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The Geography of (Concurrent) Block Proposing

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

arXiv:2610.18812v1cs.GTecon.TH

TL;DR

Geographic concentration can leave blockchain agents exposed to shared disruptions and censorship pressure. The paper analyzes proposer locations and auction competition, finding that competition rewards exclusive information and changes location incentives.

  • Problem

    Geographic concentration can expose otherwise independent blockchain agents to shared disruptions and censorship pressure, motivating study of proposer-location incentives.

    Source: §1 · pp. 1, 3

  • Method

    The paper analyzes a location game in which proposers receive delayed histories from dispersed sources and compete over information.

    Source: §1 · pp. 2–3, §2.1 · p. 4

  • Results

    Competition makes the auction outcome more informative about value than an optimally located monopolist, although equilibrium locations need not maximize informativeness.

    Source: §1 · p. 3

  • Takeaways & Limitations

    Because profits reward information received before a rival, two proposers locate at different sources when there are two sources, regardless of their weights.

    Source: §1 · pp. 2–3

    The analysis does not establish existence of a general M-bidder auction equilibrium or a pure location equilibrium for that general game.

    Source: §5 · p. 15

Abstract

from arXiv · show

The resilience and censorship resistance of a blockchain depend on where the validators and proposers of that blockchain physically operate. We study the economic incentives behind this location choice, both with a single proposer and with multiple concurrent proposers. Formally, we study a game where proposers first choose locations and then compete in a first-price auction for an opportunity whose common value is driven by information from geographically dispersed sources. A single proposer minimizes variance-weighted delay, locating at a weighted median of the sources and co-locating with any dominant one. By contrast, with two proposers, information both observe equally early earns no rent, and each proposer's expected profit is proportional to the standard deviation of the information it receives before its rival. We show that the location game admits the jointly unobserved variance as an ordinal potential, so global solutions of a weighted two-median problem are equilibria. With two sources, one proposer locates at each, however weak one is. With more proposers and at least as many sources, equilibrium locations are distinct under regular bidding. Multiple concurrent proposers thus do more than limit any single proposer's control over inclusion: by rewarding exclusive rather than shared information, they create an economic force toward geographic decentralization.

1. Introduction

The paper studies how economic incentives shape proposer locations, with geographic dispersion relevant to blockchain resilience. It models competition for information rents and situates that approach among work on geographic incentives, proposer competition, and market latency.

1. Introduction

  • With competing proposers, information received equally early creates no informational advantage, and expected profit tracks the standard deviation of information received before a rival.A proposer may profit by moving toward a less important source if that creates an exclusive informational lead.Source: §1 · p. 2
  • A single proposer chooses a variance-weighted median, while two-proposer equilibria relate to a weighted two-median problem; with two sources, proposers split across them regardless of their weights.The two-proposer location game uses jointly unobserved variance as an ordinal potential.Source: §1 · p. 3
  • With at least as many sources as proposers, regular bidding yields pairwise distinct locations, though proposers may be close; existence of such equilibria remains open.Location-dependent deadlines act as effective delays, while a single fixed auctioneer makes co-location an equilibrium.Source: §1 · p. 3
  • Competition makes auction outcomes more informative about value than an optimally located monopolist, but equilibrium locations need not be the most informative.The paper evaluates how an outside observer can infer terminal value from auction outcomes.Source: §1 · p. 3

1.1. Related literature

  • Unlike Yang et al. ’s latency-calibrated simulations and Babel et al. [2]’s transaction-market-share incentives, this paper derives location incentives from auction rents on dispersed information.It complements reduced-form modeling with an explicit information-and-auction model; it abstracts from consensus guarantees and implementation [1] [6] [7].Source: §1.1 · p. 3
  • The model treats location relative to multiple information sources and competitors as the choice variable, with propagation technology fixed, unlike latency-investment studies.Its location game uses exact and ordinal potentials, and global two-median solutions are equilibria, though coordinatewise minima may also be equilibria.Source: §1.1 · p. 4

2. Model

The model links proposers’ locations to delayed information from geographically dispersed sources and to competition for a common-value opportunity. It specifies a two-stage location-and-auction game and its interpretation for concurrent proposers.

2.1. Sources, locations, and information

  • Each of M proposers chooses a location in a compact metric space, where each of K sources’ information arrives after a continuous, bounded delay.In distance examples, delay is the metric distance from the source to the location, measured in time.Source: §2.1 · p. 4

2.2. The auction and the location game

  • Proposers choose locations simultaneously, then bid using their delayed information in a first-price auction for a right with common terminal value.Location payoffs are expected auction profits under a selected continuation equilibrium; the baseline is an unrestricted no-reserve auction.Source: §2.2 · p. 5

2.3. Interpretation

  • The bidders represent proposers working with searchers or builders to pursue a financial opportunity, while the model abstracts from consensus, collusion, and other block-building tasks.The opportunity’s monetary value may depend on an external reference price; validators enter through the bidding deadline.Source: §2.3 · pp. 5–6

3. One proposer

A single proposer chooses a location that minimizes variance-weighted delay, balancing the timing of information from the sources. In the Euclidean two-source case, the variance weights determine whether the optimum is a source or an interval between them.

3. One proposer

  • More information raises the dispersion of the posterior mean without changing its expectation, benefiting a proposer that can abstain after an unfavorable signal.In the Brownian model, expected profit is strictly increasing in observed variance, which yields the delay-minimization criterion.Source: §3 · p. 6
  • The monopolist’s optimal locations are exactly those minimizing variance-weighted delays.With Euclidean delays, the minimizers are variance-weighted geometric medians; on a feasible interval they are weighted medians.Source: §3 · p. 6
  • For two sources on a Euclidean segment, unequal weights place the optimum at the higher-variance source, while equal weights make every feasible point between the sources optimal.More generally, a source whose variance weight is at least the sum of the others’ is an optimal co-location point, but need not be the unique optimum.Source: §3 · p. 6

4. Two proposers

This section examines how information observed earlier than a rival shapes rents and location incentives with two proposers, and how a finite reserve affects expected profits and location payoffs.

4.1. A general auction equilibrium

  • With independent signals and a coordinatewise nondecreasing common-value aggregator, the quantile bidding construction gives each proposer a continuous, strictly increasing bid and a 1/2 chance of winning.The construction permits asymmetric signal distributions but depends substantively on independence; correlated signals need not satisfy the equilibrium result.Source: §4 · p. 6, §4.1 · pp. 7–8

4.2. Additive values and information rents

  • For additive values, each proposer’s expected rent depends only on its own signal distribution and equals half the expected absolute difference between two independent draws from it.The quantile equilibrium and this rent formula do not require normality.Source: §4.2 · p. 8

4.3. The Brownian signal decomposition

  • Proposer i’s ex ante profit is s_i/√π; a larger exclusive variance raises rent but not winning probability, which remains 1/2 at nondegenerate profiles.Source: §4.3 · p. 9

4.4. The location game

  • In the two-proposer no-reserve location game, −Ω0 is an ordinal potential; pure location equilibria are exactly coordinatewise minimizers of Ω0, and a pure location equilibrium exists.Source: §4.4 · pp. 10–11
  • With two sources at opposite ends of a line and positive weights, the unique pure location equilibrium places one proposer at each source, however weak either source is.More generally, co-location is an equilibrium only at a site minimizing delay to every source; for distance delays and distinct source sites, that condition fails.Source: §4.4 · p. 11
  • For sources ordered along a line, a minimizing split into two consecutive nonempty groups yields an equilibrium by placing proposers at weighted medians of the groups.The split minimizes the sum of the groups’ one-facility costs.Source: §4.4 · pp. 11–12

4.5. A finite reserve

  • With a finite reserve, the weak proposer abstains below qR while the strong proposer may pool at the reserve; in the Gaussian model the reserve atom is positive exactly when sH > sW.The general reserve result requires a two-active-bidder condition; with positive exclusive variances, that condition holds for every finite reserve in the additive Gaussian model.Source: §4.5 · pp. 12–13
  • Proposition 5 characterizes expected profits in the reserve equilibrium conditional on C = c.The supplied passage does not include the formulas themselves.Source: §4.5 · p. 13
  • As L−c → −∞, bids at every fixed interior quantile converge to (15), and profits converge to s_i/√π.Source: §4.5 · p. 13
  • With a fixed nominal reserve, location payoffs average reserve profits over C ∼ N(V0, Γ), and both exclusive variances and common-information distribution matter.Indexing the reserve as L(C) = C + ℓ removes the need for this averaging, but presumes the seller can observe or implement C.Source: §4.5 · p. 14
  • Reserve profits need not depend only on a proposer’s own exclusive variance, so the potential proof does not extend and no general two-median characterization is available.The reserve location problem is specified by (26); payoffs are also defined when an exclusive variance is zero.Source: §4.5 · p. 14

5. More than two proposers

With more than two proposers, the paper rules out duplicated locations under stated source and bidding conditions, using the profit difference between exclusive and shared information. The result does not establish that such equilibria exist in general.

5. More than two proposers

  • The argument against duplication combines positive profits from exclusive information with zero profits from duplicated information, making replacement of duplication by an exclusive increment a profitable deviation.For K ≥ M, a co-located group leaves a feasible source site unoccupied, which supplies the deviation in Theorem 5.Source: §5 · pp. 14–15
  • A proposer with positive variance from information unknown to every rival can earn positive profit against any finite rival bids, with or without a reserve.This argument does not require an explicit M-bidder equilibrium or allocation rule.Source: §5 · p. 14
  • Under regular pure bidding continuations, members of a co-located group earn zero expected profit because a member could otherwise overbid tied partners.Regularity requires deterministic bids, finite conditional expected payoffs, and no atom in the conditional highest outsider bid at the group's highest bid.Source: §5 · pp. 14–15
  • When K ≥ M, every pure location equilibrium with regular pure auction continuations has pairwise distinct proposer locations, provided delays are distances to distinct positive-weight feasible source sites.Distinct locations need not be far apart, and the existence of a general M-bidder auction equilibrium or pure location equilibrium remains unestablished.Source: §5 · p. 15
  • The distinct-location conclusion has boundaries: with one source and M ≥ 3, all proposers can co-locate there in equilibrium, and mixed bidding with unbounded subsidies is outside Lemma 2.The pairwise theorem also does not establish equilibrium existence for the general game.Source: §5 · p. 15

6. Location-dependent delays and deadlines

The paper incorporates location-dependent message delivery into effective source delays and applies its location results to that adjusted information structure. It also shows how a fixed auctioneer can make co-location sustainable.

6. Location-dependent delays and deadlines

  • Replacing propagation delays with effective delays that include location-dependent delivery time preserves the information structure used for the location results, when effective delays are at most T.The effective source delay accounts for the need to deliver a bid or message before the common deadline.Source: §6 · p. 16, §6.1 · p. 16

6.1. An attester-dependent deadline

  • An attester-dependent deadline uses the earliest delivery time that reaches fraction γ of total attester weight, and its delivery-time function is continuous under metric communication times.The two-proposer no-reserve potential and monopolist location results apply to the effective-delay formulation; a constant delivery delay leaves two-proposer no-reserve location incentives unchanged.Source: §6.1 · p. 16

6.2. A fixed auctioneer

  • With a fixed auctioneer at z, bidders minimize weighted information delay by locating at z, and every profile (z, y) or (y, z) is a two-proposer no-reserve equilibrium.Co-location at z is sustainable but not forced: against a rival at z, a proposer cannot gain an information advantage elsewhere.Source: §6.2 · p. 17
  • An auctioneer that commits to a location before proposers choose theirs cannot guarantee co-location revenue V0 without an equilibrium-selection argument.With proposer locations fixed, revenue depends on the auctioneer's location through the difference between the proposers' delivery delays.Source: §6.2 · p. 17

7. Price informativeness

The paper studies how much an outside observer can infer about terminal value from auction outcomes when the observer has delayed information. It compares monopoly and competition benchmarks and characterizes information-optimal locations for two sources.

7. Price informativeness

  • The observer's error about terminal value accounts for how information affects bids, rather than treating the winning bid itself as an unbiased estimate.The observer sees the auction outcome and has a delayed signal; the analysis assumes participants are better informed than the observer.Source: §7 · p. 17, §7.1 · p. 17

7.1. The observer’s information and the monopoly benchmark

  • For a monopolist with positive post-public innovation variance, the information-optimal reserve is L = cpub, and the information-optimal location minimizes D.The benchmark is αHW + (1 − α)D∗, where D∗ is the minimum of D over locations.Source: §7.1 · p. 18

7.2. The information in a winning bid

  • The winning bid reveals a combination of common information and the winner's standardized signal, while winner identity is independent of the maximum standardized signal and each proposer wins with probability 1/2.The observer also learns that the loser's signal fell below the winner's quantile.Source: §7.2 · p. 18

7.3. Competition versus the monopolist benchmark

  • At pure two-proposer location equilibria, competition's observer-error comparison with the information-optimal monopoly benchmark is strict whenever some feasible location receives information newer than the public signal with positive variance.The supplied passage states strictness under this condition but does not specify the direction of the comparison.Source: §7.3 · p. 19

7.4. Information-optimal locations

  • With two sources, the minimum mean-square error across locations is the lower of co-location at the stronger source, with error m, and opposite endpoints, with error Eend.The location equilibrium separates proposers even with lopsided weights, so that equilibrium geography need not be information-optimal.Source: §7.4 · p. 20

A. The regular monotone auction equilibrium

Under differentiability, monotonicity, and positive-density assumptions, the appendix establishes uniqueness of the regular monotone auction equilibrium using first-order conditions and a lower-tail rent condition. It also illustrates the bidding formula with a symmetric product example.

A. The regular monotone auction equilibrium

  • The uniqueness argument assumes continuously differentiable, strictly increasing bid functions with positive derivatives on a common open support and positive signal densities.These conditions apply to the class for which uniqueness is asserted, not to the existence proof of Theorem 2.Source: A. The regular monotone auction… · p. 22
  • The first-order conditions imply proportional inverse quantiles, which coincide at the common upper support endpoint; zero limiting rent at the lowest quantile then fixes the additive constant at zero.Without the lower-tail condition, unrestricted negative bids allow a construction that adds the same positive amount to every interim profit.Source: A. The regular monotone auction… · p. 22
  • For independent uniform signals with V = XAXB, β(q) = q^2/3, each proposer's ex ante rent is 1/24, and expected revenue is 1/6.The formula averages along matched quantiles rather than over the square of independently truncated signals.Source: A. The regular monotone auction… · p. 22

B.1. Proof of Theorem 4

The proof verifies the equilibrium by comparing deviations across the reserve, pooling region, and increasing bid branch. It also handles zero-variance information profiles with explicit reserve continuations.

B.1. Proof of Theorem 4

  • Types above qR prefer their own bid on the increasing branch, while types below qR prefer its limiting boundary bid.The branch bid is strictly increasing, starts at L, and has a finite upper endpoint.Source: §B.1 · p. 23
  • Lower W types abstain, and active W types do not prefer a deviation to the reserve bid L.The cutoff type’s limiting profit from just above L is zero; the reserve deviation is no better.Source: §B.1 · p. 23
  • The equilibrium yields no sale with probability pR qR and a sale at L with probability (qR − pR)qR.The no-sale event requires both proposers’ quantiles to fall below their respective cutoffs; a reserve sale requires H to be pooled and W inactive.Source: §B.1 · p. 24

B.2. Normal integrals and the zero-variance cases

  • When sW = 0, the reserve continuation has the uninformed proposer abstain; when both exclusive variances vanish, both abstain below L and bid c at or above L.For sH = s > 0 and c > L, H abstains below u, bids L from u to zR, then follows the increasing branch; W randomizes over abstention and branch bids.Source: §B.2 · p. 24
  • The zero-variance continuation strategies are actual mixed equilibria, not merely limits of formulas with positive sW.When c ≤ L and only H is informed, H bids L if c + XH ≥ L and otherwise abstains, while W abstains.Source: §B.2 · p. 24

C. Choosing the auctioneer’s location

With fixed proposer locations, auctioneer location affects no-reserve revenue through a scalar differential latency. A maximizing latency can be selected from the feasible endpoints and breakpoints.

C. Choosing the auctioneer’s location

  • With fixed proposer locations, no-reserve revenue depends on auctioneer location only through h = ha(z).This fixed-location calculation is a different game from one where proposers choose locations after the auctioneer commits; guaranteeing revenue V0 then requires an equilibrium-selection argument.Source: C. Choosing the auctioneer’s location · p. 25
  • A revenue-maximizing differential latency lies among the feasible interval’s endpoints and breakpoints −∆k.Between consecutive breakpoints, revenue is convex because the exclusive variances are nonnegative affine functions and their square roots are concave.Source: C. Choosing the auctioneer’s location · p. 25

D. A co-located equilibrium with one source and many proposers

With one source and multiple proposers, the specified continuations support a co-located equilibrium, so no-co-location conclusions require assumptions on sources and information advantages. The binary-threshold analysis identifies a unique optimal threshold at zero.

D. A co-located equilibrium with one source and many proposers

  • When exactly one proposer is informed, the n = M − 1 uninformed proposers can randomize quantiles so their maximum is uniform, matching the informed proposer’s opponent bid distribution.At each uninformed proposer’s support bid, expected value conditional on defeating the informed proposer equals the bid, yielding zero expected profit.Source: D. A co-located equilibrium with one… · p. 25
  • With one source and M ≥ 3 proposers, all proposers locating at 0 is a Nash equilibrium in which everyone earns zero.A unilateral move to 1 still leaves at least two informed rivals at 0, so the deviator earns zero.Source: D. A co-located equilibrium with one… · p. 25

E.1. The optimal binary threshold

  • The binary threshold uniquely maximizes explained variance at z = 0.The proof’s inequality is strict for z ≠ 0.Source: §E.1 · p. 26

E.2. The universal normal quantities

The paper derives universal normal quantities and bounds for its information criterion, then gives an integral representation for computing that criterion without simulating bidding strategies.

E.2. The universal normal quantities

  • The passage reports E[r(M2)] = 2√π, E[r(M2)^2] = 2 − κ, and Var(r(M2)) = 2 − κ − 4, without a /π term in the variance.Source: §E.2 · p. 26
  • J(η) starts at zero, is nondecreasing in η, and remains strictly below Var(r(M2)) at every finite η.At η = 0, −λ(M2) determines M2; increasing η adds independent normal noise to the observation.Source: §E.2 · pp. 26–27
  • A positive covariance establishes Var(E[r(M2) | Y]) > 0, proving the strict bound in (36).The positivity argument uses an independent copy and the product of corresponding differences.Source: §E.2 · pp. 26–27

E.3. An integral representation for computation

  • The integral representation evaluates the information criterion without simulating bidding strategies.Its denominator is positive for every finite y, and the formula is used together with J(0) = 0.Source: §E.3 · p. 27
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