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Policy Games

Policy Games. Nuclear deterrence, game theory, and subgame perfect equilibria. Games of Deterrence: Credible Threat and Restraint. War. Preferences A: Cap B SQ War FS B B: SQ FS B War Cap B. Nuke. Attack. Don’t Nuke. Cap B. FS B. Don’t Attack. Nuke.

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Policy Games

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  1. Policy Games Nuclear deterrence, game theory, and subgame perfect equilibria

  2. Games of Deterrence: Credible Threat and Restraint War Preferences A:CapB SQ War FSB B: SQ FSB War CapB Nuke Attack Don’t Nuke CapB FSB Don’t Attack Nuke Subgame Perfect Equilibrium Don’t Nuke SQ Deterrence Success!!!

  3. Preferences A:CapB SQ War FSB B:FSB SQ War CapB Games of Deterrence: Credible Threat But No Restraint War Nuke Subgame Perfect Equilibrium Attack Don’t Nuke CapB FSB Don’t Attack Nuke Don’t Nuke SQ Deterrence Fails!!!

  4. Preferences A:CapB SQ War FSB B: SQ FSB CapB War Games of Deterrence: Restraint, But No Credible Threat War Nuke Attack Don’t Nuke CapB Subgame Perfect Equilibrium FSB Don’t Attack Nuke Don’t Nuke SQ Deterrence Fails!!!

  5. Thinking in terms of “Types” – Incomplete information Define cost tolerance as degree to which possible policy gains outweigh anticipated costs – Type II is “Hawkish” compared to I, III B is Type I (Medium Cost Tolerance) A:CapB SQ War FSB B: SQ FSB War CapB B is Type III (Low Cost Tolerance) A:CapB SQ War FSB B: SQ FSB CapB War B is Type II (High Cost Tolerance) A:CapB SQ War FSB B:FSB SQ War CapB

  6. Harsanyi’s Transformation • Suppose B is known to be either Type I or Type II, but A does not know which. • This is asymmetric incomplete information – B knows B’s type but A doesn’t • Can transform into a game of imperfect information by having Nature make a move

  7. P= Pr (B is Type I)Now we have a standard game of imperfect information! War1 War2 p 1-p Nuke Nuke Don’t Nuke Don’t Nuke CapB2 CapB1 Attack Attack FSB1 Nuke FSB2 Don’t Attack Don’t Attack Nuke Don’t Nuke Don’t Nuke SQ2 SQ1

  8. Solution requires cardinal utilities over outcomes – A will be deterred if p>µ and will attack if p<µ. But µ is unknown and depends on the specific values of A’s payoffs War1 War2 p 1-p Nuke Nuke Don’t Nuke Don’t Nuke CapB2 CapB1 Attack Attack FSB1 Nuke FSB2 Don’t Attack Don’t Attack Nuke Don’t Nuke Don’t Nuke SQ2 SQ1

  9. 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.

  10. b. What determines payoffs? Six factors to consider… • General approach: utility = benefit + benefit… - cost - cost… • All numbers are positive, so that adding them means more utility and subtracting them means less utility • N is the terrorist’s demand, while n is the value of that demand to the government. Both are positive and represents what the government would have to give the potential terrorist in Negotiations. Therefore, if the government negotiates, it loses n 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. • R represents the pain of government Retaliation to the terrorist. • B represents the costs of retaliating for the government – the bombs, diplomatic efforts, etc needed. • The SQ is assumed to have a value of zero for each player

  11. c. Structure and Payoffs

  12. d. Solutions. Begin at the end:

  13. G retaliates iff -2A-2B>-2A-B-n--Add 2A+B to both sides  -B>-n--Now multiply both sides by -1  B<n

  14. G retaliates iff -A-B>-A-n--Add A to both sides  -B>-n--Now multiply both sides by -1  B<n

  15. 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!)

  16. If B>n:

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

  18. If B>n and N>P:

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

  20. If B>n and N>P:

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

  22. B>n and N<P:

  23. B>n and N<P:

  24. B>n and N<P:

  25. B>n and N<P:

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

  27. Now Suppose N is large: B<n

  28. B<n:

  29. B<n:

  30. B<n:

  31. B<n:

  32. B<n:

  33. B<n: No Terrorism! Credible threat to retaliate instead of negotiate deters attacks

  34. e. Summary of findings • Terrorism shouldn’t happen! No attacks if information is perfect and complete (both sides agree on values of N, n, P, B)  all terrorism (under these assumptions) represents sub-optimal outcomes for both sides! • Values of A and R are irrelevant! size of attacks and retaliation is less important than credibility of threats to do so

  35. 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)?

  36. 4. Key variables are N and 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

  37. 5. 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 since n includes future anticipated costs/benefits) • Terrorists may miscalculate value of n to government… • 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!

  38. Example: Both sides know B>n, but only T knows whether P>N (need Bayesian equilibrium) • How does T decide to move? Always protest first. • What about G? Key is whether G will bluff or not. • If G ignores, T “knows” to Concede • But knowing this, G should “bluff” even if P>N. T needs to know Pr (G would bluff when N>P) • Given G’s move, T updates using Bayes’ Rule and decides to Attack or Concede • No advantage for G to bluff in final round p (N>P) (N<P) 1-p

  39. 6. The mystery of prolonged terror campaigns • After a few attacks and retaliations, shouldn’t the values of B, N, 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

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