Auctions Microeconomics II Auctions Levent Koçkesen Koç University Many economic transactions are conducted through auctions treasury bills art work foreign exchange antiques publicly owned companies cars mineral rights houses airwave spectrum rights government contracts Also can be thought of as auctions takeover battles queues wars of attrition lobbying contests Levent Koçkesen (Koç University) Auctions 1 / 1 Levent Koçkesen (Koç University) Auctions 2 / 1 Auction Formats Auction Formats 1 Open bid auctions 1 ascending-bid auction aka English auction price is raised until only one bidder remains, who wins and pays the final price 2 descending-bid auction aka Dutch auction price is lowered until someone accepts, who wins the object at the current price 2 Sealed bid auctions 1 first price auction highest bidder wins; pays her bid 2 second price auction aka Vickrey auction highest bidder wins; pays the second highest bid Auctions also differ with respect to the valuation of the bidders 1 Private values each bidder knows only her own value and information that others have do not affect her value artwork, antiques, memorabilia 2 Interdependent values bidders do not know the value of the object and have different private information about that value information that others have do have an affect on the value special case: common values oil field auctions, company takeovers Levent Koçkesen (Koç University) Auctions 3 / 1 Levent Koçkesen (Koç University) Auctions 4 / 1
Equivalent Formats Independent Private Values Open Bid Strategically Equivalent Dutch Auction Same Equilibrium English Auction in Private Values We will study sealed bid auctions Sealed Bid First Price Second Price Single object for sale, n bidders Bidder i assigns value Vi to the object Vi is independently and identically distributed over [0,v] according to distribution function F F has a continuous density f = F such that f(v) > 0 for all v [0,v] Each bidder knows only her own valuation vi and that others values are independently distributed according to F Bidders are risk neutral Levent Koçkesen (Koç University) Auctions 5 / 1 Levent Koçkesen (Koç University) Auctions 6 / 1 Auctions as a Bayesian Game Set of players N = {1,2,...,n} Type set Θi = [0,v],v > 0 Strategy set, Si = R+ Beliefs Other bidders valuations are independent draws from F Payoff functions: for any b R n + and vi Θi { vi p(b) ui(b,vi) = m, if bi maxj ibj 0, if bi < maxj ibj p(b): price paid by the winner if the bid profile is b m: Number of highest bidders Strategy of player i, βi : Θi R+ Levent Koçkesen (Koç University) Auctions 7 / 1 Second Price Auctions In this case player i pays p(b) = maxj ibj if she wins. Proposition βi(v) = v for all i is a weakly dominant strategy equilibrium. Proof. Consider player i with value v and take any bid x < v. We have to show that bidding v brings at least as much payoff as x for any bid profile b i and strictly more payoff for some b i. Consider the bid profiles in which 1 maxj ibj v > x: payoffs to both v and x are zero. 2 v > x > maxj ibj: both payoffs are equal to v maxj ibj 3 v > maxj ibj > x: payoff to v is positive whereas the payoff to x is zero. 4 v > maxj ibj = x: payoff to v is v maxj ibj whereas the payoff to x is (v maxj ibj)/m, for some m 2. Exercise: Show that v weakly dominates any bid x > v. Levent Koçkesen (Koç University) Auctions 8 / 1
First Price Auctions In this case player i pays bi if she wins. Would you bid your value? What happens if you bid less than your value? You get a positive payoff if you win But your chances of winning are smaller Optimal bid reflects this tradeoff Bidding less than your value is known as bid shading 2 bidders F is uniform over [0,1] Is there an equilibrium in which βi(v) = av for some a > 0? Optimal bid cannot be greater than a, since a is the highest possible bid of the other player Your expected payoff if you bid b [0,a] (v b)prob(you win) = (v b)prob(b > av2) =(v b)prob(v2 < b/a) =(v b) b a Levent Koçkesen (Koç University) Auctions 9 / 1 Levent Koçkesen (Koç University) Auctions 10 / 1 If optimal bid is positive, FOC must hold Bidding zero yields zero payoff Expected payoff to bid v/2 b a + v b = 0 b = v a 2 v 2 2a Bidding zero is optimal only for v = 0 βi(v) = v/2 is a Bayesian equilibrium n bidders An equilibrium in which βi(v) = av for all i? Your expected payoff if you bid b This is equal to (v b)prob(you win) (v b)prob(b > av2 and b > av3... and b > avn) (v b)prob(b > av2)prob(b > av3)...prob(b > avn) = (v b)(b/a) n 1 Levent Koçkesen (Koç University) Auctions 11 / 1 Levent Koçkesen (Koç University) Auctions 12 / 1
First order condition: Solving for b (b/a) n 1 +(n 1) v b a (b/a)n 2 = 0 b = n 1 n v βi(v) = n 1 n v is a Bayesian equilibrium βi(v) = β(v) and β is strictly increasing and differentiable? β(0) = 0 Suppose β(0) = b > 0 β(b) > b Expected payoff to v = b (b β(b))prob(β(b) > β(v)) = (b β(b))f(b) < 0 Levent Koçkesen (Koç University) Auctions 13 / 1 Levent Koçkesen (Koç University) Auctions 14 / 1 Expected payoff to b is Equivalently d dv (G(v)β(v)) = vg (v) (v b)f(β 1 (b)) n 1 = (v b)g(β 1 (b)) where G = F n 1 is the distribution function of the highest value among n 1 bidders. Maximizing with respect to b yields the FOC Substituting b = β(v) (v b) G (β 1 (b)) β (β 1 (b)) G(β 1 (b)) = 0 G(v)β (v)+g (v)β(v) = vg (v) Theorem (Fundamental Theorem of Calculus) Let f : [a,b] R and F : [a,b] R be continuous on [a,b] and F (x) = f(x) for all x (a,b). Then x F(x) = f(t)dt+f(a) a By the fundamental theorem of calculus and the fact that β(0) = 0, we get β(v) = 1 v xg (x)dx G(v) 0 This is only the necessary condition. Next we will show that β constitutes an equilibrium. Levent Koçkesen (Koç University) Auctions 15 / 1 Levent Koçkesen (Koç University) Auctions 16 / 1
We can restrict ourselves to deviations b from β(v) that are in the range of β. b β(v) v : b = β(v) (since β is st ly increasing and continuous)... and b > β(v) cannot be a profitable deviation Expected payoff of player i with type v who plays β(y) We want to show Ui(y v) = (v β(y))g(y) Ui(v v) Ui(y v) for all y 0 We have y Ui(y v) = (v β(y))g (y) β (y)g(y) From the FOC before: β (y)g(y) = yg (y) G (y)β(y) Therefore y Ui(y v) = (v y)g (y) > 0 for y < v < 0 for y > v Levent Koçkesen (Koç University) Auctions 17 / 1 Proposition There is a unique symmetric Bayesian equilibrium in strictly increasing and differentiable strategies given by β(v) = 1 v xg (x)dx G(v) 0 It has also been shown to be the unique equilibrium Note that we can write this as β(v) = E[X X < v] where X is the random variable equal to the highest of n 1 values. We can also write it in terms of F v 0 β(v) = v F(x)n 1 dx F(v) n 1 β(v) < v for all v > 0: bid shading Levent Koçkesen (Koç University) Auctions 18 / 1 First price auctions: β(v) < v, highest value wins Second price auctions: β(v) = v, highest value wins If you were the seller which format would you use? Doesn t matter: they generate the same expected revenue It turns out that this is a more general result: Proposition (Revenue Equivalence Theorem) Any auction with independent private values with a common distribution in which 1 the number of the bidders are the same and the bidders are risk-neutral, 2 the object always goes to the buyer with the highest value, 3 the bidder with the lowest value expects zero surplus, yields the same expected revenue. Levent Koçkesen (Koç University) Auctions 19 / 1