Econ 805 Advanced Micro Theory 1 Dan Quint Fall 2009 Lecture 4 Today Necessary and Sufficient Conditions For Equilibrium Problem set 1 online shortly Last lecture integral form of the Envelope Theorem holds in equilibrium of any Independent Private Value auction where The highest type wins the object The lowest possible type gets expected payoff 0 Today necessary and sufficient conditions for a particular bidding function to be a symmetric equilibrium in such an auction 2 Today s General Results Consider a symmetric independent private values model of some auction and a bid function b T R Define g x t as one bidder s expected payoff given type t and bid x if all the other bidders bid according to b Under fairly broad but not all conditions everyone bidding according to b is an equilibrium b strictly increasing and g b t t g b t t tt FN 1 s ds 3 Necessary Conditions 4 With symmetric IPV b strictly increasing implies the envelope theorem If everyone bids according to the same bid function b And b is strictly increasing Then the highest type wins And so the envelope theorem holds So what we re really asking here is when a symmetric bid function must be strictly increasing 5 When must bid functions be increasing Equilibrium strategies are solutions to the maximization problem maxx g x t What conditions on g makes every selection x t from x t nondecreasing 6 When must bid functions be increasing Recall supermodularity and Topkis Strong Set Order two sets A B A SSO B if for every x x x B and x A x B and x A What this means visually A function g X x T R has increasing differences if for every x x the difference g x t g x t is nondecreasing in t Topkis if g x t has increasing differences and t t then x t SSO x t This means there exists some selection x t from x t which is monotonic But it does not guarantee that every selection is monotonic so it doesn t answer our question We need something stronger than increasing differences in some ways although what we use is weaker in others 7 Single crossing and single crossing differences properties Milgrom Shannon A function h T R satisfies the strict single crossing property if for every t t h t 0 h t 0 Also known as h crosses 0 only once from below A function g X x T R satisfies the strict single crossing differences property if for every x x the function h t g x t g x t satisfies strict single crossing That is g satisfies strict single crossing differences if g x t g x t 0 g x t g x t 0 for every x x t t When gt exists everywhere a sufficient condition is for gt to be strictly increasing in x 8 What single crossing differences gives us Theorem Suppose g x t satisfies strict single crossing differences Let S X be any subset Let x t arg maxx S g x t and let x t be any pointwise selection from x t Then x t is nondecreasing in t Proof Let t t x x t and x x t By optimality g x t g x t and g x t g x t So g x t g x t 0 and g x t g x t 0 If x x this violates strict single crossing differences Milgrom PATW theorem 4 1 or a special case of theorem 4 in Milgrom Shannon 1994 9 Strict single crossing differences will hold in most symmetric IPV auctions Suppose b T R is a symmetric equilibrium of some auction game in our general setup Assume that the other N 1 bidders bid according to b g x t t Pr win bid x E pay bid x t W x P x For x x g x t g x t W x W x t P x P x When does this satisfy strict single crossing 10 When is strict single crossing satisfied by g x t g x t W x W x t P x P x Assume W x W x probability of winning nondecreasing in bid g x t g x t is weakly increasing in t so if it s strictly positive at t it s strictly positive at t t Need to check that if g x t g x t 0 then g x t g x t 0 This can only fail if W x W x If b has convex range W x W x so strict single crossing differences holds and b must be nondecreasing e g T convex b continuous If W x W x and P x P x e g first price auction since P x x then g x t g x t 0 so there s nothing to check But if W x W x and P x P x then bidding x and x give the same expected payoff so b t x and b t x could happen in equilibrium Example A second price auction with values uniformly distributed over 0 1 2 3 The bid function b 2 1 b 1 2 b vi vi otherwise is a symmetric equilibrium But other than in a few weird situations b will be nondecreasing 11 b will almost always be strictly increasing Suppose b were constant over some range of types t t Then there is positive probability N 1 F t F t FN 2 t of tying with one other bidder by bidding b plus the additional possibility of tying with multiple bidders Suppose you only pay if you win let B be the expected payment conditional on bidding b and winning Since t t either t B or B t so either you strictly prefer to win at t or you strictly prefer to lose at t Assume that when you tie you win with probability greater than 0 but less than 1 Then you can strictly gain in expectation either by reducing b t by a sufficiently small amount or by raising b t by a sufficiently small amount In addition when T has point mass second price first price 12 So to sum up in well behaved symmetric IPV auctions except in very weird situations any symmetric equilibrium bid function will be strictly increasing and the envelope formula will therefore hold Next when are these sufficient conditions for a bid function b to be a symmetric equilibrium 13 Sufficiency 14 What are generally sufficient conditions for optimality in this type of problem A function g x t satisfies the smooth single crossing differences condition if for any x x and t t g x t g x t 0 g x t g x t 0 g x t g x t 0 g x t g x t 0 gx x t 0 gx x t 0 gx x t for all 0 Theorem PATW th 4 2 Suppose g …