability of attacking than Fatah regardless of its relative popularity. Averaging over the observed states, Hamas attacks with probability 0.42, and Fatah with probability 0.11. This maps onto our estimates. Hamas cares more about its popularity than Fatah, and it has a comparatively lower attack cost, although Fatah uses terrorism more effectively to increase its support. Also, terrorism is particularly prevalent when Hamas is relatively popular, specifically during the Second Intifada and after the group wins legislative elections in 2006.

FIGURE 3. Estimated equilibrium probability of attacking over time

Model Fit and Comparison

Before considering the substantive implications of the estimated outbidding model, we consider how well it describes the data, both on its own terms and in comparison to alternative theories. In this section, our goal is not to test a particular hypothesis but to demonstrate the validity and usefulness of the model in explaining variation in the observed terrorism data.

For the first exercise, recall that the estimated competitive incentives match the direction posited by outbidding theories: attacking is costly, groups value support, and attacks increase relative support. These restrictions were not imposed during estimation, and we would be skeptical of outbidding's ability to explain the data if they did not hold. For example, how could outbidding be a consistent theoretical explanation if groups wanted to become less popular? We can also examine the states in which the model predicts attacks well; Figure 4 does so visually. Ideally, we should see more attacks when the equilibrium choice probabilities are higher, all else equal. For the most part, this is true: observed attacks fall mostly when relative popularity s is between −7 and −3, where the equilibrium choice probabilities peak.

FIGURE 4. Estimated equilibrium attack probabilities as a function of state

Nonetheless, the data show a cluster of Hamas attacks around s ∈ [8, 9], where attack probabilities are smaller. These attacks are difficult to attribute to outbidding as Hamas was close to its nadir of popularity, and the estimates suggest that, given the relatively few number of periods, this many attacks are unlikely. Thus, it seems reasonable to suspect that another theory of terrorism may better explain these attacks. Kydd and Walter argue that some of these attacks were part of an attempt by Hamas to undermine or “spoil” the Oslo process and drive a wedge between Fatah and Israeli negotiators.Footnote ⁷³ If these attacks are indeed more associated with spoiling and less attributable to outbidding, then it is not surprising that they stand out in Figure 4. This analysis highlights an advantage of this structural approach: it allows us to identify observations that do not fit the theory's predictions.

It also raises a question: can other theories of terrorism better explain the data? It is beyond the scope of this paper, or any one paper, to consider and adjudicate among all theories of terrorism. Indeed, we think the field's understanding of the strategic forces behind terrorism will advance if scholars construct competing models of terrorism from different theories and estimate these models on the same data. This would let us compare the models and their associated theories regarding how well they explain observed terrorism. Given the lack of previous structural models of terrorism, we have no obvious prior model to compare ours to. Thus, we create some alternative structural models and acknowledge that, until more models are available, our comparison models are inherently ad hoc.

The first alternative model is a null model in which there is no competition between groups because they cannot (or do not care to) compete for popularity. We can nest such a model within our outbidding model by assuming γ _F,1 = γ _F,2 = γ _H,1 = γ _H,2 = 0. With this assumption, we cannot identify β. The only parameters left to fit the no-competition model are κ _F and κ _H—that is, groups are attacking without reference to relative popularity and only due to static incentives. Because this alternative model is nested, it can be compared to the main model using a standard likelihood ratio test. We reject the null hypothesis that the no-competition model fits as well as the main model (Table 3).

TABLE 3. Comparative model tests

The second alternative is a non-nested model based on tit-for-tat retaliation.Footnote ⁷⁴ For this model, when group i chooses to attack ($a_i^t = 1$) or not ($a_i^t = 0$) in each period, we make the following assumptions:

1. 1. The new publicly observed state variable $r^t = ( r_{\rm F}^t , \;r_{\rm H}^t ) \in { 0, \;1} \times { 0, \;1} $ is a two-dimensional variable that represents whether each actor attacked in the previous period, with $r_i^t = 1$ meaning group i attacked in period t − 1.

2. 2. The systematic utility function for group i is now

   $$u_i( a_i^t , \;r^t; \;\tau , \;\kappa ) = a_i^t ( \underbrace{\kappa _i}_{{\scriptstyle {\rm baseline} \atop \scriptstyle {\rm cost} } } + \underbrace{{\tau _ir_{^{{-}1}}^t }{),} } _{{\scriptstyle {\rm retaliation} \atop \scriptstyle {\rm benefit }}}\;$$
   where κ _i is the baseline cost of attacking and τ _i is the additional benefit or cost a group receives when attacking in response to a previous attack by its rival. We collect these parameters into the vectors κ = (κ _H, κ _F) and τ = (τ _H, τ _F).

As in the baseline model, we assume that groups’ per-period payoffs depend on privately observed action-specific payoff shocks that are IID draws from the standard T1EV. Additionally, this tit-for-tat model is a discrete dynamic game, so we can use techniques almost identical to those used with the main model to characterize Markov equilibria, except with appropriate changes to the utility functions and the state transitions, which are now deterministic, as $r_i^t = a_i^{t-1} $. Moreover, we can use a CMLE to fit the model to the same GTD data to estimate κ and τ.Footnote ⁷⁵ The goal is to compare this model with our outbidding model regarding how well they explain the attack data.Footnote ⁷⁶

The point estimates from the tit-for-tat model are all in the expected directions for a tit-for-tat theory: attacking is costly, but groups have an additional benefit if they attack in response to their rival (Appendix G.1). Comparing the tit-for-tat model against the outbidding model requires a non-nested model test. We use Clarke's test, which is a comparison of “pointwise” log-likelihood values.Footnote ⁷⁷ The null hypothesis is that the two models are equally good; we reject this, finding that the outbidding model better fits the data (Table 3). Overall, we conclude that the outbidding model explains the data better than the tit-for-tat model.Footnote ⁷⁸

Substantive Effects of Competition on Violence

What is the substantive effect of competition on violence? Does heightened competition encourage or discourage violence? Answering these questions without a structural analysis is difficult because raw attack rates—even changes in attack rates—cannot be used as evidence for either discouragement or encouragement effects. If we see a group using violence more (or less) frequently in a given time frame, then the pattern could be explained by small (or large) attack costs, the equilibrium path visiting states in which a group is likely (or unlikely) to use violence, or merely small-sample bias arising via stochastic decisions. Instead of interpreting the data atheoretically, we use the fitted structural model to quantify how a group's use of violence changes as competition changes while holding everything else equal. To do this, we warp different aspects of competition in the fitted model while keeping other parts fixed and record how its predictions concerning the groups’ use of violence would change in response. We consider this in two ways, by adjusting each actor's competitive behavior and then their competitive incentives.

Before proceeding, note that these exercises differ conceptually from those that would be computed from standard regression-based studies. Instead of focusing on observed correlations between competition and violence to quantify the effects of interest, we construct counterfactuals based on the theoretical model. Specifically, we take the competitive environment described by the estimated model and equilibrium as fixed and change specific features (re-solving for equilibrium) to find the effects of competition on violence. For example, the discouragement effects we will describe do not reflect an observed correlation between terrorist attacks and relative popularity. Rather, we are comparing the groups’ estimated attack probabilities to their attack probabilities in the counterfactual world, holding fixed relative popularity.

This approach has two main benefits. First, it does not require untested proxy variables for competition. Second, we do not have to worry that unobserved confounders related to both competition and violence create a spurious result. The reason for this is that we control exactly what is changing in the model and change only parts of the model related to competition (individual group behavior or one of the competitive incentives). In other words, after fitting the model, we can tweak it in specific ways that are related to competition only and assess the changes. Evaluating the consequence of policies or behaviors that have never been observed is one strength of structural approaches because, without a model, it is unclear how to estimate the effects of such changes that are outside the data's support.Footnote ⁷⁹

The main concern with this approach is if the model poorly describes the data or if we have done a poor job estimating its primitives; we considered these concerns in the model fit section, and say more in Appendices D and F to J.Footnote ⁸⁰ To be clear, these are not trivial concerns; however, we believe that the structural approach is complementary to more traditional empirical analyses because each comes with its own set of limitations and trade-offs.

Effect of Competitive Behavior on Violence

First, we compare how a group behaves with and without violence from its rival. That is, would Fatah use more or less violence if Hamas did not engage in terrorism, and vice versa? Specifically, we compare group i's estimated equilibrium attack probability (in Figure 3) to i's attack probability in its single-agent problem—that is, i's predicted use of violence if it expects its rival to never attack. Subtracting the latter from the former is one way to quantify the effect of competitive behavior on violence where the equilibrium attack probabilities represent violence in a competitive environment and the single-agent attack probabilities are from a noncompetitive environment. Figure 5 graphs these differences over time given the observed relative popularity s ^t. Positive values indicate a positive effect of competition on violence, where a group's equilibrium probability of attacking is higher than its probability of attacking in its single-agent problem. Negative values indicate a negative effect.Footnote ⁸¹ Thus, one interpretation of the figure is that the value in month t with popularity level s ^t indicates the effect on group i's immediate attack probability if group −i were to stop using violence in all future periods.Footnote ⁸²

FIGURE 5. Effects of competitive behavior on violence

Before interpreting this figure, we provide some additional historical context for the period after the 2006 election. This era is characterized by various reconciliation attempts and agreements, with different levels of success, as well as several moments of tension. The first post-election spike in Figure 5, for example, appears during the 2007 Battle of Gaza and the consolidation of Hamas control in the Gaza Strip.Footnote ⁸³ The relatively flat spot of this figure runs from 2009 to 2014, a period largely characterized by reconciliation talks, while the peak in 2014 is around the time of a failed coup attempt, when Hamas tried to unseat Fatah's leadership in the West Bank.Footnote ⁸⁴

Turning to the counterfactual, for Fatah, the values are entirely positive, indicating that Hamas encourages Fatah to use more violence than it would in the absence of competition. On average, competition from Hamas increases Fatah's use of violence by 34 percent relative to the counterfactual noncompetitive environment. This is the encouragement effect of competition on violence expected by the outbidding literature. Table 4 decomposes the effect into three time periods. Fatah's propensity for terrorism increases by about three percentage points due to competition from Hamas, especially after the start of the Second Intifada.

TABLE 4. Average effect of competitive behavior on violence in three eras

Note: Average difference between equilibrium and single-agent attack probabilities from different eras, with standard errors in parentheses.

For Hamas the story is different, as heterogeneous effects exist. Competition from Fatah depresses Hamas's use of violence during the Oslo era, although we find a positive effect during and after the Second Intifada. During the Oslo era, Hamas's propensity for terrorism would increase by about one percentage point, on average, in the absence of competition from Fatah. This point estimate represents an average over this period, but if we consider the largest monthly effect, then we would predict a 9 percent increase in Hamas attacks were Fatah to commit no violence. Put differently, this corresponds to a 4 to 5 percent reduction in violence by Hamas during the Oslo era compared to its counterfactual single-agent problem where Fatah never attacks. This is the discouragement effect of competition on violence, where a group uses less violence in the competitive environment than in a noncompetitive one. Substantively, this change implies two or three more months with Hamas terrorism in the counterfactual world than in the observed data. While this is a relatively small effect, the potential devastation and loss of life associated with any given attack (particularly by Hamas) means that it is likely to be substantively meaningful.

These estimates suggest a competition-based explanation for the Oslo lull. Specifically, the popularity of the peace process of the 1990s boosted Fatah's standing among the Palestinian population. Figure 1 shows that Fatah frequently leads Hamas by thirty percentage points in terms of trust and support during this time. Accordingly, relative popularity is overwhelmingly in Fatah's favor relative to the rest of the sample (Figure 4). Hence, although Hamas has incentives to use violence, it also knows that the competition is very lopsided in Fatah's favor. Furthermore, Fatah is also more effective at using violence to increase its popularity, which depresses Hamas's use of violence.

This theoretical account has anecdotal support in some contemporary understandings of the conflict. As Kristianasen writes, “While the Oslo agreement consecrated Hamas's role as a new national resistance to Israel, it ushered in a reality that progressively would tie the movement's hands.”Footnote ⁸⁵ They further argue that Hamas had trouble remaining relevant during parts of this period due to Fatah's popularity, and that delays and discontentment with the peace process (that is, negative shocks to Fatah's public approval) helped Hamas remain relevant. Others affirm this trouble, pointing out that “Hamas was swimming against a tide of popular support” during this era,Footnote ⁸⁶ and that in the mid-to-late 1990s, Hamas reduced their operations in the face of popular resistance.Footnote ⁸⁷ While Hamas still pulled off several high-profile attacks during this time, its overall public support was low enough that it was unclear to contemporary observers whether the group would continue to be a relevant actor.Footnote ⁸⁸

This explanation requires two caveats. First, it is not the only possible explanation for this period of Fatah-Hamas interactions. As mentioned, the model does not include other key aspects of this relationship, such as the efforts of Hamas to spoil the peace, outside of a desire for local support. Second, because our results are derived from counterfactual comparisons to a world where Fatah never uses violence, qualitative evidence cannot directly support the effects described here. Such evidence is inherently indirect, as the internal workings and strategic calculations of groups are often not well known in the real world and are completely unknown in the counterfactual world. However, the historical record does lend credence to the idea that Hamas was deterred by Fatah's popularity during this period; contemporary writers and conflict historians acknowledge that Fatah's popularity at that time had a notable effect on Hamas’ strategic calculus.

We see in Figure 5 and Table 4 that the presence of a rival terrorist group can depress violence. With a rival that is an effective outbidder (Fatah), a group (Hamas) may use less violence than it normally would when it falls behind in the race for public opinion and sees the competition as increasingly difficult. As Figure 5 illustrates, this discouragement effect emerges in the Oslo era, when Fatah was more popular than Hamas and at its peak of popularity. Although some argue that increasing the number of terrorist groups—a common proxy for competitiveness—can decrease violence, the mechanisms underlying that argument do not appear in this setting. For example, Nemeth argues that increasing the number of ideologically similar groups should decrease violence through free-riding dynamics.Footnote ⁸⁹ Hamas and Fatah are generally seen as ideologically opposed, however, and there are no free-riding incentives in the model. Another example comes from Conrad and Spaniel, who argue that the government may change its demands in response to a larger number of terrorist groups, leading to a negative correlation between this number and violence.Footnote ⁹⁰ We find that endogenous government demands are not necessary for competition to reduce violence. Instead, the contest itself explains these results; the discouragement effect emerges when the popularity contest becomes lopsided in favor of the more effective actor.

Effect of Competitive Incentives on Violence

Now we examine how groups’ incentives affect their attack probabilities. For example, how would overall violence levels change if group i became a more effective outbidder? The first counterfactual quantified the effects of competitive behavior on violence; this one illustrates the effects of competitive incentives on violence. To do this, we fix the transition parameters estimated from Table 1, the payoff parameters in Table 2, and the estimated equilibrium quantities. Then, for each group i, we change how effectively i can boost its popularity through terrorism by increasing and decreasing γ _i,1 by 1 percent. As the effectiveness of attacks changes, the equilibrium probabilities of attacks will change as well. Recall that γ _i,1 reflects the effectiveness of i in using terrorism to shift relative public opinion. An increase or decrease in γ _i,1 may reflect a change in tactics that the public may find more or less distasteful.

Because multiple equilibria can exist, we cannot just vary γ _i,1, compute a new equilibrium, and compare choice probabilities under the old and new parameter values. Doing so would not guarantee that the new equilibrium bears any resemblance to the estimated one. Indeed, it may be possible to change behavior even though γ _i,1 does not change, by changing the selected equilibrium. To ensure that the counterfactuals fix the equilibrium that is selected by the data and the CMLE, we use a homotopy method to map equilibria as locally continuous functions of the parameters.Footnote ⁹¹ Appendix L gives details.

Figure 6 graphs these differences given the change in γ _i,1 and observed state s ^t. Positive (negative) values indicate that violence from group i in observed state s ^t increases (decreases) in the counterfactual scenario. As in the earlier exercise, one interpretation of the figure is that values in month t with popularity level s ^t indicate the effect on the groups’ immediate attack probabilities if i were to exogenously become more or less competitive.

FIGURE 6. Relationship between terrorism and effectiveness of attacks

Focusing on the effects of Hamas's competitive incentives, we find evidence of outbidding's expected encouragement effect: when Hamas has greater incentives to compete, violence by both groups increases. We estimate that a 1 percent increase in Hamas's effectiveness results in a one percentage-point increase in the frequency of terrorism by Hamas and a 0.1-percentage-point increase in the frequency of terrorism by Fatah. On average, this implies Hamas would increase its use of violence by 2 percent, and Fatah by 1 percent. These encouragement effects are even stronger when focusing on more recent observations, after the Oslo era.

Focusing on the effects of Fatah's competitive incentives, we find evidence of outbidding's unexpected discouragement effect: when Fatah has greater incentives to compete, violence by both groups decreases. We estimate that a 1 percent increase in Fatah's effectiveness results in a one percentage-point decrease in the frequency of terrorism by Hamas and a 0.2-percentage-point decrease in the frequency of terrorism by Fatah. On average, this implies both groups would decrease their violence by 2 percent if Fatah were to have greater incentives to compete via becoming 1 percent more effective at outbidding. Again, these discouragement effects are even stronger after the Oslo era.Footnote ⁹²

Figures A.3 and A.4 illustrate the same exercise for the value of support, β _i, and the costs of attacking, κ _i, respectively. The main takeaways are similar: when Hamas becomes more competitive, both sides attack more frequently (as expected by the outbidding literature); but when Fatah becomes more competitive, both sides tend to attack less frequently (in contrast to expectations in the outbidding literature).

These discouragement effects arise from asymmetric competition. Fatah has an advantage in its effectiveness in using terrorism to increase public support; that is, |γ _F,1| is substantially larger than |γ _H,1|. When Fatah's incentive to compete increases, it more readily bears the upfront costs of attacks to increase future support. This affects Hamas's equilibrium strategy. When Fatah becomes more aggressive, Hamas generally attacks less, because it cannot efficiently compete against the more aggressive and more capable Fatah. In equilibrium, this creates a feedback loop where Fatah uses less violence as Hamas uses less violence. These discouragement effects will be the strongest in periods (or states) when the model predicts that both groups will use substantial violence—that is, after Fatah loses substantial popularity (Figures 3 and 4)—because behavior in these periods (or states) will be the most sensitive to strategic incentives.