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Game Theory

Game Theory. The Logic of Deterrence. A. Assumptions. Assumptions Rational choice (Transitive and Connected Preferences) – Note that preferences do not need to be reasonable or sensible, just consistent Strategic interaction – Inherent in IR Elements

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Game Theory

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  1. Game Theory The Logic of Deterrence

  2. A. Assumptions • Assumptions • Rational choice (Transitive and Connected Preferences) – Note that preferences do not need to be reasonable or sensible, just consistent • Strategic interaction – Inherent in IR • Elements • Players – Two or more (Nuclear: Usually two) • Strategies – The choices players have • Outcomes – The results of the players’ choices (what the world looks like afterwards) • Payoffs (Preferences) – How much each player values each Outcome (since the same outcome can be valued differently by different people)

  3. B. Games in Normal (aka Strategic) Form: The Matrix • This form is used to represent simultaneous choice

  4. 1. Solving a Normal/Strategic-Form Game Without Math • Where do the numbers come from? PREFERENCESFirst step is always rank-ordering outcomes for each player. • 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:

  5. Solving a Game Without Math c. Not every game has a Nash Equilibrium (prediction = instability / switching between strategies) • Example:

  6. Solving a Game Without Math d. Some games have multiple Nash Equilibria (prediction = one of the following outcomes…) • Example:

  7. C. Common Strategic-Form Games • Prisoners’ Dilemma • Both players end up worse, even though each plays rationally!

  8. b. Using PD to model Arms Races (The Security Dilemma) Note that payoff structure is just like a PD

  9. 2. Chicken: Who will swerve?

  10. 2. Chicken: Who will swerve? What If: You could throw your steering wheel out the window?

  11. Nuclear Crises and Chicken: The Cuban Missile Crisis Key distinction: In Chicken, each player would rather be the (nice) sucker than have both players be nasty  Not so in PD

  12. Problem 1: An India-Pakistan Nuclear Crisis • Determine preferences for each side (discussion) • If Pakistan assembles, what does India want to do? • If Pakistan doesn’t assemble, what does India want to do? • If India assembles, what does Pakistan want to do? • If India doesn’t assemble, what does Pakistan want to do? • Identify any Nash equilibria • Translate this into the real world – what does game theory predict?

  13. Problem 2: An India-Pakistan Nuclear Crisis, Phase Two • Determine preferences for each side (discussion) • If Pakistan doesn’t strike, what does India want to do? • If Pakistan strikes, what does India want to do? • If India doesn’t strike, what does Pakistan want to do? • If India strikes, what does Pakistan want to do? • Identify any Nash equilibria • Translate this into the real world – what does game theory predict?

  14. D. Games in Extensive Form: The Tree • Extensive form adds information: • What is the order of moves? Example: “If you do this, then I will do that.” • 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 player’s choices to the outcomes • Information Sets – When a player doesn’t know which node it is at • Outcomes – Terminal nodes

  15. 3. Solving an Extensive Form Game • Subgame Perfect Equilibrium – Eliminates “non-credible” threats from consideration • Process = Backwards induction – “If they think that we think…”

  16. E. 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!!!

  17. Preferences A: CapB SQ War FSB B:FSB SQ War CapB E. 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!!!

  18. Preferences A: CapB SQ War FSB B: SQ FSB CapB War E. 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!!!

  19. Problem Three: Deterring the USSR Given USSR NFU Doctrine Nuke NWarEUR Nuke Don’t Nuke CWinUS Invade Europe Don’t Nuke WinUSSR NWarCON Don’t Invade Nuke Nuke Don’t Nuke Don’t Nuke NWinUS SQ

  20. Problem Three: If the US is willing to trade New York for Bonn Nuke NWarEUR Nuke Don’t Nuke CWinUS Invade Europe Don’t Nuke WinUSSR NWarCON Don’t Invade Nuke Nuke Don’t Nuke Don’t Nuke NWinUS SQ

  21. Problem Three: If the US is NOT willing to trade New York for Bonn Nuke NWarEUR Nuke Don’t Nuke CWinUS Invade Europe Don’t Nuke WinUSSR NWarCON Don’t Invade Nuke Nuke Don’t Nuke Don’t Nuke NWinUS SQ

  22. F. Preferences as Variables • When the payoffs for each side include variables, try to establish rankings among variables (i.e. territory is always appreciated while casualties are always shunned --- so T must be positive, C must be negative, and T>C) • When no “natural” ranking exists, the solution depends on the values of the variables (i.e. as T gets bigger, war becomes more likely  key is whether T + C > 0)

  23. Problem Four: An asymmetric game of war termination Variables in payoffs D = amount demanded by player A, as a percentage of total utility in dispute. 0 < D < 1 C = costs of continued fighting (assume equal for both sides) C > 0 (subtract it) P = probability A wins the war. 0 < P < 1 d = Cost to A of making an offer that fails Constant: Assume the value of the object of struggle is 1 and scale everything else accordingly Settle Accept Demand D Reject WarD WinA Demand All Surrender Don’t Surrender War

  24. Problem Four: An asymmetric game of war termination D, 1-D Accept Demand D Reject P-C-d, 1-P-C 1, 0 Demand All Surrender Don’t Surrender P-C, 1-P-C

  25. Problem Four: An asymmetric game of war termination D, 1-D Accept Key: Is 1-D > 1-P-C? Key: Is C > D-P? Demand D Reject P-C-d, 1-P-C 1, 0 Demand All Surrender Key: 0 > 1-P-C? Key: C > 1-P? Don’t Surrender P-C, 1-P-C

  26. Effects of C: Step One • D-P < 1-P, by definition (D < 1) • Three scenarios possible: • C < D-P < 1-P • D-P < C < 1-P • D-P < 1-P < C

  27. Effects of C: Step TwoScenario 1 (C < D-P < 1-P) D, 1-D Accept Demand D Reject P-C-d, 1-P-C 1, 0 Demand All Surrender Don’t Surrender P-C, 1-P-C

  28. Effects of C: Step TwoScenario 2 (D-P < C < 1-P) D, 1-D Accept Demand D Reject P-C-d, 1-P-C 1, 0 Demand All Surrender Don’t Surrender P-C, 1-P-C

  29. Effects of C: Step TwoScenario 2 (D-P < C < 1-P) D, 1-D Accept Key: A controls D. So A always makes a demand if any D is large enough so that: D > P-C  1 > P-C P can be no larger than 1, so this condition is ALWAYS TRUE! There is always some D available that A prefers making to fighting IF C > D-P Demand D Demand All Don’t Surrender P-C, 1-P-C

  30. Effects of C: Step TwoScenario 2 (D-P < C < 1-P) D-d, 1-D Accept Key: A controls D. So A always makes a demand if any D is large enough so that: D > P-C  1 > P-C P can be no larger than 1, so this condition is ALWAYS TRUE! There is always some D available that A prefers making to fighting IF C > D-P Demand D Demand All Don’t Surrender P-C, 1-P-C

  31. Effects of C: Step TwoScenario 3 (D-P < 1-P < C) D-d, 1-D Accept Demand D Reject P-C-d, 1-P-C 1, 0 Demand All Surrender Don’t Surrender P-C, 1-P-C

  32. Problem Four: Solutions (1) • The war will end with B’s surrender if C > 1-P (costs exceed B’s expected payoff of war, considering its likelihood of winning) • If C is less than D – P (that is, the amount A is asking for above and beyond what it is expected to seize on the ground) then the war continues • If C is more than D-P but less than 1-P, then the war ends in a negotiated settlement

  33. Problem Four: Solutions (2) • Does the war ever continue? Graph of C: B Fights B Surrenders B Deals 0 D-P 1-P ∞ A can always demand less than P, which is a deal B will accept. When does A want to do so? A Won’t Deal

  34. Problem Four: Solutions (2) D, 1-D Accept The two sides both want a deal when D > P-C (A’s condition) AND D < P (B’s condition) So: P-C < D < P  There is ALWAYS some value of D that meets both conditions! The “range” of D (the zone of agreement) is given by C. Demand D Demand All Don’t Surrender P-C, 1-P-C

  35. Problem Four: Solutions (3) • Everything depends on C: Small C = negotiated settlement, large C = B surrenders (note the asymmetric outcome!) B Surrenders B Deals 0 1-P ∞ A Won’t Deal A demands more than P-C and less than P

  36. G. Bayesian Inference: Dealing with incomplete information about other player’s preferences • Problem: What if we don’t know the other side’s preferences? • USSR doesn’t know what “type” US is (will it sacrifice NY for Bonn?) • Pakistan doesn’t know what “type” India is (will it sacrifice 100,000 Indians to defeat Pakistan once and for all?) • Solution: Bayesian equilibrium • Assume 50/50 probability (or evidence-based prior judgment) • Bayes principle – When uncertain, alter prior probability estimates by averaging in new information • Note that players have incentives to misrepresent preferences (think of chicken)  signaling games (need to show resolve / test resolve)

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