October An Equilibrium of the First Price Sealed Bid Auction for an Arbitrary Distribution.

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1 October An Equilibrium of the First Price Sealed Bid Auction for an Arbitrary Distribution. We now assume that the reservation values of the bidders are independently and identically distributed according to the distribution on the interval [ ]. The probability that any bidder s value is less that the number is (). We also assume that has a density function that is nonzero on [ ], i.e., () () for [ ] We ll repeat the above steps, replacing the uniform distribution with the arbitrary distribution and the interval [ 1] with [ ]. Indivdual s expected utility when he bids and all others use an increasing function ( ) to determine their bids is ( )( ) 1 () 1 We want ( ) to be the value of the bid that maximizes bidder s expected utility. The Envelope Theorem implies ( ( )) ( ) evaluated at ( ) 1 () 1 evaluated at ( ) ( ) 1 and so Z ( ( )) () 1 + where is a dummy variable. We can conclude that as a consequence of increasing strategies and the fact that a bidder with reservation value therefore has an expected payoff of zero, i.e., (()). Unfortunately, we cannot simplify this further in the case of a general distribution. We have another formula: Z ( ( )) ( ( )) ( ) 1 () 1 and so or ( ) R R () 1 ( ) 1 () 1 ( ) ( ) 1 Weassumedinthisderivationthat( ) is an increasing function. Let s verify that the 86

2 above answer actually has this property. ( ) 1 ( ) 1 ( ) 1 R () 1 ( 1) ( ) 2 ( ) ( ) 2 2 R 1 1+ () 1 ( 1) ( ) ( ) R () 1 ( 1) ( ) ( ) andsoitisstrictlyincreasingforall. Example 4 (All-Pay Auction) Assume there are two bidders and that the reservation value of each is independently drawn from the uniform distribution on [1]. Each bidder submits a bid; the high bidder gets the item but both bidders are required to pay their bids to the seller. We seek an increasing function :[1] R such that a bidder maximizes his expected utility by using to choose his bid when he assumes that the other bidder is also using. Bidder s expected utility when is his value and he bids is ( ) 1 () where 1 () is the probability that the other bidder bids less than. Let ( ) denote his equilibrium expected utility when he bids ( ), ( )( ( )) 2 ( ) The Envelope Theorem implies ( ) ( ) at ( ) 1 () at ( ) Therefore, Z ( ) We know that a bidder with value wins with probability equal to zero. He clearly should not bid a positive amount in this case because he ll have to pay it! Therefore, () and, and so ( ) 2 2 We now solve for ( ) by equating the two formulas for ( ): 2 ( )( ) 2 2 ( ) 2 2 Exercise. Calculate an equilibrium in the all-pay auction in the case of bidders whose reservation values are drawn from the uniform distribution on [ 1]. Exercise. Calculate an equilibrium in the all-pay auction in the case of 2bidders whose reservation values are drawn from the distribution on [ 1]. Generalize your 87

3 result to an arbitrary number of bidders. Your answers will be formulas in terms of the distribution. What happens to the equilibrium bidding strategy as? Example 41 (Revenue Equivalence) We next compare the expected revenue of the seller in the all-pay auction, the second price auction, and the first price auction in the case of 2bidders and reservation values independently drawn from the uniform distribution on [ 1]. All-Pay Auction. The expected payment of bidder is Both bidders pay their bid, and so the expected revenue of the seller is 13. First Price Auction. We have derived () 1 2 as the equilibrium bidding strategy. Bidder s expected payment when is his value is 2 where 2 is his payment when he wins and is the probability that the other bidder s value is below his and so he wins. His expected payment to the seller is therefore which is exactly the same as in the all-pay auction. The seller s expected revenue is again 13. An alternative approach is to calculate the expected bid made by the higher of the two values. Let s first figure out the cumulative distribution of the higher of the two values. What is the probability that two values independently drawn from the uniform distribution lie below a number? Pr [ 1 2 ] 2 The density or derivative of this cumulative distribution is 2. The expected value of the highest bid is therefore which is the expected revenue of the seller. Second Price Auction. A bidder with value pays the expected value of the other bidder s value conditional on it being lower than. The assumption of uniformity implies that this expected value is 2. Bidder s expected payment to the seller is therefore 2 where 2 is his expected payment when he wins and is the probability that the other bidder s value is below his and so he wins. Alternatively, we can calculate the expected value of the second highest reservation value. We first determine the cumulative distribution of the second highest value. The probability that the second highest value is below some number is 1 minus the probability 88

4 that both values are above : 1 (1 ) The density of the second highest value is therefore 2 2. The expected value of the second highest value is therefore Z Ã 1! (2 2) All three auctions therefore produce the same expected revenue for the seller. With the outcome equivalence of the Dutch and the first price auctions, and the outcome equivalence of the English and the second price auctions, all five auctions produce the same expected revenue for the seller at least in the uniform case with only two bidders Revenue Equivalence in the Case of the Uniform Distribution The founding question of auction theory is, "How should a seller operate an auction so as to maximize his revenue?" It is interesting to compare the revenue or expected price received by the seller in the equilibrium of the first price auction in comparison to the second price auction in which each trader bids his true value (his unique dominant strategy). For a sample 1 2 of values for the bidders, order them as (1) (2) () The symbol () is the th order statistic for the sample. The expected revenue of the seller in the first price auction is 1 () and his expected revenue in the second price auction is ( 1) Recall that we continue to assume that the s are independent and uniformly distributed on [ 1]. We wish to evaluate these expected values so that we can compare them. The probability that () is less than some number is (i.e., all values are no more than ), and so the density of () at is 1 Therefore, µ Z () µ 1 µ Turning now to the second price auction, the probability that ( 1) is less than some number is 1 (1 )+ This sums the probabilities of two disjoint events, namely (i) exactly 1 of the values are below and one is above, plus (ii) the event in which all values are below. The 89

5 density of ( 1) at is therefore ( 1) 2 (1 ) Therefore, ( 1) 2 (1 ) ( 1) ( 1) 2 (1 ) ( 1) 1 ( 1) ( 1) ( +1) 1 +1 Interestingly, the expected revenue of both of the two auctions is the same. Notice also that because the English auction is outcome equivalent to the second price auction, and the Dutch auction is outcome equivalent to the first price sealed bid auction, we can therefore conclude that all four of these auctions produce the same expected revenue for the seller. This illustrates the Revenue Equivalence Theorem, which states conditions under which a variety of different auction procedures produce on average the same revenue for the seller. We have shown it here only in the case in which reservation values are independent and identically distributed according to the uniform distribution on [ 1]. It is in a sense perplexing, because it fails to provide insight into why some forms of auctions are more commonly used than others. In particular, the second price auction is rarely used, despite the simplicity of its equilibrium behavior. We ll need to reappraise our model of auctions A General Proof of the Revenue Equivalence Theorem. I ll provide here a shorter proof than in Campbell. You are not responsible for reading the discussion in Campbell, which I found difficult. We assume that the reservation values of the bidders are independently drawn from the distribution on [ ]. Let denote the density of, i.e., In this setting, a standard auction is defined as any auction that has the following two properties in its equilibrium: 1. A bidder whose reservation value equals has an expected utility of zero from participating in the auction; 2. The bidder with the largest reservation value buys the item. All four of the auctions that we have defined in the course are standard auctions (i.e., the Dutch, the English ascending bid, the first price sealed bid, and the Vickrey or second price auction). The "all pay" auction is also a standard auction. We can be creative and consider other possibilities; Campbell, for instance, discusses the auction in which the highest bidder is charged four times his winning bid as the price. It can be difficult to determine the equilibrium in an arbitrary auction, but if you can determine that the equilibrium has the above two properties, then on average it produces the same revenue 9

6 for the seller as any other standard auction. Theorem 7 (Revenue Equivalence Theorem) with the same expected revenue. All standard auctions provide the seller "Expected" here refers to averaging over all possible samples of reservation values. In any particular sample of reservation values, different auctions provide the seller with different prices or revenue. On average, however, it all "evens out", as long as the two auctions being compared are both standard in the sense defined by the two properties above. Do we have to figure out an equilibrium in every possible standard auction in order to prove this theorem? Thankfully, no. The theorem is proven by taking a "big picture" view of what an auction does and what equilibrium means. Recall that a standard auction is defined as any auction that has the following two properties in its equilibrium: 1. A bidder whose reservation value equals has an expected utility of zero from participating in the auction; 2. The bidder with the largest reservation value buys the item. What does an auction do? It assigns the item to one of the bidders and charges one or more bidders various prices. Let ( ) denote the expected utility or profit of bidder in the auction as a function of his reservation value. Wehave ( ) ( ) ( ) where ( ) is the probability that he receives the item and ( ) is his expected payment.,,and are all functions of because bidder s behavior in the auction (whatever its rules) may depend upon his reservation value. The theorem is proven by deriving a formula for ( ) in terms of, the distribution, and the number of bidders. In other words, given the distribution and the number of bidders, the money paid by bidder as a function of his reservation value is exactly the same for every standard auction. If the expected payments made by bidders is the same in every standard auction, then the total expected payments to the seller is the same in every standard auction. Revenue equivalence therefore holds. I first claim that for any [ ], the following inequalities hold as a consequence of equilibrium: ( ) ( ) ( ) ( ) ( ) (11) and, reversing the role of and, ( ) ( ) ( ) ( ) ( ) (12) To understand this, let s consider the role of in the formula ( ) ( ) ( ) The first on the right-hand side is the bidder s reservation value. The asavariablein ( ) and ( ),reflects the rules of the auction and the bidder s behavior in the auction. 91

7 If it were true that ( ) ( ) ( ) ( ) ( ) then the bidder could profit by behaving (or bidding) in the auction as though his reservation value equaled instead of. This contradicts the idea that the functions and reflect equilibrium behavior: if a bidder can profit by changing his behavior to bidding as though is his reservation value instead of, then we aren t discussing equilibrium. We thus have inequality (11) and its dual (12). Let s use these two inequalities: A B ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) C D I ve labeled the four terms as A, B, C, and D. We have because of the inequalities. Substituting for these terms implies ( )( ) ( ) ( ) ( )( ) for all,. Ifweassume,wehave ( ) ( ) ( ) ( ) Now take the limit as. If is a continuous function, then the left and right sides of this inequality converge to ( ) while the middle part converges to ( ). The continuity of ( ) is a technical detail that you ll have to grant me. Comparing the left and right sides of this last inequality, however, shows that ( ) is a nondecreasing function, from which continuity almost everywhere follows. We thus have ( ) ( ) and so Z ( ) () + (13) for some constant,where in the integral is a dummy variable. We have used the idea of equilibrium at this point in the proof but we have not yet used the fact that we are considering a standard auction. Equation (13) implies () Z () + andsoproperty1 of the definition of a standard auction implies that. We therefore have Z ( ) () Property 2. of this definition implies that () equals the probability that the given value is larger than the reservation values of the other 1 bidders that are independently drawn from the distribution, () () 1 (14) 92

8 Therefore, ( ) Z () 1 Lastly, we go back to the beginning of the proof and set our two formulas for ( ) equal to each other. After substituting using (14) for ( ), we then solve for ( ): Z () 1 ( ) ( ) ( ) ( ) 1 ( ) ( ) ( ) 1 Z () 1 (15) We are done! We have written each bidder s expected payment ( ) in terms of and the distribution. The right-hand side of this last inequality does not depend upon the rules of the auction in any way, shape, or form. All standard auctions collect money on average from bidders according to exactly the same formula (15). As a consequence, all standard auctions therefore generate exactly the same expected revenue for the seller. 93