Deadline Credibility, Protracted Trades andMarket Efficiency*
Masahiro Yoshida†
December 11, 2023

Abstract
Many real-world negotiations are chronically protracted until deadlines, but deadlines
are costly in generating separations. Must all deadlines be perfectly credible in one-onone market trades? To refine the institutional role of deadlines, I theoretically propose amechanism of an imperfectly credible pre-deadline to facilitate the agreement. Employing
a seller–buyer dynamic bargaining model with a deadline, I analytically show a possibility
that the complementary pre-deadline elicits earlier agreements without triggering separations and, consequently, enhances the market efficiency. Under a well-designed threat ofseparation, the seller is tempted to discount a price as intertemporal price discriminationand the buyer is more likely to compromise right before the pre-deadline, as the pricing
resembles an ultimatum. The results of a laboratory experiment broadly support the mechanism’s efficacy.
JEL Classification: C78, C91
Keywords: bargaining, one-sided incomplete information, deadline effect, durable goods
monopoly, market efficiency, laboratory experiment* I have benefited from discussions with Patrick Dejarnette, Yukihiko Funaki, Yoshio Kamijo, Kohei Kawa-

mura, and participants at Micro Workshop at Waseda. I thank Ruiyi Dong and Xianglong Kong for their research
assistance. I gratefully acknowledge the financial support from the Department of Economics at Washington University at St. Louis. The author declares that he has no relevant financial interests that relate to the research
described in this paper.
† Department of Political Science and Economics, Waseda University, Japan;
e-mail: m.yoshida@waseda.jp

1

1

Introduction

Many negotiations are inexorably delayed under asymmetric information. One classical institution used for deterring endless lag is a deadline. For example, civil and criminal pretrial
disputes entail a regulatory deadline to file a lawsuit. Consequently, 93% of medical malpractice disputes are settled by pretrial bargains (Sieg (2000)) and 94% of federal criminal cases
are resolved by plea bargains1 (Fisher (2003)) without formal lawsuits. In labor contract negotiations, holdouts (or on-the-job bargaining after the contract expiration)2 are often bounded
by a deadline of a union’s strike or an employer’ lockout, after which much bleaker disputes
may begin. In international finance, many sovereign debt renegotiations close just before theexpiration of debt repayment, which often implies de facto defaults of the country.3
Similar to Parkinson’s Law (Parkinson (1957)),4 real-world negotiations chronically delay
to deadlines. Field evidence (for plea bargaining examples, see Spier (1992); for civil litigations, see Williams (1983)) and laboratory experiments (Roth, Murnighan and Schoumaker(1988)) reported the prevalence of last-minute agreements before deadlines as deadline effects.5
At first glance, deadlines have virtues of enforcing agreements within a time limit. By contrast,
however, the same deadlines turn fatal if agreements are missed. If pretrial bargainings in civil
and criminal litigations passes its deadline, formal trials with expenses for lawyers on both sides
may occur for years. For labor contract negotiations, 12% of disputes end in strikes and lockouts (Cramton and Tracy (1992)), generating substantive social harms (See e.g., Lee and Mas(2012)). When the default happens under sovereign debt disputes, post-default renegotiations
are notoriously delayed—typically for more than 7 years—and the creditors ultimately lose,
on average, 40% of the nominal debt burden (Benjamin and Wright (2009)). Across a wide
range of negotiations in the field and laboratory settings, Roth (1995) summarized consistencyof disagreement rates at approximately 15-45%.
1 A plea bargain is a process for agreement of the prosecutor and the defendant pleading guilty to a crime

without a trial in exchange for a reduction of sentence.
2 Using data of labor contract disputes during 1970-1989, Cramton and Tracy (1992) documented that holdouts
are the most common form of disputes, running for approximately two months.
3 In 2015, Greece was faced with the maturity of debt from creditors (i.e.; the European Central Bank (ECB),
the European Commission (EC) and the IMF), July 20, as a hard deadline, which de facto would induce a defaultof the country. The negotiation narrowly closed eight days before the deadline on July 12.
4 Under the organizational context of bureaucracy, Parkinson’s Law argues that volume of tasks in one projectis dictated by its deadline.
5 Roth and Ockenfels (2002) report last-minute agreements from the last minute biddings at eBay and Amazon
Internet auctions.

2Motivated by the substantial variety of eleventh-hour agreements at the cost of separations
from hard deadlines, I sought to explore the possibility of a refined negotiation mechanism—asoft deadline—to improve the deadline institutions under a market of one-on-one trades. An
implementation of this soft deadline is illustrated as follows. Suppose a hard deadline is set up
at the start of the negotiation, either agreed upon or externally imposed. In parallel, suppose thata mechanism designer sets an earlier complementary soft deadline on a specific intermediateday. This complementary deadline serves as a stochastic “time bomb”: if the agreement is notreached on the day, the bargaining pairs might break up with a probability of α ∈ (0, 1), but
continue with a probability of 1 − α (See Figure 1). Instead of the terminal hard deadline, I
propose that the imperfect enforcement of an earlier, non-fatally soft deadline may be a catalyst
to nudge both parties into making earlier progress towards deadlines.6
While admittedly speculative, the soft deadline system might operate as a discipline tocomplement hard deadline in various inefficient trades above.7 In civil litigations, a private or
public mediator (e.g., insurance firms or courts) may intervene sometime earlier than the deadline as a preliminary injunction. Before the labor contract disputes, each party could formulate
an ex-ante negotiation rule that an intermediator (e.g. stakeholders) could potentially clears up
the bargaining at an earlier negotiation. For sovereign debt negotiations, a group of creditors
and a debtor may join a commitment in which a third party (e.g. the International Court ofJustice) may intervene in negotiations. Particularly in minor litigation cases, recent progress in
artificial intelligence (AI) may allow some models or algorithms to serve as ”robot judges” thatsuggest enforceable mediation plans, potentially simulated from historical precedents.8
To formalize the above concept, I begin with a seller–buyer bargaining model with onesided, incomplete information under a deadline (a la Fudenberg, Levine and Tirole (1985);
Sobel and Takahashi (1983)).9 Consider an uninformed proposer selling a durable good with
6 Perhaps more intuitively, outside the bargaining contexts in academics, for example, soft deadlines are

adopted as an incentive system to induce consistent efforts before hard deadlines. For Ph.D. dissertation, a midterm proposal is imposed as a soft deadline before a final defense. For college students, a midterm exam before a
final exam serves as a soft deadline to help complete the curriculum.
7 In FOA (final offer arbitration) in legal proceedings, each party submits a proposal to an arbitrator and thearbitrator selects one of the two proposals. The soft deadline mechanism may serve as an intermediate possibilityof FOA mediation.
8 A.I. is already implemented for litigation settlement in Estonia (Niler (2019)) and China (Pillai (2019)). For
recent theoretical and ethical discussions on “robot judges,” see Sourdin (2018) and Casey and Niblett (2020).
9 This bargaining framework with a deadline has been widely applicable to real world bargaining scenarios,
such as Tracy (1987), Hart (1989), Cramton and Tracy (1992) for labor disputes, Bebchuk (1984) and Silveira(2017) for plea bargaining, and Bai and Zhang (2012) for sovereign debt negotiations.

3buyers (”she”) with private values within N periods. A marginal cost is normalized to 0, anda private value ranges from 0 to 1. The marginal cost and the distribution of the values arecommon knowledge. The proposer can update a price in every period, and a buyer can react byagreement or rejection. The trade continues until the buyer accepts the price; when the deadline
arrives, both fall back to outside options, which are also 0.
In equilibrium, instead of committing to a static monopolist price, the price declines overtime and a delay occurs as a screening of private information: lower-type buyers delay agreements and lowest-end buyers reject all offers, ending in separation. In this setting, the market
efficiency could be restored through both parties’ reactions. Spoiled by multiple rounds of pricerevisions, a seller is tempted to make a compromise as intertemporal price discrimination, leading to earlier agreements; in contrast, the buyer compromises right before the deadline, as thepricing resembles an ultimatum.
Then, suppose that a market designer imposes a soft deadline characterized by a probabilityof α at an earlier period n∗ < N (below, I call this the threat period; see Figure 1). If the buyer
at period n∗ rejects the offer, they are forced to leave the table by a probability of α, and seekoutside options of 0.10 Theoretically, an imposition of the time bomb would stimulate the two

Figure 1: An imperfectly credible deadline under a deadline
Note: An intermediate deadline with a conditional separation probability α ∈ (0, 1) is imposed at the end of periodn∗ in contrast to the final deadline at period N. The negotiation might end at the end of n∗ with a probability α
but continue with probability 1 − α. In my full model, I allow for multiple threat periods for generality. For alaboratory experiment, by contrast, I use n∗ = 3 and N = 6.

forces above. Intuitively, for the buyer’s side, the buyer is more likely to make a compromise
at the soft deadline due to the opportunity loss by potential separation. Consistent with the
10 If α = 0, this is nothing beyond an original setting. If α = 1, this becomes shorter-horizon bargaining with adeadline on N = n∗ .

4experimental literature on deadline effect (Roth, Murnighan and Schoumaker (1988)), I showthat agreements disproportionately occur at the soft deadline reallocated from the hard deadline.
One could view that this buyer’s compromise as a stochastic analog of the deadline effect.
For the seller, the compromise is non-trivial. Suppose the seller knows that the buyer provides a compromise at the soft deadline. In that case, the seller has a larger incentive to exploit
the buyer, with the offer resembling a quasi-ultimatum. However, I show that a range of credibility α exists where the soft deadline facilitates the price discount. This stems from thebuyer’s deadline effect at the soft deadline; he is bound to compromise because rejection at thesoft deadline signals that the opponent’s private value is not decently high enough (see Figure
2, right). Because he foresees this intertemporal price discrimination after the threat period,a forward-looking seller would shift down the pre-threat price schedule with a lower opening
price. Alternatively put, soft deadline serves as a signaling device for low-type buyers to induce
price discrimination from sellers. Notably, in the conventional deadline framework, this pricediscounting channel is absent.
Suppose that a mechanism designer optimizes the market efficiency, defined as the sum ofboth parties’ ex-ante payoffs of both parties. Given the two channels activated, does a soft
deadline improve the market efficiency? When both parties were sufficiently patient, I findthat an optimal threat α ∗ ∈ (0, 1), which is characterized by bargaining primitives, achieves
the highest ex-ante efficiency. This finding reflects a straightforward trade-off between shorter
delays until agreements and a rising expected loss. If α is small, the risk relatively facilitates
the compromise of both parties as an effective deterrence, as described above. However, if α
is large, the game resembles a short-sighted ultimatum game. The expected separation costfrom the rise of monopoly power dominates the efficiency benefit of risk-induced compromise.
A parameterized model shows that a non-empty interval of imperfect credibility α ∈ (0, 1)
enhances the market efficiency without excessively fueling separations.
My theoretical finding, founded on mathematically equivalent modeling of multi-buyer
markets, revisits an accepted wisdom regarding bargaining horizon and market efficiency basedon the famous Coase (1972) conjecture in the durable goods monopoly. Coase contended that
one-sided asymmetric information without a deadline leads to an immediate agreement favorable to the informed party (i.e., buyers). As the bargaining horizon extends (or as a good
becomes more durable), the efficiency is retrieved. This well-known phenomenon is framed5

as “durability attenuates monopolistic distortion”.11 Two polar cases succinctly characterize
this intuition. In a one-period ultimatum game,12 bargaining undergoes the heaviest efficiency
loss under the maximized monopolist’s power. In contrast, under an infinite-horizon scenario
given unlimited pricing periods, the monopolistic market is closer to Pareto-efficient with animmediate agreement as Coase conjectured. If one generalize the deadlines’ time structure, thesoft deadline breaks the conventional link: the model suggests that the horizon appears shorter,
but the efficiency may be enhanced in expectation.
As an initial step to obtain a proof of concept, I ran a controlled laboratory experiment toempirically test the validity of the soft deadline. Elaborating on the experimental literature on aseller–buyer multi-period bargaining experiment (Rapoport, Erev and Zwick (1995); Reynolds(2000); Cason and Reynolds (2005)), I implemented a simplified version of my model in acomputer laboratory. From this experiment, I obtained approximately 1,200 pieces of trade
data from 62 subjects. The subjects were randomly chosen to engage in a bilateral bargaining
game (N = 6) under various threat levels predetermined on the middle threat period (n∗ = 3).
The experiment broadly evidences a soft deadline benefit in terms of efficiency. Expectedly, I found a highly robust agreement agglomeration on the soft deadline. As the soft deadline became harder, the magnitude of agreements would increase. Consequently, delays until
agreements shrunk on average. Intriguingly, when the soft deadline became highly credible, theefficiency gain and the buyer’s advantage were more pronounced despite a suggestive evidenceof rising separations and contrary to the model predicting efficiency losses and disadvantages
to buyers.
To discern the cause of the stronger efficacy for relatively higher credibility, I detected the
behavioral deviations of subjects from the model. I contrasted the offered prices and decisionsof buyers with theoretically-predicted ones. Under the hard deadline regime (α = 0), most(89%) of buyers’ reactions and few (12%) of the sellers’ pricing decisions were reasonably
justified. Notably, 59% of prices were framed as “demanding”—sometimes beyond the static
ultimatum price. This pricing has been canonically reported and reconciled by the prior experimental literature without soft deadlines. As a soft deadline hardened, I found that sellers drop
the mean prices on the threat period more saliently than the model—novel to this literature
11 See, for example, Güth and Ritzberger (1998). See Section 2.4 for greater detail.
12 My model with N = 1 can be framed as a variant of an ultimatum offer game under incomplete information.

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but reminiscent of an extremely well-established history of ultimatum game experiments13 .
Consequently, a significant fraction of seller’s pricing becomes theoretically reasonable, evencooperative, especially during the threat period. Founded on the discussion of potential behavioral forces at play (See Section 4) in contrast to the experimental literature, I conclude that a
relatively higher deadline credibility remedied the upward-biased pricing of sellers and fueled
the earlier agreements of buyers; consequently, it enhanced market efficiency.
Related literature: This paper proposes a mechanism of negotiations to complement theconventional deadline. Theoretically, it intersects with several streams of bargaining theories.
First, it extends deadline effects in bargaining models by enriching a deadline structure. The
finite-horizon bargaining model was formally built by Sobel and Takahashi (1983) and Fudenberg and Tirole (1983), although the role of deadlines on the market efficiency was not
explicitly featured.14
Regarding pretrial civil litigations, Spier (1992) uses a similar one-sided incomplete-information
model to mine, deriving an agglomeration of trade at the deadline of the trial date.15 More recently, Fuchs and Skrzypacz (2013) theoretically explored the impact of outside options after
breakups on deadline effects in the continuous time limit. Relative to Fuchs and Skrzypacz(2013), my model normalizes the outside options and features the efficiency implication of the
proposed pre-deadline mechanism in discrete periods, which is directly tested in the laboratory.
Using a reputation model by Abreu and Gul (2000), Fanning (2016) provided a foundation ofdeadline effects from reputation across a wide range of protocols. However, the one-sided,
incomplete information in his model delivers no delay, making the bargaining efficient, which
is not appropriate for the workforce model in my project.16
Second, my soft-deadline design is isomorphic to the random separation of bargaining (inalternating-offers game, see Rubinstein and Wolinsky (1985); in a decentralized market, see
Binmore, Rubinstein and Wolinsky (1986)). Models in most of the early literature had no dead13 On average, ultimatum game experiments show a proposer offers between 30% and 50% of the money, andmore than half of the opponents reject the proposal with his share under 20% (see Camerer (2003)).
14 Some theoretical works on deadlines explore the effect of strategic use of deadlines (Ma and Manove (1993);
Fershtman and Seidmann (1993); Özyurt (2023)) In contrast, my paper seeks for improvement of the institutionalrole of deadlines, which is only set by a market designer.
15 For different protocols with two-sided incomplete information, Ponsati (1995) and Damiano, Li and Suen(2012) derived an atom of the trade at the hard deadline in concession games.
16 See the “Related Literature” section in Fanning (2016) for a comparison with one-sided, incomplete information models (e.g., Spier (1992); Fuchs and Skrzypacz (2013)).

7

lines under complete information and, again, generated no delay (e.g., Muthoo (1999)). Morerecently, several studies (Fuchs and Skrzypacz (2010); Fanning (2016); Simsek and Yildiz(2016)) developed bargaining models with random breakdowns, where the timing of a deadline
is uncertain and where separation can stochastically occur any time. By contrast, my model
presumes a specific event (or events) of potential separation that a mechanism designer presets. Theoretically, this soft deadline signals low-type buyers to induce price discriminationfrom sellers, thereby enhancing market efficiency.
Third, and most substantially, the paper theoretically revisits the conventional wisdom of
the Coase conjecture (Coase (1972)) in the durable goods monopolist with a market of a continuum of buyers (from Stokey (1981); Bulow (1982), later formalized by Gul, Sonnenschein andWilson (1986), Ausubel and Deneckere (1992) and Thépot (1998)).17 The conjecture’s essencelies in the proposition that “durability hurts the monopoly power,” suggesting a positive association between the time horizon (interpretable as durability) and market efficiency in the monopolistic market.18 More recently, under the different, but mathematically equivalent protocolof repeated ultimatum games with private information on fairness, Fanning and Kloosterman(2022), supported by experimental evidence, predicted the Coaseian outcomes of almost immediate agreements on an equal split, conditional on sufficient players’ patience. Compared witha snapshot ultimatum case, the efficiency is again, significantly larger in the infinite horizon. As
a marked advancement in the literature literature, the soft deadline breaks this link: seemingly
shorter bargaining periods deliver higher ex-ante market efficiency (see Proposition 2). Reframing the durability of goods with players’ patience (or less bargaining friction19 ), the Coase
conjecture also implies that players’ higher patience shrinks the monopolistic distortion.20 As
varying deadline credibility corresponds with adjusting players’ patience, my primary result
may also shed new light on the proposition.
17 Departure from the conjecture is also a deep theoretical theme (e.g.; Bagnoli, Salant and Swierzbinski (1989);

Fuchs and Skrzypacz (2010); Board and Pycia (2014)). Most of these papers adopt infinite horizon without
deadlines. Once a deadline is imposed, the monopolist power is restored.
18 See Sobel and Takahashi (1983), Theorem 6, or Güth and Ritzberger (1998). Bond and Samuelson (1984)
also showed that if the good depreciates faster, the Coase conjecture can be evaded such that the monopolistic
power revives.
19 In the bargaining model, patience is often viewed as bargaining friction (e.g., the setup cost for meetings orcost of crafting a proposal and making a decision).
20 Related to this statement, Stokey (1981) showed that shorter intervals between offers (∆ → 0) harm the monopolistic power with more frequent offer revision. Observe that as discounting factor is typically formulated byδ = exp(−γ∆) (γ is an instantaneous discount rate), a limit case (∆ → 0) implies a higher patience (δ → 1). Güthand Ritzberger (1998) showed that sufficiently high patience is a requisite for intertemporal price discriminationin the Coase conjecture.

8

Employing experimental data, this paper relates to two strands of experimental literature on
bilateral bargaining. First, the paper builds on instances of experimental bargaining with deadlines in parallel to bargaining models with deadline effects. In a laboratory experiment, Roth,
Murnighan and Schoumaker (1988) documented highly robust deadline effects across different
bargaining protocols; about half of all agreements were reached within the final 30 seconds.21
Güth, Levati and Maciejovsky (2005)’s work further supports this robustness across protocols
of time horizons, decreasing pies and alternating roles. Overall, experimental deadline effects
appear to be universally established across bargaining models. My experiment, which generalizes a deadline structure, bolsters the view that deadline effect emerges even if a deadline issoft; as a soft deadline becomes more credible, more trades close on the threat period.
Second, this paper borrows from, extends, and complements the experimental literature on
seller–buyer durable goods trades with one-sided, incomplete information. Largely, the sellers’pricing in the experiments was inconsistently higher than predictions from rational selfishplayer benchmark models.22 Scholars in the literature have attempted to reconcile puzzling
price decisions (especially, the level and comparative statics of opening price) via behavioral
forces. Below, I contrast my paper with its closest precedents.
Rapoport, Erev and Zwick (1995) ran infinite horizon experiments of durable goods tradeswith changing time discounting. They found that the higher the discount factor, the higher the
average opening prices, inconsistent with the infinite-horizon model. In contrast, my experiments confirm this relationship of prices across all periods (See Table 3), consistent with afinite-horizon model. In addition, Rapoport, Erev and Zwick (1995) reported mean opening
prices that were far above the static ultimatum price and that buyers accepted prices higher
than the model, which was also observed in my study. Bounded rationality and the fairness ofbuyers were proposed as primary explanations.
Reynolds (2000) ran finite-horizon experiments similar to mine under a hard deadline (α =0) with one (bargaining) or five buyers (market), given a constant discount factor. Regarding
one buyer bargaining regime, opening prices were higher when the time horizon became longer,from one, two, to six periods, contrary to the model. My experiments showed that as a soft
21 Roth, Murnighan and Schoumaker (1988) state that “deadline effect appears to be quite robust, in that the

distribution of agreements over time appears to be much less sensitive to experimental manipulations than is the
distribution of the terms of agreement.”
22 A notable exception is Güth, Kröger and Normann (2004), who found theoretically consistent price paths in
two-period bargaining under the privately known patience.

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deadline “hardens” (or the expected horizon “shortens”), the opening prices shrink, as partiallyrationalized by my model (Lemma 1). However, aligned with Reynolds (2000), who compared
three (α = 1) and six (α = 0) periods, I find that a shorter horizon induces a lower opening
price in the pre-threat period (n ∈ {1, 2, 3}). Relative to potential behavioral factors (e.g.,risk aversion, fairness, prior beliefs), Reynolds (2000) argued that bounded rationality is apromising candidate to rationalize pricing.
Cason and Reynolds (2005) ran finite-horizon experiments with the closest formulation to
my soft deadline protocol. In my model’s language, their model is a special case, with two
periods (N = 2), one threat period (D = 1), four treatments of separation probability (α ∈{0%, 10%, 40%, 70%}), perfect patience (δ = 1) and ten restricted grids of pricing and two
types of value (v ∈ {0.18, 0.54}).23 Lemma 1 in my paper formalizes their numeric examplesof perfect Bayesian equilibrium paths. In contrast to my experiments, Cason and Reynolds(2005) reported that opening prices were not significantly responsive to separation probability.
Based on discussions of Reynolds (2000), the central interest of this prior study was in building
behavioral models with bounded rationality to reconcile the deviations from the model.
My results under a hard deadline inherit the similar pricing behavior already observed in theliterature—particularly that the mean opening price is generally higher than the prediction andeven higher than static monopolistic pricing. As a value-added finding, my experiment documents unreported behavioral regularities in the line of the literature above, which evokes, however, well-established ultimatum game experiment results (for a survey, see Camerer (2003)):
as a soft deadline gets more credible, the seller’s systematically demanding pricing is adjusted
to a reasonable or even cooperative level (see Section 4). One may view that the soft deadline
rectified the seller’s systematic demanding bias reported in the previous studies.
Layout: The paper is organized as follows: Section 2 presents a soft deadline bargaining
framework under a hard deadline and characterizes the unique equilibrium. Then, I explore theoverall market efficiency and distributional outcome’s sensitivity with higher deadline credibility. Comparative statics of the ex-ante separation probability and delay until agreement areprovided. Next, Section 3 presents the design and findings from the laboratory experiments.
I compare the actions of both players with theoretical ones and provide behavioral interpreta23 Due to their very short periods (N = 2), their model is not designed to capture the deadline effects, which areobserved in a distribution of agreements across periods.

10tions to reconcile the gap. Section 4 concludes the paper. The Appendix provides proofs oftheoretical results and delivers operational details and auxiliary analyses of experiments.

2

Bargaining with Imperfectly Credible Deadlines

To formalize the proposed mechanism, a natural candidate for a benchmark model should contain a deadline and generate an endogenous delay.24 Below, I enrich a screening-type seller–buyer bargaining model under a hard deadline (Sobel and Takahashi (1983); Fudenberg, Levine
and Tirole (1985))25 by embedding intermediate soft deadlines.

2.1

Setup

A seller (“he”) sells a durable good with a buyer (“she”) with unknown private values. The
market has no supply shortage; each good has a commonly known zero marginal cost. Eachinfinitesimal buyer has her private value v ∈ [0, 1] for the good.26 I assume that v is distributed
according to the shared cumulative distribution function F(v) = vσ (σ > 0).27 Suppose thatboth are rational and risk-neutral.
Time is measured by discrete and finite periods with n = 1, 2, 3 · · · and a length ∆ > 0 foreach bargaining round. Suppose that the seller credibly imposes a deadline at period N.28 At
the beginning of the period n, the seller proposes an offer Pn . Then, the buyer immediately
accepts or rejects. If she accepts the price at the end of period n, the game ends up with an
outcome: the seller gets δ n−1 Pn , and the buyer gets δ n−1 (v − Pn ), where δ ∈ [0, 1] is a periodic
discount factor. If the buyer keeps rejecting the price until n = N, she and the seller both receive
0 as an outside option.
24 In the context of strikes in labor disputes, the classic Hicks Paradox (Hicks (1963)) features a puzzle thatrational parties cannot reach a non-Pareto optimal outcome in a bargaining model under complete information.
Embedding asymmetric information is a standard solution.
25 The model is isomorphic to a finite-horizon protocol of durable goods monopolist model of a product market
filled with a sequence of infinitesimal buyers (Stokey (1981); Bulow (1982)).
26 A “no-gap” protocol is presumed such that marginal cost is no lower than the lower bound of the buyer’s
private value.
27 This distributional assumption is taken due to analytical convenience (See Ausubel and Deneckere (1992);
Fuchs and Skrzypacz (2013)).
28 In the durable goods monopolist context, the monopolist supposedly has larger reputational concerns thatencourage commitment to a hard deadline. As a micro-foundation of one-sided offer protocol, Ausubel and
Deneckere (1992) demonstrated the Silence Theorem under alternating-offer games with one-sided incomplete
information: the informed party endogenously never made any serious offers when ∆ was sufficiently short; only
the uninformed party made offers.

11

Suppose that a market designer embeds a series of soft deadlines at an earlier periodn∗d ∈ [1, N) (d ∈ [1, · · · , D]), which I call a threat period.29 (For simplicity, the subscript d isomitted throughout the paper, especially when only one threat period is imposed, i.e., D = 1).
The threat’s credibility is captured by a conditional separation risk αd ∈ (0, 1) at the end ofeach threat period n∗d . (See Figure 1) This implies that if a proposal is rejected at period n∗d ,bargaining ends with probability α, and both receive outside options, but proceeds to periodn∗d + 1 with probability 1 − α.
The seller’s problem is recursively defined as follows:max Γ(Pn )Pn + {1 − Γ(Pn )}ηn δVn+1
Pn

Where Γ(Pn ) is the probability of the agreement at period n if the seller offers Pn . Vn is hisvalue function at period n; ηn is a risk-adjustment factor attached to a discount factor δ suchthat ηn = 1 − αd (n = n∗d ) and ηn = 1 (n 6= n∗d ). The buyer’s problem is to choose the acceptanceperiod nb given the seller’s price schedule of {Pn } such thatnb = arg max Πηn δ n−1 (v − Pn ).
n

2.2

Equilibrium

Both players share the posterior of private value as a belief system. Let [0, Kn ) be a posteriorvaluation at period n, and both players know Kn at period n as an upper bound of private value
v. Given Kn and Pn , the buyer calculates the cutoff value Cn , satisfying:Cn − Pn| {z }

payoff of agreement today=ηn δ (Cn − Pn+1 )
|
{z
}

(1)

payoff of agreement tomorrow

Intuitively, (1) implies that a marginal buyer with a value v = Cn is indifferent between buying
today and tomorrow. Then, at period n, the buyer’s cutoff strategy is to accept Pn if v ≥ Cn andF(Kn ) − F(Cn )rejects if v < Cn . Therefore, Γ(Pn ) =(∀n ∈ {1, · · · , N}) holds. Immediately
F(Kn )
from the buyer’s cutoff strategy, the belief system of the posterior valuation is characterized by
the buyer’s cutoff such that
29 This setting corresponds to the general formulation of a random separation (e.g., Binmore, Rubinstein and

Wolinsky (1986)).

12

Cn = Kn+1 (∀n ∈ {1, · · · , N − 1}),

(2)

suggesting that cutoff at period n serves as a posterior at period n + 1, constituting a belief
system. Then, employing {Kn } and {(Pn ,Cn )}, I introduced a standard perfect Bayesian equilibrium (for theoretical foundations, see Sobel and Takahashi (1983); Fudenberg, Levine andTirole (1985)).
Definition 1. A pair of strategies {(Pn ,Cn )} and a belief system {Kn } constitutes a perfect
Bayesian equilibrium of the game if their actions maximize their expected payoffs at all information sets (sequential rationality) and if a belief system is consistent with the Bayes rule
whenever possible (consistency).

2.3

Equilibrium Paths

The model is solved via backward induction from the hard deadline. The action schedules{(Pn ,Cn )} of players are periodically determined by a pair of their bargaining powers, captured
by sequences {(An , Bn )} as follows.
Proposition 1. [Equilibrium paths and bargaining powers]
Given the state variable {Kn } at period n, the equilibrium path of {(Pn ,Cn )} (n ∈ {1, · · · , N})
is sequentially characterized as

Pn = An Kn and Cn = Bn Pn(3)

where the following difference equations recursively characterize {An } and {Bn }:

−1



An = ((σ + 1) − σ ηn δ An+1 Bn ) σ /BnBn = {1 − ηn δ (1 − An+1 )}−1





AN = (1 + σ ) −1
σ ; BN = 1(n ∈ {1, · · · , N − 1})(n ∈ {1, · · · , N − 1}) ,(4)

The seller’s and the buyer’s respective value functions, Vn and Wn , are characterized asfollows by {An }:Vn = An Kn E(v), Wn = (1 −where E(v) =σ +2
An )Kn E(v)σ +1

σis an ex-ante expected private value.
σ +1
13

(5)

[Proof] See the Appendix.
Due to the model’s recursive structure, the seller and buyer chooses Pn and cutoffs Cn based
solely on a periodic posterior Kn , regardless of historical actions under the equilibrium. Furthermore, the analytical convenience of a functional form of F(v) = vσ yields that both Pn and
Cn are linear in Kn , combined with An and Bn , which are derived as functions of primitives
δ , σ , αd , n∗d , N (see the Appendix for the explicit formula).
Intuitively, An is monopolistic and Bn is the buyer’s bargaining power, such that higher
An , Bn raises prices and cutoffs, respectively. Analogous to prices and cutoffs, the value functions Vn ,Wn of sellers and buyers are also linear in the state variable Kn . V1 and W1 capture
the ex-ante surplus of both players, derived from the ex-ante maximum gains from the tradeσE(v) =. The overall market efficiency—of central interest in the paper—is characterσ +1
ized by the sum of the value functions at the initial period, V1 + W1 . Accordingly, given theequilibrium paths, how do both players behave?
Purchase schedule:

The buyer’s purchase decision is characterized by the cutoff Cn or theminimum value she is willing to accept given the price Pn . Figure 2 (left) illustrates simulated
paths of cutoffs under a parameterized model (N = 6, D = 1, n∗ = 3, δ = 0.98, σ = 1). Its results
show that buyers with private values higher than the cutoff curve are willing to purchase. The
buyer’s cutoff curve sharply drops not only at n = 6 (the canonical deadline effect) but also
at n = 3, when α > 0. The one at n∗ = 3 may represent a deadline effect at the soft deadline.
When the soft deadline gets hardens, the magnitude of compromise expands: a buyer of a given
private value is likelier to agree on the threat period.30
Price schedule: Considering the buyer’s compromise in the face of the soft deadline, theseller would presumably jack up the price in a strategic interaction. This inference appears
justified, considering that the soft deadline generates a quasi-ultimatum offer in the last period,
which raises the question of how the price schedule appears.
Using the same parameterized model, Figure 2 (right) documents the simulated price path.
Obviously, in contrast to committed monopolistic prices, the price schedule decreases overperiods. Saliently, the seller performs a conspicuous compromise just after the threat period
30 In the context of durable goods monopoly under product market of continuum value of buyers, this cutoff

drop could be interpreted as distinctly larger share of purchase.

14

Figure 2: Equilibrium dynamics across the levels of credibility (left: cutoff; right: price)
Note: The model is simulated under an experimental setting (n∗ = 3; N = 6; D = 1) and baseline parameters(δ = 0.98; σ = 1). See Figure 1.) A threat is zero if α = 0, small if α ∈ {0.05, 0.1, 0.2, 0.3}, medium if α ∈{0.4, 0.5, 0.6}, large if α ∈ {0.7, 0.8, 0.9} and complete if α = 1. I simulate theoretical average prices within thethreat category, weighted by the number of experimental observations of each environment. (See section 3.1 foran experimental setting.) The shades feature deadline effects for a buyer at n = 3, 6 (left), and a conspicuous saleat n = 4 (right).

n = 4 (especially when δ = 0.98). This big sale is novel to my soft deadline regime as adirect consequence of the deadline effect at the soft deadline stated above. As the seller knowsthat the buyer’s cutoff drops at n∗ = 3, he infers that the remaining buyer’s value at n = 4 issignificantly bounded from above (Recall that K4 = C3 per (2)). Due to deadline effects (shown
above), after-threat prices (P4 , P5 , P6 ) monotonically decline with higher credibility. Cateringto the diminished asymmetric information, the seller drops the price as an intertemporal pricediscrimination. In other words, the soft deadline is a signaling device of low private value,
once the offer at the threat period is rejected. Therefore, the forward-looking seller offers witha cheaper opening price, shifting the total price schedule downward. Compared to the no-threat
case, Figure 2 (right) shows that the seller shifted the opening price for small and medium
credibility. In fact, the simulation showed that for most in a range of imperfectly credible soft
deadlines (α ∈ (0, 0.789]), the seller7s first-half prices (P1 , P2 , P3 ) were lowered compared tothose in the hard deadline regime (α = 0). However, when a soft deadline is close to a hard one,the canonical strategic interaction dominates: The seller raises the price at the soft deadline.
This insight is formalized as follows.
Lemma 1. [The seller’s opening price] Suppose that the players are sufficiently patient. Then,15αd∗ ∈ (0, 1) uniquely minimizes an opening price P1 s.t.

αd∗ =

δ A2n∗ +1 − (1 − δ ){(1 + σ ) − (2 + σ )An∗d +1 }d

δ (1 − An∗d +1 )(1 + σ − An∗d +1 ).

(6)

where An∗d is recursively characterized by function of primitives An δ , σ , αd , n∗d and N for
d = 1, 2, · · · , D by (4).
[Sketch of the Proof ]First, one obtains the first-order condition (F.O.C.) as

E(v)
dP1
dA1
=
=−
dαd Recall P1 = A1 dαd(σ + 1)2

dAn∗d

dA1
dAn∗
| {z d}

dαd

= 0.

(7)

(> 0) Appendix

In Appendix, I show

dAn∗d
dαd

dA1
> 0. The F.O.C. (7) is reduced todAn∗d= δ (1 − An∗d +1 )(1 + σ − An∗d +1 )αd − δ A2n∗ +1 + (1 − δ ){(1 + σ ) − (2 + σ )An∗d +1 } = 0.
d

By solving for αd , one attains the desired αd∗ in (6) (the detailed derivation and the S.O.C.
are shown in the Appendix.) Then, I show that αd∗ ∈ (0, 1). Using (6), αd∗ > 0 requires asufficiently large δ > δ ∗ ; the threshold discount factor is given byδ∗ =(1 + σ ) − (2 + σ )An∗d +1(1 + σ − An∗d +1 )(1 − An∗d +1 )Suppose αd∗ ≥ 1. Then by (6), An∗d +1 ≥

.

(8)

1+σ
must hold. Accordingly, because An∗d +1 < AN =2+σ1+σ(∀σ > 0) holds, this is a contradiction. Therefore, αd∗ < 1. 
2+σAs the opening price captures the seller’s ex-ante bargaining power (recall thatP1 = A1 ), the1(1 + σ )− σ <theorem indicates a theoretical possibility that some range of imperfect credibility suppresses
the monopoly power. The non-linearity reflects two forces at work, shaping the monopoly
power. The first is intertemporal price discrimination during post-threat periods, as statedabove. If the seller foresees that he must discount a price if the threat is unrealized, he is
tempted to reduce the price at the outset. As the soft deadline hardens and the game resembles
an ultimatum, however, a conventional exploitive force dominates, leading to the monopolist

16

power being gradually restored. 31 A parameterized simulation in Figure 2 that combines
responses of opening prices (or pre-threat prices) P1 , P2 , P3 and monotonically declining postthreat prices P4 , P5 , P6 with credibility, suggests that pricing is discounted across periods formost of the deadline credibility.

2.4

Efficiency

The central theoretical question is how the overall market efficiency responds to the intensity ofthe deadline credibility. To determine this, I define the bargaining efficiency as the sum of theplayers’ ex-ante expected payoffs at n = 1 before the private value v is realized (U ≡ V1 +W1 ).32
As an immediate consequence of Lemma 1, the paper’s key theoretical finding of the paper isgiven below.
Proposition 2. [Efficiency gain from imperfect deadline credibility]
Suppose that the players are sufficiently patient such that δ ≥ δ ∗ , as defined by (8). Then,αd∗ ∈ (0, 1) uniquely maximizes the efficiency U, as well as level W1 , and share W1 /U of buyers’
expected surplus.
[Proof ] Using (5), one can see that the efficiency U and the level of buyers’ expected surplus
W1 are shown to strictly decrease in the opening price P1 such thatU ≡ V1 +W1 = (1 −

P1
)E(v)
σ +1

The share of the buyer’s expected surplus is W1 /U =W1 = (1 −σ +2P1 )E(v).
σ +1

(9)

(σ + 1) − (σ + 2)P1
. W1 /U is also strictly(σ + 1) − P1

decreasing in P1 becaused(W1 /U)= −(σ + 1)2 < 0
dA1
holds. The desired results immediately derive from the Proof of Lemma 1. 

31 When players are not sufficiently patient, however, price discrimination force is always dominated by an

exploitive force. This is because if the bargaining gets more frictional, sellers care less for the future market
and exploit the current market myopically, aligned with Güth and Ritzberger (1998), showing that the Coase
conjecture does not hold for low patience of players. Consequently, the opening price is monotonically increasing
with credibility.
32 In the durable goods monopoly with a demand pool of buyers, V ,W is interpreted as the ex-ante monopoly
11
surplus and collective consumer surplus, respectively.

17

The theorem states that given the specific threat period n∗d , the imperfect credibility of asoft deadline maximizes the overall bargaining efficiency and ex-ante buyers’ advantage. Thisimplies that the non-zero threat of separation could enhance the market efficiency by suppressing the monopoly power compared to the hard deadline regime. In Figure 3, a simulation of amodel parameterized model finds that the overall efficiency is maximized at an imperfect credibility α ∗ = 0.28, suggesting that efficiency is enhanced in the majority of credibility range(α ∈ (0, 0.789]) compared to the hard deadline regime (α = 0). Intuitively, the non-linearity
stems from a dynamic trade-off between the deterrence effect (efficiency gain by facilitatedcompromises) and the separation effect (efficiency loss by termination). When the efficiency
benefits from avoided separation and smaller discounting outweigh the direct separation loss,
imperfectly designed credibility restores a part of efficiency.33

Figure 3: The efficiency curve in regard to deadline credibility
Note: The efficiency is computed by V1 +W1 based on (9). A model was simulated based on the analytical formulawith a baseline parameter of experiments N = 6, σ = 1, δ = 0.98, and a single threat period set on n∗ = 3. The
vertical line is the optimal threat α ∗ = 0.28 and the upper-bound value of imperfect credibility to enhance theefficiency α = 0.789.

Although this study highlights the soft deadline’s role as a countermeasure for trade inefficiency, the model provides an intriguing distributional implication. As the model relates theoverall efficiency as a share and the level of buyers’ expected payoff, the non-linear implicationis also inherited to the buyer’s advantage as well.34
33 When players are not sufficiently patient, however, the efficiency is monotonically decreasing with the threat

because the threat does not serve as an effective deterrent if both parties care less for after-threat option value.
34 Alternatively, one may interpret that the sensitivity of buyer’s advantage is conversely shaped by the response

18

The theoretical results revisit the conventional wisdom on the link between a time horizon
specified by a hard deadline and market efficiency in the literature on durable goods monopolists. Alined with the Coase conjecture (Coase (1972)), in the unlimited horizon without the
backup of the deadline, the monopolist loses the bulk of the bargaining power. In my model,
this corresponds with the extreme case where, under an infinite horizon (N → ∞), U increases
to the maximum portion of the total gains from trade E(v).35
In the other extreme of a one-period ultimatum game (N = 1), the monopolist gains thestrongest bargaining power and minimizes efficiency. Note that when a buyer’s value is uniformly distributed (σ = 1), and perhaps surprisingly high, precisely half of the buyer cannot
purchase (as the price and cutoff is both 1/2). My model’s results showed that under a multistage game (1 < N < ∞), a seemingly shorter time horizon with imperfect credibility α ∈ (0, 1)
in the soft deadline would partially recover the market efficiency to facilitate compromises fromboth parties.

2.5

Efficiency loss

The preceding section showed that a well-designed threat of separation augments the efficiency.
Nevertheless, the outcome remains below the Pareto optimality. This section complements theefficiency analysis by examining how the sources of efficiency loss varied with credibility. The
source of trade inefficiency stemmed from a deadline interacted with by asymmetric information.36 Operationally, I formally defined the pair of efficiency loss from potential separationand frictional delay as follows.
Definition 2. [Ex-ante separation probability and delay until agreement]
The ex-ante separation probability for a buyer is defined byD

αd Cn∗1| {z }

first threat period (d=1)

+∑
d=2

|

!

D

(1 − αd 0 −1 ) αd Cn∗d
∏
0

d =2

{z

following threat periods (d≥2)!

D+ ∏ (1 − αd )CN .

(10)

d=1

} |

{z

terminal period}of monopolistic power in Lemma (1) under the non-cooperative bargaining.
35 Under δ = 0.98, σ = 1, and N → ∞, the efficiency U increases to 0.469, closest to the total potential gains
from trade E(v) = 0.5. The monopolist power A1 decreases to the lowest 0.124, in contrast to the static ultimatum
maximum of 0.5 (see the Appendix for a simulation when N → ∞).
36 Without asymmetric information and a terminal deadline, the monopolist perfectly price discriminates eachbuyer. In this case, the entire pie goes to the monopolist.

19

The ex-ante delay until agreement is defined byNE(bn) = ∑ nn=1n!
(Kn −Cn )| {z }

∏ ηl

l=1

.

(11)

ratio of buyers of agreement at n

Based on Definition 2, the sensitivity of these two sources of inefficiency with the credibility
is simulated, as shown in Figure 4.

Figure 4: Ex-ante separation probability and delay until agreement (simulation)
Note: (10) and (11) are simulated with N = 6, σ = 1, δ ∈ {0.7, 0.98}, and a single threat period set on n∗ = 3.
When D = 1, (10) is reduced to Cn∗ αn∗ + (1 − αn∗ )CN .

The parameterized model simulation showed that the ex-ante separation probability would
exhibit a non-linear sensitivity. Aligned with the non-linear sensitivity of market efficiency concerning α, as shown in Proposition 2, an appropriately designed separation risk would induce
compromised agreements to avoid separations, generating a lowered probability of separation.
Intriguingly, for most of the range α ∈ (0, 0.606], the separation probability is lower than under
the hard deadline regime (α = 0). However, if the soft deadline resembles a hard deadline, theseparation occurs beyond the role of deterrence.37
Another proxy to capture trade friction is the expected duration until reaching agreements,
as shown in Figure 4 (right). As delays were only defined in the samples reaching agreements,
an ex-ante bargaining duration until an agreements is expectedly decreasing in α, regardless of
discount factors, contributing to another efficiency gain from the threat. The response of proxypairs was tested in laboratory experiments, as discussed in the following section.
37 When players are not sufficiently patient, however, the separation probability is increasing with credibility,aligned with Proposition 2, because the threat cannot serve as a strong deterrent.

20

3

Laboratory Experiments

Building on the conventional deadline regime, this study’s model delivers a theoretical possibility: embedding an intermediate, imperfectly credible deadline may restore market efficiency.
To obtain a proof of concept of the proposed mechanism, I ran a simple laboratory experiment, following the prior experiments on multi-period durable good trades (Rapoport, Erevand Zwick (1995); Reynolds (2000); Cason and Reynolds (2005)).

3.1

Setup

Experiments were conducted over four days at the Missouri Social Science Experimental Laboratory (MISSEL). The laboratory is exclusively designed for computer experiments in social
science. Each desk was partitioned for privacy, and each participant was identified by their ID.
The program in research operations was written in z-Tree, a C++-based software package by
the University of Zurich (Fischbacher (2007)).38 All the games’ actions and outcomes were
aggregated in the central host computer.
In total, 62 subjects participated in the experiments. Before the recorded experiments eachday, subjects practiced unrecorded trades as sellers and buyers that would not affect their scores.
Operationally, I divided the subjects into two groups, with each taking turns as sellers or buyers. To exclude reputation formation or potential coordination with the same opponent, subjects
were randomly matched with a different subject across groups in every trade.39 Only individual payoffs earned during the day were exchanged for their monetary compensation under alinear conversion rate: 30 points corresponded to 1 US dollar. Each daily experiment tookapproximately two hours, and subjects got 29.6 dollars on average.40
Subjects played a simplified version of my model with one threat period (D = 1, n∗ = 3)
out of six periods (N = 6). An environment for each trade was characterized by its unique
set of three game primitives {α, σ , δ }. To identify the effect of deadline credibility, I let
the credibility α vary from α ∈ {0.1m, 0.05}, (m ∈ {0, 1, · · · , 10})41 within each session of a
38 I thank Xiaozhang Pan for writing an experimental computer program, Keh-Kuan Sun for his recruiting

support, and my Washington University in St. Louis (WUSTL) colleagues for operating experiments.
39 Although I cannot reject the possibility that some sellers form reputation of becoming a demanding seller, it
is difficult to imagine the reputation was formed within two hours. Each group has no collective incentive, andplayers were random matched under perfect anonymity split in each partitioned desk.
40 Instructions used in this study are available on request.
41 α = 0.05 is intended to examine the effect of a small positive credibility, guided by a simulation in Figure 3.

21

given (σ , δ ) ∈ {(1, 0.98), (2, 0.98), (1, 0.7)}. Seven to eight sessions of different (σ , δ ) wereoperated during the day.42 Table 1A tabulates the number of trades across the environments of
1,161 trade samples.
Before each trade, both parties were informed of their role (seller or buyer) and the environment. A private value for a buyer was drawn from the shape parameter σ ∈ {1, 2}, generatinga uniformly distributed or an upward-biased distribution of private value. The history of prices
was displayed to ensure participants’ perfect memory at the start of each period at n ≥ 2.43 To
help their decisions be as consistent as possible, subjects were encouraged to record all theiractions and results on paper each time they completed a trade.
Table 1B summarizes key descriptive statistics on prices, agreements and bargaining outcomes. Three points are worth noting alongside the model’s prediction.44 First, consistent
with the model, most offered prices declined over the trade periods; stubborn commitment to asingle price or raised pricing was minor.45 However, aligned with the prior experimental literature under hard deadlines, opening prices (mean 63.7) were higher than the model’s prediction(mean 44.3) and even higher than a simulated static monopoly price (mean 52.2).
Second, despite the higher opening prices P1 , an agreement ratio in the first period (23.6%
of pairs agree in the first period) was on average far larger than the model (14.5%), suggestingthat some buyers cooperatively accepted during the opening period.46 Deviations of players’
behaviors from theoretical predictions will be tested and discussed in Section 4.
Third, the aggregate share of the buyer’s surplus including separations,47 is 35.7%—systematicallylower than 50%, indicating a monopolist’s advantage in the game. This share is even lower thanthe model’s counterpart ex-ante surplus share for buyers (44.8%), consistent with the seemingly
cooperative purchase behavior in the initial period.
42 See the Appendix for a detailed schedule of sessions each day.
43 Trades in offered prices above private values, generating negative profits, are not allowed by design.
44 The simulated value in this page (opening price, agreement ratio, and the share of buyer’s surplus) is computed

along the formula (3)-(5), weighted with samples of each environment in the experiment.
45 Out of 2,407 pairwise prices (i.e.; P and Pnn+1 (n = 1, · · · , 5)), 2,050 (85.2%) discounted prices, 270 (11.2%)
kept pricing and 87 (3.6%) raised pricing.
46 Simulation suggests that an agreement ratio in the opening period ranges in approximately 12-15%, while theratio consistently exceeds 20% in the experiment.
47 This aggregate share of buyer’s surplus is calculated as the share of sum of all buyers’ surpluses in the sum of
trade efficiencies including separations. Table 1B documents an alternative proxy, computed as an averaged shareof a buyer’s surplus within trades reaching agreements (excluding separations) as a similar level of 34.8% ,while
its simulation counterpart is 44.4%.

22

Table 1: Descriptive statistics
Note: A deadline credibility is zero if α = 0, small if α ∈ {0.05, 0.1, 0.2, 0.3}, medium if α ∈ {0.4, 0.5, 0.6},
large if α ∈ {0.7, 0.8, 0.9} and complete if α = 1. A dash denotes a value unavailable by design (α = 1). Delay
until agreement is defined within samples reaching agreements. Separations consist of cases both at n = 3 andn = 6. Efficiency is the sum of the payoffs of both players. The buyer’s surplus share (separations excluded) isan averaged share of a buyer’s surplus within trades reaching agreements. The buyer’s surplus share (separations
included) is the share of the sum of all buyers’ surpluses in the sum of trade efficiencies, including separations.

23

3.2

Testing the efficiency benefit

Guided by theoretical insights from the model and having employed the laboratory data, I
empirically assessed testable statements of four effects of an imperfectly credible deadline, asdetailed in the subsections below.
Deadline effect: The direct channel to enhance the market efficiency is buyers’ compromises
in the face of earlier separation risk. As illustrated in Figure 2, the proposed mechanism reallocated the agreement agglomeration from a hard deadline to a soft one. Figure 5 (top) showsthat as the credibility increases from 0 to 1, the agreement ratio is sensitive at each deadline, as
measured by the number of pairs reaching an agreement out of all the pairs. One can see that asthe credibility rises, the agreement ratio at n = 3 and the one at n = 6 increases and decreasing,respectively, a finding roughly consistent with the model.

Figure 5: The sensitivity of the agreement ratio and the mean delay until agreement
Note: The top shows the sensitivity of the agreement ratio at soft (n = 3) vs. hard deadlines (n = 6) (black:experiment vs. red: model); An agreement ratio is the number of pairs reaching agreements divided by total
number of pairs reaching the n = 3 or n = 6, respectively. The bottom shows the sensitivity of the mean delay
until agreement (black: experiment, red: model) The delay until agreement is only defined within samples reaching
agreements. The black dots are standard deviations.

24Motivated by the agreement agglomeration at the soft deadline, this first subsection tests
the expanding deadline effect at the soft deadline, as formalized below.
Effect 1 [The deadline effect]

In contrast to the hard deadline regime, imperfect credibil-

ity increases the agglomeration of agreements on the threat period relative to the pre-threat
periods (left in Figure 2).
Following the literature on deadline effects, I proxied deadline effect at the soft deadline
as the relative occurrence of agreements on the threat period (n = 3) relative to each pre-threat
period (n ∈ {1, 2}). More comprehensively, Table 2 (1) analyzes how an agreement distribution
is regulated by the deadline credibility.48 Setting a binary agreement outcome each period, I

Table 2: Credibility effects on agreement distributions
Note:Parentheses contain standard errors, clustered by day and sessions. Trade experience controls the orderof trades and session-fixed effects within a day. Delay until agreement is normalized to a unit interval within
878 trades reaching agreements. ****, ***, **, and * show p < 0.1%, p < 1%, p < 5% and p < 10%, respectively. AMPE (average marginal probability effect) for a multinomial logit is computed as an average of the MPE(marginal probability effect) of each observation.

jointly estimated a multinomial logit model. Of course, each of observation would not be independent: autocorrelation and individual-subject learning effects might hinder the analysis. To
address the autocorrelation, error terms were clustered across each session by day,49 along withplayer-fixed effects of seller and buyers. To capture the learning effect, I controlled the orderof trade and session-fixed effects within a day. These econometric safeguards are inherited
48 In (1), larger σ is shown to significantly facilitate agreements across all periods as intuitively interpreted;higher-type buyers buy the good earlier and are more likely to evade the separations.
49 Recall that each session gives a common (σ , δ ) with different magnitudes of credibility α.

25throughout this section.
The estimates accord with detail of Figure 5; as credibility rises, the last-minute agreements at n = 3 occur more frequently (1.03, p < 1%), while ones at n = 6 before the final deadline happen less often (−6.03, p < 0.1%). The α’s estimate at n = 3 indicates a
1.37 p.p. higher average marginal probability effect (AMPE) of agreement ratio from a 10p.p. rise in credibility. Given estimates at n ∈ {1, 2} are both smaller with less precision(0.57, p = 5.2% and 0.63, p = 10.7%) than the estimate (1.03, p < 1%) at n = 3, the results
suggest an emerging deadline effect at the soft deadline. By contrast, when the conventional
deadline effect is defined as agreements just on the deadline (n = 6) relative to each prior period(n ∈ {4, 5}), comparing −6.03 p.p. (p < 0.1%, n = 6) with −3.44 p.p. (p < 0.1%, n = 4) and−4.92 p.p. (p < 0.1%, n = 5) shows fading deadline effect at the hard deadline. As the soft
deadline becomes harder, Table 2(1) jointly illustrates the transition of deadline effects fromthe hard to soft deadlines.
An immediate consequence of higher deadline credibility is shrinking delay until agreements by mechanically reallocating the distribution of agreements from post-threat periods(n ∈ {4, 5, 6}) to pre-threat (n ∈ {1, 2}) and threat period (n = 3) . To illustrate this, Figure 5(bottom) draws a mean duration that is conditional on agreement across environments, compared with the simulation (red lines computed by (11)). Roughly consistent with the simulation,
delay until agreements monotonically decreased with higher credibility. Formally, Table 2(2)shows a response of a unit-normalized bargaining duration until an agreement with higher credibility. The coefficient of credibility was found to be significantly negative (−0.284, p < 1%),indicating that a 10 − p.p. rise in credibility shortens the delay by an average of 2.8p.p. (i.e.,
0.17 periods out of 6 periods), enhancing the market efficiency.50
Compromised offers: As an indirect channel to restore the market efficiency, the model predicted that non-zero credibility of the soft deadline might, on the sellers’ side, induce a compromise in the price schedule. Lemma 1 shows that opening prices may drop for some imperfectcredibility. Figure 2 (right) shows a case under under the base parameter (δ = 0.98, σ = 1)
where pre-threat and threat prices on n ∈ {1, 2, 3} decreased for most imperfect credibility(α ∈ (0, 0.789]), compared with the hard deadline regime (α = 0). Moreover, reflecting on
50 The coefficient on δ is found to be significantly positive, consistent with the theory; the seller’s bargaining

power rises and buyers permit a longer delay to seek a cheaper pricing.

26

the expanding deadline effects with credibility, the post-threat prices on n ∈ {4, 5, 6} monotonically decreased (Figure 2 (right)) because a remaining buyer in the post-threat period was
likelier to have lower private value. Guided by these overall pricing behaviors, I test whether asoft deadline system induces sellers’ compromises, as discussed below.
Effect 2 [Compromised offers] In contrast to the hard deadline regime, imperfect credibility
decreases .
Table 3 reports the estimated sensitivity of a periodic price with deadline credibility.51
Columns (1)-(3) shows the negative sensitivity of pre-threat and threat periods. (−0.076, −0.076,−0.174; p < 0.1%) 52 For post-threat periods, the price schedule monotonically decreases with

Table 3: Credibility effect on price schedules
Note: Parentheses contain standard errors, clustered by day and sessions. Trade experience controls the order of
trades and session fixed effects within a day. ****, ***, **, and * show p < 0.1%, p < 1%, p < 5% and p < 10%,respectively.

credibility on period n ∈ {4, 5} (−0.146; p < 0.1% in (4) and −0.089; p < 5% in (5)), consistently with the model. At n = 6, the point estimate is negative but not statistically significant in(6) (-0.059, p = 27.1%), plausibly due to limited survivors.
More rigorously, to directly capture the non-linear response of opening prices P1 that areconditional on sufficient patience (Corollary 1), I employed a less parsimonious quadratic
51 Aligned with Rapoport, Erev and Zwick (1995) (infinite horizon), I found similar inconsistencies of opening

prices (P1 ). For a higher discount factor (δ = 0.98), a mean opening price was higher than the lower discountfactor case (δ = 0.7) and above the static ultimatum price, inconsistent with the model.
52 Observe that this drop is most notable on the threat period (n = 3). To complement the analysis, I test theP3
is negatively the largest with robust
sensitivity of inter-period price ratios and find that only a coefficient ofP2
statistically significance (see the Appendix), conjuring up ultimatum offer experiments. I discuss more on thedrivers of this finding in Section 4.

27

model. However, the implications are similar to the linear model in Table 3 (see the Appendix
for additional details). I will reserve the discussion on backgrounds of monotonic sensitivity
for Section 4, after the analysis of outcomes is provided.
Efficiency and distribution: If higher deadline credibility suppresses prices (Effect 2) andfacilitates earlier agreements (Table 2), does it also enhance the market efficiency? Moreover,
who benefited from the intervention in the laboratory? Founded on Proposition 2, this subsection tests the sensitivity of efficiency as the central question and that of a buyer’s surplus.
Effect 3 [Restored market efficiency]

In contrast to the hard deadline regime, imperfectcredibility improves the market efficiency.
Effect 4 [Advantage for the responder]

In contrast to the hard deadline regime, imperfectcredibility yields a larger level and share of the buyer’s surplus.
To test Effect 3, Column (1) of Table 4 regressed the efficiency U (or the sum of the players’
realized payoffs), revealing that a coefficient of credibility α was mildly significantly positive(0.057 with p < 10%). This finding indicates that higher separation risk typically enhances
the efficiency. It suggests that an anticipated side effect of embedding a soft deadline—mean
separation cost—would not increase to the degree of harming overall market efficiency. Guided
by this inference, I investigated the potential rise of separations, as partially suggested by themodel (Figure 4 (left)), by test whether higher deadline credibility induces separations compared to the hard deadline regime. In column (2), I set a binary outcome of a separation as an
outcome variable, estimating the effect of higher credibility via a logit model. Perhaps surprisingly, a coefficient of credibility α did not yield much precision (p = 20.7%). While its point
estimate of 0.286 and 95% confidence interval [−0.158, 0.730] would suggest that higher credibility is likelier to bring separations, the effects was not large enough to harm the efficiency or
reduce separations.53
Then, I examine whether imperfect credibility contributes to advantage for the buyers. In
columns (3)–(6), I examined the sensitivity of a level and share of buyers’ surplus. I considered
53 Both an increase or decrease of breakdown probability is partially consistent with the simulation (Figure
4 (left)). Alternatively, direct testing of the model’s non-linear implication by a quadratic model found complementary evidence of a rise in separation probability, capturing an anticipated downside of the soft deadline.
Analogously, the rise in separation probability is not large enough to overturn the efficiency implication trade,
either (see the Appendix for additional detail.)28

Table 4: Credibility effect on bargaining outcomes
Note: Parentheses contain standard errors, clustered by day and sessions in (1)-(5) and by day (6). Trade experience controls the order of trades and session-fixed effects within a day. In (1), 41 samples were dropped after
including fixed effects. ****, ***, **, and * show p < 0.1%, p < 1%, p < 5% and p < 10%, respectively. The
buyer’s surplus share (separations excluded) is a mean share of As the buyer’s surplus within trades reaching
agreements. The buyer’s surplus share (separations included) is the share of the sum of all buyers’ surpluses in thesum of trade efficiencies including separations. In (6), a unit of observation is an environment by day and includes
day-fixed effects.

the surplus level and share of buyers with separation cases as included and excluded, respectively. As documented in Table 1, the buyer’s surplus share, excluding separations (addressed in(3) and (5)), is an averaged share within trades reaching agreements; the buyer’s share, including separations (addressed in (4) and (6)), is an aggregate share of all buyers in the sum of trade
efficiencies including separations. (6) adopts an environment by day as the unit of analysis andincludes day-fixed effects. As the model relates the overall efficiency with the level and shareof buyer’s surplus (see Proposition 2), we should expect that all columns (3)–(6) exhibit similar
patterns—the actual result. Columns (3)-(6) suggest that a 10-p.p. rise in credibility significantly increases the expected buyer’s surplus by 0.64 p.p. (level; p < 5%), 0.35 p.p. (level;p < 5%), 0.31 p.p. (share; p = 12%), and 0.40 p.p.(share; p < 10%), respectively. Positive
estimates of these α terms indicate that a soft deadline could also serve as a countermeasure to
the monopoly power.
So far, readers might notice that Tables 3 and 4 suggest even more robust efficacy (Effects 2–4) of the soft deadline than the benchmark model: higher deadline credibility shrinks
the monopolistic pressure and enhances the efficiency without severely inducing separations.
This result suggests that a soft deadline is an affordable deterrent albeit not an empty threat.
Contrasting with the model’s non-linear implication, where the benefits and drawbacks of the29higher threat are comparable, I directly tested the non-linearity via a quadratic model. However, the qualitative implication is largely stable (see the Appendix for an auxiliary analysis).
In the following, I further explore the root cause of the findings.

4

Discussion

As shown in Section 3, the experiments demonstrated a more straightforward efficacy of the soft
deadline than the model predicted. Higher credibility of the soft deadline suppresses offeredprices, augments the efficiency, and yields the responder’s advantage, albeit with an empiricallyambiguous rise in the separation rate. I shall next investigate the source of deviations from themodel to discuss how the results could be interpreted.
To detect the deviations from the model, I started by assessing sellers’ pricing and buyers’
purchasing decisions by the following criteria. A periodic price in some environments is reasonable if it is within ±20% range of its theoretical price of the environment and is demanding(or cooperative) if it is above (or below) its theoretical price, respectively.54 The buyer’s agreement or disagreement in period n of an environment was reasonable if she followed a cutoff
rule to accept if v ≥ Cn and reject if v < Cn , where Cn is a cutoff computed in each environment(see Section 2.2). Analogously, if v < Cn , the acceptance was cooperative, and if v > Cn , therejection was demanding.
Table 5A documents the base share of pricing across 3 pricing postures under the harddeadline regime (α = 0). Most (59%) of the pricing (especially, 74% of initial pricing) was
framed as demanding, which is even higher than the static monopoly price (consistent withprior works). Relatively fewer pricing decision—29%—were cooperative. However, the ratio of cooperative pricing gradually rose with periods from 1% (n = 1) to 12% (n = 5) andspiked to 42% (n = 6). The last-minute price discounting appears to be novel in the context ofmulti-period trade experiments with one-sided incomplete information55 but resembles ultimatum games (discussed below). For the buyers’ side, a much larger portion of decisions werecooperative (11%) than demand withholding (2.7%). This marked contrast of postures between
54 Pricing at the final period cannot be cooperative by design (v < C ). Recall that C = P holds in the terminal
6
6
6

period (n = 6). Then, cooperative pricing yields a negative profit, which is not allowed in the experiment.
55 The previous works—Reynolds (2000) in a six-period case and Rapoport, Erev and Zwick (1995) with an
infinite horizon one—reported a puzzling rise of near-end prices. To explain this, they adopted fairness but in the
opposite direction as my explanation: the seller is no longer motivated to discount prices after a series of rejections
from the buyer.

30

sellers and buyers seemingly resulted in the inferior buyer’s share in the experiment comparedto the model (35.7% in the experiment vs. 44.8% in the model, including separations). Table 1
in Section 3.1 is relevant to this finding.
To analyze the sensitivity of pricing and purchase posture with deadline credibility, Table 5B and 5D report the within-player estimate of sensitivity with deadline credibility. Eachestimate was a coefficient of a logit model, setting each posture as an outcome variable andincluding player-fixed effects56 . Consistent with Table 3, pricing in periods n ∈ {1, 2, 3} becomes less demanding, more reasonable, and even more cooperative. This result is most salientin the threat period (n = 3), as indicated by its large, robust estimates: −4.37 (p < 1%) for demanding and 3.92 (p < 1%) for cooperative pricing. The soft deadline potentially remedied
for the upward-biased opening price, consistent with Effect 4 (i.e., the soft deadline augments
the buyer’s bargaining power).
Plausibly thanks to discounted prices, the buyers enjoyed more reasonable acceptance replacing reasonable rejections in periods n ∈ {1, 2, 3}, though the buyer got slightly more cooperative (0.40, 0.64; p < 10%) in the pre-threat periods (n ∈ {1, 2}). Most saliently, in thethreat period (n = 3), the buyers became significantly less cooperative (−1.12, p < 1%), a
posture mirrored by the seller’s cooperative discount in the period. As the change in buyer’s
posture well aligned with the seller’s softening pricing, I interpret that higher deadline credibility remedied sellers’ systematically demanding biases and facilitated the buyers’ reasonableagreements, thereby enhancing the market efficiency.
Still, a question remains of why more sellers become cooperative chiefly during the threat
period (n = 3) and modestly in the pre-threat periods (n ∈ {1, 2}), as the soft deadline hardens(as documented in Table 3). Below, I discuss four behavioral mechanisms related to my context:
fairness, bounded rationality, ill-updated belief, and risk aversion.
Fairness: The threat period is straightforwardly reminiscent of the canonical ultimatum games.
In ultimatum games, rationality dictates an incredibly selfish proposal to be accepted by an opponent. Hundreds of experiments, however, show a well-known behavioral regularities: anaverage proposer offers between 30% and 50% of the money, and more than half of the opponents reject the proposal with a share under 20% ( Camerer (2003)). The literature adopts
56 In Table 5, I use fixed effects of either sellers or buyers of interest, because a part of estimates are unavailable

due to lack of variation within fixed effects.

31Table 5: Deviation from the model and their sensitivities with deadline credibility
Note: For definitions of behavioral postures (for sellers: reasonable; demanding; cooperative; for buyers: reasonably accept; reasonably reject; demanding; cooperative), see the main text. Setting each posture of players
as an outcome variable, Table 5B and 5D report within-player estimates of logit models with deadline credibility
α, controlling for other primitives (δ , σ ), the order of trades, session-fixed effects within a day, and player-fixed
effect of sellers or buyers, respectively. – shows an unavailable value by design. NA denotes an unavailable estimate from a lack of variation under fixed effects. The aggregate analysis of actions in all periods adds period-fixed
effects. Parentheses contain standard errors, clustered by day and sessions. ****, ***, **, and * show p < 0.1%,
p < 1%, p < 5%, and p < 10%, respectively.

the proposer’s fairness57 as a central explanation (e.g., Fehr and Schmidt (1999); Bolton and
57 Alternatively, experimental literature calls fairness a form of inequality aversion, equity, or reciprocity. In thispaper, I consistently use the term fairness.

32

Ockenfels (2000)). Observing the protocol’s similarity, one may regard sellers’ systematic cooperation before the threat as their display of fairness.58 Consistently, in the terminal period(n = 6) of the hard deadline regime (α = 0), similarly high cooperative pricing (42%) was
observed where a monopolistic ultimatum behavior would be optimal. Instead of strategicallyleveraging the buyer’s compromise as a first mover, the psychological bias seemingly encouraged the sellers to concede in the face of threat.
Bounded rationality:

Given the similarity with ultimatum games, the fairness explanation isappealing. However, the fairness concern could be mitigated under asymmetric information.59
Because a higher price might be unfair to lower-type buyers, but fair to higher-type buyers,
rejection in this game does not immediately imply inequality aversion (Güth, Ockenfels andRitzberger (1995)). Moreover, provided the alternating roles of sellers and buyers, the social
norms for fairness might be diluted. If subjects take turns occupying an advantaged seller
position, the seller would have less guilt in exercising his privilege.
Aside from fairness, another compelling explanation is a version of bounded rationality60 ,stemming from Selten (1978), who argued that a multi-period setting impedes subjects fromperforming rational decision-making. Particularly, in my experiment of six periods, subjects’capability to correctly infer the best responses of opponents by running backward induction
was dubious.61 In face of the soft deadline, sellers potentially did not accurately infer that thebuyers would make a compromise in response to the threat. At the same time, their attentions
was primarily drawn to more intuitive separation losses. Therefore, sellers failed to leverage
the soft deadline under strategic interaction, indicating that higher credibility monotonicallyhinders their monopoly power.62 I conjecture that this form of bounded rationality and fairness
triggered the sellers’ compromises at the soft deadline.
58 To rationalize the failure of the Coase conjecture in the laboratory, Fanning (2022) built a behavioral modelwith preference for fairness in line with Fehr and Schmidt (1999) and proposed distaste to disadvantageous pricing(i.e.; monopolists prefer not to offer unfavorable competitive pricing) as an explanation. In contrast, my usage offairness for the sellers has an opposite meaning (i.e.; monopolists do not prefer too demanding pricing).
59 Comparing experimental results of dictatorship games and ultimatum games, Forsythe et al. (1994) showedthat fairness is not an exclusive factor in rationalizing the compromises in ultimatum games.
60 Reynolds (2000) and Cason and Reynolds (2005) proposed bounded rationality to rationalize the puzzlingly
high opening price, a different focus of pricing with my paper.
61 Especially in later trades within a day, most of the subjects did not spend much time to submit each action(roughly with 5-10 seconds). As calculating a Perfect Bayesian equilibrium path in different environments musttake much more time, the role of a heuristic rule of thumb appeared dominant.
62 To put both forces in perspective, it is insightful to map each in my bargaining framework. The lower

33

Ill-updated beliefs: The previous two biases supposedly limit the monopoly power. Alternatively, the sellers might have failed to update the Bayesian posterior: Kn∗ . (c.f. “Homemade
Priors” by Camerer and Weigelt (1988)).63 The experimental protocol forced all subjects toshare the initial beliefs of private values, but subjects might have substantially lowered theirposterior in the threat period for any reason. Although I cannot decisively reject the possibility
that Kn∗ was substantially low, I suspect that the low posterior is unlikely to be the primary
driver for the compromise. Suppose that Kn∗ was substantially low to explain the price drop.
Then, because commonly shared posterior Kn should be shaped by the frequency of agreementsin the previous periods, similarly increasing the portion of trades must be agreed in n = 1, 2.
This is incongruent with the observed change in the distribution of agreements; the rise ofagreements on the pre-threat period n = 1, 2 with deadline credibility is much smaller relative
to the rise of agreements on the threat period (n = 3) (see the multinomial logit coefficients of
0.57, 0.63, 1.03 in columns (1)-(3) of Table 2). No particular reasons support the belief that theposterior drops at n = 3 instead of n = 4.64
Risk aversion: The model assumes both parties’ risk neutrality; subjects’ risk preferenceswere not controlled in the experiment. Sellers’ risk aversion in the face of increasing risk seems
to explain the findings. Although it is a tempting explanation, I suspect that risk posture itself
cannot account for monotonically shrinking seller’s power with deadline credibility. Since typical pricing is classified as demanding, pricing is consistent with risk-seeking sellers—opposite
to the preceding explanation. Moreover, as buyers’ purchasing behavior are more inclined tomonopoly power An∗ is decomposed as
monopolist’s exploitation (< 0)buyer’s compromise (< 0)dAn∗ (Bn∗ , α)
dα
|
{z
}

change in monopoly powerz }| {
dAn∗
dBn∗

=
|

z }| {dBn∗
dα{z

strategic interaction

monopolist’s compromise (< 0)+} |z }| {∂ An∗∂{zαprice discrimination

}

dAn∗
dAn∗
∂ An∗
and/or a negatively small. Because
does not depend
∂α
dBn∗
dBn∗
∂ An∗
on the risk α, fairness is supposed to work only at
by fueling the price discrimination. Bounded rationality∂αdBn∗
is captured by near-zero.
dα
63 To track the reasoning formally, recall that sellers’ pricing is formulated as P = A K , where A is a periodicnn nnmonopoly power and Kn is (the sellers’ inference of) the posterior. Therefore, lower Pn∗ stems from either lower
bargaining power An∗ or lower Kn∗ .
64 Theoretically, recall that updating the posterior induces a conspicuous price drop from n = 3 to 4. On the
fourth period (n = 4), the remaining buyer, who have rejected the offer in the face of threat, demonstrably havelower value. (See discussions on price schedule around Figure 2).
Fairness contributes to a negatively larger34

be cooperative than demanding, this purchasing behavior is consistent with risk-averse buyers.
There could be no reason to presume that the risk posture of subjects flips with their bargainingrole on the same day (see Rapoport, Erev and Zwick (1995)).
Implementation in the field
Overall, I suspect that a combination of fairness and a version of bounded rationality (i.e.; lackof strategic interaction) forms a pricing rule of thumb for sellers. Based on the behavioral
interpretations, what insights from these laboratory experiments could be exported to the realworld? Presently, the implementation of soft bargaining to the “naturally occurring” market
remains challenging (e.g., incentive compatibility of both parties65 ). However, based on thepotential behavioral mechanism at work, one may infer that the finding in the laboratory is
relatively more smoothly exportable to a peer-to-peer (P2P) bargaining scenario in the fieldcompared to the business-to-business (B2B) negotiations.
Recently, small yet burgeoning field evidence shows widespread cooperative behaviors consistent with fairness. Keniston et al. (2021) document cooperations in various bargaining contexts (e.g., automobile prices negotiations, insurance claims, and TV game shows). Backus
et al. (2020) demonstrate similar behavioral regularities in the millions of negotiations on eBay.
If fairness is a principal behavioral force, then the efficacy of the soft deadline might potentially skew the model’s benchmark prediction in favor of the responders, as in the laboratory.
Although deadline effects have been observed in some renowned B2B trade anecdotes (e.g.,a fiscal cliff at congressional negotiations, trading in professional sports), there is supposedly
little room at work in cutthroat business deals or political fields for fairness—despite thosevery settings being where collective benefits of firms or countries are at stake.66 Analogously,
if bounded rationality is a primary behavioral driver, the result is less likely to apply to B2B
trades. Such deals are typically negotiated by experienced professionals, armed with a richer
knowledge of the best responses of opponents than consumers in P2P trades.67
65 For example, imbalanced outside options (currently, normalized to zero in both a model and experiments)

might violate the participation constraints of either player. A comprehensive discussion on implementation is,
however, beyond the scope of the paper.
66 If negotiations are repeated in the long run, however, bargaining may be better framed as repeated games sothat some cooperation emerge out of rational dynamic concerns.
67 The literature on real-world games report that professionals play game-theoretic best responses. In the board
games, professional players increasingly play A.I.-suggested best moves (Strittmatter, Sunde and Zegners (2020)
for chess; Shin et al. (2023) for go). Using the data from professional games, players follow the game-theoretic
strategy founded by mutual best responses. Some examples include Chiappori, Levitt and Groseclose (2002) for35

5Concluding Remarks

Many instances of real-world bargainings are protracted until deadlines. However, conventional deadlines are far from the perfect institution, by ruthlessly generating costly separations.
In this setting, must all deadlines be perfectly credible? Guided by the disproportionate agglomeration of eleventh hour agreements on deadlines, I explore redesigning the conventional
deadline structure by embedding an earlier soft deadline to restore ex-ante market efficiency.
Enriching a seller-buyer screening-type bargaining model under a deadline with random
separation at an intermediate date, I theoretically demonstrated a potential that an imperfectly
credible earlier deadline might enhance ex-ante efficiency. Using a laboratory experiment conducted with sixty-two subjects, I show primitive evidence of the soft deadline’s benefit. The
overall results suggest an even stronger efficacy than the model’s prediction, potentially due toearlier agreements being fueled by the cooperative pricing of sellers.
Admittedly, from a mechanism design perspective, exporting the system beyond the laboratory will be challenging. The stringent control over the environment in the laboratory almost
inevitably abstracts many real-world institutions that motivate the research and may limit theexternal validity of the findings (Harrison and List (2004); List (2007)). Nevertheless, thepaper’s findings could offer a new perspective to assist market designers concerned with efficiency.

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