Notes on Auctionsy. ECON 201B - Game Theory. Guillermo Ordoñez UCLA. February 15, 2007

Transcription

1 Notes on Auctionsy ECON B - Game Theory Guillermo Ordoñez UCLA February 5, 7 Describing an auction game n these notes we will discuss auctions as an extension of Bayesian Games. We will consider auctions for a single indiisible object (say a painting, a car, a franchise, etc). Potential buyers di er in their aluations of the object and the seller does not know these aluations. Hence, since each player has priate information about the true alue of the object to him (or about priate signals of the true alue, in the case of common alues objects), this problem has basically the same structure that any Bayesian Game. n these notes we will focus on symmetric auctions (where participants are ex-ante identical), independent signals and symmetric equilibria. More formally, consider bidders. The type of the ith bidder is his true priate aluation i, drawn from a common distribution on the interal [,]. The cumulatie distribution function F i is smooth and strictly increasing, such that F () is the probability that a gien bidder s true alue is or less. A pure strategy for a player i is the bidding function f i : [; ]! b i +, where f i () is player i s bid if his true alue is. We will assume f i is a continuous and di erentiable function (this assumption only helps the exposition but in fact nothing in the de nition of equilibrium requires that strategies be di erentiable, continuous or een measurable). y These notes were prepared as a back up material for TA session. f you hae any questions or comments, or notice any errors or typos, please drop me a line at guilord@ucla.edu

2 We will analyze rst price, second price and third price auctions, generalizing the results to 4th and lower-price auctions. We will also discuss the eenue Equialence concept. First Price Auction n a rst price auction, the bidder who o ers a highest price gets the good and pays his bid. The other bidders do not pay anything. y Gien the assumptions we can de ne Pr(; y) = F (t) dt = G (y) De ne also a bid function f that maps alues to bids b, such that f is continuously di erentiable and strictly increasing f >. Also consider f() = and f() = M. This is, when the alue of the bidder is he will bids while when it is, he will bid M. The expected utility for the bidder is Eu i (bj i ; i ) = E( bjb wins) = E( i bjb f ( i )) We will change ariables de ning w = f (b) =) b = f (w). Hence we can rede ne E( i f (wi )jf (wi ) > f ( i )), being w i the alue the indiidual i will pretend to hae Because we assume f is strictly increasing, we can express the preious expression as: E( i f (wi )jw i > i ) = ( i f (wi )) Pr(w i > i ) = ( i f (wi ))G (w) where G (w) is the probability i wins (the product of probablities the other ( ) bidders alue the good by less than w). Hence, the bidder would like to sole the following problem max w Eu(f w) j; f i ) = ( f (w) )G (w) Eu(f w)j; f i ) w= =

3 By equilibrium condition we further know that f (w) = f (), and, which is the same, w ( f (w))g (w) = ( f(w) )G (w) + [ f (w)]( )G (w) G (w) = w= eplacing in the FOC. ( f () )G () + [ f () ]( )G () G () = ( )G () G () = f () G () + f () ( )G () G () being the right hand side of the equation just [f () G () ], then [f () G () ] = ( )G () G () ntegrating both sides between and (the alue for the bidder), Z f () G () f () G () = ( )G () G () d (and f () = ) f () = ( ) G () Z G () G () d Particular case: Uniform Distribution Consider is distributed uniformly between [,]. n this particular case, G (y) = Pr( < y) = y f () = ( ) Z : :d n soling this integral recall there is a constant of integration. To eliminate it we need a boundary condition. n our case this condition is gien 3

4 by the fact that no player would bid more than M or less than. Hence the constant of integration is and Hence, f () = ( ) =) f () = f st () = ( ) () This means that the bid of each person will depend on his or her own aluation and on the number of bidders. t s interesting to note that the more the quantity of competing bidders, the less the di erence between each bidder s real alue and the bid o ered. Expected eenue for the seller To obtain the expected reenue to the seller, we need to integrate oer all participants with alue less than the alue of the winner. n the case of the winner we need to integrate oer and. This multiple integral should be multiplied by the number of bidders to hae all possible winners combinations (it does not matter the order of the lossers, only who is the winner). Without loss of generality consider bidder is the winner. We use the general formulation considering that the equilibrium bid is gien by f () = ( ) Expected eenue E = > [( ) ]d :d d The limits of the integration can be written as E = E = E = [( ) ]d :d d [( ) ]d d [( ) ]d d 4

5 and following this pattern until we reach, we hae E = Z ( ) d = ( + ) + Hence, E = + () Hence, when buyers paticipate the expected reenue to the seller is 3, when 3 bidders participate the expected reenue is, when 4 bidders participate the expected reenue is 3 5 ; and so on. As can be seen, as!, E!, which means the seller almost gets the maximum possible alue players hae. This is because, when the number of participants increase, both the bidders bid are close to their real alue and the probability a buyer draw a alue arbitrarily close to, is almost one. 3 Second Price Auction n this type of auctions, the highest bidder (call him i) wins the object and pays the second highest bid (such that the winner gets a utility i max j6=i b i while other bidders pay nothing haing a utility of ). f seeral bidders win the auction, the good is allocated randomly among them. n any case this case has zero probability of ocurrence since we are dealing with continuous functions. We will show that in this enironment each player i will bid his or her own aluation. This will be a (weakly) dominance equilibrium. De ne the highest bid as b. We will analize all the possible ordering among own bids b i, own true alues i and the highest bid by others b We will compare the strategy of bidding any b i, against the strategy of bidding the own true alue i : 5

6 i Bids i i Bids bi esult Utility esult Utility b i < i < b Lose Lose i < b i < b Lose Lose 3 b i < b < i Win i b > Lose 4 i < b < b i Lose Win i b < 5 b < b i < i Win i b > Win i b > 6 b < i < b i Win i b > Win i b > Hence, bidding the true alue i weakly dominates bidding any other alue b i. f nd () = (3) This is true for all bidders, such that eerybody bids the own aluation. Expected eenue for the seller To obtain the reenue expected by the seller, we need to integrate oer all players with a alue smaller than the alue of the winner. We hae to integrate oer the runner up bid (in equilibrium the runner up true alue). Without loss of generality we will call this payment. This multiple integral should be multiplied by the number of all possible combinations of winners and runner ups, that is by ( ). (t does not matter the order of the lossers from the third place on. n order to obtain the payments only matters who is the winner and who is the runner up). Expected eenue E = ( ) > The limits of the integration can be written as E = ( ) :d :d d 6 > ; :d :d d

7 E = ( ) E = ( ) E = ( ) E = ( ) [ j ]d d [ ]d d [ ]d d [ 3 ]d d Following this calculation exercise up to, we hae E = ( ) [ ]d :d = ( ) d = ( ) + E = ( ) ( + ) Hence, E = + :d Also in this type of actions, when buyers paticipate the expected reenue to the seller is 3, when 3 bidders participate the expected reenue is, when 4 bidders participate the expected reenue is 3 5 and so on. As can be seen, as!, E!, which means the seller almost gets the maximum possible alue players can hae. (4) 4 Third Price Auction By the third price auction we mean the highest bid wins but pays the third highest price. Without loss of generality, we will assume an order in the aluations, > > >, being the highest aluation and the lowest one. The idea of the ordering is ery important to obtain the probablities of winning and the expected payments from winning. 7

8 To obtain the symmetric equilibrium of the third price auction we will follow the same method used to obtain rst price auction equilibrium. n the rst price auction we maximize the expected reenue of a bid, gien by the di erence between the own alue and the own bid, conditional on winning. Howeer, in the third price auction this is not so simple because the expected alue to maximize is the di erence between the own alue and the expected alue of the third price, conditional on being the third bid and conditional on the bidder winning the auction. n the third price auction, the expected utility for the bidder is, Eu i (b i j i ; i ) = E i ( i E(b 3 jb < b i )jb i wins) = E i ( i b 3 jb i f ( i )) being b i the own bid and E(b 3 jb < b i ) the expected alue of the highest third price that is necessary to pay in case of winning the bid (i.e. conditional on the fact that the own bid is greater than the second highest price). Now on, we will refer to the rst bidder as the winner, without loss of generality. We can rede ne E ( E(f (w3 )jf (w ) < f (w ))jf (w ) < f (w )), being w i the alue the indiidual i will pretend to hae. Subscripts, and 3 refers to the position of the bid in the auction. Bidder only wins with the highest bid f (w ), in which case he would pay the third highest bid f (w3 ). Since we are assuming f is strict increasing, we can express the preious expression as E ( E(f (w3 )jf (w ) < f (w ))jw < w ) E = E(f (w3 )jf (w ) < f (w )) Pr(w < w ) E = E(f (w3 )jf (w ) < f (w )) G (w ) Naturally, preiously to any maximization, it is important to deelop the expression E(f (w3 )jf (w ) < f (w )), which is the expected payment the winner has to face 8

9 Z E(f (w3 )jf (w ) < f (w )) = w f (w3 ):h (w3 jw <w )dw 3 (5) ecall w is the ariable oer which we are integrating and h (w3 jw <w ) is a conditional density function of an order statistic. To sole it consider that by the de nition of a conditional distribution h (w3 jw <w ) = h 3;(w3 ;w )=h (w ) (6) ecall that each w i is distributed with a density function g, then the joint density function of w ; w ; :; w, such that < w < w < : < w <, can be written as h(w ; w ; :; w ) = (!)g (w )g (w )::g (w ). t is possible to express it as the product of indiidual densities since they are independent. We also need to multiply this by the factorial of the number of indiiduals (), which shows the possible arrangements in which w s can be ordered. Now, we can obtain the marginal distribution of w, h (w ) = = = n the next step, (because G () = ) and so on h (w ) = (!) w w w (!) (!)g (w )g (w )::g (w )dw dw w g (w )dw g (w )g (w )g (w )dw ::dw (!)G (w ):g (w )g (w )g (w )dw ::dw G (w ):g (w )dw = [G (w )] ( )! g (w )g (w )dw h i G (w ) w = h i G (w ) G(w ) h (w ) = (!) g ( )! (w ) (7) 9

10 doing We can calculate any marginal density function of a k th order statistic h k(wk ) = w w 3 w k k w Hence, the result of the integral up to the (k (!)g (w )g (w )::g (w )dw dw k+ dw k dw )th ariable is equal to the case for the rst order statistic, but as if it were k bidders. So, h k(wk ) = (!) [G (wk )] k w (k )! g (wk ) g (w )g (w )::g (wk )dw k dw w 3 w k and w g (w )dw = G (w w ) = G w (w ) G (w ) w h k(wk ) = (!) [G (wk )] k w (k )! g (wk ) G(w ) G (w ) g(w )::g (wk )dw k dw w 3 w 4 w k G(w ) G (w ) g(w )dw = [G (w ) G (w )] w w 3 w 3 = [G(w) G(w) G(w) G(w3)] = [G (w ) G (w3 )] Doing the same until nishing the integrals, we obtain nally the k th order statistic. h k(wk ) = (!) k G(wk ) G(w ) G ( )!(k )! (wk ) g(wk ) (8) Now, combining both results we can nally obtain the join density of any two order statistic, i and j such that w i > w j. Now we are in conditions to nally obtain the expected payment to face in case of winning. h j;i(wj ;w i ) = w 3 w w i w i j+ j :: w i w j w j (!)g (w ):g (w )dw dw i+ dw i dw j+ dw j dw

11 Operating similarly, we can obtain, h j;i(wj ;w i ) = (!) h i G (i )!(j i )!( j)! (wj ) j h G (wi ) G (wj )i j i G(w ) G (wi ) i g(wj )g (wi ) For our speci c case, where i = and j = 3. h 3;(w3 ;w ) = (!) 3 G(w3 ) G(w ) G ( 3)! (w3 ) g(w3 )g (w ) Now, applying (6) to obtain the conditional density function, h (w3 jw <w ) = h 3;(w 3 ;w ) h (w ) = (!) ( 3)![G (w3 )] 3 [G (w ) G (w3 )]g (w3 )g (w ) (!) ( )![G (w )] g (w ) h (w3 jw <w ) = ( )( ) 3 G(w3 ) G(w ) G (w3 ) g(w3 ) (9) G(w ) Plugging (9) in (5) we hae the expected amount the bidder should pay in case he wins the auction, Z E(f (w3 )jf (w ) < f (w )) = w f (w) :( )( ) 3 G(w3 ) G(w ) G (w3 ) g(w3 ) dw G(w ) () and the expected utility for the bidder in case of winning is E ( E(f (w3 )jf (w ) < f (w ))jw < w ) = E(f (w3 )jf (w ) < f (w )) G (w ) E () = G (w ) G (w ) E(f (w 3 )jf (w ) < f (w )) E () = G (w ) G (w ) f (w3 ):( )( ) [G (w 3 )] 3 [G (w ) G (w3 )]g (w3 ) dw [G (w )] 3 earranging and canceling terms

12 E () = G (w ) h knowing ( ) G (w3 ) f (w3 )( ) G (w ) G (w3 ) h ( ) G (w3 ) i 3 g(w3 )dw 3 = dg (w 3 ), we can express. 3 g(w3 )dw 3 i Z E () = G (w ) ( ) w f (w3 ) G(w ) G (w3 ) dg (w 3 ) () Finally, this is the expression bidder has to maximize with respect to w, the control ariable or the aluation the bidder will pretend to hae. ecall in equilibrium, w = FONC fw g ( )G (w ) G (w ) +( )f (w )G (w )dg (w ) ( )f (w )G (w )dg (w ) ( )G (w ) f (w3 )dg (w 3 ) = =) ( )G ( ) G ( ) = ( )G ( ) diiding by ( )G ( ), G ( ) = f (w3 )dg (w 3 ) di erentiating with respect to dg ( ) d = f ( ) G ( ) + dg ( ) d and considering dg ( ) d = ( )G 3 f ( ) = + G ( ) ( )G 3 ( ) G ( ) ( ) G ( ) f (w3 )dg (w 3 ) f 3rd ( ) = + G ( ) ( )G ( ) () Because the equilibrium is symmetric, then this is the optimal strategy for all bidders.

13 Particular case: Uniform Distribution Consider alues are distributed uniformly between and. n this case G ( ) = and G ( ) =.Hence, f ( ) = Compare this result with rst price auctions symmetric equilibrium strategies (f ( ) = ) and with the second price symmetric equilibrium strategies (f ( ) = = ): Generalizing, a set of conclusions can be obtained from here: First, we can extrapolate the results to any lower-price auctions with symmetric equilibrium strategies for the case in which alues are distributed uniformly between and. n general, f kst ( ) = k + (3) being k the bid the winner should pay in case of winning (by haing the highest bid). n fact we could check this result is an equilibrium for any lower-price strategy. As can be seen, a lower-price auction (an increase in k) leads to higher bidding. Second, for third and lower-price auctions, bids are higher than own aluations. This is the opposite to shade aluations, as is the case in rst price auctions. Third, as in rst price auction, bids conerges to the true alue when the number of bidders increases. The di erence is that while in rst price auctions the conergence is from below, in lower price auctions the conergence is from aboe. Finally, it s posible to show also that expected reenue to the seller is the same no matter the kind of auction we are considering. t does not matter if we are talking about a rst, second, third or any lower-price auction, always the expected utility for the seller will be the same. This property is called eenue Equialence. 3

14 5 First-Price Auctions with a eseration alue n many rst-price auctions (such as the ones you can play in E-Bay), the seller is entitled to set a reseration alue and commits to not sell the item unless the winning price exceeds. We will try to nd in this case the symmetric equilibria to see how it changes with respect to the standard case without reseration alues. As we will see it does not imply that eery bidder must hold a bid surpassing this threshold (a bidder with a reseration alue lower than the implicit in the seller s reseration alue will bid below this bound leading his probabilty of winning to zero - autoexclusion scenario). We will assume that the seller reeals his true priate alue so the resere price announced to the bidders is in fact the true reseration alue (p ( ) = ) As an announced seller s reseration alue ( ) exist, the o er by the i th bidder has to full ll the following restriction, 8 < f( i ) i > f (p ( ) ) b i = : < p ( ) i 6 f (p ( ) ) The expected utility of the bidder is gien by 8 < Eu i (bj i ; i ) = : E( bjb wins) = E( i bjb f ( i )) i > f (p ( ) ) i 6 f (p ( ) ) An important feature is the reseration alue is announced beforehand, and therefore it is not necessary to compute the probability of beating this alue (as it is necessary when computing the probability of beating other participants bids). Otherwise the seller would be considered as an additional bidder whose priate resered aluation (transformed into a "seller bid" or resere price) should be beat and the conditioned expectation of the utility of the bidder should recognize this fact. 4

15 Using the same tools that the standard case without reseration alue we get, [f () G () ] = :( )G () :G () ntegrating both sides between (the public alue for the seller) and (the priate alue for the bidder) we hae Z Z [f () G () ] = :( )G () :G () Z f () G () f ( ) G ( ) = f () = G ( ) G + () :( )G () :G () d (and f ( ) = ) ( ) G () Z :G () :G () d Particular case: Uniform Distribution Consider is distributed as a uniform between [,]. f () = Z ( ) + : f () = ( ) + ( f () = ( ) + : ( ) f () = ( ) + ) :d ( ) : ( ) 5

16 f st w= () = + (4) This result is consistent with a Bayesian Nash symmetric equilibrium where each player s bid depends on his own aluation (), on the aluation of the seller ( ) and on the number of bidders ().! ; f ()! ). (in the limit, as emember also that in the simplest case, when there is no reseration alue for the seller, the optimal bid was f () = ( ) which is the case by setting = (i.e. no reseration price for the seller) in equation (4). Expected eenue for the seller when reseration alue is To obtain the expected reenue to the seller, we need to integrate oer all players with alue below the alue of the winner and, for the winner between and. Setting = without loss of generality. Using the general formulation, f () = +, the expected reenue is, E = E = E = Hence, > " + d :d d + d :d :d d!# d :d :d d :d + E = Z " +! # :d 6

17 Z E = E = Z d + ( ) :d + + ( ) + E = + + ( ) + E = ( + ) + ( + ) + (5) We can obtain the expected reenue of rst price auction without reseration alue () by replacing in (5) = Now, a natural question arises. Which is the alue of that yields the greatest expected reenue for = (+) (+) + (+) = (+) = = = Just to erify this candidate is indeed a a E (@ ) == E (@ h ( )() s < i (+) + E (@ ) < == Hence the resere alue of = for the seller is the maximizer of his expected reenue in this particular case. 7