Econ 805 Advanced Micro Theory I Dan Quint Fall 2008 Lecture 7 Sept 23 2008 Thus far we ve made four key assumptions that have greatly simplified our analysis 1 Risk neutral bidders 2 Ex ante symmetric bidders 3 Independent types 4 Private values These have bought us a lot we proved revenue equivalence and solved for the optimal auction out of every possible feasible auction mechanism But they re also restrictive assumptions and revenue equivalence fails unless we have all of them There s a significant literature devoted to what happens when you relax each of these assumptions generally one or two at a time Since most common auctions are either first price second price or ascending auctions much of this literature compares these formats to each other in terms of revenue efficiency etc when each of these is relaxed When bidders are risk averse things get complicated Revenue equivalence fails There s a paper by Maskin and Riley on the syllabus characterizing the optimal auction under risk aversion which is a pretty complicated object Basically the seller can use the bidders risk aversion in two ways First he can effectively sell the bidders insurance increase their payoffs when they lose the auction and decrease it when they win and charge them for this And second he can use risk aversion to extract more surplus from the high types Since these are at odds which each other he can t do either perfectly but the optimal auction has aspects of both There s also a general result that when bidders are risk averse first price auctions outperform second price auctions Intuition I may base a homework problem on this We already showed that when bidders are risk neutral risk averse sellers prefer first price auctions We ve already seen the optimal auction with asymmetric bidders There s also a nice paper by Maskin and Riley also on the syllabus comparing first and second price auctions when bidders are asymmetric They find that when one bidder s types are drawn from a stochastically higher distribution than the other s the strong bidder prefers second price auctions and the weak bidder prefers first price auctions but which one raises more revenue depends 1 The last two assumptions independent signals and private values are both relaxed in a fantastic paper by Milgrom and Weber 1982 Econometrica They introduce the affiliated interdependent values framework It s very general but in particular it nests two special cases that have received a lot of attention Private values which we ve been looking at already but with values allowed to be positively correlated across bidders Common values where ex post the bidders all value the object the same but this true value is uncertain and each bidder has different information about it This is commonly used as a model of auctions for natural resource rights Right to drill for oil on a tract of government owned land is likely the same for every oil company depends on how much oil is underground each company might drill a couple of test holes to sample so each has a different estimate of the value of the object up for bid This week we ll develop the Milgrom and Weber model also covered in PATW section 5 4 After we study this model we ll then look at the special case of common values some of which predated the Milgrom and Weber results Before we introduce the Milgrom and Weber model however we need one result that I had hoped to cover earlier but didn t get to We say a probability distribution F first order stochastically dominates another one G if F t G t for every t Lemma 1 Let X and Y be random variables with distributions F and G If F first order stochastically dominates G then Eu X Eu y for any increasing function u When u is differentiable there s an elegant proof similar to the one we used for second order stochastic dominance Define the step functions 1 if t k 1k t 0 if t k Note that if F first order stochastically dominates G then Z Z k Z E1k X 1k t f t dt 0f t dt 1f t dt 1 F k 1 G k E1k Y k so we have the result when u is one of these step functions To prove it for general differentiable u write Z t Z 0 u t u u s ds K u0 s 1s t ds so Eu x R u t f t dt R R K u0 s 1s t ds f t dt R R K u0 s 1s t f t dt ds R R K u0 s 1s t g t dt ds Eu y So if X first order stochastically dominates Y that is F t G t for all t then Eu X Eu Y for any increasing function u 2 Affiliated Interdependent Values Model The general setup for the Milgrom Weber model is that there are N risk neutral bidders Each bidder i gets a signal ti the value of the object to bidder i given these signals is vi t1 t2 tN t0 where t0 indicates information that is not available to any of the bidders This could be information the seller has or information that nobody has t0 is allowed to be multi dimensional that is it could consist of several different attributes of the good but this doesn t end up making a difference so we ll treat it as a single variable for simplicity As before individual bidders signals ti must be one dimensional Milgrom and Weber make the following assumptions about vi vi is nonnegative and continuous vi is nondecreasing in all its arguments so good news for one bidder is good news for all the bidders vi is symmetric in the following way vi ti x t i y t0 z vj tj x t j y t0 z and vi ti x t i y t0 z vi ti x t i y t0 z where is any permutation That is each bidder s valuation responds in the same way to t0 responds in the same way to his own signal responds in the same way to the other bidders signals and does not respond to which of his opponents had which signal just what they all are Given this symmetry we can rewrite bidder i s valuation as vi ti x t i y t0 z v x y 1 y 2 y N 1 t0 where y i is the ith highest of the N 1 elements of y That is we can rewrite bidder i s valuation as a function of his own signal t0 and the order statistics of his opponents signals and by the symmetry assumption this function is the same for every bidder Note that private values is simply the special case where v x y 1 y 2 y N 1 t0 g x for some function g and pure common values is the special case where vi ti x tj y t ij z t0 …