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 ).
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).)
You have one item to sell and you need to post a price for it. There is a single potential buyer and the value of the item for the buyer is distributed according to a known probability distribution.
It is quite easy to compute which posted price will maximize your revenues. You need to maximize the price multiplied by the probability that the value of the item is greater or equal to that price.
1) When the value of the item for the buyer is 10 with probability 1/2 and 15 with probability 1/2. The optimal price is 10 and the expected revenue is 10.
2) When the value of the item for the buyer is 10 with probability 1/2 and 40 with probability 1/2. The optimal price is 40 and the expected revenue is 20.
Now you have two items to sell and as before a single potential buyer. For each of the items, the buyer’s value behaves according to a known probability distribution. And these distributions are statistically independent. The value for the buyer of having the two items is simply the sum of the individual values.
Now we allow the seller to post a price for the bundle of two items and he posts the price that maximizes his revenues.
In summary: The values are additive, the distributions are independent.
Test your intuition:
1) Can the revenues of a seller for selling the two items be larger than the sum of the revenues when they are sold separately?
2) Can the revenues of a seller for selling the two items be smaller than the sum of the revenues when they are sold separately?
PS: there is a new post by Tim Gowers on the cost of Elseviers journals in England. Elsevier (and other publishers) are famous (or infamous) for their bundling policy. The movement towards cheaper journal prices, and open access to scientific papers that Gowers largly initialted two years ago is now referred to as the “academic spring.”