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Introduction to Game Theory

Introduction to Game Theory. Lecture 8: Bayesian Games. Preview. Simultaneous games with complete information Normal form Iterative elimination Nash Equilibrium Sequential games – perfect and complete information Extensive form Subgame perfect Nash equilibrium

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Introduction to Game Theory

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  1. Introduction to Game Theory Lecture 8: Bayesian Games

  2. Preview • Simultaneous games with complete information • Normal form • Iterative elimination • Nash Equilibrium • Sequential games – perfect and complete information • Extensive form • Subgame perfect Nash equilibrium • Today – Incomplete information (one side) • Bayesian Games • Nash equilibrium of Bayesian game

  3. Bayesian Games • Preferences are not known to all players • Players have believes over preferences • Examples • Auction • Bidder knows the value of object to him but does not know other players’ valuation • Duopoly • Cost function is private information • Street fight • Worker’s skills/ability

  4. Dating in Grease • Danny Zuko is strongly attracted to Sandy Olsonn which is publicly known. • Danny does not know whether Sandy feels the same however he has believes ½ to ½ about that. • If Sandy is then she wants to meet Danny. • If Sandy is not then she wants to avoid him. • Friday evening – two places to go: • Moe’s – smoky pub, preferred by Danny • Jack Rabbit Slim's – retro restaurant, preferred by Sandy

  5. Dating in Grease • Two types of Sandy: Attracted ½ Not Attracted ½ • Sandy knows her type. • Need to keep “two” Sandies in order to track Danny’s perspective. • vNM preferences (Danny faces lotteries)

  6. Dating in Grease • Danny faces 4 possible combination of actions done Sandies: • (M,M), (M,JRS), (JRS,M), (JRS,JRS) • Danny computes expected payoff • Danny plays M’s and Sandies (M,M): ½*2+½*2=2 • Danny plays JRS’s and Sandies (M,M): ½*0+½*0= 0

  7. Danny’s Preferences • Expected payoffs for all possible combinations of Sandies

  8. Bayesian Game Nash Equilibrium • Given the Dannies believes, action of Danny is optimal given the actions of two Sandies. • The action of each Sandy is optimal given action of Danny. • Be very careful which numbers you compare. • NE – (M’s,(M’s,JRS’s))

  9. Dating – Forest and Jenny • Everything is the same, just Forest is not Macho as Danny is • Forest believes Jenny is attract to him with probability 0.1 Attracted – 0.1 Not Attracted – 0.9

  10. Dating – Forest and Jenny • No Nash equilibrium in pure strategies • Forest assigns higher value to Jenny who is not attracted to him • Mixed strategy equilibrium • At least one of the three players (Forest and two Jennies) is mixing.

  11. Cournot’s Duopoly • Two firms: Firm A, Firm B • Firms A has publicly known production costs: CA (qA)=10 • Firm B has costs known only to her. • Firm A believes with p=½ that CBH (qBH)=10 qBH or CBL (qBL)=4 qBL • The first case – Firm B is low cost company • The second case – Firm B is high cost company • Firm maximize their expected profit • Inverse demand function: P(Q)=100-Q, where Q= qA+qB • Nash equilibrium?

  12. Grease with Two Sided Info. Assym. • Sandy is not sure whether Danny is attracted to her or not. • Sandy believes that Danny is Attracted to her with q=2/3.

  13. Dannies’ perspectives • Dannies do not know which Sandy they face

  14. Sandies’ perspectives • Sandies do not know which Danny they face

  15. Table of Expected Payoffs • Expected payoffs for two Dannies and two Sandies.

  16. Nash Equilibrium • Dannies cannot do better by deviation given the action of Sandies (and Dannies’ believes) • Sandies cannot do better by deviation given the actions of Dannies (and Sandies believes)

  17. Nash Equilibrium • Best response of attracted Danny • Best response of not attracted Danny • Best response of attracted Sandy • Best response of not attracted Sandy

  18. Nash Equilibrium • Two Nash Equilibria • ((M’s,M’s),(M’s,JRS’s)) and ((JRS’s,M’s),(JRS’s,JRS’s))

  19. Summary • Bayesian games • Incomplete information • Several types of agents • Limited information • Nash equilibrium • Every type of every player does the best given his believes about the distribution of opponents and combination of their actions. • Consider every type of the player as individual player.

  20. Midterm – Notes • Best response correspondence: • is not extensive game • Describe in mixed strategies as well • Sequence of players from to root to leaves – from the first decision maker to the last • Expected payoff • If player mixes – not necessary assign equal probabilities to his actions • Probability have to sum up to one • Yes: p and 1-p, NO: p and p-1!!! • SPNE – specify actions taken in all decision nodes

  21. Results • Will be publish once I find out how to do that  • If you would like to see, stop by office in NB 339 - Friday 16:15 – 17:15 or agree on appointment by email (Politickych Veznu 7 – CERGE-EI, NHÚ AV, office 331) • 5 – above 70% - good, keep the pace • 4 – 50% - 70& - sufficient, be careful • 2 – below 50% - need to speed up!

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