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# A Game-Theoretic Analysis of the Empirical Revenue Maximization Algorithm with Endogenous Sampling

NIPS 2020, (2020)

EI

摘要

The Empirical Revenue Maximization (ERM) is one of the most important price learning algorithms in auction design: as the literature shows it can learn approximately optimal reserve prices for revenue-maximizing auctioneers in both repeated auctions and uniform-price auctions. However, in these applications the agents who provide inputs...更多

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简介

- The auctioneer can set the price to be p = ERM0(v) to maximize revenue if bids are equal to values.
- Goldberg et al [18] show that this auction is not incentive-compatible as bidders can lower the price by strategic bidding.

重点内容

- In auction theory, it is well-known [31] that, when all buyers have values that are independently and identically drawn from a regular distribution F, the revenue-maximizing auction is the second price auction with anonymous reserve price p∗ = arg max{p(1 − F (p)}: if the highest bid is at least p∗, the highest bidder wins the item and pays the maximum between the second highest bid and p∗
- Many works (e.g., [12, 16, 24]) on sample complexity in auctions have studied how to obtain a near-optimal reserve price based on samples from the distribution F instead of knowing the exact F
- One of the most important price learning algorithms in those works is the Empirical Revenue Maximization (ERM) algorithm, which outputs the reserve price that is optimal on the uniform distribution over samples
- Huang et al [24] show that a one-bidder auction with posted price set by ERMc and with N samples from the value distribution is (1 − ) revenue optimal with = O((N −1 log N )2/3) for Monotone Hazard Rate (MHR) distributions and = O( DN −1 log N ) for bounded distributions
- A natural question would be: what is the largest class of value distribution we can consider? Note that for non-regular distributions, Myerson [31] shows that revenue optimality cannot be guaranteed by anonymous reserve price, so ERM is not a correct choice

结果

- Azevedo and Budish [4] show that, uniform-price auctions are incentive-compatible in the large in the sense that truthful bidding is an approximate equilibrium when there are many bidders in the auction.
- The authors say that a mechanism is (1 − ) revenue optimal, for some 0 < < 1, if its expected revenue is at least (1 − ) times the expected revenue of Myerson auction.4 Huang et al [24] show that a one-bidder auction with posted price set by ERMc and with N samples from the value distribution is (1 − ) revenue optimal with = O((N −1 log N )2/3) for MHR distributions and = O( DN −1 log N ) for bounded distributions.
- As the incentive-awareness measure of P becomes lower, the price learning function becomes more incentive-aware in the sense that bidders gain less from non-truthful bidding: Theorem 2.1.
- The algorithm, which the authors call “two-phase ERM”, works as follows: in the first T1 rounds, run any truthful, prior-independent auction M; in the later T2 = T − T1 rounds, run second price auction with reserve p = ERMc(b1, .
- In the uniform-price auction, suppose that any m bidders can jointly deviate from truthful bidding, no bidder can obtain more utility (the authors call this (m, )-group BIC), where,
- Theorem 2.2 shows that, in the two-phase model, approximate incentive-compatibility and revenue optimality can be obtained simultaneously for bounded distributions and for MHR distributions.

结论

- Note that for non-regular distributions, Myerson [31] shows that revenue optimality cannot be guaranteed by anonymous reserve price, so ERM is not a correct choice.
- For any α > 0, the authors obtain bounds similar to MHR distributions on ∆wNo,mrst and on approximate incentive-compatibility in the two-phase model and the uniform-price auction.

引用论文

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- 2. If q(t) ∈ I1 ∪ · · · ∪ Il, then A(t−1) does not decrease.
- 3. If q(t) ∈ Il+1, A(t−1) does not change.
- 2. For the second step, first note that the qi ∈ [q∗ − γ, q∗] in the first step will be considered by ERMc since i
- 3. Finally, if qj
- 2. For the second term, we claim that
- 3. For the third term, we claim that
- 2. For the second step, first note that the qi ∈ [q∗ − γ, q∗] in the first step will be considered by ERMc since i ≥ (qi − γ)N ≥ (q∗ − 2γ)N ≥ (e(α) − 2γ)N > cN. Then suppose ERMc chooses another quantile qj instead of qi, we must have j i
- 2. For the second term, we claim that
- 3. For the third term, we claim that
- 1. However, an argument similar to above shows the same lower bound directly on 1. Consider the two-point distribution F
- 1. However, if bidder i deviates to bI, then P (bI, v−I ) becomes 1, because
- 1. Thus, the reserve price is decreased from D to 1 with probability at least:
- 0. The proof of (34) is separated into two parts. Firstly, Pr[TN−1] =

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