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    Multi-attribute Reverse Auctions with the Imaras

    1. Single-attribute reverse auctions
    2. Multi-attribute reverse auctions
      1. Example
      2. Bid determination
      3. Bid limits

    1. Single-attribute reverse auctions

    Many auctions are single attribute; most often the bidding is over price. In this situation a buyer sets up an auction in which sellers engage in a bidding process to sell the buyer a product or service. The sellers bid until the auction reaches its deadline or no seller is willing to make a bid. This type of auctions relies on the following two principles:

    When graphically displayed, the result is that every participant can precisely determine if a particular bid is winning or not. Because a seller cannot submit a bid that is worse than the current bid, each bid is a winning bid when it is made.

    The auction may be continuous or divided into earlier announced rounds of a pre-set length. They may have a fixed deadline, a moving deadline (e.g., the deadline is postponed by a fixed amount of time when a bid is made), or no deadline, but the auction ends when there is no new bid for a pre-set amount of time.

    Single1Single2

    The distinction between continuous and round-based auctions is introduced because such a distinction is also possible in multi-attribute auctions and it can be implemented in Imaras. (Note that in Fig. 1, every bid is winning at a given time because every next bid has to be better than the previous one. In Fig. 2, the bids are assessed at the end of the round; therefore a bid other than “your bid” may be the winning one.)

    2. Multi-attribute reverse auctions

    In multi-attribute auctions the situation is complicated because participants do not know if a new bid is better for another participant than an earlier bid. The bidder also does not know if her bid is better for the buyer than another bid made at the same time or earlier. This may be simplified if the buyer’s criteria or value function is revealed to the bidders.

    a. Example
    Consider a very simple case in which the revenues (or profits) are very simple functions of two attributes: price (p) and lead time (l). Price is measured in millions of dollars. The lead time is measured in weeks.

    Figure 3 illustrates selected revenue functions of the buyer and of the two sellers. Every combination of price and lead time which corresponds to a point lying on the same revenue curve (line) yields the same revenue (e.g., points a and b yield the same profit of 20 for B). Revenue

    There are three important differences between multi-attribute auctions and single-attribute auctions. In the case of multi-attribute auctions:

    BiddingDThe difficulties of making bids in multiattribute auctions are illustrated in Figure 4. Let’s assume that S1 made bid b1=(p=8; l=5) with revenue S1 = 26.6. For the buyer this offer yields revenue 19.5 and for seller S2 the revenue is 24.9.  

    Bid b1 may be shown to seller S2 but he may have difficulties in making a bid that is better for the buyer than b1. For S2 to be sure that his bid is better than the previous bid, he has to decrease the value of both attributes. (Note that this requires an assumption that the buyer’s preferences on the attributes are opposite, albeit not necessarily strictly opposite, to the seller’s preferences.)

    b. Bid determination

    Making a bid in which all attributes are worse for the bidder is not an efficient strategy because the bidder concedes on all attributes when he could concede on one or a few attributes.
    After S1 made bid b1 = (8; 5), S2 may consider the following three potential bids: b2 = (7; 5); b3 = (7.5; 4.5) and b4 = (8; 4). These three bids differ from b1 in that either one is subtracted from only one attribute, or 0.5 is subtracted from two attributes.

    Table 1. Bids by Seller 1 and Seller 2

    Bider (bid) Price Lead time Revenue B Revenue S1 Revenue S2
    S1 (b1) 8.0 5.0 12.0 26.6 24.9
    S2 (b2) 7.0 5.0 13.0 25.4 22.6
    S2 (b3) 7.5 4.5 14.4 23.9 22.6
    S2 (b4) 8.0 4.0 15.6 22.4 22.6
    S2 (b5) 3.7 1.8 25 8.0  7.7
    S1 (b6) 2.0 2.2 26.1 7.0  4.7
    S1 (b7) 5.0 1.2 24.4 7.0  9.3

    Seller S2 is indifferent between the three bids but the buyer is not. If S2 knew about the buyer’s revenue function, then he should bid b2. This bid may allow S2 to bid later b3 or b4, depending on what S1 bids.

    The problem arises when the sellers do not know the buyer’s revenue function. Every seller has to rely solely on the knowledge of his own revenue function and the bids made by other sellers. In such a situation the bidders may not only be unable to optimize their strategies, but even make bids that are worse than their earlier bids.

    For example, seller S2 made bid b5. Seller S1 considers decreasing his revenue to 7 from 8 obtained from b5. He could make a small change on both price and lead time; instead he wants to offer either a much larger decrease in price or in lead time. There are two possible bids he can choose from: b6 and b7. Bid b6 indeed would be welcomed by the buyer because her revenue would increase from 25 to 26.1. However, bid b7 should not be made because it is worse for the buyer than b5. Unless seller S1 has additional information (e.g., about the buyer’s revenue), he is not aware of this situation.

    There are several options in which the additional information about the buyer’s interests may be passed to the sellers. The simplest and most controversial is to give the sellers the buyer’s revenue function. It is simple because then the sellers precisely know what is better and what is worse for the buyer—the auction reverts then to a single-attribute in which the attribute is the buyer’s revenue. This option is controversial because in many real-life situations the buyers want to keep their revenue function secret.

    c. Bid limits

    In our multiattribute auction the problem of making inadmissible bids like bid b7 is resolved by the introduction of limits. Limits are bounds set up for the attribute values and they help sellers to make only progressive bids, that is, bids which are better for the buyer. In this way we give the bidders information which makes bidding in multiattribute auction similar to bidding in the single-attribute auction.

    limitsAt each point in time (or round) there may be one or more sets of limits. One set defines limits on every attribute. In Figure 5, three rounds are indicated. In round 1 the best bid is b1; in round 2 it is b2; and in round 3 the best bid is b5.

    The limits for each round are determined based on the best bid in the previous round and additional parameters set up by the buyer (e.g., the number of limit sets calculated for each round). This means that there may be more than one set of limits set in each round.

    For Rounds 2 and 3, two sets per round are shown in Figure 5; they are indicated by rectangles bounded by arrows; every bid in the rectangle is admissible. Three limit sets are determined for bidding in Round 4.
    Bids outside the limit set are inadmissible. Observe that the limits assure that, after S2 made the winning bid b5 in Round 3, S1 cannot make b7 because this bid is outside of the limits. Bid b6, however, may be made. The buyer’s revenue function is now shown with dotted lines to indicate that the bidders do not know it; for them it is sufficient to know the limits.
    In every round, every bidder may choose the set of limits that best suits his interests. Every bidder, however, has to observe at least one set of limits. This means that at the beginning of Round 2, the sellers obtain two limit sets (see Table 2) which they have to follow. Their bids have to be such that: (1) either the price is lower or equal to 8 and the lead time lower or equal to 5; or (2) the price have to be lower or equal to 10.3 and the lead time lower or equal to 3.8.  

    Table 2. Limit sets

    Round no. Set no. Price Lead time
    Round 2 Set 1   p ≤ 8   p ≤ 5
    Set 2   p ≤ 10.3   p ≤ 3.8
    Round 3 Set 1   p ≤ 7   p ≤ 5
    Set 2   p ≤ 5.8   p ≤ 3.3
    Round 4 Set 1   p ≤ 3.7   p ≤ 1.8
    Set 2   p ≤ 4.3   p ≤ 1.1
    Set 3   p ≤ 2   p ≤ 3.7

    (Note. For simplicity in the above description every limit set determined for one round is associated with the same value of the buyer’s revenue (i.e., it lies on the same revenue curve). This feature allows the sellers to discover the buyer’s revenue function. The algorithm implemented in the Imaras system does not need to follow so that the buyer can keep her revenue function private.)