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2015 | OriginalPaper | Chapter

4. Assigning Multiple Homogeneous Items in a Single Auction

Authors : Asunción Mochón, Yago Sáez

Published in: Understanding Auctions

Publisher: Springer International Publishing

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Abstract

In previous chapters, the main single-unit auction models were discussed. However, in many instances auctions are used to award multiple related units. For example, if an olive oil factory wants to sell part of its stock, it can choose to conduct an auction. The items to be auctioned may be homogeneous (oil bottles of the same size and acidity) or heterogeneous (different sizes and acidities) and may be awarded in a single auction or in different auctions.

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previous chapter Other Single-Unit Auctions

next chapter Sequential and Simultaneous Auctions

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In multi-unit auctions, items can be substitutes or complements. If the items are substitutes, the value of winning the second unit is less than the value of winning the first. Items are complements (synergies) if the value of winning the second unit is higher than that of winning the first, if the first item has already been purchased. In Chap. 6, we will explain these concepts in more detail.

In this chapter, we assume substitute items: decreasing marginal values, \(v_{i,1} \geq v_{i,2} \geq \cdots \geq v_{i,M}\).

The aggregate demand in round T could exceed the remaining supply, Σ _t = 0 ^T Q ^t∗ > M. To solve this tie, the seller can set different rules to allocate the items among the bidders that have submitted offers in this last round. There are many options such as: allocating the items to those who bid first or in a random way.

In multi-unit descending auction bidders will not bid sincerely because this strategy implies zero surplus. Bidders tend to wait until the price is less than the valuation to obtain a positive surplus: underbidding (p ^t < v _i, j).

According to the rules of the auction, when the price increases, bidders are required to maintain or decrease their bid, but not increase it. Furthermore, bidders who decide to stop bidding in a round cannot bid again in later rounds.

It could happen that at a certain increase of the price, the aggregate demand decreases below the supply (Q ^T < M). To avoid unsold items, the seller can set a rationing rule to allocate all items, see Appendix at the end of this chapter.

This procedure sequentially implements the Vickrey rule, which establishes that each bidder pays an amount equal to the opportunity cost of the item won.

See demonstration in [4].

We have assumed that the marginal values are decreasing (substitute items). Therefore, the bid vector must satisfy the following condition: \(b_{i,1} \geq b_{i,2} \geq \cdots \geq b_{i,M}\).

To calculate the winners, the aggregate demand function must first be obtained by horizontally adding the N individual demand functions of the bidders. The supply will then determine the winner bidders.

If there is a tie among bidders, the seller can set a tie-breaking rule such as: by order of submission and randomly.

The specific auction chosen affects the bidding strategy of the bidders, which makes it difficult to determine the model that generates the greatest revenue for the seller.

Cramton and Sujarittanonta [19] analyzed the effect of choosing the HRB or the LAB in a multi-unit ascending auction.

If the price were calculated according to the LAB, it would be equal to p = 4 euros (b _1, 2 = 4); therefore, the first bidder would have paid \(P_{1}^{{\ast}} = 4 \times 2 = 8\) euros for the two items and the second bidder \(P_{2}^{{\ast}} = 4 \times 1 = 4\) euros. In this case, the seller’s revenue would have been \(R^{{\ast}} = 4 \times 3 = 12\) euros.

The Vickrey auction is a VCG mechanism, in which each winning bidder pays an amount equal to the opportunity cost of the acquired item.

According to the rules used for the 3G spectrum auction in the UK, each bidder could not earn more than one item.

To explore other possible rationing rules, see [34] or [69], among others.

If an integer number is not obtained, the seller may round up.

L.M. Ausubel. An efficient ascending-bid auction for multiple objects. American Economic Review, 94:1452–1475, 2004.CrossRef

19.

P. Cramton and P. Sujarittanonta. Pricing rule in a clock auction. Decision Analysis, 7(1): 40–57, 2010.CrossRef

34.

T.A. Gresik. Rationing rules and european central bank auctions. Journal of International Money and Finance, 20(6):793–808, 2001.CrossRef

38.

P. Klemperer. Auctions with almost common values: The wallet game and its applications. European Economic Review, 42:757–769, 1998.CrossRef

41.

V. Krishna. Auction Theory. Elsevier, 2009.

69.

Y. Saez, D. Quintana, P. Isasi, and A. Mochon. Effects of a rationing rule on the ausubel auction: a genetic algorithm implementation. Computational Intelligence, 23(2):221–235, 2007.CrossRef

Title: Assigning Multiple Homogeneous Items in a Single Auction
Authors: Asunción Mochón
Yago Sáez
Publisher: Springer International Publishing
Book: Understanding Auctions
Print ISBN: 978-3-319-08812-9

Electronic ISBN: 978-3-319-08813-6

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-3-319-08813-6_4