ec459_lec25.pdf - Lecture 25 Review 1 Overview The&nal...

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Lecture 25: Review 1 Overview The °nal exam will emphasize topics from the second half of the semester. I will not ask purely 1st-semester topics, such as maxmin, minmax, and rationalizable strategies. However, you should still know the concepts of NE and SPNE. If I ask about static games, I might start with complete info (NE) and go to a game with incomplete info (BNE). If I ask about dynamic games, I might start with complete info (most likely to ask about SPNE) and then go to incomplete info (PBE). It is not inconceivable that strictly/weakly dominated strategies will come up. So for today, I will just review these concepts and go through some examples. 2 Recap of the Equilibrium Concepts ° in Bayesian games (static and dynamic), a player knows his own type, but his strategy must specify a plan of action for every type that the other players think he could possibly be ° in dynamic games, a strategy must specify what do do at every possible history, including those that can only be reached by deviations ° with all of our concepts, checking whether something is an equilibrium amounts ot checking whether there are any pro°table deviations: there is no way to answer this question without a speci°cation of what will happen if someone deviates ° for the concepts SPNE and PBE, solve the game by backward induction : go to the last decision node, and °gure out what is optimal for that player. Take this as given. Then go to the stage before this, and °nd optimal behavior. Keep working back to the start of the game. For these concepts, optimality requires optimal behavior at every point: i.e., at every history (including those that should never be reached), the player moving there must feel that he could not improve his payo/ in the rest of the game by deviating ±given his beliefs about his opponents²types, and given the opponents²strategies. ° for the PBE concept, you must specify updated beliefs (about opponent types) at each decision node: this belief must be Bayesian after stu/ that happens with positive probability (under the equilibrium strategies), but may be speci°ed freely after things that were never supposed to happen. For example, suppose you think ³normal´ P1 will choose A, ³crazy´ P1 will choose B, and then he chooses C. Bayes²rule doesn²t provide a formula for what you should think about P1²s type at this point, so you can specify any probability of crazy that you like (but must specify something, and your behavior must be optimal given this belief). 3 Example 1: Auction Consider a 1st-price auction in which P1²s type v 1 is drawn from a uniform distribution on [0 ; 1] , similarly for P2²s type v 2 , but the value of the object to both players is v 1 + v 2 2 . Is there an equilibrium in which each player bids °
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