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Game Theory

Game Theory. Formalizing Strategic Interaction. A. Assumptions. Background Assumptions Rational choice: Connected and Transitive preferences between Outcomes Strategic interaction: Each side affects the other Key Elements Players – Two or more Strategies – The choices players have

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Game Theory

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  1. Game Theory Formalizing Strategic Interaction

  2. A. Assumptions • Background Assumptions • Rational choice: Connected and Transitive preferences between Outcomes • Strategic interaction: Each side affects the other • Key Elements • Players – Two or more • Strategies – The choices players have • Outcomes – The results of the players’ choices • Payoffs (Preferences) – How much each player values each Outcome

  3. B. Games in Normal (aka Strategic) Form: The Matrix

  4. 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:

  5. Solving a Game Without Math c. Not every game has a Nash Equilibrium • Example:

  6. Solving a Game Without Math d. Some games have multiple Nash Equilibria • Example:

  7. C. Common Strategic-Form Games • Prisoners’ Dilemma • Both players end up worse, even though each plays rationally! • Reflects Snyder and Jervis “predation” argument

  8. C. Common Normal/Strategic-Form Games • Chicken • Equilibria: Someone swerves – but who? • Used to model “going nuclear” – manipulating the threat of something both sides wish to avoid (i.e. conquest by external enemies). • “Tied Hands” strategy – throw away the steering wheel!

  9. 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

  10. 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 player’s choices to the outcomes • Information Sets – When a player doesn’t know which node it is at • Outcomes – Terminal nodes

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

  12. Incumbent ( 0, m ) No enter ( d, d ) Accommodate Entrant Enter ( w, w ) Fight Profit Implications: m > d > w and m > d > 0 4. Example: Monopolist’s Paradox: The Threat

  13. Incumbent ( 0, m ) No enter ( d, d ) Accommodate Entrant Enter ( w, w ) Fight Profit Implications: m > d > w and m > d > 0 4. Example: Monopolist’s Paradox: Threat Not Credible!

  14. Incumbent ( 0, m ) No enter ( d, d ) Accommodate Entrant Enter ( w, w ) Fight Profit Implications: m > d > w and m > d > 0 4. Example: Monopolist’s Paradox: The Equilibrium Subgame Perfect Equilibrium

  15. 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!!!

  16. 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!!!

  17. 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!!!

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