U.U.D.M. Project Report 2016:27 Auction Theory Philip Selin Examensarbete i matematik, 15 hp Handledare: Erik Ekström Examinator: Veronica Crispin Quinonez Juni 2016 Department of Mathematics Uppsala Uniersity
Auction Theory Philip Sehlin May 23, 2016 Abstract We study auction theory where bidders hae independent priate alues. We describe different auction types, and derie Nash equilibria for the symmetric case in which all bidders hae alues drawn from the same distribution. We also study a case with uncertainty about the number of bidders, and examples with asymmetric distributions. 1
Contents 1 Introduction 3 1.1 Some Common Auction Forms.............................. 3 1.1.1 English Auction.................................. 3 1.1.2 Dutch Auction................................... 3 1.1.3 Sealed-Bid First Price Auction (FPA)...................... 3 1.1.4 Sealed-Bid Second Price Auction (SPA)..................... 3 1.2 Nash Equilibrium...................................... 4 2 Indepentent Priate Values (IPV) Auctions with Symmetric Bidders 5 2.1 First Price Auction Nash Equilibrium.......................... 5 2.2 Second Price Auction Nash Equilibrium......................... 7 2.3 The Enelope Theorem.................................. 8 2.4 Reenue Equialence.................................... 8 2.4.1 First Price Auctions................................ 9 2.4.2 Second Price Auctions............................... 10 2.4.3 Expected Reenue from FPA and SPA...................... 10 3 Uncertain number of bidders 11 3.1 First Price Auctions.................................... 11 3.2 Second Price Auctions................................... 13 4 Asymmetric Auctions 13 4.1 Strong and weak bidders.................................. 13 4.1.1 Equilibrium for strong and weak bidders..................... 13 4.1.2 Reenue Equialence in Asymmetric Auctions.................. 15 4.2 Different length of the interal of alues, with uniform distributions.......... 16 4.3 Same length of interal of alues, but different distributions.............. 16 4.3.1 Uniform and Normal, both with interal [0,1].................. 16 4.3.2 Uniform and Exponential, both with interal [0,1]............... 17 4.4 Different settings with three players........................... 17 4.4.1 Uniform, Normal and Exponential, eeryone with interal [0,1]........ 17 4.4.2 Two players uniform distributed oer [0,1] and one player uniform distributed oer [0,2]...................................... 18 4.4.3 Uniform oer [0,1], Uniform oer [0,2] and Normal oer [0,1].......... 18 5 References 19 2
1 Introduction We consider auctions with independent priate alues (IPV). We let the number of bidders be n, and the set of possible bids is [0, ). A generic bid by player i is denoted b i, and the alue of player i is denoted i. The distribution function of a player s alue is denoted by F. 1.1 Some Common Auction Forms 1.1.1 English Auction The English auction is an open ascending price auction. This auction has a low start alue, and increases with small increments until there is only one interested bidder left. This will be the same as the Second Price Auction, since when the second highest bidder drops off the highest bidder wins and pays the second highest price. i max Bidders payoff function [1][2] : u i (b) = u i (b 1,..., b n ) = b j, if b i > max b j. j i j i 0, otherwise. 1.1.2 Dutch Auction The Dutch auction is the open descending price counterpart of the English auction. Here the auction has a start alue higher than eery interested bidder, and gradually lowered until there is a bidder that is interested at that price. This will be the same as the First Price Auction, since the highest bid wins and pays his own bid. Bidders payoff function [1][2] i b i, if b i = max : u i (b) = u i (b 1,..., b n ) = b i. i N 0, otherwise. 1.1.3 Sealed-Bid First Price Auction (FPA) In this form the bidders submit their bids in sealed enelopes. The bidder that has the highest bid when the auction has ended wins the auction and pays the price that he bids. Bidders payoff function [1][2] i b i, if b i = max : u i (b) = u i (b 1,..., b n ) = b i. i N 0, otherwise. 1.1.4 Sealed-Bid Second Price Auction (SPA) This auction is also known as a Vickrey Auction. Here also the bidders submit their bids in sealed enelopes, but in this case if you hae the highest bid, you will pay the second highest bid. i max Bidders payoff function [1][2] : u i (b) = u i (b 1,..., b n ) = b j, if b i > max b j. j i j i 0, otherwise. 3
1.2 Nash Equilibrium Nash Equilibrium is a stable state of a system inoling the interaction of different participants, in which no participant can gain by a unilateral change of strategy if the strategies of the others remain unchanged. Definition 1.2.1. Nash equilibrium of a strategic game [4] (N, (A i ) i N, (u i ) i N ) is a profile a A of actions with the property that for eery player i N, we hae where u i is the payoff function. u i (a i, a i) u i (a i, a i) for all a i A i a i is the strategy of player i, and a i if the strategies of all players except for player i. 4
2 Indepentent Priate Values (IPV) Auctions with Symmetric Bidders In the IPV case there are two important features that define an IPV auction, such as Bidder i s information is independent of j s information. Bidder i s alue is independent of j s information. 2.1 First Price Auction Nash Equilibrium Proposition 2.1.1. In a First Price Auction the Nash Equilibrium is to bid [1] where is the lowest possible alue F (x) is the CDF of each player s alue ( ) n 1 F (x) β() = dx F () Proof. N = {1,..., n} : The set of bidders. β i () = [0, ) for each i N : The set of possible bids by player i. A generic bid by player i is denoted b i () β i. Assume that β i () β() since this is a symmetric equilibrium, and all bidders will use the bid function b = β(). i b i, if b i = max Bidders payoff function: u i (b) = u i (b 1,..., b n ) = b i. i N 0, otherwise. A player will bid b if i β 1 (b). ( n 1 A player with aluation i and who bids b i expects to earn ( i b i ) P r(b j i b i )). P r( i β 1 (b)) = F ( β 1 (b) ) and ( ) n 1 ( P r( j i β 1 (b)) = F ( β 1 (b) )) n 1. ( A player with aluation i and who bids b then expects to earn ( i b) F ( β 1 (b) )) n 1. To calculate the Nash Equilibrium we will take the deriatie with respect to b. 5
( ( b) (F ( β 1 (b) )) ) n 1 = (F ( β 1 (b) )) n 1 ( b)(n 1) (F ( β 1 (b) )) n 2 F ( β 1 (b) ) + b β ( β 1 (b) ) = 0 Integraton by parts gies us: This gies us the Nash Equilibrium: ( ) n 2F ( ) n 1 β() (n 1) (F ()) () F () + β () ( ) n 1 ( n 2F β () F () + β()(n 1) F ()) () = ( n 2F = (n 1) F ()) () = 0 ( [ ( ) n 1 ] ( n 2F β() F () = x(n 1) F (x)) (x)dx ( ) n 1 [( n 1 ] dx β() F () = x F (x)) ( ) n 1 ( ) n 1 ( n 1dx β() F () = F () F (x)) β() = ( ) n 1 F (x) dx F () ) (1) 6
Example 2.1.1. If the alue of each player is uniformly distributed oer [0, 1] we hae We use P roposition 2.1.1. that β() = And by soling this integral we get that ( F (x) F () F (x) = x ) n 1 dx = 0 β() = 1 (n 1) = n n So if we use 2 players for this equation we get that β() = 1 2 ( ) x n 1 dx Figure 1: Nash Eqiulibrium for p 1 and p 2, (where both is drawn from the same distribution Uni[0,1]) for any gien alue at [0, 1] 2.2 Second Price Auction Nash Equilibrium Proposition 2.2.1. In a Second Price Auction the Nash Equilibrium [1] is to bid his true alue b i = i Proof. We want to show that the strategy (b 1,..., b n ) = ( 1,..., n ) i.e. a truthful bid. We aluate the bids as ( 1,..., n ) = (b 1,..., b n ). Then b 1 = 1 wins the auction, he will pay the second highest bid b 2 = 2. The payoff will then be u 1 = 1 2 0. Likewise, for all other bidders i 1 i will need to change her payoff from 0 and will bid higher than 1, in which case the payoff will be u i = i 1 < 0 and will result a negatie payoff. Therefore no one will make a profit when deiate from the strategy. 7
2.3 The Enelope Theorem Suppose that b i = b( i ) is the symmetric equilibrium, then i s eqiulibrium payoff gien alue i is: since i is playing his best response we get U( i ) = ( i b( i ))F n 1 ( i ) U( i ) = max b i ( i b i )F n 1 (b 1 (b i )) And if we take the deriatie of this with respect of, we get: d d U() = F n 1 (b 1 (b( i ))) = F n 1 ( i ) =i and i U( i ) = U() + F n 1 ( s)d s and since a bidder with alue neer will win the auction gies us that U() = 0. And by combining the equations aboe we can sole for the equilibrium strategy [2][4] as 2.4 Reenue Equialence b() = i F n 1 (ṽ)dṽ F n 1 () Theorem 2.4.1. Reenue Equialence Theorem (RET) [2][4] Suppose n bidders hae alues 1,..., n identically and independently distributed with cumulatie distribution function F ( ). Then any equilibrium of any auction game in which 1. The bidder with the highest alue wins the object. 2. The bidder with alue gets zero profits. With these settings the different auction types generate the same expected reenue, such that the seller will generate the same profit from any type of auction he choose. Proof. We consider a general auction where bidders will place there bids b 1,..., b n. The auction rule specifies for all i, such that x i : B 1... B n {0, 1} t i : B 1... B n R, (2) where x i is the probability that player i will win the object and t i is players i s payment as a function of (b i,..., b n ). Gien this rule, bidder i s expected payoff as a function is: E(U i ) = U i ( i, b i ) = i E b i [x i (b i, b i )] E b i [t i (b i, b i )]. b i, b i is the equilibrium of the auction, so bidder i s equilibrium payoff is therefore: U i ( i, b i ) = i F n 1 ( i ) E i [t i (b i ( i ), b i ( i )]. 8
and here we use condition (1) from T heorem 2.4.1. where the highest alue wins the object, so we can write E i [x i (b i ( i ), b i ( i )] = F n 1 ( i ). d d U i() = E b i [x i (b i ( i ), b i ( i )] = F n 1 ( i ), =i and also that i i U i ( i ) = U i () + F n 1 (ṽ)dṽ = F n 1 (ṽ)dṽ where we can use (2) from T heorem 2.4.1. to write U i () = 0 which uses the condition that the bidder with alue gets zero profits. If we combine our expressions for U i ( i ), we get bidder i s expected payment as: E i [t i (b i, b i )] = i F n 1 ( i ) i F n 1 (ṽ)dṽ = i ṽdf n 1 (ṽ), and since x i is not a part of this expression, bidder i s expected equilibrium payment gien his alue is the same under eery auction that satisfies both (1) and (2) of T heorem 2.4.1., then the expected payment is: E[V 1:n 1 V 1:n 1 i ] = E[V 2:n V 1:n = i ]. So therefore the expected reenue is: E[Reenue] = ne i [i s expected payment i ] = E[V 2:n ]. Just to show how the expected reenue is for the different auctions we show this for the example where there are two bidders with alues drawn from the same distribution U ni[0, 1]. Then the expected reenue is 1 2 3 and the expected profit for bidder i with alue is 2 2.4.1 First Price Auctions Bidder i has the alue i b i = (n 1) n i = i 2 The probability that b i wins the auction is P ( i ) = i The expected payoff if b i wins is then i 2 So therefore the expected profit is ( E[U i ( i )] = i i ) i = 2 i 2 2 9
2.4.2 Second Price Auctions Bidder i has the alue i b i = i The probability that b i wins the auction is P ( i ) = i The expected payoff if b i wins is then i 2 So therefore the expected profit is ( E[U i ( i )] = i i ) i = 2 i 2 2 2.4.3 Expected Reenue from FPA and SPA Since the bidder with lowest alue has the expected profit E[U( i )] = 0, therefore: E[U( i )] = E[U( i )] + U (x)dx = 0 + xdx = 2 0 0 2 And the expected profit for each bidder is: E[U 1 ()] = E[U 2 ()] = 1 0 U()d = This gies us the total expected profit for both bidders as: 1 0 2 2 d = 1 6 E[Total bidder profit] = 1 3 And hence expected reenue is gien by: E[Reenue] = E[Surplus] E[Total bidder profit] where E[Surplus] = E[max{ i, j }] = 2 3 So this gies us that the expected reenue is E[Reenue] = 2 3 1 3 = 1 3 10
3 Uncertain number of bidders 3.1 First Price Auctions Example 3.1.1. If we let m be the number of possible bidders, and set the possibilities that player i will attend as (1 p i ) and p i else. Let m = 2 and p 1 (1) be the probability that player 1 beliees that there only will be 1 player (himself) in the auction. Since this is a symmetric auction with only two buyers we can write this as the following equation p 1 (1) = p and p 1 (2) = (1 p) F () = p( b) + (1 p)f (β 1 (b))( b) Maximize by taking the deriatie df () db If we use that =β 1 (b) we get = p (1 p)f (β 1 (b)) + (1 p)( b)f(β 1 1 (b)) β (β 1 (b)) = 0 df () (1 p)( β()f()) = p (1 p)f () + = 0 db β df () = (p + (1 p)f )β + (1 p)f()β() = (1 p)f() db df () ( (p ) ) = + (1 p)f β = (1 p)f() db So the equilibrium function is β = (1 p) 0 uf(u)du p + (1 p)f () (3) So to see the price difference between full information about the number of players who will attend we can take the example with two players in both situations with p 1 (1) = p 1 (2) = 0.5 for the equation with uncertain number of bidders. Uncertain number of bidders, FPA case gies us the Nash Equilibrium at b if P A = (1 p) 2 2(p + (1 p)) and then will gie us b if P A = (0.5) 2 2(0.5 + (0.5)) = 2 2(1 + ) 11
Example 3.1.2. If we here instead of Example 3.1.1. take three players instead of two where p 1 (1) = p, p 1 (2) = q and p 1 (3) = (1 p q) we will get F () = p( b) + qf (β 1 (b))( b) + (1 p q)f (β 1 (x)) 2 ( b) Maximize by taking the deriatie df () db If we use that =β 1 (b) we get = p qf (β 1 (b)) + q( b)f(β 1 1 (b)) β (β 1 (b)) (1 p q)f (β 1 (b)) 2 + + 2((1 p q)( b)f (β 1 (b))f(β() 1 1 β (β 1 ()) ) = 0 df () 1 = p qf () + q( β()f()) db β () (1 p q)f ()2 + 1 + 2(1 p q)( b)f ()f() β () = 0 df () = p qf () (1 p q)f () 2 q( b)f() + 2(1 p q)( b)(f ()f()) + db β () df () = (p + qf () + (1 p q)f () 2 )β (qf()β() + 2(1 p q)f ()f()β()) = db = (q + 2(1 p q)f ()f()) df () = ((p + qf () + (1 p q)f () 2 )β) = q + 2(1 p q)f ()f() db So the equilibrium function is β = q 0 udu + 2(1 p q) 0 uf (u)f(u)du p + qf () + (1 p q)f () 2 (4) And here we use as aboe that p 1 (1) = p 1 (2) = p 1 (3) = 1/3 for the equation with uncertain number of bidders. Uncertain number of bidders, FPA case gies us the Nash Equilibrium at b if P A = q 2 3 2 + 2(1 p q) 3 p + q + (1 p q) 12
3.2 Second Price Auctions For the Second Price Auction with uncertain number of bidders we still hae the same strategy and Nash equilibrium since it will not change with the uncertainty of players attendance. So the weakly dominant strategy is to bid his true alue b i = i. 4 Asymmetric Auctions Eery auction we hae studied so far has been symmetric in the way that all players hae the same distribution and the same interal of alues, so that all players draw eery alues equal likely. The scenario where it is an symmetric auction is not that common in practice, that is why I also want to study the asymmetric auctions where players are randomly drawn to hae different distributions and alues. We want to see how the Nash Equilibrium differ if we hae this conditions in different type of auctions. 4.1 Strong and weak bidders [3] When we talk about asymmetric auctions it is common to refer to strong and weak bidders, such that a strong bidder dominated the weak bidder with the alues, for example The strong bidder has a shifted distribution such that F S Uni[2, 3] and F W Uni[0, 1] where F S is the strong bidders distribution and F W is the weak bidders distribution. The strong bidder has a stretched distribution such that F S Uni[0, 2] and F W Uni[0, 1]. 4.1.1 Equilibrium for strong and weak bidders If we let b S and b W be the equilibrium bid functions, then we hae the problem For the strong bidder as which hae the first-order condition max F W (b 1 b W (b))( S b) f W (b 1 W (b)) F W (b 1 W (b))(b 1 W ) (b) 1 S b and if we set this equal to 0 at S = b 1 (b) we get For the weak bidder as which gies the first-order condition S f W (b 1 W (b)) F W (b 1 W (b))(b 1 W ) (b) 1 b 1 S (b) b max F S (b 1 b S (b))( W b) f S (b 1 S (b)) F S (b 1 S (b))(b 1 S ) (b) 1 W b 13
and if we set this equal to 0 at W = b 1 W (b) we get f S (b 1 S (b)) F S (b 1 S (b))(b 1 S ) (b) 1 b 1 W (b) b Theorem 4.1.1. Suppose F S conditionally first-order stochastically dominates F W, such that F S (x) F W (x) [2]. Then comparing a first price auction and a second price auction, both who are uniform distributed, Eery type of strong bidder prefers the second price auction since the expected payoff is higher in the second price auction for the strong bidder. Eery type of weak bidder prefers the first price auction since the expected payoff is higher in the first price auction for the weak bidder. Proof. For this proof b S () and b W () hae the same range, so we define a matching function as m() b 1 W (b S()) as a weak bidder who bids equal as the strong bidder in the first price auction. Since b S () < b W () we know that m() <. We calculate the strong bidders expected payoff as E(U( i )) = P r(b W ( W ) < b)( b) and by taking the deriatie with respect to this gie us and by replacing b = b S () this gie us E (U( i )) = P r(b W ( W ) < b) E (U( i )) = P r(b W ( W ) < b S ()) = P r( W < m()) = F W (m()) since P r( < a) = F (a) when uniform distributed. By the enelope theorem we get VS F P () = F W (m())d For the second price auction, both bidders bid their true alue, so and so E (U( i )) = P r( W < ) = F W () VS SP () = F W ()d Since m() < and F W is strictly increasing, the strong bidder prefers the second price auction. By the exact same logic, the weak bidders expected payoff for the first price auction is and for the second price auction V F P W () = F S (m 1 ())ds VW SP () = F S ()d Since m 1 () > the expected payoff is higher with the first price auction for the weak bidder. 14
4.1.2 Reenue Equialence in Asymmetric Auctions Proposition 4.1.1. With asymmetric bidders, the expected reenue in a first price auction may exceed that in an English auction [1][2]. 1 Example 4.1.1. Suppose that the weak buyers aluation is distributed as b W Uni[0, 1+z ] and the 1 strong buyers aluation is distributed as b S Uni[0, 1 z ]. So the strong buyer has a wider interal than the weak buyer. If z = 0, both b W, b S Uni[0, 1] and b 1 i (b) = 2b is buyer i s equilibrium inerse bid function in the first price auction. A buyer with aluation 2b has a probability to win P r(win i = 2b) = 2b and the expected payment is therefore P r(win i = 2b)( i b i ) = 2b(2b b) = 2b 2 When z becomes positie, in the English auction the weak buyer with aluation 2b wins with probability 2b(1 z) and the expected payment is 2b 2 (1 z) In a high-bid auction buyers do not use b 1 1 i (b) = 2b, if they did the strong buyer would outbid the weaker buyer by 2(1 z) to 1 2(1+z), and so can reduce his bid and still win with probability 1. So for equilibrium the strong buyer must reduce his bid as a function of his aluation. A reduction would make the weak buyer to bid more aggressiely than with b 1 i (b) = 2b, since the strong buyers bids are distributed more densely than before. In equilibrium the weak and strong inerse bid functions are therefore b 1 W (b) = 2b 1 + z(2b) 2 and b 1 S (b) = 2b 1 z(2b) 2 The cumulatie distribution function for the winning bid of the first price auction is F F P A (b) = F S (b 1 S (b))f W (b 1 W (b)) = (1 z)b 1 S (b)(1 + z)b 1 W (b) = (1 z2 )(2b) 2 1 z 2 (2b) 4 For the English auction the second aluation is less than b iff it is not the case that both aluations are higher. So F EA (b) = 1 (1 F S (b))(1 F W (b)) = F S +F W F S F W = (1 z)b+(1+z)b (1 z 2 )b 2 = 2b (1 z)b 2 The cumulatie distribution function in the open auction is increasing in z. If z = 0 the two distributions yield the same expected reenue. When z > 0 the expected reenue is strictly greater for the first price auction than for the English auction. 15
4.2 Different length of the interal of alues, with uniform distributions In this case we compare uniform distributions with two players where p 1 Uni(0, 1) and p 2 Uni(0, 2) Figure 2: Nash Equilibrium for p 1 and p 2 for any gien alue at [0, 2] So here the Nash Equilibrium would be gien by b i = 2 3 for both p 1 and p 2 if p 1 alue is 1, and p 2 alue is 2. 4.3 Same length of interal of alues, but different distributions When using normal and exponential distribution I hae scaled these distributions such that the area under the interal is equal to 1. The normal distribution is scaled to the interal [0, 1], so C 2π e (x 0.5)2 2, C > 1 The exponential distribution is scaled to the interal [0, 1], so Ce x, C > 1 4.3.1 Uniform and Normal, both with interal [0,1] In this first case we compare uniform distribution with normal distribution, both with the interal [0, 1]. Figure 3: Nash Equilibrium for p 1 Normal(0.5, 1) and p 2 Uni(0, 1) for any gien alue at [0, 1] 16
4.3.2 Uniform and Exponential, both with interal [0,1] In this second case we compare uniform distribution with exponential distribution, both with the interal [0, 1]. Figure 4: Nash Equilibrium for p 1 Exp(1) and p 2 Uni(0, 1) for any gien alue at [0, 1] 4.4 Different settings with three players We now just hae looked at asymmetric auctions with two players, but how will the equilibrium functions look when we instead uses three players? Here I also hae used the scaled distributions for normal and exponential, such that the normal distribution is scaled to the interal [0, 1], so C 2π e (x 0.5)2 2, C > 1 the exponential distribution is scaled to the interal [0, 1], so Ce x, C > 1 4.4.1 Uniform, Normal and Exponential, eeryone with interal [0,1] Figure 5: Nash Equilibrium for p 1 Exp(1), p 2 Normal(0.5, 1) and p 3 Uni(0, 1) for any gien alue at [0, 1] 17
4.4.2 Two players uniform distributed oer [0,1] and one player uniform distributed oer [0,2] Figure 6: Nash Equilibrium for p 1 = p 2 Uni(0, 1) and p 3 Uni(0, 1) for any gien alue at [0, 2] 4.4.3 Uniform oer [0,1], Uniform oer [0,2] and Normal oer [0,1] Figure 7: Nash Equilibrium for p 1 Normal(0.5, 1), p 2 Uni(0, 1) and p 3 Uni(0, 2) for any gien alue at [0, 2] 18
5 References [1] Krishna, Vijay. Auction Theory. San Diego: Academic, 2002. [2] Lein, J. Stanford. Auction Theory, 2004. [3] Maskin, E. and Riley, J. Harard Uniersity. Asymmetric Auctions. 1st ed, 1998. [4] Milgrom, Paul R. Putting Auction Theory to Work. Cambridge, UK: Cambridge UP, 2004. 19