Game Theory and Terrorism

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Game Theory and Terrorism

• Evaluating Policy Responses

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A. Assumptions of Game Theory

• Assumptions
• Rational choice
• Strategic interaction
• Elements
• Players Two or more (our examples use two)
• Strategies The behavioral choices players have
  (examples counterterrorism policies or decision
  to attack or not attack)
• Outcomes (Consequences) The results of the
  players choices (examples casualties, costs,
  reputation, territory status)
• Payoffs (Preferences) How much each player
  values each Outcome

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B. Games in Normal (aka Strategic) Form The
Matrix
Player 2 Player 2 Player 2
Player 1 Strategy A Strategy B
Player 1 Strategy A Outcome 1 Player 1 Payoff, Player 2 Payoff Outcome 2 Player 1 Payoff, Player 2 Payoff
Player 1 Strategy B Outcome 3 Player 1 Payoff, Player 2 Payoff Outcome 4 Player 1 Payoff, Player 2 Payoff
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1. Solving a Normal/Strategic-Form Game Without
Math

• Nash Equilibrium ? Neither player could do any
  better by unilaterally changing its strategy
  choice
• 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 Player 2 Player 2
Player 1 Strategy A Strategy B
Player 1 Strategy A 2,3 3,4
Player 1 Strategy B 0,0 4,2
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Solving a Game Without Math

• c. Not every game has a Nash Equilibrium
• Example

Player 2 Player 2 Player 2
Player 1 Strategy A Strategy B
Player 1 Strategy A 2,3 3,4
Player 1 Strategy B 0,5 4,2
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Solving a Game Without Math

• d. Some games have multiple Nash Equilibria
• Example

Player 2 Player 2 Player 2
Player 1 Strategy A Strategy B
Player 1 Strategy A 2,5 3,4
Player 1 Strategy B 0,0 4,1
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C. Common Strategic-Form Games

• Prisoners Dilemma
• Both players end up worse, even though each plays
  rationally!
• Enders and Sandler May apply to unilateral
  actions against terrorism by two states
  (displacement)

Player 2 Player 2 Player 2
Player 1 Remain Silent Confess
Player 1 Remain Silent Misdemeanor, Misdemeanor Life, Walk Free
Player 1 Confess Walk Free, Life Felony, Felony
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c. The Displacement Dilemma

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

State 2 State 2 State 2
State 1 Do Nothing Unilateral Security
State 1 Do Nothing Terror, Terror More Terror, No Terror - Costs
State 1 Unilateral Security No Terror - Costs, More Terror Terror Costs, Terror - Costs
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C. Common Normal/Strategic-Form Games

• Chicken
• Equilibria Someone swerves but who?
• Used to model all-or-nothing crises (think Beslan
  siege)
• Credible commitment throw away the steering
  wheel!

Player 2 Player 2 Player 2
Player 1 Swerve Drive Straight
Player 1 Swerve Status Quo, Status Quo Wimp, Cool
Player 1 Drive Straight Cool, Wimp DEAD, DEAD
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C. Common Strategic-Form Games

• Stag Hunt, aka the Assurance Game, aka
  Mixed-Motive PD
• Equilibria depends on trust Nobody wants to be
  the only one looking for a stag!
• Used to model non-predatory security dilemma,
  driven by fear instead of aggression (need for
  international cooperation)

Player 2 Player 2 Player 2
Player 1 Deer Rabbit
Player 1 Deer Deer, Deer Nothing, Rabbit
Player 1 Rabbit Rabbit, Nothing Rabbit, Rabbit
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D. Games in Extensive Form The Tree

• Extensive form adds information
• What is the order of moves?
• What prior information does each player have when
  it makes its decision?
• Elements
• Nodes Points at which a player faces a choice
• Branches Decision paths connecting a players
  choices to the outcomes
• Information Sets When a player doesnt know
  which node it is at
• Outcomes Terminal nodes

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3. Solving an Extensive Form Game

1. Subgame Perfect Equilibrium Eliminates
   non-credible threats from consideration
2. Process Backwards induction If they think
   that we think

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4. Example Monopolists Paradox The Threat
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But Threat Not Credible!
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Equilibrium is Accommodate Shows problem of no
negotiation strategy (difficult to make credible)
Subgame Perfect Equilibrium
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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.

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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

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c. Structure and Payoffs
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d. Solutions. Begin at the end
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G retaliates iff -2A-2Bgt-2A-B-N--Add 2AB to
both sides ? -Bgt-N--Now multiply both sides by
-1 ? BltN
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G retaliates iff -A-Bgt-A-N--Add A to both sides
? -Bgt-N--Now multiply both sides by -1 ? BltN
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We now know that equilibrium depends on relative
values of B and N. If N is small (terrorists
dont ask for much, then no retaliation occurs!)
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If BgtN
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If BgtN Now we need to know if N-P gt 0 (which
means NgtP) ? if then, T attacks
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If BgtN and NgtP
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If BgtN and NgtP (add A N to both)
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If BgtN and NgtP
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If BgtN and NgtP No Terrorism! (Fear of terror is
enough to get G to listen to protests)
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BgtN and NltP
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BgtN and NltP
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BgtN and NltP
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BgtN and NltP
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BgtN and NltP No Terrorism! Terrorist threat
isnt credible because the stakes are small
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Now Suppose N is large BltN
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BltN
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BltN
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BltN
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BltN
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BltN
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BltN No Terrorism! Credible threat to retaliate
instead of negotiate deters attacks
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e. Summary of findings

1. Terrorism shouldnt 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

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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)?

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4. Key variable is N

• Very large N means NgtB Government would rather
  retaliate than negotiate. The terrorists are
  simply asking for too much
• Very small N means BltN and NltP Government
  doesnt 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

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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 PltNltB, they gain
  concessions. Note that this is impossible unless
  PltB.
• Government has an incentive to make retaliation
  cheaper for itself and to make acquisition of
  materials more expensive for terrorists if PgtB
  government can ignore protests

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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 gt B from the governments 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!

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7. The mystery of prolonged terror campaigns

• After a few attacks and retaliations, shouldnt
  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