Abstract
We study a dynamic model of procurement auctions in which the agents (sellers) will abandon the auction if their utility does not satisfy their private target, in any given round. We call this “abandonment” and analyze its consequences on the overall cost to the mechanism designer (buyer), as it reduces competition in future rounds of the auction and drives up the price. We show that in order to maintain competition and minimize the overall cost, the mechanism designer has to adopt an inefficient (per-round) allocation, namely to assign the demand to multiple agents in a single round. We focus on threshold mechanisms as a simple way to achieve ex-post incentive compatibility, akin to reserves in revenue-maximizing forward auctions. We then consider the optimization problem of finding the optimal thresholds. We show that even though our objective function does not have the optimal substructure property in general, if the underlying distributions satisfy some regularity properties, the global optimal solution lies within a region where the optimal thresholds are monotone and can be calculated with a greedy approach, or even more simply in a parallel fashion.
Emmanouil Pountourakis was partially supported by NSF grant CCF-2218813. Part of the research was conducted when the authors were hosted by the Simon’s Institute for the Theory of Computing.
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Notes
- 1.
We conjecture that threshold mechanisms are optimal for this setting among mechanisms that satisfy ex-post incentive compatibility.
- 2.
In the case of ties we need to slightly adjust the description of the mechanism. We refer the reader to the full version of the paper for the general version of the mechanism. Since we assume continuous distributions, we can assume no ties for optimizing our objective, without loss of generality.
- 3.
Assuming no ties.
- 4.
For consistency of notation, we define \(\mu _{2:1}=1\). This is because when we save i agents in round one, the expected cost of the second round would be \(\mu _{2:i}\) for \(i\ge 2\), and 1 if \(i=1\).
- 5.
Note that whenever \(t_{i-1}=1\), the previous thresholds \(t_2,...,t_{i-2}\) are irrelevant. Therefore, this lemma holds even if we only had \(t_{i-1}=1\). However, we state the lemma as is for the sake of the next lemma.
- 6.
This is similar to revenue maximization where if we condition F to be above a certain value v and obtain the conditional distribution \(\tilde{F}\), we have that \(1-\tilde{F}(x)= \frac{1-F(x)}{1-F(v)}\) and \(\tilde{f}(x)=\frac{f(x)}{1-F(v)}\). This implies that the inverse hazard rate and as a result the virtual value functions of these distributions remain the same.
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Khodabakhsh, A., Nikolova, E., Pountourakis, E., Horn, J. (2023). Threshold Mechanisms for Dynamic Procurement with Abandonment. In: Deligkas, A., Filos-Ratsikas, A. (eds) Algorithmic Game Theory. SAGT 2023. Lecture Notes in Computer Science, vol 14238. Springer, Cham. https://doi.org/10.1007/978-3-031-43254-5_22
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