1. 1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAA A AA A

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3.  There is a distribution D i on the types T i of Player i  It is known to everyone  The actual type of agent i, t i 2 D i T i is the private information i knows  A profile of strategis s i is a Bayes Nash Equilibrium if for i all t i and all t’ i E d -i [u i (t i, s i (t i ), s -i (t -i ) )] ¸ E d -i [u i (t’ i, s -i (t -i )) ]

4.  First price auction for a single item with two players.  Private values (types) t 1 and t 2 in T 1 =T 2 =[0,1]  Does not make sense to bid true value – utility 0.  There are distributions D 1 and D 2  Looking for s 1 (t 1 ) and s 2 (t 2 ) that are best replies to each other  Suppose both D 1 and D 2 are uniform. Claim: The strategies s 1 (t 1 ) = t i /2 are in Bayes Nash Equilibrium t1t1 Cannot winWin half the time

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6.  Bayes Nash equilibria (assumes priors) ◦ Today: characterization  Special case: Dominant strategy equilibria (VCG), problem: over “full domain” with 3 options in range (Arrow? GS? New: Roberts) – only affine maximizers (generalization of VCG) possible.  Implementation in undominated strategies: Not Bayes Nash, not dominant strategy, but assumes that agents are not totally stupid 6

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11. 11 The supremum of a family of convex functions is convex Ergo, is convex

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13.  Since 13

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17.  Dominant strategy truthful: Bidding truthfully maximizes utility irrespective of what other bids are. Special case of Bayes Nash incentive compatible. 17

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20. Expected Revenue: ◦ For first price auction: max(T 1 /2, T 2 /2) where T 1 and T 2 uniform in [0,1] ◦ For second price auction min(T 1, T 2 ) ◦ Which is better? ◦ Both are 1/3. ◦ Coincidence? Theorem [Revenue Equivalence]: under very general conditions, every two Bayesian Nash implementations of the same social choice function if for some player and some type they have the same expected payment then ◦ All types have the same expected payment to the player ◦ If all player have the same expected payment: the expected revenues are the same

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