Earlier in 2011, Prof. Silvio Micali, Alessandro Chiesa and I discovered a new area in game theory, that is to study auctions (or more general games) when the participating players do not exactly know their valuations. This page is my attempt to keep track of all of our published papers and technical reports, and to resolve potential confusions on the model.
Our Knightian Valuation Model
Through detailed notations and motivations can be found in our paper, the one-line version of our model is as follows.
Our Knightian Valuation Model. We focus on auctions where the designer has no information about the players, and in addition, the only information a player \(i\) has about the profile of true valuations, \(\theta^*\), consists of a set of distributions \(\mathcal{K}_i\), from one of which \(\theta^*_i\) has been drawn.
We refer interested readers to our paper for motivations and examples for such setting. Clearly, when \(\mathcal{K}_i\) is a singleton for each \(i\), this setting collapses to that of the classical second-price, Vickrey, and VCG mechanisms. We observe in our paper that, at least in auctions, this model is mathematically equivalent to the following distribution-free version.
Our Knightian Valuation Model (distribution-free). Each player \(i\)'s sole information about \(\theta^*\) consists of a set of valuations \(K_i\) such that \(\theta^*_i \in K_i \subseteq \Theta_i\). We call \(K_i\) the candidate set of player \(i\) (or approximate valuation of player \(i\) in our earlier version of the paper).
Our goal is to study the social-welfare efficiency of auction mechanisms, as a function of \(\delta\), the maximum inaccuracy of the candidate sets \(K_i\).
Our Results
We have presented our results in two computer science conferences, ITCS 2012 and ACM-EC 2014. We refer these two papers as [CMZ12] and [CMZ14]. Our journal version of the paper [CMZ15] is published in Econometrica. We have also produced 4 related technical reports [CMZ-bridge], [CMZ-vcg-regret], [CMZ-vcg-utility] and [CMZ-single-param] that we may choose to submit independently in the future. The table below provides a visual summary of these papers.
| Results | Solution Concept | [CMZ12] | [CMZ14] | [CMZ15] | [CMZ-single-param] | [CMZ-vcg-utility] | [CMZ-vcg-regret] |
| (1). dominant-strategy and ex-post-Nash mechanisms fail to work even for single-good auctions | dominant-strategy ex-post Nash |
included | included | ||||
| (2). the second-price mechanism produces great social-welfare under undominated strategies for single-good auctions | undominated strategies |
included | |||||
| (2'). 2 is essentially optimal among all finite mechanisms for single-good auctions | undominated strategies |
included | |||||
| (3). there is a probabilistic mechanism that slightly outperforms the second-price one for single-good auctions | undominated strategies |
included | |||||
| (4). the Vickrey mechanism produces great social-welfare under undominated strategies for multi-unit auctions | undominated strategies |
included | |||||
| (4'). 4 is essentially optimal among all finite mechanisms for multi-unit auctions | undominated strategies |
included | |||||
| (5). classical DST mechanisms continue to work under Knightian uncertainty in undominated strategies for single-parameter domains | undominated strategies |
included | |||||
| (6). the VCG mechanism performs poorly in undominated strategies for unrestricted combinatorial auctions | undominated strategies |
included | included | ||||
| (7). the VCG mechanism performs well in regret-minimizing strategies for unrestricted combinatorial auctions | regret-minimizing strategies |
included | included | ||||
| (7'). 7 continues to hold for utility maximizers who only resort to regret to break ties | regret-elimination
after undominated strategies see [CMZ-bridge] |
included | included |
References
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