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Game Theory and Terrorism Evaluating Policy Responses Slide 2 A. Assumptions of Game Theory 1. Assumptions a. Rational choice b. Strategic interaction 2. Elements a. 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 Slide 3 B. Games in Normal (aka Strategic) Form: The Matrix Player 2 Player 1 Strategy AStrategy B Strategy A Outcome 1 Player 1 Payoff, Player 2 Payoff Outcome 2 Player 1 Payoff, Player 2 Payoff Strategy B Outcome 3 Player 1 Payoff, Player 2 Payoff Outcome 4 Player 1 Payoff, Player 2 Payoff Slide 4 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 Player 1 Strategy AStrategy B Strategy A 2,33,4 Strategy B 0,04,2 Slide 5 Solving a Game Without Math Player 2 Player 1 Strategy AStrategy B Strategy A 2,33,4 Strategy B 0,54,2 c. Not every game has a Nash Equilibrium Example: Slide 6 Solving a Game Without Math Player 2 Player 1 Strategy AStrategy B Strategy A 2,53,4 Strategy B 0,04,1 d. Some games have multiple Nash Equilibria Example: Slide 7 C. Common Strategic-Form Games 1. Prisoners Dilemma a. 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 Player 1 Remain SilentConfess Remain Silent Misdemeanor, Misdemeanor Life, Walk Free Confess Walk Free, LifeFelony, Felony Slide 8 c. The Displacement Dilemma If unilaterally increasing security just displaces terrorism, states may over- provide unilateral security: State 2 State 1 Do NothingUnilateral Security Do Nothing Terror, Terror More Terror, No Terror - Costs Unilateral Security No Terror - Costs, More Terror Terror Costs, Terror - Costs Slide 9 C. Common Normal/Strategic- Form Games 2. Chicken a. 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 Player 1 SwerveDrive Straight Swerve Status Quo, Status Quo Wimp, Cool Drive Straight Cool, WimpDEAD, DEAD Slide 10 C. Common Strategic-Form Games 3. Stag Hunt, aka the Assurance Game, aka Mixed-Motive PD a. 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 Player 1 DeerRabbit Deer Deer, Deer Nothing, Rabbit Rabbit Rabbit, Nothing Rabbit, Rabbit Slide 11 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. Elements a. Nodes Points at which a player faces a choice b. Branches Decision paths connecting a players choices to the outcomes c. Information Sets When a player doesnt know which node it is at d. Outcomes Terminal nodes Slide 12 3. Solving an Extensive Form Game a. Subgame Perfect Equilibrium Eliminates non- credible threats from consideration b. Process = Backwards induction If they think that we think Slide 13 4. Example: Monopolists 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 Slide 14 But Threat Not Credible! Incumbent Entrant No enter Enter Fight Accommodate ( 0, m ) ( d, d ) ( w, w ) Profit Implications: m > d > w and m > d > 0 Slide 15 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 ) Profit Implications: m > d > w and m > d > 0 Subgame Perfect Equilibrium Slide 16 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. Slide 17 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 Slide 18 c. Structure and Payoffs Slide 19 d. Solutions. Begin at the end: Slide 20 G retaliates iff -2A-2B>-2A-B-N --Add 2A+B to both sides -B>-N --Now multiply both sides by -1 B G retaliates iff -A-B>-A-N --Add A to both sides -B>-N --Now multiply both sides by -1 B If B>N: Slide 24 If B>N: Now we need to know if N-P > 0 (which means N>P) if then, T attacks Slide 25 If B>N and N>P: Slide 26 If B>N and N>P: (add A + N to both) Slide 27 If B>N and N>P: Slide 28 If B>N and N>P: No Terrorism! (Fear of terror is enough to get G to listen to protests) Slide 29 B>N and N B>N and N 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