Auctions with Frictions: Recruitment, Entry, and Limited Commitment

This paper develops an auction model that jointly incorporates three frictions pervading informal price-formation processes: (1) costly recruitment by the seller, (2) costly participation by bidders, and (3) the seller’s inability to commit to a recruitment level or reserve price. The authors argue these frictions are especially prevalent in markets for idiosyncratic assets such as mergers and acquisitions, real estate, and home repair contracting, where auction houses like Christie’s and Sotheby’s command fees of 20–30% of revenues precisely because they reduce the underlying inefficiencies.

The model features a single seller who exerts recruitment effort gamma at cost gamma*s, generating a Poisson-distributed number of contacted bidders with mean gamma. Each contacted bidder independently decides whether to pay entry cost c > 0 to learn their private value and participate in a first-price auction (FPA). The seller cannot commit to gamma (which is unobservable to bidders) or to a reserve price. Two scenarios are analyzed: PO (participation-observable, where bidders observe the number of entrants before bidding) and PU (participation-unobservable).

The central tension is between the seller’s incentive to recruit more bidders to intensify competition and raise revenue, and bidders’ rational concern that excessive recruitment makes entry unprofitable. Because the seller cannot commit, this tension generates several novel inefficiency results.

In the PO scenario, the seller’s marginal revenue from recruitment Ro’(lambda) is single-peaked, meaning there is a minimum profitable participation scale lambda_o below which the seller will never recruit. Combined with a maximum participation level lambda-bar_c above which bidders will not enter (defined by U(lambda-bar_c) = c, where U is the bidder’s expected payoff), no-trade equilibrium is the unique outcome whenever lambda-bar_c < lambda_o — even for arbitrarily small recruitment cost s. This result holds because with unobservable effort, bidders correctly anticipate the seller will target participation above lambda_o, making entry unprofitable. When lambda-bar_c > lambda_o, three regimes arise: (i) no trade if s exceeds a threshold s-bar_o; (ii) an interior equilibrium with full entry (q* = 1) and lambda* = lambda_o(s) for intermediate s; and (iii) for small s, an equilibrium with lambda* = lambda-bar_c and partial entry q* = Ro’(lambda-bar_c)/s < 1. In regime (iii), total recruitment cost lambda*(s/q*) equals the constant lambda-bar_c * Ro’(lambda-bar_c) regardless of s — so even as s approaches zero, wasteful recruitment costs do not vanish, because they are determined by incentive constraints rather than by technology.

In the PU scenario, a no-trade equilibrium always exists for all parameter values, because the seller cannot credibly disclose participation, creating self-reinforcing expectations of zero competition. The seller’s recruitment incentive xi(lambda) is strictly weaker than Ro’(lambda) in the PO scenario (proven via revenue equivalence: Ro’(lambda) = xi(lambda) + a positive term reflecting how greater participation induces more aggressive bidding). This yields ranking reversals: for intermediate s and small c, the PO scenario dominates PU; but for small s or large c, the PU scenario’s weaker recruitment incentive reduces wasteful over-recruitment, making PU preferable. These comparisons translate directly to a comparison of FPA and SPA with unobservable participation: the two formats are not equivalent in the presence of recruitment and entry frictions because they generate different recruitment incentives.

A sampling-curse mechanism drives near-complete market unraveling when sellers have privately known recruitment costs drawn from a continuous uniform distribution on [0, s_o]. Because low-cost sellers recruit more, a contacted bidder believes the seller is more likely to have low costs — and hence to have recruited many other bidders — making entry unprofitable. Proposition 3 establishes a threshold c-hat such that for c in (c-hat, c-bar), as the lower bound of the cost distribution approaches zero, the fraction of seller types that remain inactive approaches one — near-complete unraveling — even though each type would be active if its cost were commonly known.

Q: What is the paper’s main modeling innovation relative to the existing literature? A: The paper’s central novelty is combining all three frictions — costly recruitment by the seller, costly participation by bidders, and limited seller commitment — in one model. The existing literature had studied entry and recruitment separately; Szech (2011) examined costly recruitment with costless entry; McAfee and McMillan (1987) and Levin and Smith (1994) studied costly entry with an exogenously given number of potential bidders; Milgrom (1987) and McAfee and Vincent (1997) studied limited commitment to a reserve price with a fixed bidder set. None combine all three.

Q: What is the “minimum profitable scale” result and why does it arise? A: Because the seller cannot commit to a reserve price, the first few bidders are complementary — they stimulate competitive bidding, causing the seller’s marginal revenue Ro’(lambda) to be initially increasing, then decreasing (single-peaked). This means the seller’s profit Pi_o(lambda, q) is maximized either at zero or at a participation level above a minimum scale lambda_o, defined by Ro’(lambda_o) = s-bar_o. The seller will never choose a participation level between 0 and lambda_o.

Q: Under what conditions does the market completely shut down in the PO scenario? A: No-trade is the unique equilibrium outcome whenever lambda-bar_c < lambda_o, where lambda-bar_c is defined by U(lambda-bar_c) = c (the participation break-even level) and lambda_o is the seller’s minimum profitable scale. This condition arises when entry costs c are large enough relative to the competitive dynamics. Importantly, no trade occurs for every recruitment cost s > 0, including arbitrarily small s — commitment failure alone can cause complete market breakdown even when recruiting bidders is nearly costless.

Q: What is the inefficiency in regime (iii) of Proposition 2 (small s, PO scenario)? A: When s < Ro’(lambda-bar_c), equilibrium has lambda* = lambda-bar_c and q* = Ro’(lambda-bar_c)/s < 1. The total recruitment cost is lambda* * (s/q*) = lambda-bar_c * Ro’(lambda-bar_c), a strictly positive constant independent of s. As s approaches zero, total recruitment effort and its cost do not vanish — they are pinned by incentive constraints. This waste could be avoided if the seller could commit to an effort level below lambda-bar_c, illustrating that commitment failure creates persistent inefficiency even when the technology of recruitment is inexpensive.

Q: Why does a no-trade equilibrium always exist in the PU scenario but not always in the PO scenario? A: In the PU scenario, if bidders expect zero participation, they bid zero conditional on being contacted; the seller then has no incentive to recruit, validating the expectation. This equilibrium is self-sustaining for all parameter values (Claim 2). In the PO scenario, the equilibrium refinement (requiring that off-path beliefs not support negative seller payoff at lambda = 0 when trade equilibria exist) rules out no-trade equilibria when lambda-bar_c > lambda_o and s is not too large; specifically, Proposition 2 shows that no-trade equilibrium is unique only when s > s-bar_o or lambda-bar_c < lambda_o.

Q: What drives the ranking reversal between PO and PU scenarios? A: The core result is Claim 3: Ro’(lambda) > xi(lambda) for all lambda > 0, meaning the marginal incentive to recruit is strictly stronger under PO than PU. This follows from revenue equivalence: Ro’(lambda) = xi(lambda) + (d/d lambda-hat) Ru(lambda, beta_{lambda-hat})|_{lambda-hat=lambda}, and the second term is strictly positive because greater expected participation induces more aggressive bidding. For intermediate s and small c, stronger PO recruitment incentives support higher participation and revenue. For small s or large c, those same stronger incentives generate wasteful over-recruitment in PO, making PU preferable.

Q: How does the paper connect its PO/PU comparison to a comparison of first- and second-price auctions? A: In any standard auction where the highest-value bidder wins, payoff and revenue equivalence imply that the bidder payoff function U(lambda) and seller revenue Ro(lambda) are identical. In particular, the dominant-strategy equilibrium of the SPA (where bidders bid their true values regardless of participation) generates the same outcomes as the PO equilibrium, because with truthful bidding the observability of participation is irrelevant. Therefore, comparing PO and PU with an FPA is equivalent to comparing the SPA and FPA with unobservable participation. The two formats are not revenue-equivalent when recruitment and entry frictions are present: their ranking depends on s and c in exactly the way described for PO vs. PU.

Q: What is the “sampling curse” and how does it cause market unraveling? A: The sampling curse arises when sellers have privately known recruitment costs. Because a lower-cost seller optimally recruits more bidders, the probability of any given bidder being contacted is higher when the seller has a lower cost. Conditional on being contacted, a bidder therefore believes the seller more likely has a low cost and thus has recruited many competitors, reducing the value of entry. In the binary-type case (Claim 8), if sL is sufficiently small relative to sH, the low-cost seller must recruit so many bidders that entry becomes unattractive; the resulting low q* makes the marginal recruitment cost sH/q* prohibitively high for the high-cost type, driving it out (lambda*_H = 0).

Q: What does Proposition 3 establish about near-complete unraveling with a continuum of seller types? A: With seller costs uniformly distributed on [s-bar, s_o], Proposition 3 establishes a threshold c-hat strictly between 0 and c-bar such that: (i) for c in (c-hat, c-bar), as s-bar approaches zero, the fraction of seller types with zero recruitment approaches one — near-complete market unraveling; (ii) for c < c-hat, all seller types remain active regardless of how small s-bar is. This is striking because for any commonly known s in (0, s_o), the PO scenario supports trade for all c < c-bar; unraveling arises purely from the interaction of private cost information and the sampling curse, not from any type’s cost being intrinsically too high.

Q: What does the welfare analysis say about equilibrium efficiency? A: The welfare-maximizing participation level lambda_w satisfies U(lambda_w) = c + s (equating the marginal bidder’s surplus to the full social cost of one more participant), with full entry q_w = 1. In equilibrium under PO, q* < 1 in some cases (wasted recruitment) and lambda* differs from lambda_w for almost all (s, c) pairs — both excessive participation (lambda* > lambda_w) and deficient participation (lambda* < lambda_w) can arise. Full efficiency requires Ro’(lambda*) = s and U(lambda*) = s + c simultaneously, but since both U and Ro’ are independent of s and c as parameters, these equalities generically fail.

Q: Does the seller benefit from being able to commit to recruitment effort? A: Claim 10 shows that with observable effort in the PO scenario, the seller commits to gamma-hat = min{lambda-bar_c, lambda_o(s)} when lambda-bar_c >= lambda_o, and to lambda-bar_c (if profitable) when lambda-bar_c < lambda_o. Commitment strictly improves the seller’s profit whenever gamma-hat = lambda-bar_c: it enables positive trade when lambda-bar_c < lambda_o and Ro(lambda-bar_c) > lambda-bar_c * s (otherwise impossible without commitment), and it saves recruitment costs when lambda-bar_c > lambda_o and Ro’(lambda-bar_c) > s. However, the commitment outcome is always welfare-inefficient: lambda-bar_c > lambda_w whenever s > 0.

Q: What anecdotal evidence do the authors cite for the model’s relevance? A: Subramanian (2010) and Boone and Mulherin (2004, 2009) show that the majority of merger and acquisition auctions are “informal” — mixtures of auctions and negotiations rather than structured processes with rules laid out in advance — and that sellers are typically unable to credibly commit to participation levels. Milgrom (2003) states from consulting experience that marketing an auction is often more critical than clever mechanism design. Fees of 20–30% of revenues paid to intermediaries like Christie’s and Sotheby’s are offered as quantitative evidence of the magnitude of the inefficiencies that such intermediaries reduce. Home repair contracting is cited as a familiar informal-auction setting where both recruitment and entry costs are material.

Recruitment effort (gamma): The seller’s costly action of contacting potential bidders, modeled as a Poisson process with mean gamma at cost gamma*s; unobservable to bidders in the baseline model.

Participation-observable (PO) vs. participation-unobservable (PU) scenarios: The two variants of the model; in PO, bidders observe the total number of entrants n before bidding; in PU, they do not observe n and the seller cannot credibly disclose it.

Minimum profitable scale (lambda_o): The smallest positive participation level the seller will ever choose in equilibrium, defined as the value where Ro’(lambda_o) equals the peak of the average revenue curve s-bar_o. The seller always recruits either zero bidders or at least lambda_o, due to the initial complementarity of bidders (they stimulate each other’s bids) under no-commitment-to-reserve-price.

Break-even participation level (lambda-bar_c): The maximum participation level at which a bidder’s expected gross payoff U(lambda) equals the entry cost c; bidders will not enter if they expect participation above lambda-bar_c.

Sampling curse: The adverse-selection mechanism arising when sellers have privately known recruitment costs: because low-cost sellers recruit more, a contacted bidder infers the seller is more likely to have a low cost and thus to have recruited many competitors, making entry less attractive and potentially driving higher-cost seller types out of the market.

xi(lambda): The seller’s marginal revenue with respect to recruitment in the PU scenario, defined as the total derivative of Ru(lambda, beta_{lambda-hat}) evaluated where actual and expected participation coincide (lambda-hat = lambda). Strictly less than Ro’(lambda) for all lambda > 0, reflecting that in PU the seller loses the ability to leverage bidder aggression via observable competition.

Wasteful recruitment: The equilibrium phenomenon in which total recruitment cost lambda*(s/q*) remains at the positive constant lambda-bar_c * Ro’(lambda-bar_c) even as s approaches zero, because incentive constraints — not technology — pin the equilibrium effort level.