Mathematik, Arbeitsrichtung Diskrete Optimierung
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Mathematik, Arbeitsrichtung Diskrete Optimierung

Tropical Mechanism Design

Project heads
Michael Joswig, Max Klimm

Project member
Sylvain Spitz

Project duration
01/2019 - 12/2021

We study the design of revenue-maximizing mechanisms for selling multiple items. Applying a duality framework, there is a one-to-one correspondence between optimal mechanisms and certain tropical polynomials and rational functions that we want to study via ideas from algebraic statistics.

Topic and Background of the Project
The question of how to sell one item to multiple bidders in a revenue-maximizing auction is well understood since the work of Roger Myerson in the 1980s. It is used in many applications and on online platforms such as eBay. But if we want to sell multiple items at once, the problem becomes substantially harder and not much is known about the optimal auction mechanism.

Deterministic Auction Mechanisms
Typically, an auction mechanism asks the participants for their bids and then allocates the items to the bidders as well as a price they have to pay. This method is called a deterministic auction mechanism. In contrast, we could introduce a lottery for some (or all) items and, after collecting the bids, allocate the items to the bidders up to a certain probability. Observations suggest that in general a revenue-maximizing auction has to introduce such lotteries. All the more so it is interesting to find cases, in which a deterministic auction is optimal.

Giannakopoulos and Koutsoupias examined the case, where the valuations, that each bidder has for the items, are drawn according to an uniform distribution. They showed, that the optimal auction mechanism for this setting and up to 6 items is deterministic and they conjectured that this holds true for an arbitrary number of items. We will explore their conjecture as well as their approach by applying other methods, especially from tropical geometry. In fact, the utility function of a deterministic auction corresponds to a tropical polynomial. Moreover, we want to use the insights we get by this method, to better understand the general case.

Straight Jacket Auction and Generalized Permutahedra
The deterministic mechanism given by Giannakopoulos and Koutsoupias is called the Straight Jacket Auction (SJA). It consists of a certain price schedule (p1,…,pn), where pk designates the price for any bundle of k items. Such a price schedule divides the n-cube in regions depending on the term which maximizes the utility function u(x) = max{ Σj∈J x– p|J] | J ⊂ {1,…,n} }. The regions are marked on the right by UJ, where J is the bundle of items that attain the maximum utility. Their volumetric properties are essential for the computation of the prices and the proof of optimality of the mechanism.

The regions are in fact covector cells of the tropical polynomial u(x) intersected with the n-cube. They can also be classified as generalized permutahedra, which is useful because those polytopes come up in a variety of applications and are therefore well studied. In this way, we can link their volume to the number of bipartite graphs including a perfect matching.