Example: Optimal Auctions for Beta Distributions
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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:
In the regions where the probability for an item is between 0 and 1, the probability is given by the slope of the corresponding curve at that point.
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 L
1 distance to the curve.