Abstract
We consider a revenue-maximizing seller with multiple items for sale to a single population of buyers. Our main result shows that for a single population of additive buyers with independent (but not necessarily identically distributed) item values, bundling all items together achieves a constant-factor approximation to the revenue-optimal item-symmetric mechanism.
We further motivate this direction via fairness in ad auctions. In ad auction domains the items correspond to views from particular demographics, and recent works have therefore identified a novel fairness constraint: equally-qualified users from different demographics should be shown the same desired ad at equal rates. Prior work abstracts this to the following fairness guarantee: if an advertiser places an identical bid on two users, those two users should view the ad with the same probability [27, 34]. We first propose a relaxation of this guarantee from worst-case to Bayesian settings, which circumvents strong impossibility results from these works, and then study this guarantee through the lens of symmetries, as any item-symmetric auction is also fair (by this definition). Observe that in this domain, bundling all items together corresponds to concealing all demographic data [23].
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Notes
- 1.
- 2.
A valuation function \(v(\cdot )\) is additive if for all sets S of items, \(v(S)=\sum _{j \in S} v(\{j\})\).
- 3.
To quickly see why the unlimited supply setting is equivalent to the single bidder setting: Because there are no supply constraints, it is feasible to pick any single-bidder mechanism and just use it for every bidder.
- 4.
This follows as the optimal auction when all items are i.i.d. is in fact item-symmetric.
- 5.
For example, it could be that \(D_i = D_{i'}\) for all \(i, i'\), and each bidder is drawn from the same population. This represents settings where the platform cannot price-discriminate based on properties of the advertiser. It could also be that \(D_i \ne D_{i'}\). In such settings, perhaps \(D_i\) is the population of ‘big’ advertisers, and \(D_{i'}\) is the population of ‘small’ advertisers, and the platform knows from which population each individual advertiser is drawn.
- 6.
We use the standard notation \(\vec {v}_{-i}\) to refer to the vector of bids excluding bidder i, and \(D_{-i}\) to refer to the distribution over valuation profiles, excluding bidder i.
- 7.
To see this, recall that \(\sigma (\vec {x}(\vec {v}_i;\vec {v}_{-i}))\) is a vector that puts \(x_{ij}(\vec {v}_i;\vec {v}_{-i}))\) in the \(i,\sigma (j)\) coordinate.
- 8.
We also note that it is an interesting open direction to extend our main results from an additive bidder to a ’scaled additive’ bidder so that this connection holds even for non-uniform demographic distributions.
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Essaidi, M., Weinberg, S.M. (2022). On Symmetries in Multi-dimensional Mechanism Design. In: Feldman, M., Fu, H., Talgam-Cohen, I. (eds) Web and Internet Economics. WINE 2021. Lecture Notes in Computer Science(), vol 13112. Springer, Cham. https://doi.org/10.1007/978-3-030-94676-0_4
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