How to bid in unified second-price auctions when requests are duplicated (Q2661498)

The author obtains, in a simplified setting of unified second-price auctions with duplicated requests, that the optimal solution for the bidder is to randomize the bid; determines the optimal distribution of the random bid; and argues against the use of unified second-price auctions as the randomization implies a loss of social welfare. The unified second-price auction is defined, as follows: \begin{itemize} \item[1.] The seller sends a bid request to the ad exchanges. \item[2.] Each ad exchanges sends a bid request to its potential buyers. \item[3.] The potential buyers answer the bid requests of the ad exchanges. \item[4.] Each ad exchange \(E_{i}\) discovers its highest bid \(x_{i}\) and its second highest bid \(y_{i}\). \item[5.] The highest bids \(x_{i}\) are sent to the seller as bid. \item[6.] The seller hosts a second-price auction: the highest \(x_{i_{1}}\) makes ad exchange \(E_{i_{1}}\) win the item, and the channel is billed the second highest bid \(x_{i_{2}}\). \item[7.] The highest bidder in ad exchange \(E_{i_{1}}\) gets the item, and is billed \(\max\{x_{i_{2}}, y_{i_{1}}\}\). \end{itemize} The author assumes that a unique second-price auction is resolved and the exchanges are not taking any margins; the bidder sees the bid requests received at the same time as identical; the bidder is able to estimate the economic value \(v\) of the bidding opportunity. Then it is obtained that the optimal cumulative density function \(K\colon [0,v] \rightarrow [0,1]\) for the bidder's random bids is \[K(t) = \frac{(n-1)G(t)}{g(t)(v-t)+(n-1)G(t)}~(t \in [0,v])\] provided \(K\) is non-decreasing, where \(n\) is the number of identical bid requests, \(g\) and \(G\) are the density and cumulative density of the highest bid in the consolidated auction, which are assumed to be known and independent of the bidder's bidding strategy. This is obtained by solving \[\max_{K}\mathbb{E}\left[\mathbb{I}_{\left[b_{1}>b\right]}\cdot\left(v-\max\{b,b_{2}\}\right)\right]\] where \(b\) is distributed according to \(g\), and \(b_{1},b_{2}\) are the highest and second highest bids of the bidder. The author notes that in the \(n=2\), \(v=1\), \(g(t)=1\) (\(G(t)=t\)) special case, the bidder bidding the randomized bit according to \(K(t)=t\) would pay \(1/3\) on average, which is smaller than the payment 1 that would be made in absence of duplication, and also smaller than the expected payment \(1/2\) that would be made in a real second-price auction.