Example: Optimal Auctions for Beta Distributions
This example considers the case of 2 items and a single additive buyer whose values 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.