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The Combinatorial Multi-Round Ascending Auction

Bernhard Kasberger, Alexander Teytelboym

TL;DR

The paper presents the first game-theoretic analysis of the Combinatorial Multi-Round Ascending Auction (CMRA), a spectrum-auction format that combines clock prices with non-linear additional bids. It shows that CMRA-truthful bidding yields efficient allocations across general valuations, but the equilibrium is not robust to bidder asymmetries and can enable risk-free collusion, motivating an alternative activity rule to deter such behavior. Through a two-bidder, single-divisible-good model, it characterizes equilibria including a simple constant-strategy outcome and a collusion mechanism, and discusses how the proposed rule preserves desirable equilibria while reducing collusive incentives. The paper also connects theory to Danish CMRA auctions, finding patterns consistent with the theory and offering practical guidance for implementing CMRA with reduced collusion risk in real-world spectrum sales.

Abstract

The Combinatorial Multi-Round Ascending Auction (CMRA) is a new auction format used in recent European spectrum auctions. We show that an auction-specific version of truthful bidding leads to an efficient allocation. We then characterize different ex-post equilibria that feature truthful bidding, demand expansion, and demand reduction. The truthtelling equilibrium is fragile to small asymmetries in the bidders' caps. Moreover, if bidders are sufficiently symmetric, the CMRA is vulnerable to risk-free collusion. We propose an alternative activity rule that prevents such collusive strategies while keeping other equilibria intact. We discuss outcomes of several Danish CMRAs in light of our equilibrium predictions.

The Combinatorial Multi-Round Ascending Auction

TL;DR

The paper presents the first game-theoretic analysis of the Combinatorial Multi-Round Ascending Auction (CMRA), a spectrum-auction format that combines clock prices with non-linear additional bids. It shows that CMRA-truthful bidding yields efficient allocations across general valuations, but the equilibrium is not robust to bidder asymmetries and can enable risk-free collusion, motivating an alternative activity rule to deter such behavior. Through a two-bidder, single-divisible-good model, it characterizes equilibria including a simple constant-strategy outcome and a collusion mechanism, and discusses how the proposed rule preserves desirable equilibria while reducing collusive incentives. The paper also connects theory to Danish CMRA auctions, finding patterns consistent with the theory and offering practical guidance for implementing CMRA with reduced collusion risk in real-world spectrum sales.

Abstract

The Combinatorial Multi-Round Ascending Auction (CMRA) is a new auction format used in recent European spectrum auctions. We show that an auction-specific version of truthful bidding leads to an efficient allocation. We then characterize different ex-post equilibria that feature truthful bidding, demand expansion, and demand reduction. The truthtelling equilibrium is fragile to small asymmetries in the bidders' caps. Moreover, if bidders are sufficiently symmetric, the CMRA is vulnerable to risk-free collusion. We propose an alternative activity rule that prevents such collusive strategies while keeping other equilibria intact. We discuss outcomes of several Danish CMRAs in light of our equilibrium predictions.
Paper Structure (40 sections, 10 theorems, 50 equations, 4 figures, 2 tables)

This paper contains 40 sections, 10 theorems, 50 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Illustrative Example: Clock Auction vs. CMRA
  3. The Combinatorial Multi-Round Ascending Auction
  4. Clock and headline demands.
  5. Additional bids.
  6. Closing rule.
  7. Price adjustment.
  8. Activity rule.
  9. Outcomes under non-strategic truthful bidding
  10. Efficiency
  11. Revenue
  12. Outcomes under strategic bidding
  13. Further assumptions for the strategic analysis
  14. Strategies and equilibrium.
  15. Information.
  16. ...and 25 more sections

Key Result

Theorem 1

CMRA-truthful bidding leads to the efficient allocation.

Figures (4)

  • Figure 1: Headline demands and additional bids under CMRA-truthful bidding strategy in the illustrative example.
  • Figure 2: Predicted bidding patterns under different bidding strategies and marginal valuations.
  • Figure SA.1: CMRA-truthful bidding with decreasing marginal values.
  • Figure SA.2: CMRA-truthful bidding with non-decreasing marginal values

Theorems & Definitions (28)

  • Theorem 1
  • Proposition 1
  • Remark 1
  • Proposition 2
  • Remark 2
  • Remark 3
  • Theorem 2
  • Theorem 3
  • Example 1: Quadratic utility
  • Example 2: Linear utility
  • ...and 18 more