We are interested in designing agents for Mth-Price Auctions--a subclass of synchronized double auctions [Wurman, Walsh, & Wellman1998]. When there is a single seller (M=1), the Mth-price auction is equivalent to the standard first-price auction, with the provision that the seller is allowed to specify a reservation price.
Consider a set of bids, of which 
are sell bids, N are buy bids. The Mth-price auction sets the clearing price at the Mth highest among all M+N bids. All buy bids at prices greater than or equal to the clearing price can be matched to all sell bids at prices less than the clearing price. The order of matching is arbitrary.
The Mth-price setting and can be seen more clearly with the example in Figure 3. In the example, the sell bids are {1, 2.5, 3} and the buy bids are {2, 4, 7}. The clearing price is therefore 3, which is the third highest price among all bids. The sellers with bids in {1, 2.5} are then transacted with the buyers having bids in {4, 7}.
