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Bidding in Simultaneous Auctions with a Constraint on Exposure

We consider the problem of determining a profit maximizing set of sealed bids in simultaneous auctions that are independent except for a restriction on the total of all bids. Although the objective function is not concave, we show that a Lagrangian approach will often yield an optimal answer and, when it does not, will yield a good solution and a bound on possible further improvements. Characteristics of optimal sets of bids are proven. The methodology employed will be generally attractive for any problem that involves allocating a single resource among a large number of competing activities with convex-concave profit functions.