A Bayesian equilibrium for simultaneous first-price auctions for complementary goods and quasi-linear bids

This article shows a Symmetrical Bayesian Nash Equilibrium in a context of $ m $ simultaneous first-price sealed-bid auctions and $ n $ bidders for complementary goods. We consider that the individual valuations of the $ m $ goods are common knowledge and identical among bidders and if the whole set of goods is gained, a private independent extra profit is obtained by the winner. For relaxing and solving the so-many mathematical complications involved in the general case we followed a classical methodology and proposed a particular bidding function that implies linear separability. Under this assumptions we obtain a Symmetric Bayesian Nash Equilibrium which functional form implies the classic quasi-linear property for bivariate functions. On addition, we compare the seller expected revenue between auctioning the complete set in one single first-price sealed-bid auction versus auctioning each item in $ m $ simultaneous first-price sealed-bid auctions.