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Latency Advantages in Common-Value Auctions

Ciamac C. Moallemi, Mallesh M. Pai, Dan Robinson

arXiv:2504.02077v1econ.TH

TL;DR

The paper asks how later action and its informational advantage affect common-value auctions, a question with implications for financial trading and on-chain protocol design. It characterizes equilibrium strategies and connects last-mover payoffs to option values. The analysis also identifies timing incentives as a potential pressure on protocol decentralization.

  • Problem

    The economics and comparative statics of latency advantages in auctions remain unexplored, although such advantages can give later movers additional information and may deter participation.

    Source: §1 · p. 2

  • Method

    The paper characterizes sequential-bidder equilibrium and represents last-mover profits through option values to study comparative statics, including timing sensitivity under Geometric Brownian Motion.

    Source: §1 · pp. 2–3

  • Results

    The equilibrium is characterized, and the last mover’s profit—and equivalently the seller’s loss—is expressed as a portfolio of options on the underlying asset.

    Source: §1 · p. 2, §3 · p. 7

  • Takeaways & Limitations

    Potential gains from increasing timing advantage can create timing pressure and incentives that may reduce effective decentralization of blockchain protocols.

    Source: §1.1 · p. 4

    The equilibrium analysis focuses on the case where equation (2) has a solution in the support of the value distribution; when no solution exists, the passage states a separate equilibrium case.

    Source: §3.1 · p. 8

Abstract

from arXiv · show

In financial applications, latency advantages - the ability to make decisions later than others, even without the ability to see what others have done - can provide individual participants with an edge by allowing them to gather additional relevant information. For example, a trader who is able to act even milliseconds after another trader may receive information about changing prices on other exchanges that lets them make a profit at the expense of the latter. To better understand the economics of latency advantages, we consider a common-value auction with a reserve price in which some bidders may have more information about the value of the item than others, e.g., by bidding later. We provide a characterization of the equilibrium strategies and study the welfare and auctioneer revenue implications of the last-mover advantage. We show that the auction does not degenerate completely and that the seller is still able to capture some value. We study comparative statics of the equilibrium under different assumptions about the nature of the latency advantage. Under the assumptions of the Black-Scholes model, we derive formulas for the last mover's expected profit, as well as for the sensitivity of that profit to their timing advantage. We apply our results to the design of blockchain protocols that aim to run auctions for financial assets on-chain, where incentives to increase timing advantages can put pressure on the decentralization of the system.

1. INTRODUCTION

Latency advantages let later bidders act on additional information, yet their economics and comparative statics remain unexplored. The paper characterizes equilibrium in common-value auctions and connects last-mover profits to option values, with implications for on-chain auction design.

1. INTRODUCTION

  • Later action can give bidders additional information and profit opportunities, while potentially deterring participation and reducing market efficiency.The paper studies the largely unexplored economics and comparative statics of these advantages.Source: §1 · p. 2
  • The model considers a volatile common-value asset auction with a reserve price, where later bidders observe more about its likely settlement value but not earlier bids.The winner pays their bid, and the asset’s value is its prevailing market price at settlement.Source: §1 · p. 2
  • In the two-bidder equilibrium, Alice bids a conditional expected value based on a random draw, while Bob bids the higher of his conditional expected value and the reserve when profitable.With zero reserve, the strategy reduces to Engelbrecht-Wiggans, Milgrom, and Weber (1983).Source: §1 · p. 2
  • The last mover’s profit, equivalently the seller’s loss, is represented as a portfolio of options, enabling comparative statics in the reserve, value distribution, and timing advantage.Under Geometric Brownian Motion, the paper measures timing pressure as marginal profit from increasing the timing advantage, interpretable as the option’s theta.Source: §1 · pp. 2–3

1.1. Applications

  • On-chain protocols face last-mover timing pressure: the potential profit can encourage last-moment messaging and incentives for validator consolidation, collusion, or corruption, potentially reducing decentralization.The paper identifies last-mover advantage as especially difficult to mitigate in decentralized protocols, where delays may reflect network conditions.Source: §1.1 · p. 4

1.2. Literature Review

  • The paper extends the no-reserve equilibrium of Engelbrecht-Wiggans, Milgrom, and Weber (1983) to allow a reserve price and adds option-based interpretations and timing comparative statics.It contrasts this dynamic setting with Hausch (1987), which uses a finite type space and a static setting without timing comparative statics.Source: §1.2 · p. 5

2. MODEL

The model studies sequential bidding for a common-value good when the later bidder learns its realized value but cannot observe the earlier bid. It extends to different bidder counts and timing structures, and maps the setup to on-chain asset auctions.

2. MODEL

  • The baseline is a two-period, two-bidder common-value auction, with a single good whose value is drawn from a distribution with finite expectation.Alice bids before the value is realized; Bob observes the realized value and bids without seeing Alice’s bid, and the winner pays their bid subject to a limit price.Source: §2 · p. 5
  • The specified tie-breaking rules favor Bob in ties with Alice or the limit price, while an Alice-only tie with the limit price leaves the good unsold.The equilibrium is studied as a function of the value distribution and limit price.Source: §2 · p. 5
  • The framework extends to more than two bidders, uncertain last-mover identity, and continuous-time bidding with stochastic timing and settlement.In the uncertain-timing extension, Alice may be the last mover with probability α; the continuous-time setup has Alice move at 0 and Bob at T > 0.Source: §2 · p. 6
  • The analysis compares seller revenue and bidder surplus across model primitives and timing parameters, including the probability and magnitude of the last-mover advantage.Source: §2 · p. 6

2.1. Application to On-Chain Auctions

  • For the on-chain application, one order is auctioned to fill a token sale at a minimum price, with the highest single bid winning and partial fills excluded.The token’s reference-market price evolves as a zero-drift geometric Brownian motion; Bob sees the price when bidding, while Alice knows only its initial price.Source: §2.1 · p. 6
  • A two-block sealed-bid implementation lets independent proposers commit bids without seeing each other’s bids, while the later proposer may gain additional price information.This setup also supports analysis of the incentive to delay bidding for further information.Source: §2.1 · p. 7

3. RESULTS

The results characterize equilibrium bidding under a reserve price, derive bidder payoffs and auctioneer revenue, and extend the analysis to uncertainty about whether the later bidder can act. The late bidder's informational advantage yields positive expected payoff, while the earlier bidder's equilibrium strategy involves randomization.

3.1. Equilibrium in the Baseline Model

  • Alice randomizes by drawing a value from Fv and bidding βL of that draw, while Bob bids as a function of the observed value; the equilibrium extends to multiple early bidders.For n bidders, the early bidders’ strategies need only preserve the distribution of Alice’s maximum bid, and the later bidder follows Bob’s strategy.Source: §3.1 · pp. 7–8

3.1.1. Proof of Theorem 1.

  • No equilibrium has Alice bidding a single amount: she must randomize, and in any equilibrium her expected payoff is zero for every bid she submits.The characterization assumes a solution to the reserve-price condition in the support of Fv; if E[˜v] < L, Bob instead bids L when v > L and zero otherwise.Source: §3.1.1 · pp. 8–9, §3.1 · p. 8

3.2. Equilibrium payoffs

  • With L = 0, Bob’s expected profit is an exchange option worth more than an at-the-money call or put; with L > 0, his payoff has additional option components.The reserve-price case includes an exchange option, a range-option component, and a short binary-call component.Source: §3.2 · pp. 10–11
  • The auctioneer’s expected revenue is represented as an option over two assets distributed as bidders’ equilibrium bids.Revenue is calculated ex ante over the distribution of the item’s value.Source: §3.2 · p. 11

3.3. Extension: Uncertain Timing Advantages

  • When Bob may fail to act with probability α and L = 0, Alice earns positive equilibrium profit proportional to α; Bob benefits from her additional bid shading, and seller revenue is lower even when Bob acts.The equilibrium structure remains similar to the certain-timing case, with Alice’s mixed strategy formed by drawing a value and bidding according to β.Source: §3.3 · pp. 11–12

4. CONTINUOUS-TIME SETTING AND COMPARATIVE STATICS

In the Black-Scholes setting, Bob bids after Alice observes the asset price, giving Bob a value tied to the price when he bids. As Bob’s latency advantage grows, his payoff rises while auctioneer revenue falls.

4. CONTINUOUS-TIME SETTING AND COMPARATIVE STATICS

  • The model assumes the token price follows geometric Brownian motion, consistent with standard Black-Scholes assumptions.The price has volatility σ, and B_t is standard Brownian motion.Source: §4 · p. 12
  • Alice bids at time 0, Bob at T, and settlement occurs later at τ; under risk neutrality and a zero risk-free rate, value at time t is p_t.Thus the auction value when Bob bids is p_T.Source: §4 · p. 12
  • As Bob’s latency advantage T increases, his payoff grows while the auctioneer’s revenue decreases.With p0 normalized to 1 and no reserve price, total surplus stays constant, making the auctioneer’s revenue plus Bob’s payoff a constant-sum game.Source: §4 · p. 13, Figure 1 · p. 13

5. VALUE OF TIMING GAMES

The section compares the value of waiting in competitive and monopolist settings under the Black-Scholes model. In the no-reserve case, competitive timing pressure exceeds the monopolist’s; with a reserve price, the pressures appear to converge as the reserve approaches the initial price.

5. VALUE OF TIMING GAMES

  • The section compares timing incentives in the competitive auction model with Bob bidding alone as a monopolist.In the latter setting, Bob bids L when v > L and otherwise bids zero.Source: §5 · pp. 13–14
  • In the monopolist setting, Bob’s expected profit is the value of a call option on v with strike price L.When L = 0, the monopolist has zero incremental value from delay because the zero-strike option is always exercised and has only intrinsic value.Source: §5 · p. 14, §5.1 · pp. 14–15

5.1. Assuming L = 0.

  • For L = 0, Bob’s competitive-auction profit is represented by an exchange option, and its timing sensitivity is measured by its time derivative, or theta.The section uses Margrabe’s formula under the Black-Scholes model.Source: §5.1 · p. 14
  • In the no-reserve competitive auction, increasing Bob’s timing advantage increases his informational advantage and the value he can extract.The incremental value of a delay Δt is of order Δt, whereas the monopolist has zero incremental value from delay.Source: §5.1 · pp. 14–15

5.2. Assuming L > 0.

  • With prices normalized so p0 = 1, Corollary 3 gives Bob’s profit, and the next passage gives its sensitivity with respect to timing advantage T; neither excerpt includes the formula.Source: §5.2 · pp. 15–16
  • With L > 0, last-mover timing pressure appears higher than the monopolist’s, but the two converge as L approaches p0 = 1.The monopolist’s timing pressure is the theta of a call option with strike price L.Source: §5.2 · p. 16

A.1. Proof of Lemma 3

The proof sets up an inverse bidding function for Bob and bounds the bids of both players using Alice’s payoff considerations.

A.1. Proof of Lemma 3

  • Assuming Bob’s bid βL(v) is non-decreasing, the proof defines its inverse as the lowest value v whose bid reaches b.The inverse is βL^-1(b) = inf{v : βL(v) ≥ b}.Source: §A.1 · p. 18
  • Alice never bids above E[v], so her bid set lies in [0, E[v]] and Bob’s bid is bounded above by E[v].Source: §A.1 · p. 18
  • For Alice bids above the reserve L, the proof next examines her payoff.Source: §A.1 · p. 18

A.2. Proof of Lemma 4

The proof first handles the zero-reserve case by constructing Alice’s distribution to support Bob’s bid rule, then extends the best-response argument to a positive reserve.

A.2. Proof of Lemma 4

  • The proof first treats L = 0 and writes Bob’s bid β0(v) in integral form.Source: §A.2 · p. 19
  • To support Bob’s conjectured bidding rule, Alice’s strategy is represented by H(v) = Fa(β0(v)), with density h(v) = fa(β0(v))β′0(v).The proof implicitly solves for H(v).Source: §A.2 · p. 19
  • After substitution and simplification, the resulting H(v) is exactly the distribution from Alice’s conjectured strategy.Source: §A.2 · p. 20
  • For L > 0, Alice can use the same strategy as in the no-limit-price case, and Bob’s best response follows by comparing whether the no-limit best response is above or below L.Source: §A.2 · p. 20

A.3. Proof of Theorem 2

The proof expresses Bob’s expected profit through his probability of winning, then simplifies it using the equilibrium distribution and integration by parts.

A.3. Proof of Theorem 2

  • The proof applies integration by parts to simplify the expression for Bob’s expected profit.Source: §A.3 · p. 20
  • Bob’s winning probability is H(v)=F_a(β₀(v)), which the proof substitutes using H(v)=F_v(v) and β₀(v) from (9).The proof identifies H(v) as the probability Bob wins conditional on observing value v.Source: §A.3 · p. 20
  • It uses that F_v(·)^2 is the CDF of the maximum of two independent draws from F_v.Source: §A.3 · p. 21

A.4. Proof of Theorem 3

The proof rewrites Bob’s expected profit using his conditional probability of winning, then simplifies its first term by integration by parts.

A.4. Proof of Theorem 3

  • The proof represents Bob’s expected profit using the conditional winning probability and the equilibrium bid function.Source: §A.4 · p. 21
  • Bob’s probability of winning given value v is F_a(β_L(v)); the proof then substitutes H(v)=F_v(v) and β₀(v) from (9).Source: §A.4 · p. 21
  • Integration by parts simplifies the first term in the profit expression.Source: §A.4 · p. 21

0 Fv(x) dx

As the reserve price L approaches zero, the reserve-related terms vanish, leaving the first term evaluated at v=0, the exchange-option payoff described in Theorem 2.

0 Fv(x) dx

  • As L approaches zero, the latter terms vanish, leaving the first term evaluated at v=0 as the payoff of an exchange option in Theorem 2.Source: §0 · p. 21
  • The proof applies integration by parts to the first term before combining and cancelling terms.Source: §0 · p. 22
  • The proof rewrites the second term and then uses the definition of v in the resulting expression.Source: §0 · p. 22

A.5. Proof of Theorem 5

Except for a degenerate zero-bid case, Alice has no pure-strategy equilibrium; the proof characterizes mixed-strategy support and checks Bob’s response.

A.5. Proof of Theorem 5

  • There is no pure-strategy equilibrium except the degenerate case where Alice bids zero with probability one.For a positive pure bid, Bob can match when profitable; Alice then loses when he acts in time, making a zero bid strictly better.Source: §A.5 · p. 22
  • In any mixed-strategy equilibrium, Alice’s lowest bid in her support is zero, and her payoff there is αE[v].That payoff must equal her payoff from every bid in her support.Source: §A.5 · pp. 22–23
  • The proof derives a cutoff v̄_α such that Bob bids zero for all v≤v̄_α.Source: §A.5 · p. 23
  • Alice’s strategy is represented by a distribution H(v)=F_a(β(v)), with density h(v)=f_a(β(v))β′(v).Source: §A.5 · pp. 23–24
  • The derived H(v) matches the distribution implied by Alice’s conjectured strategy, and the proof then checks Bob’s optimality.Source: §A.5 · pp. 23–24

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