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Unending Sequential Auctions

Amir Ban

TL;DR

This paper studies unending sequential auctions for identical items with private values, incorporating a per-round exit probability to model bidder uncertainty and analyzing the resulting Markov dynamics. In the zero-uncertainty case, the steady state induces an emergent posted-price mechanism with price $X_{(\lambda)}$, allocating items to high-value bidders and excluding low-value bidders. Introducing uncertainty ($\delta>0$) preserves a unique stationary distribution but renders the threshold fuzzy, with low-value bidders gaining winning chances and some high-value bidders benefiting from lower equilibrium bids; auctioneer welfare, however, tends to decline. The authors derive a bidding function via a first-order condition, characterize the stationary winner distribution, and discuss robustness extensions, including multiple winners and second-price variants, offering insights into practical persistence of auctions in digital and financial contexts.

Abstract

Sequential auctions for identical items with unit-demand, private-value buyers are common and often occur periodically without end, as new bidders replace departing ones. We model bidder uncertainty by introducing a probability that a bidder must exit the auction in each period. Treating the sequential auction as a Markov process, we demonstrate the existence of a unique steady state. In the absence of uncertainty, the steady state resembles a posted-price mechanism: bidders with values above a threshold almost surely win items by repeatedly bidding the threshold price, while those below the threshold almost surely do not. The equilibrium price corresponds to the threshold value that balances supply (bidders with values above the threshold) and demand (auction winners). When uncertainty is introduced, the threshold value persists but becomes less precise, growing "fuzzier" as uncertainty increases. This uncertainty benefits low-value bidders, those below the threshold, by giving them a significant chance of winning. Surprisingly, high-value bidders also benefit from uncertainty, up to a certain value limit, as it lowers equilibrium bids and increases their expected utility. On the other hand, this bidder uncertainty often reduces the auctioneer's utility.

Unending Sequential Auctions

TL;DR

This paper studies unending sequential auctions for identical items with private values, incorporating a per-round exit probability to model bidder uncertainty and analyzing the resulting Markov dynamics. In the zero-uncertainty case, the steady state induces an emergent posted-price mechanism with price , allocating items to high-value bidders and excluding low-value bidders. Introducing uncertainty () preserves a unique stationary distribution but renders the threshold fuzzy, with low-value bidders gaining winning chances and some high-value bidders benefiting from lower equilibrium bids; auctioneer welfare, however, tends to decline. The authors derive a bidding function via a first-order condition, characterize the stationary winner distribution, and discuss robustness extensions, including multiple winners and second-price variants, offering insights into practical persistence of auctions in digital and financial contexts.

Abstract

Sequential auctions for identical items with unit-demand, private-value buyers are common and often occur periodically without end, as new bidders replace departing ones. We model bidder uncertainty by introducing a probability that a bidder must exit the auction in each period. Treating the sequential auction as a Markov process, we demonstrate the existence of a unique steady state. In the absence of uncertainty, the steady state resembles a posted-price mechanism: bidders with values above a threshold almost surely win items by repeatedly bidding the threshold price, while those below the threshold almost surely do not. The equilibrium price corresponds to the threshold value that balances supply (bidders with values above the threshold) and demand (auction winners). When uncertainty is introduced, the threshold value persists but becomes less precise, growing "fuzzier" as uncertainty increases. This uncertainty benefits low-value bidders, those below the threshold, by giving them a significant chance of winning. Surprisingly, high-value bidders also benefit from uncertainty, up to a certain value limit, as it lowers equilibrium bids and increases their expected utility. On the other hand, this bidder uncertainty often reduces the auctioneer's utility.

Paper Structure

This paper contains 29 sections, 20 theorems, 33 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Model
  4. An Emergent Posted-Price Mechanism
  5. The Unending Auction with Uncertainty
  6. Calculating a Bidding Function
  7. Probability of a Bidder's Success
  8. The Bidding Function
  9. Bidder Expectation
  10. Is Uncertainty Good or Bad for Bidders?
  11. Discussion
  12. Multiple Winners
  13. A Few Impatient Bidders
  14. A Change in Bidder Arrival Rates
  15. Unending Second Price Auctions
  16. ...and 14 more sections

Key Result

proposition 1

If $\delta = 0$ and $\lambda > 1$, the pool size grows without limit: $\lim \sup N_t = +\infty$. I.e., there is a positive probability that, starting from $N_t = k$, $N_T \neq 0$ for every $T > t$, and this probability limits at $1$ as $k \to \infty$.

Figures (10)

  • Figure 1: A snapshot of transaction fees in the mempool with expected waiting times (from bitcoinfees.net)
  • Figure 2: Bidding function with U[0,1] and $x^2$ power-law distribution
  • Figure 3: Stationary distribution $\{p_n\}$ with $\lambda=2, \delta=0.01$
  • Figure 4: Mean pool size and time to win with values $F^{-1}(\frac{\lambda - \lambda^*}{\lambda})$
  • Figure 5: Stationary winner density $w(g)$ by percentile
  • ...and 5 more figures

Theorems & Definitions (36)

  • proposition 1: $\lambda > 1$
  • proposition 2: $\lambda < 1$
  • proposition 3: $\lambda=1$
  • theorem 1: Winners' Threshold
  • remark 1
  • theorem 2: Bidding without Uncertainty
  • corollary 1
  • proposition 4
  • proposition 5
  • proposition 6
  • ...and 26 more