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Efficient Equilibrium Computation in Symmetric First-Price Auctions

Aris Filos-Ratsikas, Yiannis Giannakopoulos, Alexandros Hollender, Charalampos Kokkalis

Abstract

We study the complexity of computing Bayes-Nash equilibria in single-item first-price auctions. We present the first efficient algorithms for the problem, when the bidders' values for the item are independently drawn from the same continuous distribution, under both established variants of continuous and finite bidding sets. More precisely, we design polynomial-time algorithms for the white-box model, where the distribution is provided directly as part of the input, and query-efficient algorithms for the black-box model, where the distribution is accessed via oracle calls. Our results settle the computational complexity of the problem for bidders with i.i.d. values.

Efficient Equilibrium Computation in Symmetric First-Price Auctions

Abstract

We study the complexity of computing Bayes-Nash equilibria in single-item first-price auctions. We present the first efficient algorithms for the problem, when the bidders' values for the item are independently drawn from the same continuous distribution, under both established variants of continuous and finite bidding sets. More precisely, we design polynomial-time algorithms for the white-box model, where the distribution is provided directly as part of the input, and query-efficient algorithms for the black-box model, where the distribution is accessed via oracle calls. Our results settle the computational complexity of the problem for bidders with i.i.d. values.

Paper Structure

This paper contains 30 sections, 16 theorems, 74 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Our Results and Techniques
  3. Algorithms for the CCFPA.
  4. Algorithms for the CDFPA.
  5. Related Work and Discussion
  6. Preliminaries
  7. Symmetric Auctions
  8. Value Distributions.
  9. Bidding Strategies.
  10. Utilities and Equilibria.
  11. Distribution Representation
  12. Continuous vs Discrete Bids
  13. Continuous Bidding Space (CCFPA)
  14. Black-Box Input
  15. White-Box Implicit Input
  16. ...and 15 more sections

Key Result

Theorem 3.1

An $\varepsilon$-BNE of a CCFPA can be computed via $O(1/\varepsilon)$ oracle calls to the cdf of its value distribution. More precisely, given $\varepsilon>0$, there exists a continuous, no-overbidding and monotone, (symmetric) bidding strategy $\beta:V\to B$ such that:

Figures (2)

  • Figure 1: A monotone bidding strategy $\beta$, succinctly represented by its jump points $0=s_0\leq s_1 \leq \cdots \leq s_m=1$.
  • Figure 2: Black-box lower-bound construction of \ref{['th:black-box-lower']}: $F_1$ is uniform; $F_2$ matches $F_1$ everywhere except on $(v_1,v_2)$, where it is “flattened then steepened” (kink at $v_2-\xi$) to remain a valid continuous, strictly increasing cdf.

Theorems & Definitions (33)

  • Definition 1: $\varepsilon$-approximate symmetric Bayes-Nash equilibrium of the FPA
  • Remark 1
  • Theorem 3.1: Black-box Upper Bound
  • proof
  • Theorem 3.2: Black-box Lower Bound
  • proof
  • Corollary 3.3: White-box FPTAS
  • Theorem 3.4
  • Theorem 4.1
  • Lemma 4.2: chwe1989discrete
  • ...and 23 more