Example: Optimal Auctions for Beta Distributions
This example considers the problem of selling of 2 items to a single additive buyer whose values for the items are distributed independently according to Beta distributions. Note that in the special case with Beta(1,1), the value for each item is distributed uniformly in [0,1].
The figure shows how the framework in the papers "Mechanism Design via Optimal Transport" and "Strong Duality for a Multiple-Good Monopolist" can be applied to compute the optimal mechanism that maximizes the seller's expected revenue:
- The square represents the region [0,1]2 where the measure lies, the dark shaded area is where the measure is negative while the light shaded area is where it is positive.
- The red dashed line gives the position of the first 0 when integrating from right to left, while the blue dashed line gives the position of the first 0 when integrating from top to bottom.
- The thick black line gives the optimal price for the grand-bundle of both items.
The solid black, blue and red lines partition the square in at most 4 regions:
- The region where the buyer gets both items with probability 1.
- The region where the buyer gets no items
- The region where the buyer gets item 1 with probability 1 and item 2 with probability strictly less than 1.
- The region where the buyer gets item 2 with probability 1 and item 1 with probability strictly less than 1.
- The buyer's utility is 0 below the thick solid curve. Above the thick solid curve, it is given by the L1 distance to the curve.