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Optimal Diffusion Auctions

Yao Zhang, Shanshan Zheng, Dengji Zhao

TL;DR

This work studies revenue maximization in diffusion auctions where the seller's revenue depends on the underlying social network structure. It shows that Myerson's mechanism with $n$ buyers provides an upper bound on diffusion revenue and proves that a universally optimal IC/IR diffusion mechanism does not exist. To address structure dependence, it introduces $k$-Partial Winner of Myerson's ($k$-PWM) mechanisms that are optimal for specific structure classes and proves a general impossibility of universal optimality, motivating robust approximations. The authors propose the Closest Winner of Myerson's (CWM) and a Shifted Reserve Prices variant (CWM-SRP) to achieve bounded approximations to the structure-specific optimum, with numerical experiments on small-world networks showing that CWM-SRP significantly improves revenue over prior diffusion mechanisms as network size grows.

Abstract

Diffusion auction design is a new trend in mechanism design for which the main goal is to incentivize existing buyers to invite new buyers, who are their neighbors on a social network, to join an auction even though they are competitors. With more buyers, a diffusion auction will be able to give a more efficient allocation and receive higher revenue. Existing studies have proposed many interesting diffusion auctions to attract more buyers, but the seller's revenue is not optimized. Hence, in this study, we investigate what optimal revenue the seller can achieve by attracting more buyers. Different from the traditional setting, the revenue that can be achieved in a diffusion auction highly relies on the structure of the network. Hence, we focus on optimal auctions with given classes of underlying networks. We propose a class of mechanisms, where for any given structure, an optimal diffusion mechanism can be found. We point out that it implies an idea of "reserve structure". Moreover, we show that an optimal mechanism that handles all structures does not exist. Therefore, we also propose mechanisms that have bounded approximations of the optimal revenue in all structures.

Optimal Diffusion Auctions

TL;DR

This work studies revenue maximization in diffusion auctions where the seller's revenue depends on the underlying social network structure. It shows that Myerson's mechanism with buyers provides an upper bound on diffusion revenue and proves that a universally optimal IC/IR diffusion mechanism does not exist. To address structure dependence, it introduces -Partial Winner of Myerson's (-PWM) mechanisms that are optimal for specific structure classes and proves a general impossibility of universal optimality, motivating robust approximations. The authors propose the Closest Winner of Myerson's (CWM) and a Shifted Reserve Prices variant (CWM-SRP) to achieve bounded approximations to the structure-specific optimum, with numerical experiments on small-world networks showing that CWM-SRP significantly improves revenue over prior diffusion mechanisms as network size grows.

Abstract

Diffusion auction design is a new trend in mechanism design for which the main goal is to incentivize existing buyers to invite new buyers, who are their neighbors on a social network, to join an auction even though they are competitors. With more buyers, a diffusion auction will be able to give a more efficient allocation and receive higher revenue. Existing studies have proposed many interesting diffusion auctions to attract more buyers, but the seller's revenue is not optimized. Hence, in this study, we investigate what optimal revenue the seller can achieve by attracting more buyers. Different from the traditional setting, the revenue that can be achieved in a diffusion auction highly relies on the structure of the network. Hence, we focus on optimal auctions with given classes of underlying networks. We propose a class of mechanisms, where for any given structure, an optimal diffusion mechanism can be found. We point out that it implies an idea of "reserve structure". Moreover, we show that an optimal mechanism that handles all structures does not exist. Therefore, we also propose mechanisms that have bounded approximations of the optimal revenue in all structures.
Paper Structure (13 sections, 11 theorems, 9 equations, 4 figures)

This paper contains 13 sections, 11 theorems, 9 equations, 4 figures.

Table of Contents

  1. Other Related Work
  2. Preliminaries
  3. The Model
  4. Myerson's Mechanism
  5. Optimal Diffusion Mechanisms
  6. Local Myerson's Mechanism
  7. The Partial Winner of Myerson's
  8. Impossibility of Universal Optimality
  9. Optimal Approximation Mechanisms
  10. The Closest Winner of Myerson's
  11. Shifted Reserve Prices
  12. Numerical Experiments
  13. Discussion and Future Work

Key Result

Lemma 1

If for any $i$, the hazard rate of her valuation distribution $f_i(z)/(1-F_i(z))$In the rest of the paper, we always assume monotone non-decreasing hazard rates without explicitly stating it for convenience. is monotone non-decreasing, then Myerson's mechanism is IR, IC and optimal in the single-ite

Figures (4)

  • Figure 1: Three example networks with 2, 3, and 4 buyers respectively, where buyers' virtual bids are listed beside them.
  • Figure 2: The above-left graph is an example of the network in a diffusion auction. The following three graphs (a), (b), and (c) show the algorithm to find the closest potential winner. In each step, the blue node represents who has the highest virtual bid among the buyers connected with solid lines.
  • Figure 3: A structure profile where the majority of buyers in $r_1\setminus r_s$.
  • Figure 4: The left graph is the average of the estimated expected revenue over all sampled structures and the right graphs give the estimation of the expected revenue of 3 samples for each number of nodes.

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Lemma 1
  • Proposition 2
  • proof
  • Definition 7
  • ...and 24 more