# Is the expected utility from entering for a bidder

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) is the expected utility from entering for a bidder with value v . Thus, rF ( r ) n - 1 = δ. Now, let us compare the second-price auction with entry fee of δ = rF ( r ) n - 1 to a second-price auction with a reserve price of r . Clearly in both cases, a ( v ) = F ( v ) n - 1 { v r } . Moreover, the expected payment for a bidder in both cases is the same: If v < r , the expected payment is 0 for both. For v r , and the auction with entry fee, p ( v ) = δ + F ( v ) n - 1 - F ( r ) n - 1 E h max i n - 1 V i r max i n - 1 V i v i , whereas with the reserve price, p ( v ) = rF ( r ) n - 1 + F ( v ) n - 1 - F ( r ) n - 1 E h max i n - 1 V i r max i n - 1 V i v i . This means that u ( w | v ) = va ( w ) - p ( w ) is the same in both auctions. Hence the threshold strategy is a Bayes-Nash equilibrium in the entry fee auction, since a bidder with value below r has negative expected utility from entering. Notice, however, that the payment of a bidder with value v can differ in the two auctions. For example, if bidder 1 has value v > r and all other bidders have value less than r , then in the entry fee auction, bidder 1’s payment is δ = rF ( r ) n - 1 , whereas in the reserve price auction it is r . Moreover, if bidder 1 has value v > r , but there is another bidder with a higher value, then in the entry fee auction, bidder 1 loses the auction, but still pays δ , whereas in the reserve price auction he pays nothing. Thus, when the entry-fee auction is over, a bidder may regret having 10 since u E ( v ) 0 for v > r and u E ( v ) 0 for v < r , and u E ( · ) is weakly increasing.
248 14. AUCTIONS participated. This means that this auction is ex-interim individually rational , but not ex-post individually rational . See the definitions in § 14.5.4. 14.5.3. Evaluation fee. Example 14.5.2 . The queen is running a second-price auction to sell her crown jewels. However, she plans to charge an evaluation fee: A bidder must pay φ in order to examine the jewels and determine how much he values them prior to bidding in the auction. Figure 14.5. The queen has fallen on hard times and must sell her crown jewels. Assume that bidder i ’s value V i for the jewels is a random variable and that the V i ’s are i.i.d. Prior to the evaluation, he only knows the distribution of V i . He will learn the realization of V i only if he pays the evaluation fee. In this situation, as long as the fee is below the bidder’s expected utility in the second-price auction, i.e. φ < E [ u ( V i | V i )] , the bidder has an incentive to pay the evaluation fee. Thus, the seller can charge an evaluation fee equal to the bidder’s expected utility, minus some > 0. The expected auctioneer revenue from bidder i ’s evaluation fee and his payment in the ensuing second-price auction is E [ u ( V i | V i )] - + E [ p ( V i )] = E [ a i ( V i ) · V i ] - . Since, in a second-price auction, the allocation is to the bidder with the highest value, the seller’s expected revenue is E h max i ( V 1 , . . . , V n ) i - n, which is essentially best possible.
14.6. CHARACTERIZATION OF BAYES-NASH EQUILIBRIUM 249 14.5.4. Ex-ante versus ex-interim versus ex-post. An auction, with an associated equilibrium β , is called individually rational (IR) if each bidder’s ex- pected utility is non-negative. The examples given above illustrate three different
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