Answer to Test Your Intuition (22)



Indeed, most people got it right! Bundling sometimes increases revenues, sometimes keeps revenues the same, and sometimes decreases revenues. In fact, this is an interesting issue which was the subject of recent research effort. So here are a few examples as told in the introduction to a recent paper Approximate Revenue Maximization with Multiple Items by Sergiu Hart and Noam Nisan:

Example 1: Consider the distribution taking values 1 and 2, each with probability 1/2.
Let us first look at selling a single item optimally: the seller can either choose to price it
at 1, selling always and getting a revenue of 1, or choose to price the item at 2, selling it
with probability 1/2, still obtaining an expected revenue of 1, and so the optimal revenue
for a single item is 1. Now consider the following mechanism for selling both items:
bundle them together, and sell the bundle for price 3. The probability that the sum of
the buyer’s values for the two items is at least 3 is 3/4, and so the revenue is 3·3/4 = 2.25
– larger than 2, which is obtained by selling them separately.

Example 2:  For the distribution taking values 0 and 1, each with probability 1/2,
selling the bundle can yield at most a revenue of 3/4, and this is less than twice the
single-item revenue of 1/2.

Example 3 (and 4): In some other cases neither selling separately nor bundling
is optimal. For the distribution that takes values 0, 1 and 2, each with probability 1/3,
the unique optimal auction turns out to offer to the buyer the choice between any single
item at price 2, and the bundle of both items at a “discount” price of 3. This auction
gets revenue of 13/9 revenue, which is larger than the revenue of 4/3 obtained from
either selling the two items separately, or from selling them as a single bundle.  (A similar situation happens for the uniform distribution on [0, 1], for which neither bundling nor selling separately is optimal (Alejandro M. Manelli and Daniel R. Vincent [2006]).

Example 5: In yet other cases the optimal mechanism is not even deterministic and must offer lotteries for the items. This happens in the following example from a 2011 paper “Revenue Maximization in Two Dimensions” by Sergiu Hart and Phil Reny: Let F be the distribution which takes values 1, 2 and 4, with probabilities 1/6, 1/2, 1/3, respectively. It turns out that the unique optimal mechanism offers the buyer the choice between buying any one good with probability 1/2 for a price of 1, and buying the bundle of both goods (surely) for a price of 4; any deterministic mechanism has a strictly lower revenue. See also Hart’s presentation “Two (!) good to be true” Update: See also this paper by Hart and Nisan:  How Good Are Simple Mechanisms for Selling Multiple Goods?

Update: See also Andy Yao’s recent paper An n-to-1 Bidder Reduction for Multi-item Auctions and its Applications. The paper is relevant both to the issue of bundling and to the issue of using randomized mechanisms for auctions. (Test your intuition (21).)




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