Counterterrorism, a game-theoretic analysis (Daniel G

COUNTERTERRORISM: A GAME-THEORETIC ANALYSIS

Abstract

This paper establishes the prevalence of deterrence over preemption when targeted governments can choose between either policy or employ both. There is a similar proclivity to favor defensive counterterrorist measures over proactive policies. Unfortunately, this predisposition results in an equilibrium with socially inferior payoffs when compared with proactive responses. Proactive policies tend to provide purely public benefits to all potential targets and are usually undersupplied, whereas defensive policies tend to yield a strong share of provider-specific benefits and are often oversupplied. When terrorists direct a disproportionate number of attacks at one government, its reliance on defensive measures can disappear. Ironically, terrorists’ can assist governments in addressing coordination dilemmas associated with some antiterrorist policies by targeting some countries more often than others.

COUNTERTERRORISM: A GAME-THEORETIC ANALYSIS

Terrorism is the premeditated use or threat of use of violence by individuals or subnational groups to obtain political, religious, or ideological objectives through intimidation of

a large audience usually beyond that of the immediate victims.1 By simulating randomness, terrorists create an atmosphere of fear where everyone feels vulnerable, thereby extending their sphere of influence as far as possible. Suicide missions can heighten this air of anxiety and place greater pressures on governments to capitulate to terrorist demands owing to the greater casualties on average associated with such events – 13 deaths per suicide attack compared with less than one death per nonsuicide attack (Pape 2003). In this regards, the events of 11 September 2001 (henceforth, 9/11) made the public and governments painfully aware of the risks posed by the new breed of suicide and other terrorists bent on maximal casualties. Following 9/11, governments have spent tens of billions of dollars on a variety of antiterrorist policies. Since 2002, the budget supporting the newly created US Department of Homeland Security grew by over 60% to $36.2 billion for fiscal year 2004 (Office of Management and Budget 2003).

Counterterrorist policies may involve taking direct actions against terrorists or their sponsors. Such proactive policies may include destroying terrorist training camps, retaliating against a state-sponsor, infiltrating terrorist groups, gathering intelligence, or freezing terrorist assets. Preemption is the quintessential proactive policy in which terrorists and their assets are attacked to curb subsequent terrorist campaigns. More defensive or passive counterterrorist measures include erecting technological barriers (e.g., metal detectors or bomb sniffing equipment at airports), fortifying potential targets, and securing borders. These defensive policies are intended to deter an attack by either making success more difficult or increasing the likely negative consequences to the perpetrator. Efforts to deter terrorist events often displace the attack to other venues, modes of attack (e.g., from a skyjacking to a kidnapping), countries,

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or regions, where targets are relatively softer (Drakos and Kutan 2003; Enders and Sandler 1993, 2004; Sandler 2003).

By protecting all potential targets, preemption provides public benefits; in contrast, by deflecting the attack to relatively less-guarded targets, deterrence imposes public costs. The irony arises because nations have a pronounced proclivity to resort to deterrence rather than preemption despite the greater social gains associated with the latter. In general, there is a marked tendency for governments to engage in defensive rather than proactive antiterrorist policies. The primary purpose of this article is to apply elementary game theory to explain this proclivity. Unlike a recent paper (Sandler and Siqueira 2003) that examines deterrence and preemption in isolation, the current exercise allows governments to choose between these policies or to employ them together. Although we allow alternative game forms to characterize both policies, deterrence is shown to have an unfortunate dominance over preemption, consistent with what is observed in the real world. A second major purpose is to investigate the game structure of other proactive and defensive counterterrorist policies. A variety of game forms are relevant, not only among different antiterrorist measures but also for the same measure under alternative scenarios. The analysis also identifies the circumstances when preemption may result owing to payoff asymmetries between targets or policies.

Game theory is an appropriate tool for investigating counterterrorism because it captures the strategic interactions between terrorists and targeted governments whose choices are

interdependent.2 In so doing, game theory permits a rich range of strategic environments and policy choices in keeping with modern-day terrorist threats. Moreover, game theory assumes that each player is rational and must second guess its adversaries; thus, a government must place itself in its opponents’ position before deciding the appropriate strategic response. To decide the best strategy, a government must not only anticipate the actions of terrorists, but also those of

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other governments that might work at cross-purposes or take advantage of another government’s action. In today’s world of networked terrorists where the threat is global, accounting for these reactions of other governments is essential to the formulation of effective counterterrorist policies.

PREEMPTION OR DETERRENCE: SPECIFIC EXAMPLE

Any strategic analysis of the choice between preemption or deterrence must account for preemption’s purely public benefits to all potential targets: direct action against terrorists or their sponsors makes everyone safer. In contrast, a strategic representation of deterrence must account for the costs imposed on the nation that deters an attack as well as the public costs incurred by others from the increased likelihood of having an attack deflected to their soil. The deterrer must not only expend resources to make its territory a less attractive venue but also suffers costs from having its people or property targeted abroad. Deterrence spending is analogous to an insurance policy that is paid regardless of the outcome, but in bad states (when an attack ensues) deterrence curbs the expected damage at home, which is the deterrer’s private benefit.

To illustrate the dilemma posed by counterterrorist policies, we use a specific example where two targets – the United States (US) and the European Union (EU) – must choose preemption or deterrence. Both players can also do nothing, denoted by the status quo. The passive player is the terrorist group that is bent on attacking the weaker of the two targets or flipping a coin if neither is weaker. For illustration, each preemption action provides a public benefit of 4 for the US and EU at a private cost of 6 to the preemptor. In Figure 1, the preemption game is captured by the northwest 2 × 2 bold-bordered matrix, where each government can preempt or maintain the status quo. If, say, the US preempts, then it gains a net benefit of –2 (= 4 – 6) while conferring a free-rider gain of 4 on the EU. These payoffs are

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reversed if the roles are switched. When both governments preempt, each receives a net gain of 2 as its preemption expense of 6 are deducted from gains of 8 (= 4 × 2) derived from both targets’ preemption efforts. Neither country acting gives 0 net benefits to both players. This 2 × 2 preemption game is a Prisoner’s Dilemma (PD) with inaction as the dominant strategy since 4

> 2 and 0 > − 2. The resulting Nash equilibrium is mutual status quo – (Status quo, Status quo) – from which neither target would unilaterally move. This equilibrium is Pareto dominated by mutual action (preemption).

Next, we turn to the 2 × 2 deterrence game displayed by the southeast 2 × 2 boldbordered matrix in Figure 1, where the players have two strategies: do nothing or deter an attack at home. In this stylized symmetric example, we assume that deterrence is associated with a public cost of 4 experienced by the deterrer and the other country. The deterrer’s costs arise from the action and its potential losses from a deflected attack (say, from its citizens residing abroad), while the nondeterrer suffers the deflection costs of being the target of choice. A deterrer is motivated by private gains of 6 prior to costs being deducted. If, say, the EU deters

alone, then it nets 2 (= 6 − 4) while the US loses 4 by becoming the target of choice. Payoffs are switched when the US deters alone. Net benefits are zero if neither act, while each receives a net payoff of –2 [= 6 – (4 × 2)] from mutual deterren ce as costs of 8 are deducted from a private gain of 6. Once again, a PD game results. Now, the dominant strategy is action (Deter) rather than inaction and the Nash equilibrium is (Deter, Deter).

The real issue is which counterterrorism policy dominates if each target can choose either policy or the status quo. This can be addressed by consulting the 3 × 3 matrix of Figure 1 with its two embedded PD games. If one player deters and the other preempts, then the deterrer gains 6 (= 6 + 4 – 4), while the preemptor receives –6 (= 4 – 6 – 4), The deterrer gets a private benefit of 6 from its deterrence and a public benefit of 4 from the other player’s preemption, but must

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cover its deterrence cost of 4. In contrast, the sole preemptor suffers a cost of 4 from the other player’s deterrence and a cost of 6 from its own efforts, but only receives a private benefit of 4 from preempting. The other payoffs in the matrix remain as before.

The dominant strategy is for both players to deter, since the payoffs are higher than the corresponding payoffs associated with the other two strategies. As both targets exercise their dominant strategy, the Nash equilibrium of (Deter, Deter) follows. This outcome is like a double PD game where the smallest summed payoff results – every other strategic combination is socially preferred from an aggregate payoff viewpoint. Of the two Nash equilibria of the embedded and overlapping 2 × 2 PD games, the Pareto-inferior equilibrium reigns. In the 3 × 3 game, the sum of the payoffs decrease when moving down the columns, but the strategy in the bottom row dominates; similarly, the sums of the payoffs decrease when moving rightward along a row, but the strategy in the right-most column dominates. This outcome is rather fascinating and disturbing; it implies that deterrence wins out over preemption when payoffs mirror one another in that the public versus private roles of benefits and costs are switched. There is an implied resilience to deterrence in this stylized example. Is this example reflective of more generalized situations where benefits and costs do not merely switch roles in terms of values for the two policies, or alternative game forms apply, or players are asymmetric? We now analyze more general representations to address this question.

GENERALIZED DETERRENCE-PREEMPTION ANALYSIS

In this section, the generalized game no longer assumes that the public costs of deterrence equal the public benefits of preemption, or that the private benefits of deterrence equal the private costs of preemption. In terms of notation, B denotes the public benefits of preemption and c represents the private costs of preemption; while C denotes the public costs of deterrence

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and b represents the private benefits of deterrence. The generalized game is in Figure 2, where the overlapping 2 × 2 preemption and deterrence games are again highlighted by boldfaced borders. In the northwest 2 × 2 preemption ga me, the pure publicness of preemption is a key feature. If there is a sole preemptor, then that player affords a free ride worth B to the other nation and receives a net benefit of B – c. We initially assume that the costs of preemption exceed the associated benefits so that c > B. If both countries preempt, then each receives 2B – c as costs are deducted from the benefits derived from the combined actions of the two governments – i.e., B from each preemption for a total of 2B. To ensure that a PD game results, we must assume that B > 2B – c, which again implies that c > B. Also, we assume that the

payoff from mutual preemption is greater than that from mutual inaction so that 2B – c > 0.3

Taken together, the inequality 2B > c > B is sufficient to ensure that a PD game characterizes the northwest 2 × 2 preemption game. The dominant strategy for this embedded game is to do nothing, which results in the Nash inactivity equilibrium with a payoff of (0, 0). The southeast 2 × 2 bold-bordered matrix indicates the deterrence game, where payoffs are computed as before. To ensure a PD game, we assume that 2C > b > C. The dominant strategy of this embedded game is to deter and the Nash equilibrium is mutual deterrence.

For the 3 × 3 game, there are also the two strategic combinations where one player deters and the other preempts. The deterrer then receives B + b – C from the associated public gain from preempting, B, and the private gain from deterring, b, minus the public deterrence costs, C.

The preemptor gains the public preemption benefit but must cover the private costs of preempting and the public costs of its counterpart’s deterrence for a payoff of B – c – C. The two sets of inequalities ensure that the dominant strategy is to deter for both governments so that the Nash equilibrium of mutual deterrence results, which is Pareto inferior to doing nothing or mutual preemption. Thus, this first generalization of the game, denoted as the baseline game,

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does not eliminate the persistence of deterrence.

Because targeted countries may face the same situation in each period, we examine when

mutual preemption can be supported in an infinitely repeated game with a discount factor of δ . Suppose that a grim-trigger strategy is employed where a player begins by preempting but will henceforth switch to deterrence if the other player ever fails to preempt, so that a “deterrence

trigger” is employed. Preemption is then sustained if 4

This inequality can be satisfied in a couple of ways. If the net costs of unilateral preemption,

B − c, outweigh the return from unilateral deterrence, b − C, then the numerator on the right-

hand side of (1) is negative and the inequality holds trivially. This implies that the smaller is the incentive for unilateral deterrence compared with the net costs of unilateral preemption, the

which indicates that mutual preemption has a better chance when the sum of public costs and benefits are large compared with the sum of private costs and benefits. Of course, the sustainability of mutual preemption also hinges on the discount factor being large so that the future is valued sufficiently.

Two problems must be resolved to secure mutual preemption through a threat-based trigger. First, mutual deterrence remains an equilibrium; thus, parties must agree to coordinate and move from deterrence to preemption. Second, sustaining mutual preemption may conflict with the short-run viewpoint taken by governments owing to election periods that limit tenure. The results for the infinitely repeated version of Figure 2 are equivalent to those for a finite, but

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indefinitely, repeated version where δ is the probability that the current period is not the last

(Shubik 1970).5 If δ represents the probability of reelection, then this may be quite low owing to term limits or other considerations, which then work against sustaining preemption through threat-based triggers. Thus, we must look elsewhere for supporting preemption. Ironically, the lifetime tenure of terrorist leaders supports the widespread cooperation that characterizes terrorist

groups since the 1960s.6

MUTUAL INACTION IS DISASTROUS

Next, we change one payoff combination in Figure 2 – that of mutual status quo where the two embedded 2 × 2 games overlap. In particular, we change this payoff to (–D, –D) (not shown in Figure 2) so that mutual inaction is disastrous in the sense that this payoff is less than the sucker payoffs of the embedded preemption and deterrence games. Thus, the payoff, Π, from mutual status quo must satisfy:

This inequality ensures that the 2 × 2 preemption game is now a chicken game with pure-strategy Nash equilibria of unilateral preemption. The dominant strategy for the associated 3 × 3 game is still deterrence with a single Nash equilibrium of (Deter, Deter) that makes at least one player

worse off than the equilibria of the embedded 2 × 2 preemption chicken game. 7 The Nash equilibrium of the embedded deterrence game wins out over those of the embedded preemption game. Thus, the persistence of deterrence is again demonstrated.

DETERRENCE AND THRESHOLD PREEMPTION GAME

To further study the resilience of the mutual deterrence equilibrium, we now tie the countries’ interests for preemption closer together by assuming a threshold preemption game in