This paper presents a methodology for designing trading agents for complex games. We compute, for the first time, Bayes-Nash equilibria for firstprice single-unit auctions and mth-price multi-unit auctions, when the auction has a set of possible closing times, one of which is chosen randomly for the auction to end at. To evaluate this approach we used our analysis to generate strategies for the International Trading Agent Competition. One of these was evaluated as the best overall and was subsequently used very successfully by our agent WhiteBear in the 2005 competition.