| Project: | Characterizing revenue non-monotonicity for two identical, independent items |
|---|---|
| Mentor: | Ariel Schvarzman |
| Coworker: | Jachym Mierva |
A multi-item auction exhibits some rather non-intuitive properties. Already in the simples example of a two item auction, the optimal revenue may behave in astrange way. Given buyers whose valuation for the items in drawn from some known distribution D, a seller can design an optimal mechanism which maximizes his revenue. Now consider two such distributions D and D', where D stochastically dominates D', which basically means the buyers value the items more on average. In that case, one would expect the optimal revenue to be larger, but [1] shows it is not always the case and provide a counter example.
Our REU project will focus around this counter example, we will try to understand it, construct more counter examples. The goal would be to describe how they are distributed in the space of all distributions and if they are rare.
After meeting our surpervior and discussing the goals of the project, we spent the rest of the week researching literature. We also worked on the opening presentation.
We agreed with our supervisor that the first step to understanding the problem is to look more closely at the strange example given in [1]. We compared revenue of two artificial distributions and concluded that such wierd examples can only come from non-convex payment rules. We plan to formalize our findings and read more relevant literature.
This week, we wrote down and formalized our findings from the last week. We also formulated a continous version of the problem and looked briefly into solving it for some special distribution of valuations.
Using numerical simulations, we studied distributions "between" the two distributions given in [1]. Our results suggest that the revenue changes dramatically even for small changes in the input distribution.
Focusing on a simple case, we used numerical simulations to study the case where our distribution has support of size two. Our results show that in this simple case, revenue monotonicity holds.
Along with our supervisor, we developed a mechanism which introduces a new element into the support of our distribution, which is lower then all other elements. In some special cases, this can produce a revenue non-monotonic examples.
This week we focused on writting our findings. We also came up with a way to test, for a given distribution and optimal mechanism, if there exists a pair of valuations with break revenue monotonicity. In other words, if there is a pair with if probability is shifted to lower valuation, the revenue increases.
[1] Maximal revenue with multiple goods: Nonmonotonicity and other observations, Hart, S., Reny, P.J., Theoretical Economics, 2015
This work is supported by CoSP, a project funded by European Union's Horizon 2020 research and innovation programme, grant agreement No. 823748.
My REU 2021 page.