trong một cuộc bán đấu giá một chiều nhà thầu chỉ quan tâm về số lượng hàng hoá mà họ nhận được và giá họ phải trả, và hồ sơ dự thầu có thể đề cập đến số lượng chỉ có giá và (trong trường hợp đấu giá nhiều đơn vị). Một mặt nổi tiếng nhất, một chiều gia đình bán đấu giá là đấu giá tiếng Anh và bán đấu giá kín thầu, theo dõi chặt chẽ bởi các gia đình Hà Lan và Nhật Bản | . COMBINATORIAL AUCTIONS 173 by the presence or absence of another good. The first is complementarity. A valuation function v has complementarities if there exist two sets of goods S T c X for which v Su T v S v T . The second property is substitutability. A valuation function v has substitutability if there exist two sets of goods S T c X such that S n T 0 for which v S u T v S v T . Before we define specific bidding language let us consider some types of bids that we may commonly want to express. We can divide these bids into symmetric and asymmetric valuations. Symmetric valuations are those in which all goods are identical from the point of view of the bidder and for this reason we sometimes use the term multiple units of good. A few common symmetric valuations are the following. Additive valuation. The bidder s valuation of a set is directly proportional to the number of goods in the set so that vị S c S for some constant c. Single item valuation. The bidder desires any single item and only a single item so that vị S c for some constant c for all S 0. Fixed budget valuation. Similar to the additive valuation but the bidder has a maximum budget of B so that vị S min c S B Majority valuation. The bidder values equally any majority of the goods so that Vị S Í1 if S 0 otherwise We can generalize all of these symmetric valuations to a general symmetric valuation. General symmetric valuation. Let P1 P2 . Pm be arbitrary nonnegative prices so that Pj specifies how much the bidder is willing to pay of the jth item won. Then l l Vị S Pj j i Downward sloping valuation. A downward sloping valuation is a symmetric valuation in which Pl P2 pm. Many common types of bids are not symmetric however. Often there are different classes of goods and valuations of sets of goods are a function of the classes of goods in the set. For example imagine that our set X consists of two classes of goods some red items and some green items and the bidder requires only items of the same .
