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# Class_16 - GameTheory Lecture 16 Outline Today Brief review...

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Econ 171 – Introduction to  Game Theory  Lecture 16

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Outline Today: Brief review of last Thursday Signaling and pooling equilibria Ex. Gift game Ex. Job-signaling game Tomorrow: Brief review Wrap-up of class Thursday: Final
Perfect Bayesian Equilibrium Culmination of many concepts in this class. Games of incomplete information, often with private information determined by Nature. A PBE is defined for pairs of strategy profiles and beliefs. Given beliefs, each player must be sequentially rational at all their information sets. Players' beliefs are updated and consistent with Bayes' rule.

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The gift game Let’s add q and ( 1-q) ; P2’s beliefs about being at either node. Beliefs must be updated following Bayes’ rule. 1,-1 -1,0 -1,0 0,0 0,0 1,1 2 Enemy (1-p) A A R R Friend (p) 1 1 Gf Nf Ge Ne (q) (1-q)
The gift game Before we look at the perfect Bayesian equilibrium, analyze the game. How many strategy profiles are there in this game? a) 4 b) 6 c) 8 d) 16 e) ? 1,-1 -1,0 -1,0 0,0 0,0 1,1 2 Enemy (1-p) A A R R Friend (p) 1 1 Gf Nf Ge Ne (q) (1-q)

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The gift game Strategies: P1={NfNe,NfGe,GfNe,GfGe}; P2={A,R} Strategy profiles: {(NfNe,A) (NfGe,A) (GfNe,A) (GfGe,A) (NfNe,R) (NfGe,R) (GfNe,R) (GfGe,R)} How many Bayesian NE are there if p=3/4 ? a) 1 b) 2 c) 3 d) 4 e) ?
The gift game What does subgame perfection tell us about the Bayesian NE. Single subgame → All SPNE. Sequential rationality doesn’t help either. 1,-1 -1,0 -1,0 0,0 0,0 1,1 2 Enemy (1-p) A A R R Friend (p) 1 1 Gf Nf Ge Ne (q) (1-q)

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The gift game But what if you believe that P1 almost only gives gifts when she is a friend, (Gf,Ne)? If P2 receives a gift, he chooses Accept. 1,-1 -1,0 -1,0 0,0 0,0 1,1 2 Enemy (1-p) A A R R Friend (p) 1 1 Gf Nf Ge Ne (q) (1-q)
The gift game What if P1 always gives gifts (GfGe)? P2 how do we determine beliefs at each node q and ( 1-q) ; no clear strategy for P2. 1,-1 -1,0 -1,0 0,0 0,0 1,1 2 Enemy (1-p) A A R R Friend (p) 1 1 Gf Nf Ge Ne (q) (1-q)

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The gift game Steps for calculating perfect Bayesian equilibria: 1. Start with a strategy for P1. 2. When possible, calculate updated beliefs q and 1-q using Bayes’ rule. 3. Given updated beliefs, calculate P2’s optimal action. 4. Check whether P1’s strategy is a best response to P2’s strategy. If it is and neither deviates, then it’s a PBE.

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Class_16 - GameTheory Lecture 16 Outline Today Brief review...

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