Electronic Companion — “ Temporary and Permanent Buyout Prices in Online Auctions ” by Jérémie Gallien and

EC.1. Proof of Theorem 1 Consider a bidder A with type v t in an auction where every other bidder uses strategy , where is an arbitrary threshold function. If A is not the first bidder, the first bidder has either placed a bid or exercised the option immediately (following strategy ), so that the buyout option is no longer available to bidder A. In that case bidder A’s weakly dominant strategy is to bid his true valuation v, as shown in Vickrey (1961). His bid submission time in t T will not affect his utility in any way, so that bidding v at any time in t T constitutes then a best response. Suppose now that A is the first bidder, so that the buyout option is available to him. We introduce the following notation for the three possible actions he may take at time t: bid t : Bid in the auction at time t (in which case it is a dominant strategy for him to bid his valuation v); buy t : Buyout at time t; wait t : Wait for − t time units before deciding to bid (if the auction is still open) or buy out (if the option is still available). We define the utility of bidder A with type v t and taking action a ∈ bid t buy t wait t as Ua v t . If bidder A chooses bid t , i.e. bids immediately, the buyout option disappears. Following strategy , all subsequent bidders will bid their true valuation. Denoting by N t T the random number of bidders arriving in interval t T and N t the cumulative number of arrivals up to t, this implies: E Ubid t v t Nt = 0 N t T = e− T−t ∫ v