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On the Computation of Equilibria in Discrete First-Price Auctions

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

TL;DR

It is proved that the problem of deciding their existence is NP-complete, even for approximate equilibria, and it is shown that correlated equilibria of the auction can be computed in polynomial time.

Abstract

We study the computational complexity of computing Bayes-Nash equilibria in first-price auctions with discrete value distributions and discrete bidding space, under general subjective beliefs. It is known that such auctions do not always have pure equilibria. In this paper, we prove that the problem of deciding their existence is NP-complete, even for approximate equilibria. On the other hand, it can be shown that mixed equilibria are guaranteed to exist; however, their computational complexity has not been studied before. We establish the PPAD-completeness of computing a mixed equilibrium and we complement this by an efficient algorithm for finding symmetric approximate equilibria in the special case of iid priors. En route to these results, we develop a computational equivalence framework between continuous and discrete first-price auctions, which can be of independent interest, and which allows us to transfer existing positive and negative results from one setting to the other. Finally, we show that correlated equilibria of the auction can be computed in polynomial time.

On the Computation of Equilibria in Discrete First-Price Auctions

TL;DR

It is proved that the problem of deciding their existence is NP-complete, even for approximate equilibria, and it is shown that correlated equilibria of the auction can be computed in polynomial time.

Abstract

We study the computational complexity of computing Bayes-Nash equilibria in first-price auctions with discrete value distributions and discrete bidding space, under general subjective beliefs. It is known that such auctions do not always have pure equilibria. In this paper, we prove that the problem of deciding their existence is NP-complete, even for approximate equilibria. On the other hand, it can be shown that mixed equilibria are guaranteed to exist; however, their computational complexity has not been studied before. We establish the PPAD-completeness of computing a mixed equilibrium and we complement this by an efficient algorithm for finding symmetric approximate equilibria in the special case of iid priors. En route to these results, we develop a computational equivalence framework between continuous and discrete first-price auctions, which can be of independent interest, and which allows us to transfer existing positive and negative results from one setting to the other. Finally, we show that correlated equilibria of the auction can be computed in polynomial time.
Paper Structure (51 sections, 31 theorems, 100 equations, 2 figures, 3 tables)

This paper contains 51 sections, 31 theorems, 100 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Our Results and Discussion
  3. Pure equilibria
  4. Mixed Equilibria
  5. Positive Results
  6. Further Related Work
  7. Preliminaries
  8. Bayesian priors
  9. The Auction Game
  10. Equilibria
  11. No overbidding
  12. Representation
  13. Efficient Computation of Bidder Utilities
  14. Pure Equilibria
  15. Mixed Equilibria
  16. ...and 36 more sections

Key Result

Lemma 2.1

Fix a DFPA. For any bidder $i\in N$ and any true value $v_i\in V_i$, the utilitySee eq:DFPA-utility-interim-mixed-randomized-bidding and eq:DFPA-utility-interim-mixed for the utility definition.$u_i(\bm\gamma,\bm{\beta}_{-i};v_i)$ of $i$, given as input any distribution $\bm{\gamma}$ over her bids a

Figures (2)

  • Figure 1: Discrete $\to$ Continuous
  • Figure 2: Continuous $\to$ Discrete

Theorems & Definitions (62)

  • Definition 1: $\varepsilon$-approximate mixed Bayes-Nash equilibrium of the DFPA
  • Remark 1
  • Definition 2: $\varepsilon$-approximate pure Bayes-Nash equilibrium of the DFPA
  • Lemma 2.1
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof : Proof sketch
  • Definition 3: Monotone mixed strategies in a DFPA
  • Definition 4: $\varepsilon$-well-supported mixed Bayes-Nash equilibrium of the DFPA
  • ...and 52 more