The material in this chapter is motivated by Experiment 9.

Transcription

1 Chapter 5 Optimal Auctios The material i this chapter is motivated by Experimet 9. We wish to aalyze the decisio of a seller who sets a reserve price whe auctioig off a item to a group of bidders. We begi by lookig at a simpler, but as it turs out, very closely related problem. We cosider a seller who faces a sigle bidder. We begi with the decisio of a seller who chooses a optimal reserve for a secod-price auctio with oe bidder. Clearly the seller who faces a sigle bidder should set a positive reserve, otherwise the sale price will be zero. I fact, the optimal reserve is equal to the moopoly price. It ca be obtaied by solvig max(1 F (r))r r The first-order-coditio is 1 F (r) f(r)r = 0 So, the optimal reserve (or moopoly price) is defied implicitly as the value of r that solves r = 1 F (r). f(r) We assume that F is such that this solutio is a maximum (i.e., we assume icreasig hazard rate). Example 5.1 Suppose F is uiformly distributed o the iterval [0, 100]. The r = 50. Note that if the seller had her ow use value for the item, v s, the her optimizatio problem becomes The first-order-coditio is max(1 F (r))r + F (r)v s r 1 F (r) f(r)r + f(r)v s = 0 31

2 3 CHAPTER 5. OPTIMAL AUCTIONS So, the optimal reserve (or moopoly price) is defied implicitly as the value of r that solves r = 1 F (r) + v s. f(r) Example 5. Suppose F is uiformly distributed o the iterval [0, 100]. The r = 50 + vs. Exercise 5.1 Cosider two scearios. I sceario 1 the seller sells a item usig a first-price auctio with o reserve to bidders. I sceario the seller sells a item to a sigle bidder, but sets the optimal reserve. Assume the bidder(s) value is draw from the uiform distributio o [0, 100]. Compute the expected seller reveue i each case. Discuss. Aswer: I sceario 1, the expected reveue to the seller is (See Chapter 3). I sceario, the optimal reserve is 50, ad seller s expected reveue is (sice there is oly oe bidder) (50) = 5. So the expected reveue i the case where there are two bidders is greater. There are two strikig facts which we will ow metio ad later verify. Both facts apply to the situatio where both the seller ad the bidders are risk eutral. 1. The optimal reserve does ot deped o the umber of bidders.. The optimal reserve is the same for the first- ad secod-price auctios. Usig the above calculatios ad takig these two facts for grated, oe ow already kows what the optimal reserve is for both first- ad secod price auctios, for ay umber of bidders! 5.1 Optimal Reserve with 1 Bidders. Of course, we should ever take thigs for grated. We will ow prove that the optimal reserve satisfies the equatio i the geeral case of bidders. r = 1 F (r) f(r) + v s.

3 5.1. OPTIMAL RESERVE WITH N 1 BIDDERS. 33 First, cosider a secod-price auctio. Whe there is o reserve, the expected paymet of a bidder with value v is m(v) = v 0 yg(y)dy Why? The bidder bids her value ad her paymet is the highest value amog the other 1 bidders, which is distributed accordig to G( ). If the seller sets a positive reserve price r > 0, we kow that the domiat strategy for each bidder is still to bid b(v) = v. However, a bidder oly wis if v > r. So, for v r, m(v, r) = rg(r) + v r yg(y)dy. I fact, it follows from the reveue equivalece theorem, that this is the expected paymet of a bidder with value v i the first-price auctio with reserve r. The seller does ot kow bidders values. However, she kows each bidder s value is distributed accordig to F ( ). Hece, she ca compute the expected paymet of a bidder i either the first- or secod-price auctio E[m(ṽ, r)] = = 1 r 1 r = rg(r) m(v, r)f(v)dv rg(r)f(v)dv + 1 r f(v)dv + = r(1 F (r))g(r) + 1 r 1 r 1 r ( v r ( 1 y ) yg(y)dy f(v)dv ) f(v)dv yg(y)dy (1 F (y))yg(y)dy. Note that the third equality is obtaied by chagig the order of itegratio. Sice this is the expected paymet of 1 bidder, the expected payoff to the seller of settig a reserve r is times this amout plus her payoff if she keeps the item. That is, her expected payoff, as a fuctio of r, is E[m(ṽ, r)] + F (r) v s [ = r(1 F (r))g(r) + 1 r ] (1 F (y))yg(y)dy + F (r) v s (5.1) We ca see how this varies with r by takig the derivative with respect to r. The result (after some maipulatios) is [ ] f(r) 1 (r v s ) (1 F (r))g(r). 1 F (r) This shows immediately that it is beeficial to set r > v s, sice the this derivative is positive. If we assume that F is such that is icreasig i f(r) 1 F (r)

4 34 CHAPTER 5. OPTIMAL AUCTIONS r (this is the stadard icreasig hazard rate assumptio) the the maximum expected payoff occurs whe the term i the square brackets is zero (up to that poit expected payoff icreases i r ad after that it declies). Hece, the optimum reserve is give by the value of r that solves or f(r) 1 (r v s ) 1 F (r) = 0 r = 1 F (r) + v s. f(r) This is the same as the expressio we derived for the case of oe bidder! Example 5.3 I the case of F uiform (where G(y) = y 1 ad g(y) = ( 1)y ) expected seller reveue as a fuctio of her value ad her reserve price choice is, from (5.1), equal to [ r r ( + 1) ] + r v s I-class discussio of Experimet 9: Reserve choice for ad 5 bidder auctios. The formula for computig the optimal reserve values for each treatmet is foud i Sectio 6.1 of the textbook. The experimet covers four cases: =, v s = 0 r = 50; =, v s = 30 r = 65; = 5, v s = 0 r = 50; ad = 5, v s = 30 r = 65. Begi by comparig average observed reveue i each of these four cases to the theoretical predictios. The examie four plots of 0 rouds of data from the reserve price experimet. Plots are =, = 5 for v s = 0; =, = 5 for v s = 30; v s = 0, v s = 30 for = ; v s = 0, v s = 30 for = 5. Exercise 5. Cosider two scearios. I sceario 1 the seller sells a item usig a first-price auctio with o reserve to bidders. I sceario the seller sells a item to 1 bidders, but sets the optimal reserve. Assume the bidders values are draw from the uiform distributio o [0, 1]. Compute the expected seller reveue i each case. Discuss. Aswer: I sceario 1, the expected reveue to the seller is 1 (See Chapter +1 3). I sceario, the optimal reserve is.5, ad seller s expected reveue is computed as follows. First, the expected paymet of ay oe bidder with value v >.5 is.5 (.5) 1 + = (.5) + ( 1) v = 1 v + (.5).5 v y( 1)y dy.5 y 1 dy

5 5.. SALIENCY OF THE RESERVE PRICE DECISION 35 Hece, a bidder s ex-ate expected paymet i the case of F uiform is v + (.5) dv = v (.5) dv dv = 1 1 (.5) +1 + (.5) (1 r) + 1 = (.5) (.5) ( + 1). So, overall, the expected payoff for the seller with value 0 is [ (.5) (.5) ] ( + 1) = (.5) + 1 (.5) ( + 1) The expressio (.5) +1 (.5)+1 > 0 for all ad is very close to zero (less tha.005) for > 5. Hece, for auctios with large umbers of bidders (where here large meas more tha 5) the seller would be idifferet betwee havig the ability to set a reserve ad the additio of oe more bidder. 5. Saliecy of the Reserve Price Decisio I our evaluatio of the experimetal data for the reserve price experimet we might wish to ask whether or ot subjects had sigificat icetive to behave accordig to the theory. I particular, we coducted 0 rouds of the experimet ad we might ask whether subjects reserve choices improved over time. If the aswer is o, it could be because subjects are ot respodig to icetives or it could be because the icetives are very small. I what follows, we evaluate how costly it is to make mistakes i the reserve price decisio. We do so be computig the expected payoff of the seller as a fuctio of her reserve choice. We preset these calculatios for each treatmet cosidered i Experimet 9. We also provide the rage of reserve price choices that result i a loss of less tha 50 cets (or.005 o the 0 to 1 scale) relative to the optimal reserve choice i each case. This iformatio is useful i evaluatig the icetives uderlyig the experimetal data. From the previous sectio, we kow that the expected payoff for the seller with value v s ad reserve price r is [ r r ( + 1) ] + r v s We ow compute expected payoff of the seller as a fuctio of the reserve choice for each of the experimetal treatmets we coducted i class. Values are multiplied by $100 to coform with the values used i the experimet.

6 36 CHAPTER 5. OPTIMAL AUCTIONS = ad v s = 0 Seller expected payoff is 100(r 4 3 r ). 1. The seller s expected payoff as a fuctio of the reserve is show i Figure Seller's Expected Payoff Reserve Figure 5.1: The seller s expected payoff as a fuctio of the reserve whe = ad v s = 0. For the optimal reserve, r = $50, seller expected payoff is $ Reserve choices that are lower or higher tha $50 cause a sigificat reductio i expected payoff. The 50 cet optimal bouds are the solutio to ( r 4 3 r3 + 1 ) =.50, 3 i.e., r = $4.58 ad r = $ = 5 ad v s = 0 Seller expected payoff is 100(r r6 + 3 ). The seller s expected payoff as a fuctio of the reserve is show i Figure. For the optimal reserve, r = $65, seller expected payoff is $ Note, that i the case of 5 bidders it is ot costly to choose a reserve that is too low.

7 5.. SALIENCY OF THE RESERVE PRICE DECISION 37 Seller's Expected Payoff Reserve Figure 5.: The seller s expected payoff as a fuctio of the reserve whe = 5 ad v s = 0. However, reserves that are too high ca be quite costly. This is reflected i the 50-cet optimal bouds. The 50 cet optimal boud is the solutio to ( r r6 + ) =.50, 3 i.e., r = $0.40 ad r = $59.8. = ad v s = 30 Seller expected payoff is 100(1.3r 4 3 r ). The seller s payoff as a fuctio of the reserve is show i Figure 3. For the optimal reserve, r = $65, seller expected payoff is $ Reserve choices that are lower or higher tha $50 cause a sigificat reductio i expected payoff. The 50 cet optimal bouds are the solutio to ( r 4 3 r3 + 1 ) =.50, 3 i.e., r = $58.57 ad r = $ = 5 ad v s = 30 Seller expected payoff is 100(1.3r r6 + 3 ).

8 38 CHAPTER 5. OPTIMAL AUCTIONS 60 Seller's Expected Payoff Reserve Figure 5.3: The seller s expected payoff as a fuctio of the reserve whe = ad v s = The seller s expected payoff as a fuctio of the reserve is show i Figure Seller's Expected Payoff Reserve Figure 5.4: The seller s expected payoff as a fuctio of the reserve whe = 5 ad v s = 30. For the optimal reserve, r = $65, seller expected payoff is $ Note, that i the case of 5 bidders it is ot costly to choose a reserve that is too low.

9 5.. SALIENCY OF THE RESERVE PRICE DECISION 39 However, reserves that are too high ca be quite costly. This is reflected i the 50-cet optimal bouds. The the the 50 cet optimal boud is the solutio to ( r r6 + ) =.50, 3 i.e., r = $55.93 ad r = $ I-class discussio of saliecy of Experimet 9.

10 40 CHAPTER 5. OPTIMAL AUCTIONS

11 Chapter 6 Commo Value Auctios The material i this chapter is related to Experimets 10 ad 11. So far we have studied auctios for which bidders have private values. I private value auctios each bidder kows how much she values the item, ad this value is her private iformatio. I this chapter we will discuss commo value auctios. I commo value auctios, the actual value of the item for sale is the same for everyoe, but bidders have differet private iformatio about what that value is. May importat auctios are commo value auctios. Examples iclude Treasury bill auctios, auctios of timber, spectrum auctios, ad auctios of oil ad gas leases. I each case, the value of the item is the same to all the bidders, but differet bidders have differet iformatio about what that value actually is. I commo value auctios the bidders are ofte subject to the wier s curse. The wiers curse has bee described i may ways. The simplest defiitio is the followig: Wier s curse: I a commo value auctio the bidder with the best (most optimistic) iformatio wis. A bidder who fails to take this ito accout pays, o average, more tha the item is worth. Example: Pey Jar Experimet However, the wier s curse also results i commo value auctios whe bidders fail to accout for the way private iformatio iflueces the biddig behavior of their oppoets. This idea is best illustrated by the followig example. Example: Auctioig a Oil Lease Player A ad Player B are each biddig to purchase the rights to develop a oil field. The field had two parts, Part A ad Part B, each of which cotais either $0 or $3 millio worth of oil. Each possibility is equally likely ad idepedetly determied. Player A is privately iformed about the amout 41

12 4 CHAPTER 6. COMMON VALUE AUCTIONS of oil i Part A. Player B is privately iformed about the amout of oil i Part B. The two players participate i a first-price auctio to purchase the rights to both parts of the field. We will discuss the results of this experimet i class. Here we derive the theoretical predictios. It is useful to go over the iformatioal aspects of this game before we begi. Player A kows the amout of oil i part A, but does ot kow the amout of oil i part B. She oly kows that with probability.5 it is $0 ad with probability.5 it is $3 millio. Likewise for player B. How should the players bid i a first-price auctio to buy both parts? No doubt the aswer should deped o a bidder s private iformatio. If a bidder sees that her part of the field is worth $0, she kows the value of both parts is either $0 or $3 millio. If see sees $3 millio, she kows the value of both parts is either $3 or $6 millio. Suppose she bids the expected value of the field coditioal o her private iformatio. That is, suppose she bids $1.5 millio if she sees $0, ad $4.5 millio if she sees $3 millio. Soud good? Well, it turs out that would be a bad idea. I fact, eve if she bids amouts less tha these, say to build i a small profit margi, she will most likely be subject to, what ecoomists call, the wier s curse. The problem with the biddig the expected value of the field coditioal o your private iformatio (or eve some positive amout less tha that) is most evidet i the case where the bidder sees $0. Suppose a bidder sees $0 ad bids $1 millio. This is well below the expected value of the field, which is $1.5 millio. Now ask yourself, how should this bidder feel if she fids out she has wo the auctio? She would probably be happy at first (we all like to wi). But the, she might thik, if my bid of $1 millio wo, how much oil is likely to be i the field. Note that this is a differet questio tha, how much oil is i the field coditioal o my private iformatio. Now we are askig how much is the expected value of the field coditioal o the value of my wiig bid. This of course depeds o the strategy of the oppoet ad so we ca t aswer this formally util we itroduce the idea of equilibrium strategies. However, it is safe to assume at this poit that if the oppoet saw $3 millio (ad kew there was a 50% chace that the field was worth aother $3 millio o top of that) she probably would have bid more tha $1. I other words, if a bidder wis with a bid of $1 millio the field is probably worth $0! Ay bidder who bids a positive amout whe she sees $0 will probably lose moey if she wis the auctio. It turs out that a bidder should also bid less tha $4.5 millio whe she sees $3 millio, but this is hard to explai without coductig the full equilibrium aalysis. So, let s do just that. We ow verify that the uique symmetric equilibrium of this game is for each bidder to bid accordig to the bid fuctio: b(0) = 0, b(3) = x [0, 3] where x is distributed accordig to B(x) = Suppose bidder B bids accordig to the proposed equilibrium strategy. Cosider the case where bidder A sees 0, ad cosider a arbitrary bid b A > 0 by x. 6 x

13 43 bidder A. The expected payoff for bidder A is 1 (0 b A) + 1 B(b A)(3 b A ) b A = 1 b A + 1 (3 b A ) < 0, 6 b A sice 3 b A 6 b A < 1. Hece, b(0) = 0 is a best respose to player B s strategy. Next, cosider the case where bidder A sees $3 millio. Give the strategy of bidder B, ad bid for bidder A betwee 0 ad 3 have a expected payoff of 3 : 1 (3 b A) + 1 B(b A)(6 b A ) b A = 1 (3 b A) + 1 (6 b A ) 6 b A = 1 (3 b A) + 1 b A = 3. Suppose she bids b A > 3. The, her expected payoff is 1 (3 b A) + 1 (6 b A) = 4 1 b A < 3. Hece, b(3) = x [0, 3], where x is distributed accordig to B(x) = x is a 6 x best respose to player B s strategy. We have show that if player B follows the equilibrium strategy it is a best respose for player A to do so as well. If we switch the roles of player A ad B, the same argumet shows that if player A follows the equilibrium strategy it is a best respose for player B to do so as well. That s it. We have verified the Nash equilibrium. I-class discussio of Experimetal 10: Oil lease experimet. Exercise 6.1 Assume that bidders i the oil field experimet ca oly place whole umber bids. Compute the pure strategy Nash equilibrium. Aswer: The symmetric Nash equilibrium bid fuctio is b(0) = 0, b(3) =. We will ow discuss two, related models of commo value auctios. The pey jar example fits the first model, the oil lease example is best captured by the secod model.

14 44 CHAPTER 6. COMMON VALUE AUCTIONS 6.1 Model I The true value of the item beig auctioed is v, but v is ukow to all bidders. Each bidder i receives a sigal, s i, about the true value, which is give by the sum of the true value v ad a radom variable ẽ i, which you should thik of as a private oise term: s i = v + ẽ i, We assume that ẽ i satisfies E[ẽ i ] = 0 ad hece each bidders sigal has the property that E[s i ] = v. That is, the expected value of each bidder s sigal is equal to the true value. Reality Check: Where do these sigals come from? The aswer depeds o the auctio eviromet. For example, i the case of a auctio to buy a gold mie, each bidder has to place a estimate o the amout of gold. This estimate ca be derived from private tests of rock samples ad these tests have errors associated with the predictios. All we are sayig i our requiremet that E[ẽ i ] = 0, or equivaletly that E[s i ] = v, is that o average peoples estimates are correct. I other words, people do ot systematically over estimate or uderestimate the true value. If they always over estimated it, for example, it would ot be surprisig that the wier eds up payig too much. Our goal i illustratig the wier s curse is to show that it arises eve if peoples estimates of the true value are correct o average. We will use this model to illustrate the wier s curse. I particular, we will look at what happes to the wiig bidder (i terms of her payoff) if she bids too large a fractio of her sigal. We will discuss how this depeds o the umber of bidders ad o how big the oise parameter ca be relative to the true value. For cocreteess, we will cosider a specific radom variable ẽ i, that meets our requiremet that E[ẽ i ] = 0. Namely, we assume that the each bidders realized sigal e i is determied by a draw from the uiform distributio o [, A]. Hece, the probability that bidder i s sigal is less tha or equal to some value e [, A], is H(e) = A + e A. H( ) is the distributio fuctio for the radom variable ẽ i. It has desity fuctio h(e) = 1. Note that H( 1) = 0 ad H(1) = 1, as required. Moreover, A A E[ẽ i ] = x 1 [ ] x A A dx = = A 4A 4 A 4 = 0, as required. Now suppose all bidders bid a fractio, m, of their observed sigal (We will estimate this fractio usig data from a class experimet). We wat to show that if m is too close to 1 the wier loses moey i expectatio. I say i expectatio because it is ot the case that the wier will always lose

15 6.1. MODEL I 45 moey; sometimes improbable thigs happe ad bad decisios ca tur out well. However, if a perso played the game may times ad always bid too close to their value they would lose moey. Sice everyoe is assumed to bid the same fractio of their sigal, the wier will be the bidder with the highest private sigal, which is the bidder with the highest realizatio of the radom variable ẽ i. We deote the radom variable for the highest oise term by ẽ (1). I order to compute the wier s expected payoff, we eed to compute the expected value of ẽ (1). To do this, we first eed to kow its desity fuctio. Sice H(e) is the distributio of ay oe oise term (i.e., the realizatio of ay oe radom variable ẽ i ), the probability that all realizatios of the oise term are less tha a value e [, A] is ( ) A + e H (1) =, A with desity Hece ( ) 1 A + e 1 h (1) (e) = A A. E[ẽ (1) ] = = A 1 (A) e ( ) 1 A + e de A A A e(a + e) 1 de Suppose =. The E[ẽ (1) ] = = = = = A (A) (A) (A) (A) (A) A e(a + e)de (Ae + e )de [ Ae + e3 3 [ 3Ae + e 3 6 [ 3A 6 A 6 ] A ] A ] = A 3. Note that eve though E[ẽ i ] = 0, the expectatio of E[ẽ (1) ] > 0. Cosequetly, E[ṽ (1) ] = v + E[ẽ (1) ] > v. Recall that we assume bidders bid some fractio m of their sigal. The implicatio is that the wier will lose moey (i expectatio) if m(v + E[ẽ (1) ]) v = m(v + A 3 ) v > 0

16 46 CHAPTER 6. COMMON VALUE AUCTIONS This occurs whe m > v v + A 3. For example, suppose that v = 3000 ad A = bidder loses moey i expectatio oly if m > The the wiig Now let s see what happes if there are more bidders. Does the wier s curse problem get better or worse? Cosider = 3. With three bidders, the expected value of the wier s sigal is greater because the expected value of ẽ (1) is greater. Namely, E[ẽ (1) ] = (A) = 3 8A 3 = A. A A e(a + e) 1 de e(a + e) de Hece, E[ṽ (1) ] = v + E[ẽ (1) ] = v + A. Now, assumig all bidders bid a fractio m of their sigal, the wier will lose moey (i expectatio) if m(v + E[ẽ (1) ]) v = m(v + A ) v > 0, which occurs whe m > v v + A. If v = 3000 ad A = 1000, the the wiig bidder loses moey i expectatio if m > 6 7. Now lets look at what happes whe gets large. Does the wier s curse get better or worse? Put simply, it gets worse. Ituitively, it should be clear that as gets large the expected value of ẽ (1) will be very close to A. Thik of takig 10,000 radom draws of umbers betwee ad A ad ask yourself, What is the value of the highest draw likely to be? Hopefully, you said A. This ca also be show formally. If we further evaluate the itegral i the

17 6.. MODEL II 47 expressio for E[ẽ (1) ] we get E[ẽ (1) ] = = = = = A 1 (A) 1 (A) 1 (A) 1 (A) e ( ) 1 A + e de A A A e(a + e) 1 de [e(a + e) A A [ = A A + 1 = A A(A) [ A(A) (A)+1 ] (A + e) de (A + e)+1 ] Clearly, as, we have 1 +1 A. The poit is that for large, E[ṽ (1) ] = v + E[ẽ (1) ] = v + A. If v = 3000 ad A = 1000, the the wiig bidder loses moey i expectatio if m > 3 4. So eve a very cautious bidder, perhaps eve oe that has bee wared about the wier s curse, might lose moey! Geeral Facts: 1. As the umber of bidders icreases, a bidder must bid a smaller fractio of her sigal to avoid the wier s curse.. For ay give umber of bidders, if the rage of the oise parameter is smaller, relative to the true value, the wier s curse results less ofte. (Not show) A ] I-class discussio of Experimet 11: pey jar experimet. 6. Model II Commo values ca also be modelled as a special case of iterdepedet values. I the iterdepedet values model v 1 = αs 1 + γs v = αs + γs 1

18 48 CHAPTER 6. COMMON VALUE AUCTIONS where s 1 ad s are private sigals of bidders 1 ad, α 0 is the weight a bidder puts o her ow sigal ad γ 0 is the weight she puts o her oppoet s sigal. We cosider the case where α = γ = 1. I this case, v 1 = v, which is a case of commo values. Note that the oil lease example fits this model whe each sigal s i is determied idepedetly ad s i = 0 or 3 with equal probability. I what follows we will cosider a more geeral treatmet of the private sigals. Suppose that the sigals s i are draw idepedetly form the uiform distributio o [0, 100]. Claim 3 The first-price auctio has a symmetric Nash equilibrium i which each bidder bids s i. Proof. Suppose bidder bids s ad cosider a arbitrary bid b 1 for bidder 1. We eed to write dow bidder 1 s expected payoff as a fuctio of her bid b 1 ad show that this is maximized at b 1 = s 1. We derive bidder 1 s expected payoff as a fuctio of her bid i three steps. Step 1. Compute the probability that bidder 1 wis with bid b 1. Bidder 1 wis oly if her bid is higher tha bidder s bid, i.e., b 1 > s. Sice we assume that sigals are uiform o [0, 100], this happes with probability b 1. Thus, bidder 1 wis the auctio with bid b with probability b 1 Step. Compute bidder 1 s expected value of the item if she wis at bid b 1. Remember that each bidder s commo value for the good it equal to the sum of the private sigals. Bidder 1 kows s 1, but she does ot kow s. Before the auctio begis, the expected value of s is simply 50. But this assumes the radom variable s ca take o ay value betwee 0 ad 100. Bidder 1 is iterested i the expected value of s oly i the evet that she wis the auctio with a bid b 1. As we metioed i step 1, give the proposed biddig strategy of bidder, this oly happes if s < b 1. Hece, we wat the expected value of the radom variable s coditioal o s < b 1. We kow that bidder s sigal is betwee 0 ad b 1. Sice all these possibilities are equally likely, the expected value of bidder s sigal is simply half its maximum value, or b 1. Hece, bidder 1 s expected value of the item if she wis with bid b 1 is 100. s 1 + b 1. (6.1) Let s sped more time makig sure you uderstad what we just did. Istead of usig Suppose bidder 1 wis with a bid of 0. That meas the sigal of bidder must have bee below 0 (because we assume he is biddig his sigal ad his bid was less tha 0). Sice, we have determied that bidder s sigal is betwee 0 ad 0, ad sice all these possibilities are equally likely, the expected value of bidder s sigal is half its maximum value, or 10, ad hece the expected value of the item for bidder 1 is s If, o the other had bidder 1 were to wi with a bid of 90, the bidder s sigal could be

19 6.. MODEL II 49 as high as 90. However, the expected value of bidder s sigal would be half that, or 45, ad the expected value of the item for bidder 1 is s I both cases, the expected value of the item to bidder 1 s 1 + b 1, just like i the geeral formula. Step 3. Compute the expected price bidder 1 pays if she wis with bid b 1. Sice this is a first-price auctio, the price she pays if she wis is her ow bid, b 1. That s it! We are ready to write dow the expected payoff. It is b 1 (s 1 + b 1 b 1 ) }{{} 100 }{{ }{{}} Step1 Step Step 3 = b (s 1 b 1 ). (6.) To maximize this with respect to the choice of b 1 we take the derivative of 6. with respect to b 1 ad set it equal to 0: s b = 0. The solutio is b 1 = s 1, as required. The same argumet ca be used to show that b = s is a best respose to b 1 = s 1. That completes the proof. Claim 4 The secod-price auctio has a symmetric Nash equilibrium i which each bidder bids s i. Proof. Suppose bidder bids s ad cosider a arbitrary bid b 1 for bidder 1. We eed to write dow bidder 1 s expected payoff as a fuctio of her bid b 1 ad show that this is maximized at b 1 = s 1. We derive bidder 1 s expected payoff as a fuctio of her bid i three steps. Step 1. Compute the probability that bidder 1 wis with bid b 1. Bidder 1 wis oly if her bid is higher tha bidder s bid, i.e., b 1 > s. Or, equivaletly, bidder 1 wis with bid b 1 if s < b 1. Sice we assume that sigals are uiform o [0, 100], this happes with probability b 1. Thus, bidder 1 wis 00 the auctio with bid b 1 with probability b Step. Compute bidder 1 s expected value of the item if she wis at bid b 1. Remember that each bidder s commo value for the good it equal to the sum of the private sigals. Bidder 1 kows s 1, but she does ot kow s. She eeds to compute the expected value of s coditioal o the evet that she wis the auctio with a bid b 1, that is, coditioal o s < b 1. We kow that

20 50 CHAPTER 6. COMMON VALUE AUCTIONS bidder s sigal is betwee 0 ad b 1. Sice all these possibilities are equally likely, the expected value of bidder s sigal is simply half its maximum value, or b 1 4. Hece, bidder 1 s expected value of the item if she wis with bid b 1 is s 1 + b 1 4. (6.3) Oce agai, let s sped a little more time makig sure you uderstad the role bidder 1 s wiig bid plays i determiig her expected value of the item. Suppose bidder 1 wis with a bid of 60. That meas the sigal of bidder must have bee below 30 (because we assume he is biddig twice his sigal ad his bid was less tha 60). Hece, the value of the item, which is the sum of the sigals, is at most s But, we do t really care about the most it ca be, we wat to base our decisio o its expected value. Sice, we have determied that bidder s sigal is betwee 0 ad 30, ad sice all these possibilities are equally likely, the expected value of bidder s sigal is half its maximum value, or 15, ad hece the expected value of the item for bidder 1 is s Step 3. Compute the expected price bidder 1 pays if she wis with bid b 1. Sice this is a secod price auctio, the price she pays if she wis is bidder s bid, which is s. The expected value of bidder s bid depeds o the bid b 1 because bidder 1 does ot wi the auctio uless b 1 > s. We eed to compute the expected value of s coditioal o b 1 > s. This is doe most easily by usig a little trick; we use the value x = s as the ruig variable i the itegratio: E[ s s < b 1 ] = = 1 b 1 [ x = b 1. b1 0 ] b1 0 x 1 b 1 dx That s it! We are ready to write dow the expected payoff. It is b 1 (s 1 + b 1 b 1 ) }{{} 00 }{{ 4}}{{} Step1 Step Step 3 = b 1 00 (s 1 b ). To maximize this with respect to the choice of b 1 we take the derivative ad set it equal to 0: 00 b = 0. The solutio is b 1 = s 1, as required. s 1

21 6.. MODEL II 51 The same argumet ca be used to show that b = s is a best respose to b 1 = s 1. That completes the proof. The equilibrium we just verified is appealig because it is symmetric, but it is ot uique. While we will ot prove it, there are other asymmetric equilibria to this auctio. Namely, for ay λ > 0, b 1 (s 1 ) = (1+λ)s 1 ad b (s ) = (1+ 1 λ )s is a Nash equilibrium. See Osbore s 004 itroductio to game theory text for a proof Reveue Equivalece Give these equilibrium bid fuctios, the expected paymet of a bidder is the same i both auctio formats. I the first-price auctio, bidder i wis with bid s i with probability s i, ad pays s 100 i. Hece, bidder i s expected paymet i the first-price commo value auctio is s i. 100 I the secod-price auctio, bidder i wis with probability ad she pays Pr ob[s i > s j ] = Pr ob[s i > s j ] = s i 100 E[ s j s j < s i ] = = s i [ x = s i. si 0 ] si 0 s j 1 s i ds j Hece, bidder i s expected paymet i the secod-price commo value auctio is s i. 100 So, i both the first ad secod-price, commo-value auctio expected reveue is 100 [ ] E[ s i 100 ] = s i 1 s ds i = i = Reveue equivalece holds i this settig!

22 5 CHAPTER 6. COMMON VALUE AUCTIONS