General Information

Proud member of Charles University group consisting of Jan Soukup (Our coordinator), Jan Bronec, Guillermo Gaboa, Svetlana Ivanova, Gaurav Kucheriya, Jáchym Mierva, David Miksanik, David Sychrovsky, Tung Anh Vu, Ilia Zavidnyi.

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Project Description

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 a strange 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.

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Research Log

Week 1:

During the first week, we met our mentor and were introduced to our project. Then we started reading the literature relevant to the topic of mechanism design and worked on the opening presentation.

Week 2:

On Monday we introduced other participants of REU to our research project. Later during the week, our mentor gave us two lectures to deepen our knowledge of mechanism design. After a discussion with him, we came up with quite reasonable assumption, that prices should be monotone with respect to the valuations of the buyer. While assuming this, and also assuming price convexity, we managed to prove the revenue monotonicity. This lead us to the conclusion that revenue non-monotonicity can happen only in mechanisms with non-convex payment rules. However, the price monotonicity assumption turned out not to be generally true, I was able to construct a counter example. We plan to investigate this example next week.

Week 3:

This week we focused primarily on writing our findings down. We formally defined the version of the problem for continuous distributions and reformulated the IC conditions in terms of partial derivatives of the allocation and price functions. We also made some numerical simulations on discrete distributions which suggested that the optimal revenue changes chaotically when continuously changing the valuation distribution.

Week 4:

We were studying a simple case of distributions with support of size 2 using numerical techiques and concluded that revenue monotonicity holds. Apart from that, we tried to study the continuous case.

Week 5:

Ariel had an idea on an algorithm, that would take a distribution (with some special properties) and create a dominated distribution by moving the lowest element, which would have greater revenue. We were then filling the details the whole week.

Week 6:

We continued investigating the algorithm from the last week. Apart from that, we tried to move with the continuous case, but it doesn't look promissing. We also studied a paper Mechanism Design via Optimal Transport that touches this problematics.

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Presentations

• Non-monotonicity (initial)

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References

1. Maximal revenue with multiple goods: Nonmonotonicity and other observations, Hart, S., Reny, P.J., Theoretical Economics, 2015
2. Mechanism Design via Optimal Transport, Daskalakis, C., Deckelbaum, A., Tzamos, C., 2015

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Acknowledgment

This work was carried out while the author was a participant in the 2022 DIMACS REU program at Rutgers University, supported by supported by CoSP, a project funded by European Union’s Horizon 2020 research and innovation programme, grant agreement No. 823748.