Technical Note for “ Multi-stage Intermediation in Display Advertising ”

In this technical report, we first characterize an optimal mechanism for intermediaries which are positioned between their captive buyers and a seller who runs a second-price auction when there is no restrictions on the set of mechanisms. This technical report also contains an analysis of the game between intermediaries and a seller, who are organized in a general tree network when agents select mechanisms within the class of strategy-proof (appropriately adapted to our setting) mechanisms where truthful reporting is a dominant strategy. We show that focusing on second-price mechanisms is i) without loss of optimality for the single-stage intermediation, and ii) without loss of optimality within the set of strategy-proof mechanisms for k-trees.