Multi-Object Auctions

I find it convenient to start a discussion of multi-object auction with case of a seller who wants to sell two dissimilar objects, an apple and an orange, to two bidders.

Each bidder has values VA,i for the apple and VO,i for the orange, i=1,2. The first question is whether the seller should bundle the two objects and hold only one auction for the fruit basket. The answer depends on the objective of the auctioneer, but that has nothing to do with the prediction of the outcome if the seller has already determined an auction rule.

As a start let us suppose that the seller had committed to use the SPA rule in the following way. When the auctions are separate each bidder submits separate bids, one for apple and one for orange. The bids are then evaluated separately for each object and the objects are sold to the bidder with the highest bid for that object following the SPA rule. If the objects are being sold as a bundle, then the SPA is conducted similarly as before with the only point to note is that the value for the object in that case is Vi = VA,i + VO,i.

The bundling decision

The best way to understand what bunding of objects does to revenue in auction is to consider a simple example. Consider the revenue if the auctioneer had decided to hold separate auctions and that when it decided to hold a bundle auction.

Two bidders

We start with the case where there are two bidders. The revenue in each case depends on the actual values for the objects to the bidders. Consider the following example:

Bidder 1’s values for the objects: VA,1 = 10, VO,1 = 5

Bidder 2's values for the objects: VA,2 = 12, VO,2 = 6

If the objects are sold as a bundle then bidder 1 has value 15 and bidder 2 has a value 18 for the bundle. When the auctioneer sticks to the SPA rule for its sales the revenue in any auction is simply the value of the object to the second highest bidder.

So, if the objects are sold separately, the revenue to the auctioneer will be 15. When the objects are sold as a bundle the revenue will be 15, as well! Regardless of which method the auctioneer took the revenue ended up being the same.

Now consider another example:

Bidder 1’s values for the objects: VA,1 = 10, VO,1 = 5

Bidder 2's values for the objects: VA,2 = 12, VO,2 = 4

In this case the revenue from the separate sales is 9, but the revenue from a bundled sale is 15!

The explanation for this phenomenon is simple. If the objects are sold separately, the auctioneer is condemned to receive the lowest value for each object as the price for that object. (The revenue is the sum of the lowest value for each object.)

If, however, the objects are sold as a bundle, it is possible to have the sum of the lowest value for the objects as the revenue. (This happens when the same bidder has the lowest value for each object.) But it is also possible to have the sum of the lower value for one object and the higher value for another object as revenue.

Since the values of the objects are independent, any of the two cases can happen: revenue in the bundle auction could be the same or equal to that from separate auctions. So, on average, the revenue from the bundle auction is higher than revenue from separate auctions.

Many bidders

When there are many bidders, say 20, things become different. The price of an object is set equal to the second highest value of an object which is much higher than the 10th highest or the 18th highest value. So while with separate auctions the auctioneer is guaranteed to receive the second highest value for each object as the price, under bundle auction it is very likely that the price for the bundle will be set by a value that is much lower than the 2nd highest value for one of the objects. That makes the revenue from a bundle auction lower than that from separate auctions.

In fact, under some broad conditions it was found that there is a number n* > 2 such that if the number of bidders is lower than n* then bundle auction generates more expected revenue but when there are more than n* bidders separate auctions generate a higher expected revenue.