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Game Theory and Terrorism

Evaluating Policy Responses

A. Assumptions of Game Theory

1. Assumptionsa. Rational choiceb. Strategic interaction

2. Elementsa. Players – Two or more (our examples use two)b. Strategies – The behavioral choices players

have (examples: counterterrorism policies or decision to attack or not attack)

c. Outcomes (Consequences) – The results of the players’ choices (examples: casualties, costs, reputation, territory status)

d. Payoffs (Preferences) – How much each player values each Outcome

B. Games in Normal (aka Strategic) Form: The Matrix

Player 2

Player1

Strategy A Strategy B

Strategy A

Outcome 1Player 1 Payoff, Player 2 Payoff


Strategy B



1. Solving a Normal/Strategic-Form Game Without Math

a. Nash Equilibrium Neither player could do any better by unilaterally changing its strategy choice

b. To Solve: Examine each cell to see if either player could do better by unilaterally choosing a different Strategy, given that its opponent does nothing different.

Example: Player 2

Player1


Strategy A 2,3 3,4

Strategy B 0,0 4,2

Solving a Game Without Math

Player 2

Player1


Strategy A 2,3 3,4

Strategy B 0,5 4,2

c. Not every game has a Nash Equilibrium Example:

Solving a Game Without Math

Player 2

Player1


Strategy A 2,5 3,4

Strategy B 0,0 4,1

d. Some games have multiple Nash Equilibria Example:

C. Common Strategic-Form Games

1. Prisoners’ Dilemmaa. Both players end up worse, even though each plays

rationally!b. Enders and Sandler: May apply to unilateral actions

against terrorism by two states (displacement)

Player 2

Player1

Remain Silent Confess

Remain Silent

Misdemeanor, Misdemeanor

Life, Walk Free

Confess Walk Free, Life Felony, Felony

c. The Displacement Dilemma

If unilaterally increasing security just displaces terrorism, states may over-provide unilateral security:

State 2

State1

Do Nothing Unilateral Security

Do Nothing

Terror, TerrorMore Terror, No Terror - Costs

Unilateral Security

No Terror - Costs, More

Terror

Terror – Costs, Terror - Costs

C. Common Normal/Strategic-Form Games

2. Chickena. Equilibria: Someone swerves – but who?b. Used to model all-or-nothing crises (think Beslan siege)c. Credible commitment – throw away the steering wheel!

Player 2

Player1

Swerve Drive Straight

SwerveStatus Quo, Status Quo

Wimp, Cool

Drive Straight

Cool, Wimp DEAD, DEAD

C. Common Strategic-Form Games

3. “Stag Hunt”, aka the Assurance Game, aka Mixed-Motive PDa. Equilibria: depends on trust – Nobody wants to be the only one

looking for a stag!b. Used to model non-predatory security dilemma, driven by fear instead

of aggression (need for international cooperation)

Player 2

Player1

Deer Rabbit

Deer Deer, DeerNothing, Rabbit

RabbitRabbit, Nothing

Rabbit, Rabbit

D. Games in Extensive Form: The Tree

1. Extensive form adds information:a. What is the order of moves?

b. What prior information does each player have when it makes its decision?

2. Elementsa. Nodes – Points at which a player faces a choice

b. Branches – Decision paths connecting a player’s choices to the outcomes

c. Information Sets – When a player doesn’t know which node it is at

d. Outcomes – Terminal nodes

3. Solving an Extensive Form Gamea. Subgame Perfect Equilibrium – Eliminates “non-

credible” threats from consideration

b. Process = Backwards induction – “If they think that we think…”

4. Example: Monopolist’s Paradox: The Threat

Incumbent

Entrant

No enter

Enter

Fight

Accommodate

( 0, m )

( d, d )

( w, w )

Profit Implications: m > d > w and m > d > 0

But Threat Not Credible!

Incumbent

Entrant

No enter

Enter

Fight

Accommodate

( 0, m )

( d, d )

( w, w )


Equilibrium is Accommodate: Shows problem of “no negotiation” strategy (difficult to make credible)

Incumbent

Entrant

No enter

Enter

Fight

Accommodate

( 0, m )

( d, d )

( w, w )


Subgame Perfect

Equilibrium

5. A Simple Game of Terror

a. Story: The first player is labeled T for potential Terrorist, and the second player is labeled G for Government.

The potential terrorist disagrees with existing government policy, and faces a choice of carrying out a terrorist attack or resorting to peaceful protest. If the terrorist attacks, the government may retaliate or negotiate

with the terrorists, making some form of concession in exchange for peace. If the government retaliates, the terrorist may either attack again or give up the struggle. If the terrorist attacks again, then the government may decide to retaliate or negotiate.

If the terrorist uses peaceful protest, the government may choose to ignore the demands or negotiate. If the government ignores the demands, the terrorist may choose to attack or give up on its cause. If the terrorist attacks, the government gets a chance to retaliate or negotiate.

b. What determines payoffs? Five factors to consider…

N is positive and represents what the government would have to give the potential terrorist in Negotiations. Therefore, if the government negotiates, it loses N (thus the -N in its payoffs) and the terrorist gains N.

-P represents the oPportunity cost to the terrorist of an attack – the resources, personnel, etc needed to carry out the operation.

-A represents the pain of a terrorist Attack to the government, and is always negative.

-R represents the pain of government Retaliation to the terrorist, and is also always negative.

-B represents the costs of retaliating for the government – the bombs, diplomatic efforts, etc needed to successfully retaliate against the terrorists. -B, too, is always negative. The status quo is assumed to have a value of zero for each player

c. Structure and Payoffs

d. Solutions. Begin at the end:

G retaliates iff -2A-2B>-2A-B-N--Add 2A+B to both sides -B>-N--Now multiply both sides by -1 B<N

G retaliates iff -A-B>-A-N--Add A to both sides -B>-N--Now multiply both sides by -1 B<N

We now know that equilibrium depends on relative values of B and N. If N is small (terrorists don’t ask for much, then no retaliation occurs!)

If B>N:

If B>N: Now we need to know if N-P > 0 (which means N>P) if then, T attacks

If B>N and N>P:

If B>N and N>P: (add A + N to both)

If B>N and N>P:

If B>N and N>P: No Terrorism! (Fear of terror is enough to get G to listen to protests)

B>N and N<P:

B>N and N<P:

B>N and N<P:

B>N and N<P:

B>N and N<P: No Terrorism! Terrorist threat isn’t credible because the stakes are small…

Now Suppose N is large: B<N

B<N:

B<N:

B<N:

B<N:

B<N:

B<N: No Terrorism! Credible threat to retaliate instead of negotiate deters attacks

e. Summary of findings

1. Terrorism shouldn’t happen! No attacks if information is perfect and complete (both sides agree on values of N, P, B) all terrorism (under these assupmtions) represents sub-optimal outcomes for both sides!

2. Values of A and R are irrelevant! size of attacks and retaliation is less important than credibility of threats to do so

3. Policy inconsistency should be rare

If G ever retaliates, it always retaliates If T ever attacks, it always attacks What explains observed inconsistency

(e.g. Israel and US negotiating with terrorists)?

4. Key variable is N

Very large N means N>B: Government would rather retaliate than negotiate. The terrorists are simply asking for too much

Very small N means B<N and N<P: Government doesn’t believe terrorists will spend resources on attacks for such a small demand

If N is big enough to be worth making a bomb or two, but smaller than the cost of a counterterror campaign to the government, then governments should simply concede the demands of protesters before things turn violent

5. Expansion: N is chosen by the terrorists

Terrorists have an incentive to not ask for too much or too little. If terrorists can choose a value of N such that P<N<B, they gain concessions. Note that this is impossible unless P<B.

Government has an incentive to make retaliation cheaper for itself and to make acquisition of materials more expensive for terrorists: if P>B government can ignore protests

6. Sources of misperception Government may worry that concessions

future attacks (reputation concerns). Note that this should NOT cause terrorism, but rather should bolster the government deterrent (because it makes N > B from the government’s perspective)

Terrorists may miscalculate value of N to government but without further miscalculation, this simply leads to concession by terrorists…

Both T and G have incentives to portray themselves as violent (that is, to make P and B appear small) key to continued terror campaign is misperception of these variables!

7. The mystery of prolonged terror campaigns

After a few attacks and retaliations, shouldn’t the values of B, N, and P be clear to both sides? What explains continued violence?

Possibility: Assumption is that bombing is always costly (-B and –P are negative). What if one or both terms were positive? (Political incentives) Equilibria include a steady-state terror-retaliation

campaign… Values of R and A now matter a great deal, since they can

offset the “profits” of attacks Since R and A matter, should see escalation of violence

up to the point they become unprofitable (-P or –B are negative again). Pattern: small attacks larger ones steady state

Suggests key to ending prolonged campaign is to eliminate political incentives (profits) from attacks