Volume Discount

We have assumed up to this point that the monetary bids,pi j, submitted by buyerito each unit of good of typetj, is independent of the size of the bid (order). Often, however, the seller is willing to consider charging less per unit for larger orders. The purpose of the discount is to encourage the buyers to buy the goods in large batches. Such volume discounts are common in inventory theory for many consumer goods. We believe that volume discounts are suitable, in a natural way, in many auction models as well.

There are many different types of discount policies. The two most popular are: all units and incremental. In each case, there are one or more breakpoints defining change points in the unit cost. There are two possibilities: either the discount is applied to all of the units in the bid (order), this is theall-unitdiscount, or it is applied only to the additional units beyond the breakpoint - theincrementaldiscount. Here we consider the incremental case. We assume further that each buyer defines his own breakpoints.

Consider the following example. Two buyers are interested in trash bags and their bids are of the following price policy. The first buyer is willing to pay 30 cents for each bag for quantities below 500; for quantities between 500 and 1,000, he is willing to pay 30 cents for each of the first 500 bags and for each bag of the remaining amount he is willing to pay 29 cents each; for quantities over 1,000, for the first 500 bags he is willing to pay 30 cents each, for the next 500 bags 29 cents each, and for the remaining amount he is willing to pay 28 cents each. The second buyer has the following similar policy. He is willing to pay 31 cents for each bag for quantities below 600; for quantities over 600 bags he is willing to pay 31 cents for each of the first 600 bags and for the remaining amount he is willing to pay 28 cents for each bag. The breakpoints in this example are: 500 and 1,000 for the first buyer and 600 for the second one.

Formally the incremental volume discount constrained multi-unit auction problem is defined as follows. Letri j denote the number of breakpoints defined by buyer i for a good of typetj, and letd_{i j}^s denote the values of the breakpoints, withd_{i j}⁰ =0.

Also, let p_{i j}^s denote the unit cost for any additional unit beyond the breakpointd_{i j}^s−1, for 1≤s≤ri j. Then, each bid is composed of a 2-dimensional vector of lengthri j, (p_{i j}^s,d_{i j}^s). That is, buyeri will pay p¹_{i j} for each one of the firstd_{i j}¹ goods of type tj

allocated to him. He will further pay p²_{i j} for each one of the next d_{i j}² −d_{i j}¹ goods, p_{i j}³ for each one of the next d_{i j}³−d_{i j}² goods, etc. The above mentioned problem is denoted as theincremental volume discount constrained multi-unit auction problem with entrance fee. The following assumptions are assumed:p_{i j}¹ > p_{i j}² >· · ·> p^r_{i j}^{i j}and 0=d_{i j}⁰ <d_{i j}¹ <d_{i j}² <· · ·<d_{i j}^{ri j}.

Using, yet, even more complicated construction, an appropriate bipartite graph can be obtained so that running a general capacitatedb-matching algorithm on this graph will result a solution to the incremental volume discount constrained multi-unit auction problem with entrance fee. Hence, the following theorem follows.

Theorem 4.3 The incremental volume discount constrained multi-unit auction prob-lem with entrance fee is computationally tractable.

5. Conclusion

In this chapter we discussed the use ofb-matching techniques in order to handle non-trivial multi-object multi-unit auctions. We have shown thatb-matching techniques are highly applicable for both combinatorial auctions and constrained auctions, two fundamental and complementary classes of multi-object and multi-unit auctions. While

the more standard approach for handling complex auctions, such as combinatorial auctions, typically relies on straightforward techniques such as LP relaxation of the corresponding IP problem, b-matching techniques can be used in order to address additional types of multi-object auctions. By combining standard matching techniques, more generalb-matching techniques, and capacitatedb-matching techniques, tractable solutions are found to basic combinatorial and constrained auctions.

The research reported in this chapter can be extended in various directions. First, combinatorial bids and constraints are two basic properties of multi-object auctions, and in future work one can combine them in a shared framework. Given that framework, a study of the applicability ofb-matching techniques to the more general setting should be introduced. In addition, we believe that the applicability ofb-matching techniques goes beyond auctions, and may be useful in other economic settings. For example, it seems that these techniques can be used in the context of double auctions and exchanges.

Finally, we remark that matching is a basic ingredient of markets in general; therefore, a more general theory connecting algorithmic matching theory and economic interactions may be a subject of future research.

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