Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

How competitive are pay-as-bid auction games?

Martina Vanelli, Giacomo Como, Fabio Fagnani

TL;DR

The paper analyzes pay-as-bid auction games in electricity markets, showing that restricting strategy spaces to Lipschitz-continuous supply functions guarantees pure Nash equilibria and a tractable structure. By introducing an activation-price reformulation, it reduces the infinite-dimensional problem to a one-dimensional action space, enabling exact characterization of equilibria and proving existence under standard concavity assumptions. In the affine-demand, quadratic-cost case, it derives closed-form equilibria and demonstrates convergence to perfect competition as the Lipschitz bound grows, yielding efficient allocation and lower prices than uniform-price supply-function equilibria. The study also provides comparative statics against Cournot, Bertrand, and SFE benchmarks, highlighting that the pay-as-bid framework can achieve closer-to-competitive outcomes, especially under increasing strategy-space restriction and cost heterogeneity. Overall, the results illuminate how pricing rules, demand structure, and cost heterogeneity shape market efficiency and prices in pay-as-bid settings, with practical implications for market design and regulation.

Abstract

We study the pay-as-bid auction game, a supply function model with discriminatory pricing and asymmetric firms. In this game, strategies are non-decreasing supply functions relating pric to quantity and the exact choice of the strategy space turns out to be a crucial issue: when it includes all non-decreasing continuous functions, pure-strategy Nash equilibria often fail to exist. To overcome this, we restrict the strategy space to the set of Lipschitz-continuous functions and we prove that Nash equilibria always exist (under standard concavity assumptions) and consist of functions that are affine on their own support and have slope equal to the maximum allowed Lipschitz constant. We further show that the Nash equilibrium is unique up to the market-clearing price when the demand is affine and the asymmetric marginal production costs are homogeneous in zero. For quadratic production costs, we derive a closed-form expression and we compute the limit as the allowed Lipschitz constant grows to infinity. Our results show that in the limit the pay-as-bid auction game achieves perfect competition with efficient allocation and induces a lower market-clearing price compared to supply function models based on uniform price auctions.

How competitive are pay-as-bid auction games?

TL;DR

The paper analyzes pay-as-bid auction games in electricity markets, showing that restricting strategy spaces to Lipschitz-continuous supply functions guarantees pure Nash equilibria and a tractable structure. By introducing an activation-price reformulation, it reduces the infinite-dimensional problem to a one-dimensional action space, enabling exact characterization of equilibria and proving existence under standard concavity assumptions. In the affine-demand, quadratic-cost case, it derives closed-form equilibria and demonstrates convergence to perfect competition as the Lipschitz bound grows, yielding efficient allocation and lower prices than uniform-price supply-function equilibria. The study also provides comparative statics against Cournot, Bertrand, and SFE benchmarks, highlighting that the pay-as-bid framework can achieve closer-to-competitive outcomes, especially under increasing strategy-space restriction and cost heterogeneity. Overall, the results illuminate how pricing rules, demand structure, and cost heterogeneity shape market efficiency and prices in pay-as-bid settings, with practical implications for market design and regulation.

Abstract

We study the pay-as-bid auction game, a supply function model with discriminatory pricing and asymmetric firms. In this game, strategies are non-decreasing supply functions relating pric to quantity and the exact choice of the strategy space turns out to be a crucial issue: when it includes all non-decreasing continuous functions, pure-strategy Nash equilibria often fail to exist. To overcome this, we restrict the strategy space to the set of Lipschitz-continuous functions and we prove that Nash equilibria always exist (under standard concavity assumptions) and consist of functions that are affine on their own support and have slope equal to the maximum allowed Lipschitz constant. We further show that the Nash equilibrium is unique up to the market-clearing price when the demand is affine and the asymmetric marginal production costs are homogeneous in zero. For quadratic production costs, we derive a closed-form expression and we compute the limit as the allowed Lipschitz constant grows to infinity. Our results show that in the limit the pay-as-bid auction game achieves perfect competition with efficient allocation and induces a lower market-clearing price compared to supply function models based on uniform price auctions.

Paper Structure

This paper contains 19 sections, 17 theorems, 118 equations, 8 figures.

Table of Contents

  1. Introduction.
  2. Related literature
  3. Structure of the paper
  4. Model and problem formulation
  5. Pay-as-bid auction games
  6. Best responses and Nash equilibria
  7. Existence of Nash equilibria
  8. Reduction to non-expansive supply functions
  9. Characterization of best responses
  10. Properties of the activation price games
  11. Existence and characterization of Nash equilibria
  12. Uniqueness of Nash equilibria
  13. All-active games
  14. The affine demand and quadratic costs case
  15. Comparative statics
  16. ...and 4 more sections

Key Result

Proposition 1

Consider the $\mathcal{C}^{\uparrow}_0$-PAB auction game with set of producers $\mathcal{N}$, production cost functions $C_i$ for every $i$ in $\mathcal{N}$, and demand function $D$. Then, for every producer $i$ in $\mathcal{N}$ and strategy profile $S_{-i}$, either $\mathcal{B}_i(S_{-i}) = \emptyse

Figures (8)

  • Figure 1: On the left, the total uniform-price remuneration, on the right, the total pay-as-bid remuneration (gray-shaded regions).
  • Figure 2: The market-clearing price (on the left) and the pay-as-bid remuneration (on the right).
  • Figure 3: Graphical illustration of Proposition \ref{['pr:br_form']}.
  • Figure 4: Two Nash equilibria of the $\hbox{Lip}_1$-PAB auction game corresponding to the same Nash equilibrium of the activation price game (see Example \ref{['ex:1_nonlin']})
  • Figure 5: Two Nash equilibria of both the activation price game and the $\hbox{Lip}_K$-PAB auction game (see Example \ref{['ex:2_nonlin']})
  • ...and 3 more figures

Theorems & Definitions (49)

  • Definition 1: $\mathcal{A}$-pay-as-bid auction game
  • Remark 1
  • Definition 2: Best responses and Nash equilibria
  • Proposition 1
  • proof
  • Remark 2
  • Proposition 2
  • proof
  • Lemma 1
  • proof
  • ...and 39 more