Econ 805 Advanced Micro Theory 1 Dan Quint Fall 2008 Lecture 4 Sept 11 2008 Today Necessary and Sufficient Conditions For Equilibrium Today necessary and sufficient conditions for a particular bidding function to be a symmetric equilibrium 2 Tuesday s big result was the Envelope Theorem Theorem Suppose that For all t x t is nonempty For all x t gt x t exists For all x g x is absolutely continuous gt has an integrable bound supx X gt x t B t for almost all t with B t some integrable function Then for any selection x s from x s V t V 0 0t gt x s s ds Then we applied this to auctions with symmetric independent private values 3 Another fairly general necessary condition monotonicity In symmetric IPV auctions equilibrium bid strategies will generally be increasing in values how to prove 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 Recall supermodularity and Topkis If g x t has increasing differences in x t then the set x t is increasing in t in the strong set order For g differentiable this is when g x t 0 But let t t if x is not single valued this still allows some points in x t to be above some points in x t so it wouldn t rule out equilibrium strategies which are decreasing at some points 4 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 but not necessary condition is for gt to be strictly increasing in x 5 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 6 So now given a symmetric equilibrium with the bid function b Define g x t as the expected payoff given bid x and value t when everyone else uses the equilibrium bid function If g satisfies strict single crossing differences then b must be weakly increasing And from Tuesday if g is absolutely continuous and differentiable in t with an integrable bound then V t V 0 0t gt b s s ds 7 All these conditions are almost always satisfied by 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 So g x t is absolutely continuous and differentiable in t with derivative bounded by 1 What about strict single crossing differences 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 8 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 g will satisfy strict single crossing differences so we know b will be nondecreasing 9 In fact 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 F N 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 10 b will almost always be strictly increasing In first price auctions equilibrium bid distributions don t have point masses even when type distributions do When there is a positive probability of each bidder having a particular value they play a mixed strategy at that value Otherwise there d be a positive probability of ties and the same logic would hold either prefer to increase by epsilon to win those ties or decrease by epsilon if you re indifferent about winning them In second price auctions if the type distribution has a point mass bidders still bid their valuations Still a dominant strategy So in that case there are positive probability ties 11 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 hold Next when are these sufficient conditions for a bid function b to be a …