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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
Outcome 2Player 1 Payoff, Player 2 Payoff
Strategy B
Outcome 3Player 1 Payoff, Player 2 Payoff
Outcome 4Player 1 Payoff, Player 2 Payoff
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 Strategy B
Strategy A 2,3 3,4
Strategy B 0,0 4,2
Solving a Game Without Math
Player 2
Player1
Strategy A Strategy B
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 Strategy B
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 )
Profit Implications: m > d > w and m > d > 0
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
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
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!)
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