March 3, 215 Steven A. Matthews, A Technical Primer on Auction Theory I: Independent Private Values, Northwestern University CMSEMS Discussion Paper No. 196, May, 1995. This paper is posted on the course webpage. Why Auctions? Why do economists (and increasingly, engineers and computer scientists) study auctions? 1. Economics studies allocation and auctions are commonly used for trading. The problem that most fundamentally motivates auctions is as follows: A seller has an item to sell. What procedure should he follow in order to maximize the revenue that he receives for selling the item? This is the problem of revenue maximization. (a) the answer is clear if the item is readily for sale elsewhere. Auctions are most commonly used for a unique item or an item for which there is a great deal of uncertainty about its value to prospective bidders. (b) Also: the fairness or the efficiency of the allocation. (c) additional allocation problems that may be adddressed using auctions: 1. reverse or procurement auctions (multiple sellers competing to sell an item to a single buyer) 2. multi-unit auctions (single seller with multiple units to sell) 3. double auctions (multiple buyers interacting with multiple sellers) 2. Auctions provide explicit models of price formation, which are sorely lacking in economic theory. 3. Economists believe that markets aggregate into prices information known privately by traders. This is captured by the theories of rational expectations and efficient markets. Auctions provide a way to test these theories. Auctions are also used explicitly for the sake of aggregating information in prediction markets. Environment Studied in this Paper: 1 seller with an indivisible item to sell 1 bidders (commonly known). Assumptions about the Bidders: Some bidders, however, may choose not to bid. Private values: Bidder privately knows the maximal amount that he is willing to pay for the item sometimes referred to as bidder s type or reservation value. key point: a bidder s value would not change if he somehow learned (say, through the process of the auction) something about the values of the other bidders Example: art and oil tracts Bayesian assumption: beliefs Independence Symmetry: Each has the same distribution ( ) Therefore, IID Risk Neutrality: each bidder acts so as to maximize his expected profit Quasilinear utility 88
Digression on Order Statistics first order statistic e (1) (1) ( ) is the probability that all values are below. This is an abuse of the common notation, where the subscript (1) denotes the smallest value in a sample. (1) ( ) ( ) density (1) ( ) ( ) 1 ( ) second order statistic e (2) (2) ( ) is the probability that e (2), i.e., at least 1 values are below. We can add the probabilities of +1disjoint events: all values are for 1, e and e for 6 We therefore have and The Seller (2) ( ) ( ) + X (1 ( )) ( ) 1 1 ( ) + (1 ( )) ( ) 1 ( ) + ( ( ) 1 ( ) ) ( ) 1 ( 1) ( ) (2) ( ) ( 1) ( ) 2 ( 1) ( ) 1 ( ) ( 1) ( ) 2 [1 ( )] ( ) is the minimal amount she is willing to accept for the item (commonly known) Variables in Auction Design Tie-breaking rule reserve price entry fee 4 auctions that we ll consider: second price sealed-bid auction, the English auction, the Dutch auction, and the first-price sealed bid auction We assume that buyer values are independently drawn from [ 1] according to the cumulative distribution Second Price Sealed-Bid Auctions ( ): second price auction with reserve price and entry fee bidder s strategy is a function :[ 1] { } [ ) where indicates the decision to not participate and is the reserve price (or the minimum bid allowed by the seller). Suppose a bidder with value bids. Let denote the maximum of the other bids (if there are any) and. The bidder s profit functionis if ( ) ( ) if if where 1 is the probability that this bidder wins with the bid of. 89
No Entry Fee: Inthecaseof : if ( ) ( ) if if Claim: The following strategy is a weakly dominant strategy of the bidder in the ( ): ½ if ( ) if is a best response (i.e., it maximizes the bidder s profit) regardless of the value of, and hence it maximizes expected profit given the probability distribution of. Existence of other equilibria: Bidder bids 1 for all of his possible values and all other bidders choose. This equilibrium is not trembling-hand perfect. How does a positive entry fee c affect these results? Recall that if ( ) ( ) if if but the bidder gets by choosing and not participating. Clearly, he should not participate if. In fact, he shouldn t participate if + becausehecannotprofit in this case (the maximum possible gain from winning is less than the cost of entry). Even if +, however,hemaywanttochoose if he believes that will be large (say, larger than his ). The point to be emphasized is that his choice of whether or not to participate depends upon his beliefs about. Consequently, his decision to participate depends upon his beliefs about the decisions of the other bidders (what he thinks they may do). This is an important observation and it s where the theory of auctions starts to get interesting (and complicated!). If a bidder chooses to enter, then the same argument as before shows that he should bid. We seek to determine a cutoff or marginal value ( + ) above which the bidder chooses to enter and below which he does not. It s reasonable to guess that he ll receive the same profit from entering as from staying out when is his value. All of this depends, of course, on the distribution ( ) of (which reflects his beliefs about the bidding behavior of the other bidders). Let s assume all bidders are using the same strategy. Each bidder therefore bids his value if it is above and stays out ( )if. The profit of the bidder with value should equal, regardless of whether he enters and bids or chooses. This should allow us to characterize in terms of, and. Once we "outline" this form of equilibrium, we ll actually prove in a formal sense that it is an equilibrium. The bidder with value wins only if the others don t bid, i.e., each of their values are below. In this event, he pays as his price. This occurs with probability ( ) ( ) 1 The bidder with value who enters therefore has expected utility ( ) ( ) 1 and utility zero if he does not enter. Equating these two expressions, ( ) ( ) 1 ( ) ( ) 1 Three points should be noted: 1. Because and ( ) 1 are both increasing in, this has at most one solution. 9
2. If, then ( ) () 1 3. If (1 ) (1) 1 1 + then no solution exists (the bidder stays out for all [ 1]). If + 1, then a unique solution exists. This defines ( ). Notice that ( ) + : Theorem 124 Assume 1 1. equilibrium of the ( ): ( ) ( ) ( ) 1 + The use of the following strategy by each bidder defines an ½ if ( ) if ( ) Remark 125 Notice that ( ), which is the dominant strategy equilibrium described above. Proof. As noted above, the same argument as before shows that a bidder who bids should bid his value. The only issue in verifying equilibrium is therefore to validate the optimality of the entry decision in the strategy ( ). The bidder with value who enters and bids his value has expected payoff no more than max ( ) ( ) 1 The first term in the brackets represents the payoff if so that the bidder never wins; the second term ( ) ( ) 1 is the expected payoff if. This "max" is strictly negative if and zero if. This supports the decision of such a bidder to stay out. The bidder with value has expected payoff from bidding his value equal to ( ) ( ) 1 which is strictly positive for. This supports his decision to enter. Remark 126 Formally, the theorem describes a Bayesian-Nash equilibrium. It is important to recognize how our definition of equilibrium has changes since we discussed the dominant strategy equilibrium in thecaseof ( ). Remark 127 Formally, the theorem describes a Bayesian-Nash equilibrium. It is important to recognize how our definition of equilibrium has changes since we discussed the dominant strategy equilibrium in thecaseof ( ). 1. In each case, a bidder s bid must be optimal for each distinct value. 2. In the second case, the optimality of a bidder s bid depends upon a specification of the strategies used by the others. We had a stronger notion of equilibrium above, i.e., a bidder s bid was optimal regardless of how the other bidders chose to bid. 3. This second concept of equilibrium involves expected value calculations, reflecting its dependence on the behavior of the other bidders. Exercise 128 Consider ( ) with. reserve price on the seller s expected revenue. The purpose of this exercise is to explore the effect of the 1. The first question assumes that there are 2bidders whose values are IID according to the distribution on [ 1]. Consider ( 1). 91
(a) Provide a formula for the seller s expected revenue in terms of and. (b) Reduce this to a formula in terms of in the case of uniform on [ 1]. (c) In the uniform case, what value of maximizes the seller s expected revenue? Exercise 129 Answer a., b. and c. from 1. in the case of 3bidders. Exercise 13 Consider ( ) with and. effect of the entry fee on the seller s expected revenue. The purpose of this exercise is to explore the 1. The first question assumes that there are 2bidders whose values are IID according to the distribution on [ 1]. Consider ( 1). (a) Provide a formula for the seller s expected revenue in terms of and. Hint: Recall our formula that determines ( ). (b) Reduce this to a formula in terms of inthecaseof uniform on [ 1]. (c) In the uniform case, what value of maximizes the seller s expected revenue? 2. Answer a. from 1. in the case of 3bidders. English Auctions Note the enormous variety of different "extensive form" models of an English "outcry" auction. Considered in Matthews: Japanese "button" auction in which bidders do not know how many or which other bidders are still holding down their buttons. set of possible auctions the same as sealed bid auction: { } [ ). means never pressing the button. Otherwise, a bidder s strategy is the time at which he releases his button. Notice that because he does not observe the actions of others, he has no basis on which he might change his behavior as the auction proceeds. (If he observed the actions of others, then his strategy could be contingent on the behavior of others that he observes as time passes; we would have to consider a complicated class of strategies). Suppose. Claim: is a weakly dominant strategy for each bidder. The use of this strategy by each bidder results in the bidder with the highest value receiving the item and paying the second highest value as his price. We thus have a sense in which ( ) and ( ) produce the same outcome. Remark 131 The equivalence of ( ) and ( ) is all that Matthews claims on p. 16 of his manuscript. I think the same argument shows that ( ) and ( ) are also equivalent. If I am wrong on this point, then I ll correct the matter later on. Exercise 132 Suppose that we modify Matthews s version of the Japanese button auction by posting alongside the price the number of active bidders (i.e., the number who are still holding down their buttons). A bidder can thus make his decision on when to release his button contingent not only on the price that has been reached but also upon the number of active bidders. Describe the set of all possible strategies. Hint: Do not describe what a bidder should do, or surely should not do describe all plans of action (or strategies) that he can choose among. First Price Auction FPA(r,c) (sealed bid) A bidder s "optimal" strategy clearly depends upon the bidding strategies of the other bidders. Bidding one s true value is a weakly dominated strategy (it is weakly dominated by any bid below your value). 92
The Model strategy: :[ 1] { } [ ) b ( ) [ wins bids and each 6 uses the strategy ( )] s "probability of winning" function given the strategies of the other bidders ignoring the entry fee, bidder s expected profit when he enters and bids is ( )( ) b ( ) A Bayesian-Nash equilibrium (BNE) is a profile of strategies ( 1 ) such that, for each player and each [ 1], the choice ( ) is a best response to ( ) 6 : 1. ( ) ( ) for all 2. ( ) 6 ( ( )) ( ) for all Symmetric BNE: the assumption that all bidders use the same strategy ( ) We ll focus now on ( ) and show that the following function defines a symmetric BNE: ( ) ( ) (18) ( ) where ( ) is the cumulative distribution of (1) in a sample of 1 values from, i.e., ( ) ( ) 1 We shall prove that this is an equilibrium and that it is in fact the only symmetric equilibrium. It is also true that no asymmetric equilibria exist (though we won t prove it). Notice that ( ) is clearly an increasing function of. Assuming that all bidders use this strategy, the item will be awarded to the bidder with the highest value (who will bid the most). This equilibrium therefore efficiently awards the item to the bidder who values it the most. Interpretation ( ) (1) (1) i.e., ( ) is the expected value of the highest value among the 1 opponents given that this highest value is below. The event (1) is the event in which the bidder with value has the highest value (don t worry about the possibility of ties). The probability that (1) is ( ) ( ) 1, and so the probability that (1) conditional on (1) is ( ) ( ) The density of (1) conditional on the event (1) is therefore ( ) ( ) which implies the interpretation of ( ). More interpretation will come later. The Dutch Auction A "wheel" indicating the price starts at a high price and steadily turns toward lower prices. The first bidder to yell "stop" buys the item and pays the price at which the wheel stops. We can have an entry fee and areserveprice at which the wheel stops. Set of actions for a bidder: { } [ ) This presumes that the seller knows enough to start the wheel at a price above anyone s choice of a time at which he intends to stop it. If we assume that the seller knows and its support, then he can start at a 93
price where ( ) 1. In the Matthews manuscript, the support of is [ 1] and so the auctioneer could startattheprice 1. Let ( 1 ) denote the actions chosen by the bidders. Notice that the action set of each bidder is exactly the same as in the first price auction ( ). We claim that the consequences for a given profile of actions ( 1 ) is exactly the same in the two auctions. If each, then no one pays the entry fee in either auction and the item is unsold. If at least one 6,thenlet max{ 6 } and let # denote the number of bidders who bid. In either auction, the bidders who bid each receive the item with probability 1 # and pay if they receive it. The outcome is thus exactly the same in each auction. ( ) and ( ) are thus equivalent in a very strong sense: they have exactly the same action sets and the consequence of each profile of actions is exactly the same in each auction. We have thus solved for the unique symmetric Bayesian-Nash equilibrium strategy of either ( ) or ( ), where ( ) ( ) 1. ( ) ( ) ( ) Returning to FPA(,): Underbidding and Competition The interpretation implies that ( ) for. ( ) (1) (1) Alternatively, integration by parts implies ( ) Notice that the amount of underbidding converges to as. ( ) ( ) ( )( 1) ( ) 2 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 Revenue Equivalence Let e 1 denote the equilibrium sales price in ( ), e the equilibrium sales price in ( ), e 2 the equilibrium sales price in ( ), and e the equilibrium sales price in ( ). We know that e 2 e and e 1 e Our next result is that [ e 1 ] [ e ] [ e 2 ] [ e ] i.e., on average, all four auctions (with the equilibria that we have focused on) produce the same revenue for the seller. This is the first revenue equivalence theorem that we will discuss. The underbidding by bidders in ( ) thus on average compensates exactly for the fact that ( ) determines a higher price for a given set of bids than ( ). 94
ProofofRevenueEquivalence We ll show here that the expected revenue for the seller in the "honest" equilibrium of ( ) is the same as the expected revenue in the equilibrium ( ) of ( ). The price e 2 in ( ) is the second highest order statistic (2) in a sample of values from the distribution on [ 1]. We have previously determined the density of the second highest order statistic. From this we obtain the following formula for the expected revenue of ( ): e 2 (2) ( 1) (1 ( (2) )) ( (2) ) 2 ( (2) ) (2) ( 1) (1 ( )) ( ) 2 ( ) (19) The second line simply drops the subscript "(2)" in order to simplify the notation. Turning to ( ), we re assuming that each bidder uses the strategy ( ) to choose his bid. Because ( ) is increasing, the winning bidder is the bidder with the highest value (1). The price 1 therefore equals ( (1) ). The expected revenue is therefore the expected value of ( (1) ) computed with respect to the density of (1), whichwehavediscussedbefore: e 1 ( (1) ) ( (1) ) 1 ( (1) ) (1) Again, we have dropped the subscripted number to simplify the notation. ( ) ( ) 1 ( ) (2) Recall that Substitution for ( ) in (2) produces ( ) ( ) ( ) ( 1) ( ) 2 ( ) ( ) 1 e 1 e 2 ( 1) ( ) 2 ( ) ( ) 1 ( ) 1 ( ) ( 1) ( ) 2 ( ) ( ) ( 1) ( ) 2 ( ) ( ) [ ( ) 1 ( 1) ( ) 2 ( ) (1 ( )) ( 1) ( ) 2 ( ) (1 ( )) ( 1) ( ) 2 ( ) In this last derivation, we changed the order of the integration and then replaced " " witha" " inthelast line. Formal Analysis of Equilibrium in FPA(,) Theorem 133 In FPA(,), 1. the strategy ( ) defines a symmetric BNE, and 95
2. this is the only symmetric BNE. This proof takes 9-1 pages in Matthews s notes! We ll break the formal analysis into series of steps, or lemmas. We ll start with the following 3 guesses concerning an equilibrium ( ): 1. a bidder with bids; 2. ( ) is strictly increasing on [ 1]; 3. ( ) is differentiable. Lemma 134 If ( ) is a symmetric BNE equilibrium strategy of ( ) and satisfies 1.-3., then ( ) ( ) Proof. We ll show that 1.-3. implies that ( ) is differentiable in, which implies the necessary first order condition ( ( )) We ll then interpret this as a differential equation whose solution (given an initial condition) is ( ). It is worth emphasizing that in verifying that ( ) defines an equilibrium, we assume that all bidders except a selected bidder uses ( ), and then show that the selected bidder maximizes his expected payoff for each by bidding ( ). Let denote the inverse function of ( ). We have b ( ) ( ( )) 1 ( ( )) i.e., the probability that a bidder wins with the bid equals the probability that the values of all other bidders are below the value at which the bid is. The expected profit of a bidder with type who bids is therefore ( ) ( ) ( ( )) The assumption that ( ) is differentiable implies that ( ) is differentiable on its domain. We have ( ) ( ( )) + ( ) ( ( )) ( ) which equals zero at ( ): using ( ( )) and ( ( )) 1 ( ), this becomes Rewrite this as Taking antiderivatives implies for any. ( ( )) ( )+ ( ( )) ( ) ( ) ( ) ( )+ ( ) ( ) ( ) ( ) ( ) ( ) ( ) Take, which gives the equation ( ) ( ) ( ) ( ) ( ) We ve shown that if strategies are restricted to a particular class (given by 1-3), then the only strategy in this class that can determine a symmetric BNE is ( ). Next step: Lemma 135 The use of ( ) by every bidder defines an equilibrium. 96
Proof. We firstnotethatbecause ( ), a bidder s payoff ( ) ( ( )) b ( ( )) is nonnegative, and hence "NO" is not preferable to bidding. The remaining issue is to show that ( ) is the optimal bid for a buyer with value. here is a bit simpler than Matthews s. We know that (). Let My argument sup ( ) (1) [ 1] which exists because ( ) is continuous. Notice that 1. First, note that no bidder could benefit by changing his bid to (1). It could never be sensible to choose a bid larger than is (1) because (i) the probability of winning is 1 from bidding (1) and (ii) bidding more than is (1) only increases the price that one pays. We therefore need to show that a bidder with value cannot benefit by changing from ( ) to some other [ ] For any such, let ( ) i.e., is the value such that ( ). Recall that ( ) ( ( )) + ( ) ( ( )) ( ) and ( ( )) ( ) becausethisishowwesolvedforthestrategy ( ). If and ( ),then i.e., ( ) ( ( )) + ( ) ( ( )) ( ) ( ( )) + ( ) ( ( )) ( ) ( ) for ( ) A similar argument shows that ( ) for ( ) It is therefore clear that ( ) is maximized at ( ). We have thus solved for the unique symmetric Bayesian-Nash equilibrium strategy of either ( ) or ( ), ( ) ( ) ( ) where ( ) ( ) 1. It remains to be shown that any symmetric equilibrium necessarily satisfies: 1. a bidder with bids; 2. ( ) is strictly increasing on ( 1]; 3. ( ) is differentiable. The arguments here address "first principles" concerning equilibrium (i.e., we aren t making any assumptions about the properties of the equilibrium we ll deduce all required properties from the definition of equilibrium and the assumption that each bidder uses the same strategy ( )). We ll prove the following 4 lemmas: Lemma 136 (Property 1.) If ( ) defines a symmetric equilibrium of ( ), then for all : ( ) 6, b ( ( )), and ( ). 97
Lemma 137 Any symmetric equilibrium ( ) of the ( ) weakly increases on ( 1]. Lemma 138 (Property 2.) Any symmetric equilibrium ( ) of the ( ) strictly increases on ( 1]. Lemma 139 (Property 3.) Any symmetric equilibrium ( ) of the ( ) is differentiable on ( 1]. Let sstartwiththemiddletwo,whichareeasier. Lemma 14 Any symmetric equilibrium ( ) of the ( ) weakly increases on ( 1]. Proof. Let. We wish to show that ( ) ( ). The bidder with value does at least as well with the bid ( ) as with the bid ( ), ( ( )) ( ( )) b ( ( )) ( ( )) b and the bidder with value does at least as well with the bid ( ) as with the bid ( ), ( ( )) ( ( )) b ( ( )) ( ( )) b (21) It s clear that i.e., and hence ( ) b ( ( )) ( ) b ( ( )) b ( ( )) b ( ( )) The previous lemma implies b ( ( )) and ( ). Inequality (21) then implies 1 b ( ( )) b ( ( )) ( ) ( ) and so ( ) ( ) ( ) ( ) whichisthedesiredresult. Digression: Revealed Preference The argument that we make in Lemma 14 is fundamental in microeconomics and merits some additional discussion. We ll present a graphical argument that a bidder s bid and his probability of winning must be nondecreasing in his bid. A bidder s expected payoff is ( ), where is his bid and is the probability that he wins. A bidder (of course) would prefer a low and a high ; the strategies of the other bidders, however, determines a function ( ) b that determines his probability of winning as a function of his bid. We can imagine a bidder s decision problem for each value of as a constrained optimization problem, max ( ) s.t. b ( ) We re not assuming anything here about ( ) b except that it is determined by an equilibrium ( ). The behavior of the other bidders determines a curve ( ( )) b that a bidder can choose along to maximize his 98
payoff. We re not going to draw it because we don t know what it looks like. We do know something about the preferences of a bidder over values of and. First, the indifference curves of a bidder for fixed slopes upward. Second, consider. The indifference curves of a bidder with value are flatter than the indifference curves of a bidder with value, reflecting the fact that at any point the bidder with the higher value is willing to pay more ( ) for an increase in the probability of winning than the bidder with lowervalue. Thisisthesingle-crossing property, i.e., the indifference curves for distinct values cross at most once. Let ( ( ) ( ( )) b denote the point that the bidder with value selects as best (it s optimality is not depicted below because we don t know how to sketch the curve ( ( )). b Consider the indifference curve of the bidder with value that passes through this point. The point ( ( ) ( ( )) b he chooses must lie in the closed 4 shaded region between the two indifference curves, reflecting the fact that the type bidder prefers ( ( ) ( ( )) b to ( ( ) ( ( )), b and vice-versa for the type bidder. From the picture, we can see that for,wemusthave ( ) ( ) and ( ( )) b ( ( )). b Q ^ (b(z),q(b(z)) (z-b)qconstant (v-b)qconstant b Question: Does the preceding discussion depend on the assumption that the other bidders are using thesamestrategy ( ) as the bidder whose bids are being studied, or does it only depend on the assumption that the strategies of the other bidders determine some function ( )? b In other words, is symmetry of the equilibrium required in this argument to conclude that a bidder s strategy and his probability of winning is nondecreasing in his value? Lemma 141 (Property 2.) Any symmetric equilibrium ( ) of the ( ) strictly increases on ( 1]. Proof. We know ( ) is weakly increasing. that: If it is not strictly increasing, then there exists such (b ) for b (b ) for b ( ) (b ) for b This means there is a positive probability of ties at the highest bid of. We ll show that this contradicts equilibrium in the sense that a bidder with value b ( ) will want to raise his bid to + to insure that he wins in this event. Specifically, there is a jump discontinuity upwards at in the probability of winning, lim ( b + ) ( ) b + because by bidding slightly more (any positive ) one receives the item with probability 1 in the event in which the highest value of the other bidders is in ( ), whereas with the bid one receives the item with probability at most 1 2 in this event. Consequently, for b ( ), 4 "Closed" if we restrict so that 1. lim b b ( + ) b b ( ) + 99
and so for small a bidder with value b ( ) would benefit by raising his bid above. Note:Weareusingherethefactthat ( ) for, which is established in Lemma 1. This insures that canbechosensothat b. You should work the exercises at the back of Matthews paper. 1