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Pirates of the Mediterranean: An Empirical
Investigation of Bargaining with Transaction Costs
Attila Ambrus∗ Eric Chaney† Igor Salitskiy‡
December 22, 2011
Abstract
This paper uses ransom prices and time to ransom for over 10,000 captives rescued
from two Barbary strongholds to investigate the empirical relevance of dynamic bar-
gaining models with one-sided asymmetric information in ransoming settings. We ob-
serve both multiple negotiations that were ex ante similar from the uninformed party’s
(seller’s) point of view, and information that only the buyer knew. Through reduced-
form analysis, we test some common qualitative predictions of dynamic bargaining
models. We also structurally estimate the model in Cramton (1991), to compare ne-
gotiations in different Barbary strongholds. Our estimates suggest that the historical
bargaining institutions were remarkably efficient, despite the presence of substantial
asymmetric information.
∗Harvard University and NBER.†Harvard University.‡Stanford University. We thank Ran Abramitzky, Gillian Weiss, seminar participants at Stanford and
NYU Abu Dhabi, and especially Mauricio Drelichman, Avner Greif and Greg Lewis for helpful commentsand discussions. The library staff at the Biblioteca Nacional de Madrid and the Archivo Histórico Nacionalfacilitated the data collection. Judith Gallego provided outstanding research assistance. Any remainingerrors are our own.
1
1 Introduction
In recent years, Somali pirates have hijacked hundreds of ships and demanded sizeable ran-
soms for the release of captured crews and cargoes. Although the uptick in piracy off the
coast of Somalia has focused international attention on ransom payments, such payments
have long provided an important source of income for criminal organizations. Despite their
economic importance, little empirical work has been done investigating the determinants of
ransoms.1
We begin to address this gap in the literature by investigating the empirical relevance
of theoretical work related to many ransoming situations. To this end, we use a historical
data set on over 10,000 captives ransomed from the Barbary Corsairs, during rescue missions
conducted by Spanish negotiating teams between 1575 and 1739. These North-African-based
Corsairs preyed on Christian European shipping in the Mediterranean and the Atlantic for
centuries. Like Somali pirates today they derived important revenues from ransoms paid for
captured crews and individuals.
Conceptually, we think about negotiations for the release of different captives as a dy-
namic bargaining game with asymmetric information. In particular, we assume that the
relevant uncertainty is one-sided, regarding the exact value of each captive for the rescuers.
This value contained a term that was impossible for the corsairs to observe: the amount of
money raised at home for the release of the given captive. For this reason, the captors could
only know the probability distribution of the value for the buyer, conditional on observables.
In reality, negotiations during each rescue mission were part of a meta-game, in which the
same captive could be bargained for during multiple missions. However, we do not observe
the set of captives who were present and feasible to bargain for at a given rescue trip, only
the set of rescued captives. For this reason, we focus on negotiations, and in particular the
time path of prices, within rescue missions. Qualitative evidence suggests that after each
1For the growing literature investigating piracy and privateering from an economic standpoint, see Leeson(2007, 2009) and Gathmann and Hillmann (2009).
2
day of bargaining there was a nontrivial chance that negotiations end for exogenous reasons,
which introduced a randomness in the duration of the rescue trip, and urgency in reaching
an agreement.
Our data set provides a unique opportunity to investigate the empirical relevance of dy-
namic bargaining models with asymmetric information for three reasons. First, contrary to
laboratory experiments examining dynamic bargaining, in our setting it is reasonable to as-
sume that the participants only cared about their own physical payoffs. Several experimental
papers, including Neelin et al. (1988) and Ochs and Roth (1989), show that the observed
play of subjects in sequential bargaining situations is far from the perfect equilibrium pre-
diction of corresponding models.2 These papers compellingly argue that the main reason for
this is that many subjects exhibit other-regarding preferences, and in particular reject offers
that would give them less than what they regard as a fair share of the surplus. This can
be clearly observed in ultimatum games, which are essentially one-period bargaining games,
as first shown in Guth et al. (1982). In contrast, the negotiations we examine were con-
ducted by professional bargaining teams on the Spanish side (with clearly laid out directives
from the court), and experienced traders on the corsairs’ side, for whom buying and selling
captives was a regular business activity.3
Second, as opposed to field data previously used in empirical research on dynamic nego-
tiations, we observe multiple negotiations by the same parties for captives who for practical
purposes were identical for the seller. This plausibly addresses endogeneity issues that have
hampered many previous empirical investigations of asymmetric bargaining models (Kennan
2For experiments on bargaining with asymmetric information, see Mitzkiewitz and Nagel (1993), Strauband Murnighan (1995), Croson (1996), Guth et al. (1996), Rapoport et al. (1996), and Schmitt (2004).The qualitative conclusion that play is far away from equilibrium predictions remains valid in the aboveexperiments, although incomplete information (in particular on the responder side) tends to lower bothaverage offers and the likelihood of acceptance of low offers.
3Another reason why observed behavior in the lab differs from theoretical predictions is that subjectsin the lab might be risk-averse, while most of the theoretical literature assumes risk-neutral parties. Giventhat the parties in our setting were typically engaged in negotiating for the release of many similar captives,it is reasonable to assume risk-neutrality within any given negotiation (with the possible exception of a fewhigh-ranked captives).
3
and Wilson (1989)).4
Third, we directly observe a portion of the buyer’s privately known evaluation: the
amount of earmarked money raised in Spain by his or her family, which could only be used
for the release of the earmarked captive. Our empirical use of information that only one of
the parties possessed adds to a growing empirical literature on adverse selection that aims
to collect and utilize such information (Finkelstein and McGarry (2006), Finkelstein and
Poterba (2006), Abramitzky (2009)). To the best of our knowledge, ours is the first paper to
empirically use such information in the context of bargaining under asymmetric information.
We conduct our investigation in two waves. First, we test reduced-form predictions that
are implied by a large class of bargaining models with one-sided asymmetric information, to
check the empirical relevance of these models. Second, we structurally estimate a particular
bargaining model, to derive comparative statics results and conduct welfare analysis.
In the reduced-form analysis, we investigate the relationships between the length of ne-
gotiations and release price, and how an observed increase in the buyer’s evaluation (in the
form of earmarked money for a captive) affects release price and the length of negotiations.
We find that in Algiers, where qualitative evidence suggests that the negotiating teams faced
higher time costs of bargaining, the data are consistent with the theoretical predictions. In
contrast, in Tetuan, where the time costs for negotiating were much more modest, we do not
find empirical support for the theory.
While our reduced form investigation allows us to test some general predictions of dy-
namic bargaining models with asymmetric information, we need a different approach to be
able to compare the bargaining process in the two main Corsair strongholds, Algiers and
Tetuan, and identify the main reasons why observed bargaining outcomes differ in these
two locations. This is because the effects of differences in the distributions of buyer values,
4Card (1990), for example, found virtually no relationship between agreed upon wage and the length ofnegotiations analysing Canadian employment contract data for the period 1964-1985 despite the fact that allrational models of bargaining (when the relevant private information is on the firm side) predict a negativerelationship between agreed upon wage and the length of negotiations. McConnell (1989), using US contractdata for the period 1970-1981, finds a statistically significant negative relationship between average wagesettlements and average strike duration, but this relationship is sensitive to the model specification.
4
and in time preferences of the bargaining parties (partly governed by the institutional back-
ground of negotiations) are complicated and intertwined in dynamic bargaining models. For
this reason, in the second half of the paper we further analyze our data on ransom negotia-
tions by structurally estimating the model in Cramton (1991). This approach enables us to
disentangle the effects of differences between Algiers and Tetuan along multiple dimensions.
We focus on the model in Cramton (1991) for two reasons. First, besides standard
exponential discounting, it allows for transaction costs, that is fixed costs that bargaining
parties had to pay for each additional time unit of negotiations. Second, the presence of
transaction costs, together with the delay arising from asymmetric information, implies that
in equilibrium there can be failed negotiations, as a buyer with low evaluation prefers quitting
negotiations to engaging in a long and costly bargaining process. This model is unique among
models of bargaining between a buyer and a seller since it facilitates both costly delays in
reaching agreement and failed negotiations, both of which are widespread in our data.5
Structural results indicate that transaction costs were modest and roughly equal in the
two strongholds. In contrast, discount rates were substantial in both settings, and in most
professions much higher in Algiers. This suggests that the dynamics of negotiations were
mainly driven by the relatively rapidly deteriorating physical conditions of hostages in captiv-
ity, as well as by the possibility of exogenously terminated negotiations. Qualitative evidence
suggests that both of these factors were more important in Algiers, in accordance with our
structural estimates.
Our structural approach also allows us to conduct welfare analysis and counterfactual
investigations.
Although in both settings the estimated proportion of failed negotiations is high, because
negotiations failed predominantly for captives with low evaluations, the resulting welfare
losses are relatively small: they tend to be around 10% of total potential surplus. Delay
5Dynamic bargaining models with no transaction costs imply that negotiations are never terminated. Atthe same time, a model with no discounting, only transaction costs, predicts immediate agreement, as shownin Perry (1986).
5
costs tend to be 7-8% of total potential surplus, with transaction costs only around 2%
of potential surplus. This means that according to our estimates, despite the presence of
substantial asymmetric information, the bargaining mechanism in this historical setting was
remarkably efficient, with realized surplus being roughly 80% of potential surplus. The
distribution of the realized surplus is estimated to be roughly equal in both settings, with
some variation across professions.
In the counterfactual investigations we investigate alternative market mechanisms that
could have been used by the parties: posted prices (take it or leave it offers from sellers),
and bundled negotiations (for released captives for whom we can confidently rule out that
they were part of bundled negotiations). We find that posted prices would have significantly
increased the surplus appropriated by the captive holders, but would have decreased real-
ized social surplus through an increased number of failed negotiations. Further bundling
would have increased social efficiency, without hurting either of the bargaining parties. This
suggests the possibility of coordination problems among sellers of different captives.
The remainder of the paper proceeds as follows. Section 2 provides a brief historical
background of the Barbary Corsairs, the process leading to captivity and the bargaining
process. Section 3 provides a theoretical overview of models of dynamic bargaining with
one-sided private information, and conducts reduced form tests of some common qualitative
predictions of these models. In Section 4 we provide a structural analysis of the model
developed by Cramton (1991).
2 Historical Background
The history of the Barbary Corsairs begins with the “Red Beard” (Barbarossa) brothers.6
These Muslim Greek-born brothers developed a reputation for military prowess, and even-
tually founded a corsairing state in Algiers (modern-day Algeria) in the early 16th century.
6For a detailed treatment of the history of the Barbary Corsairs see among a large literature Julien (1970),Abun-Nasr (1977), Bono (1998), Davis (2003), Panzac (2005) and Weiss (2011).
6
Soon after its foundation, this corsair state became part of the Ottoman Empire.
Algiers and the other Ottoman-controlled North African provinces (which controlled the
territories that roughly correspond to present day Tunisia and Libya) developed corsair fleets
that targeted Christian European shipping and coastal populations. In addition to these
Ottoman centers, Muslim refugees from Spain developed corsairing fleets in Salé, Morocco
(on the Atlantic coast) and Tetuan, Morocco (on the Mediterranean).7
At their height, the Barbary Corsairs significantly disrupted commercial routes in the
Mediterranean (and to a lesser extent the Atlantic) striking as far afield as Iceland. In
addition to their exploits at sea, the Corsairs made their mark on coastal settlement patterns
in regions in close proximity to Corsair strongholds (like Spain’s Mediterranean coast). Many
areas in these regions became depopulated as the local populations fled continuous corsairing
raids.8 One scholar estimates that between 1530 and 1780 the Corsairs enslaved over one
million European Christians (Davis 2001, 2003).
By the end of the 17th century the Corsairs began a slow decline. In Algiers, con-
temporaries attributed this decline to more determined reprisals by European powers and a
decreasing demand for slave labor (Davis 2001, p. 106). In Morocco, Mawlay Ismail (reigned
1672-1727) moved to curtail corsair activities in order to encourage legitimate commercial
activity and to centralize political power (Friedman 1983, p. 29). By the start of European
colonization in the 19th century, corsairing activities had largely ceased.
The remainder of this paper focuses on the corsairs that operated out of Algiers, Algeria
and Tetuan, Morocco since the captives in the data set were ransomed from these centers.
2.1 Bargaining for Captives in North Africa
Corsair centers across North Africa derived important revenues from the capture and sale of
captives. These captives were most often taken in land raids or at sea, although some captives
7See Coindreau (1948, p. 27) and Maziane (2007, pp. 200-201).8For the impact of piracy on Venetian commerce, see Tenenti (1967). For the impact of piracy on Spanish
coastal settlement patterns, see Friedman (1983, pp. 48-49).
7
were prisoners of war from the numerous military confrontations between the Corsairs and
European powers. After the Corsairs finished a raiding expedition, they brought home the
captured merchandise and captives. Captives were sold in local slave market after the leader
of the corsair center had taken his cut.9
A captive’s price in the slave market seems to have been primarily determined by two
factors. The first of these factors was related to the present value of a captive’s marginal
product. Older captives were valued less and captives with special skills (such as carpentry)
commanded higher prices. The second factor was a slave’s potential for ransom. Captives
believed to be desired for ransom by European powers were kept in special quarters. Slave
owners (throughout the text we refer to slave owners and corsairs interchangeably for ex-
positional ease) were eager to have their slaves ransomed and encouraged captives to write
letters home. These letters were carried by merchants and the ransoming missions back to
Europe where they were given to family members.
The Barbary Corsairs particularly valued captives from the Spanish Empire. Spanish
ransoming missions to Algiers and Tetuan occurred frequently and Spaniards were known
to command higher than average ransom prices (Friedman 1983, p. 57). Although the
corsairs were generally able to identify a captive’s nationality and rank in a general sense
(and used this information to divide captives between those more and less likely to be
ransomed), the exact amount that Spaniards were willing to pay for each captive was not
known. One historian has stressed that it was usually difficult for high-ranking captives to
pass as “normal” captives since the corsairs were “very crafty at learning if their captives
were nobles or ecclesiastics” (Friedman 1983, p. 151). Despite this fact, there are examples
of high-ranking captives successfully hiding their identity (Friedman 1983, p. 151). Those
who were of a lower rank often were indistinguishable from the corsairs view.10
Catholic religious orders ran virtually all of these ransoming missions in the period covered
9This section draws on Martínez (2004), Bono (1998) and Friedman (1983).10Friedman’s (1983, p. 149) statement that “the North Africans usually knew both the identities of their
captives” refers to elite captives. There is substantial evidence that the corsairs did not know the exactidentity of lower ranking captives (see, for example, Martínez 2004, p. 41; Friedman 1983, pp. 57, 143).
8
by the data. These religious orders negotiated (and paid if the negotiation was successful)
the captured individual’s ransom on behalf of the family. The funds for ransoming came
from the money that was earmarked for individual captives, money that had been donated
by the crown or wealthy individuals for the ransom of a specific type of captive (such as
women, children or soldiers) and a general fund for the ransoming of Spanish captives in
North Africa that came from alms throughout the year and bequests. These general funds
were often restricted, however, to be used to ransom captives from the areas in which these
alms were collected.
Consequently, Spanish ransoming missions to the Barbary States concentrated on freeing
two groups of individuals. The first group included “favored” captives such as women,
children, clerics, individuals who were captured in the “service of the crown” (such as soldiers
and royal emissaries), individuals from specific geographic areas and others in danger of
converting to Islam. The exact identity of the captives did not matter inasmuch as s/he
belonged to one of these groups. The second group contained the group of captives for
whom family and friends had donated money. Here, the exact identity of the captives
was extremely important since the ransoming missions were required to use these funds to
ransom the individual for whom they had been earmarked. The amount of earmarked money
sent for a captive often did not cover the ransom cost (the differences were made up from
the non-earmarked general funds) and these individuals do not seem to have been readily
distinguishable from other ransomed captives. The ransoming team gave special priority to
earmarked captives and attempted to rescue these captives as soon as possible (Friedman
1983, p. 146).
From the reign of Phillip II (reigned 1556-1598) onwards, royal authorities appointed
a notary to accompany these rescue missions or “redemptions”. This notary was required
to record all financial transactions and often provided anecdotes relevant to the bargaining
procedures. We used these records to construct our data set and historians have generally
stressed their quality and meticulous detail (Friedman 1983, p. 107). Summary statistics
9
are presented in table 1, and a detailed discussion of both the data construction and the
summary statistics is provided in the appendix.
The identity of the captives for which the ransom missions had earmarked donations
(known in Spanish as adjutorios) and the size of these donations constituted a source of
uncertainty for slave owners. Miguel de Cervantes, the renowned Spanish author, was captive
in Algiers from 1575 until his ransom in 1580. Cervantes —from a family of relatively modest
means— was captured and ransomed long before his rise to fame. His adjutorio provides a
good example of how the system of earmarked funds worked. In the notary’s records from
the ransoming expedition of 1580 to Algiers we read that:
In the town of Madrid on the 31st day of July of the said year [1579] [...] the
redemptors received 300 ducados [a money of account in early modern Spain...]
250 of these came from the hand of Leonor de Cortinas, the widow of Rodrigo
de Cervantes. The remaining 50 came from Andrea de Cervantes [... to aid] in
the ransom of Miguel de Cervantes from the said town [Alcala de Henares] [...]
who is captive in Algiers in power of Ali Mami, captain of the ships of the navy
of the king of Algiers. [The said Miguel] is 33 years old and has lost use of his
left arm. The redemptors then gave them 2 receipts [i.e. proofs of payment].11
In this trip, Miguel de Cervantes was rescued, although if he had not been rescued the
redemptors (i.e. the negotiating team) would have had to return the 300 ducados to his
family. It is important to note that individuals were only allowed to send funds with the
redemption if they had hard evidence (such as a letter) that the individual was being held
captive in the redemption’s destination.12
The surviving ransom books provide detailed bargaining instructions given to the ransom-
ing team. These instructions stress the importance of keeping the identity of these captives
secret and note that11AHN, códices, legajo 120, f. 3212See Friedman (1983, p. 112) and BNM, MSS 3819, f. 3.
10
the redemptors should feign indifference to the captives they most want to rescue,
and should pretend to only care about silver and not the captives. Since the Turks
are greedy and love silver more than the good service of their slaves, they don’t
want to miss the chance to make money [...so that] every day the redemptors
delay, they will save 20%.13
Qualitative evidence suggests that the redemption team largely followed these instruc-
tions, carefully guarding the identities of earmarked captives.14 This evidence is important
for the empirical analysis below, because it demonstrates that despite the corsairs’ attempts
to ascertain the value of captives to the ransoming teams, the corsairs could not distinguish
between many captives. The redemptors could obtain lower prices for these captives by
hiding their true valuations.
The earmarked and general funds were transported together. In Tetuan, the ransom
funds were kept in Ceuta (a Spanish enclave in North Africa) and were not observed by
the corsairs. In Algiers, the corsairs unloaded the ransom funds and knew the amount of
funds available (although some members of the rescue team successfully hid funds on their
persons) although they did not know what percentage of the treasury was earmarked nor
for whom they were destined (Friedman 1983, pp. 120-121). If the rescue team ransomed
a non-earmarked captives with earmarked funds, they would return the earmarked funds to
the donors using funds raised for subsequent ransoming missions.
We have not been able to find any direct references to the incentive schemes to motivate
the redemptors. Despite this fact, we do know that experienced redemptors were highly
valued (BNM, MSS 3870, f. 10; Martínez 2004, p. 94) and drawn from the highest level
of the redemption orders.15 This suggests that whatever the exact incentive scheme used,
skillful bargainers that could both obtain lower ransom prices and were willing to endure
13BNM, MSS 2974, ff. 5-6.14For examples see Garí y Simuell (1873, p. 288) or BNM, MSS 647, f.6.15At least for the Mercederian order during the late medieval era. One historian speculates this might
have been because “such senior brothers possess[ed] the prudence and experience necessary for the successof the mission” (Brodman 1986, p. 109).
11
hardship were highly valued.
In sum, rescue missions to the Barbary strongholds focused on rescuing two types of
captives. First, they used general funds to rescue captives with certain characteristics (such
as women, children or soldiers). Second, the redemptors sought to rescue “earmarked”
captives (those for whom money had been given).
2.2 Differences in the Bargaining Conditions between Algiers and
Tetuan
Historical evidence suggests differences in the bargaining conditions between Algiers and
Tetuan. We divide the historical evidence regarding these differences into those that af-
fected the discount rate of the redemptors and those that affected their transaction costs.
Transaction costs are independent of the value of the captive, whereas a higher discount rate
has a differential effect on the cost of waiting across captives of different values.
2.2.1 Differences that affected the discount rate
Algiers enjoyed greater military power —and poorer relations with Spain— than Tetuan. These
differences led to a larger captive population in Algiers and to a worse treatment of the slave
population in this stronghold (Martínez 2004, p. 60).16 Friedman (1983, p. 71) suggests
that differences in how captives were treated could lead to both higher discount rates for
the redemption team as well as to shorter negotiations. She remarks that in both corsair
strongholds even the most valuable captives could be treated quite poorly since “their captors
thought it would be easier to obtain high prices for important captives if they were kept in
miserable surroundings.” The captors presumably did this because the redemption team
incurred greater costs by leaving captives who were subject to more severe treatment in
16He notes that “it was not the same to be captive in Algiers or Morocco. From the start of the 16th
century until well into the 19th century, Algiers maintained hostile relations with the Spanish crown whichresulted in a larger number of Spanish captives. It was logical, then, that their treatment was more harshthan that faced by captives in Morocco.”
12
captivity.
In addition to differences in the treatment of captives between the two strongholds,
greater political instability in Algiers seems to have led to a higher discount factor for the
redemption team in this stronghold. Data on ruler durations in Algiers shows that the
average ruler remained in power for roughly three years (Stewart 2006, pp. 14-15). Although
we have not been able to locate ruler duration data for Tetuan qualitative evidence suggests
that this stronghold was subject to less ruler turnover, and its ruling oligarchy appears to
have been more stable. Martínez (2004, p. 105), for example, noted that Tetuan’s “relatively
more stable political situation made [it] a more desirable destination to ransom captives.”17
This increased political instability could abruptly lead to the end of negotiations (for one
example see Martínez 2004, p. 99).
2.2.2 Differences that affected transaction costs
In Tetuan, transaction costs seem to have been lower than in Algiers. When the redemption
team went to Tetuan, they hired a boat in Gibraltar to cross the straits to Ceuta. This boat
did not have to remain in port for the length of the negotiations and thus the ransoming
team did not have to pay for its rental during the negotiations. In addition, the proximity
of a Spanish enclave in Ceuta (and the possibility of rapid military intervention) greatly
reduced the risk of physical harm to the redemptors.18
In Algiers, the redemption team had to hire a ship that remained in port until the
redemption was over. In addition, there is qualitative evidence that the ransoming team
paid higher prices for items such as accommodation than they did in Tetuan. Finally, the
redemption team in Algiers faced a higher probability of capture or bodily harm than it did
in Tetuan. Thus an additional day of negotiations “increased the risk of capture of the entire
17See Friedman (1983, p. 131) for additional evidence of political instability in Algiers.18There are no documented examples (to the best of our knowledge) of the redemption team being held
captive in Tetuan. In Algiers, however, the redemption team was held captive on at least one ocassion(Friedman 1983, pp. 136-138) and was threatened with death by fire on another (Garí y Simuell 1873, p.326). See also the discussion in Friedman (1983, p. 142).
13
bargaining team” (Martínez 2004, p. 99).
3 Reduced form analysis of predictions of dynamic bar-
gaining models
In this section we conduct a reduced form analysis of predictions that are shared by a
large class of dynamic bargaining models. In Subsection 3.1 we provide a brief overview
of the theoretical models we consider, and highlight their common testable predictions. In
Subsection 3.2 we report the empirical results.
We note that the theoretical models that we take to the data are crude approximations of
the bargaining situation we examine, abstracting away from potentially important features
of negotiations. Below we mention a few of these aspects, calling attention to the importance
of incorporating them into models of dynamic bargaining in future work.
First, we treat every negotiation in isolation. In reality, the thousands of negotiations
in our data set were part of a meta-game. If negotiating for the release of a given captive
failed at a bargaining trip, there was a possibility of resuming negotiations at the next trip.
However, we only observe information on successful negotiations, therefore we have to base
our analysis on the negotiation within a given rescue mission (the one during which the
captive got released). We also abstract away from issues such as competition between the
Corsair strongholds to attract ransoming trips, as well as reputation-building, reputation-
erosion, or inventory-building by the negotiating parties.19
Second, the models we consider do not incorporate either budget constraints (upper
bound on the amount of money spent) or spending constraints (lower bound on the amount
of money spent). In reality, the rescuers faced a soft form of both constraints, at least
in Algiers, where they had to unload their money at the beginning of a trip. Spending
19This can be partly justified by the fact that the identities of the Spanish negotiators and the set of sellersoften changed from trip to trip.
14
more money than they had brought was costly, as it involved borrowing extra money at
high interest rate. But spending less than the total amount involved costs as well, as any
remaining money would have to be “wrestled back” from the political leaders of Algiers.
Third, in this section we do not take up the issue of negotiating for each captive separately,
or for a bundle of captives together. Although we do not directly observe whether the release
of a given captive was part of a bundled negotiation, there are many instances in our data
when similar captives were released at the same time and at the same price, suggesting that
bundling was commonly used. We do not know any theoretical work investigating this issue
in a dynamic bargaining context, although there is a clearly related literature on bundling
and pricing decisions of a monopolist seller in a static context.20 The main takeaway from this
literature is that some form of bundling is likely to be optimal for the seller. Our conjecture
is that similar results are likely to extend to dynamic bargaining situations.
3.1 Theoretical background and predictions
We consider negotiations for ransom between the Spanish rescue team and the captive holders
as a dynamic bargaining game with asymmetric information. In particular, the relevant
private information is the exact value of a given captive for the rescuers. Our motivation
here is that the value of a particular captive for the Spaniards always had a component not
known by the slave owners: the amount of earmarked money that was collected for a given
captive. Over time, the captors could learn the distribution of this private value conditional
on observables of a captive, but not the exact value for individual captives. In contrast, other
important parameters for the bargaining process, such as the parties’ time preferences and
transaction costs, or reservation values of different types of captives for the holders, could
either be observed by the parties through public information (such as interest rates charged
by money lenders, or the price that a certain type of captive could be sold at slave markets)
20See for example Adams and Yellen (1976), McAfee et al. (1989), Armstrong and Rochet (1999) andFang and Norman (2006). There is also a literature on bundling decisions in a mechanism design context,reaching similar conclusions. See Palfrey (1983), Chakraborty (1999), Armstrong and Rochet (1999), Averyand Hendershott (2000), Armstrong (2000) and Jehiel et al. (2007).
15
or learned over time.21
Besides considerations based on qualitative evidence, our data also rejects some of the
main predictions of an alternative class of models in which the relevant private information
is the bargaining parties’ patience. Bargaining in such contexts takes the form of a war-of-
attrition game in which one of the parties end up acquiring the whole surplus, which we do
not see any indication of in our data. Moreover, these models predict that settlement rates
over time should be monotonically declining. In the Supplementary Appendix we show that
this is not the case in our data.
3.1.1 Dynamic models of bargaining with one-sided asymmetric information
Dynamic bargaining models that assume one-sided private information regarding the value
of the object for the buyer are incomplete information extensions of the models proposed by
Stahl (1972) and Rubinstein (1982). Below we briefly summarize the most standard models
used in the literature. All the models reviewed here assume that there is one buyer and
one seller, bargaining for one indivisible object, and that all parameters besides the buyer’s
evaluation are commonly known by participants.
One set of models assume that only the uninformed party (the seller) can make offers,
and there is a fixed amount of time that has to elapse between offers. They are referred
to as screening models, as the seller’s offers at different points of time are accepted by only
certain buyer types, hence the seller screens different types of buyers throughout the bar-
gaining process. Sobel and Takahashi (1983) introduced a finite version of the model, while
Fudenberg at al. (1985) and Gul et al. (1986) extended the analysis to infinite-horizon. The
equilibrium dynamics in these models are simple: the seller proposes a decreasing sequence
of prices. In each round, there is a cutoff buyer type such that all remaining buyer types
above this level accept the proposal, while buyers with evaluation below this level reject it.
21Captive-holders might have privately known individual-specific evaluation for a certain type of captive,exceeding market price. However, for common type captives, the thickness of the market implies that theycould purchase additional captives of the same kind until the marginal benefit became equal to the marketprice.
16
The main intuition is that buyer types with higher evaluation are more impatient and willing
to pay higher prices to receive the object earlier.22
A major alternative to the above models is a class of models in which a player whose turn
it is to make an offer can wait before doing so, endogenizing the timing of the offer. They
are usually labeled as signalling models, because in such a framework a low valuation buyer
can credibly signal his type by waiting longer before making a (counter-)offer. The version
of the model investigated in Admati and Perry (1987) assumes alternating-offer bargaining,
with the seller making the first offer. After a rejected offer, there is a minimal required time
that has to lapse before the other player can make an offer, but the player can decide to wait
any additional amount of time. An important assumption in these models is that bargainers
can commit to not revising an offer until a counteroffer is made.23
Below we summarize two common implications of signaling and screening models de-
scribed above.
Relationship between length of negotiations and price
The main prediction, shared by all rational models of dynamic bargaining with one-sided
private information on the buyer side is that there is a negative relationship between the
buyer’s value and the length of negotiations, otherwise high value buyers would find it worth
to imitate low value ones. A higher value of the buyer implies larger costs of time before
reaching an agreement, thus willing to wait longer becomes a credible proof that a buyer’s
value is low. This prediction in our setting implies that given a fixed setting of negotiations
and a homogenous set of captives (from the captors’ point of view) both the amount of
22Versions of the above models in which the buyer and the seller make offers alternatingly are screeningmodels with an element of signaling. The latter is because proposals by the buyer can reveal private infor-mation about his valuation of the object. These models typically have a severe multiplicity of equilibria, asthere is a lot of freedom in specifying the updated beliefs of the seller regarding the buyer’s valuation after anout-of-equilibrium proposal by the buyer. Rubinstein (1985), Grossman and Perry (1986), and Bikhchandani(1992) propose refinements of sequential equilibrium in related models that lead to a similar analysis as inpure screening models.23However, if one allows for nonstationary strategies, the signaling equilibrium outcome can be approx-
imated as a perfect Bayesian Nash equilibrium in the alternating-offer bargaining game with a fixed timebetween offers when time between offers is small (see Ausubel and Deneckere (1994)).
17
earmarked money and the release price of a captive should be negatively associated with the
length of the negotiations for the captive.
Effect of amount of uncertainty
Another common feature of different bargaining models, although one that proved to be
difficult to formalize analytically, is that more uncertainty leads to longer negotiations.24
This is easy to see in the limit case of no uncertainty: the unique perfect equilibrium in-
volves immediate agreement (see Rubinstein (1982)). Similar conclusions apply when the
uncertainty about the buyer’s value is small (see Rubinstein (1985)). For larger amounts
of uncertainty, because of the relative complexity of bargaining models with asymmetric
information, it is only possible to show that more uncertainty tends to lead to longer nego-
tiations in restricted classes of games or through numerical computations, as in Grossman
and Perry (1986) and Kennan and Wilson (1989). With the above caveats, the prediction in
our setting is that given two otherwise comparable homogenous group of captives, and the
same mean earmarked money, the group with higher variability of earmarked money should
be associated with lower prices and longer negotiation times.
3.2 Testing common predictions of models with one-sided asym-
metric information
This section uses data on privately owned individuals to investigate the empirical relevance
of the theoretical predictions shared by models outlined in the previous subsection.25
24Tracy (1986, 1987) conducts an empirical investigation along these lines, suggesting that the extent ofthe union’s uncertainty about the firm’s profits might be associated with variability of the firm’s marketvalue. He finds a significant positive empirical association between this measure of uncertainty and bothstrike incidence and duration, consistent with the theoretical prediction.25We limit the sample to privately-owned individuals because these individuals were not able to coerce the
redemption team to the same extent as the government was able to (Friedman 1983, pp. 130-131). Resultsfor government-owned captives, however, are qualitatively similar to those reported below although generallynot statistically significant. Pooling all captives generally increases significance.
18
3.2.1 Delay and Prices
As noted above, bargaining theory predicts that all else equal the release price of a captive is
negatively related to the length of negotiations. Below we focus on the relationship between
release price and the length of negotiations within a rescue trip. Recall that the possibility of
an exogenous breakdown of negotiations (leading to the end of the rescue trip) could imply
a low discount factor even over short time horizons, justifying a significant drop of prices
over the time horizon of a rescue trip.
We focus on the effect of within-trip delay on price, as opposed to the effect of a captive
not being rescued during previous rescue missions (between-trip delay) because while within-
trip delay is directly recorded in our data set, we only have a noisy measure of between-trip
delay. This measurement error stems from the fact that we only observe a captive’s time
spent in captivity before ransom. We do not observe the true measure of interest, namely
how many times a captive was negotiated for before his or her ransom. In addition to this
source of measurement error, the trade in slave across the Islamic world meant that many
captives spent much of their captivity in regions where the redemption missions did not go
before they were sold to an owner in a location where they could be ransomed. For these
captives, a long wait before ransom had nothing to do with strategic delay. Thus, time in
captivity is at best a noisy estimate for between-trip delay and its use is likely to result in
biased (attenuated) estimates of the causal effect of between-trip delay on prices. Consistent
with this discussion, we find evidence that our estimates of between trip delay suffer from
substantial attenuation bias.26
To estimate the effect of within-trip delay on equilibrium prices we use the specification:
100 ∗ ln(priceij) = αj + βdayselapsedij + γ X+ εij (1)
26In practice, we find that the coefficient on time in captivity -while statistically significant- is smaller inabsolute value than that on days elapsed. In a previous version of the paper we developed an instrumentalvariable strategy to address this problem and found evidence suggesting that the OLS estimates suffer fromattenuation bias.
19
where priceij is the ransom price of captive i in ransoming trip j, dayselapsedij measures
the days elapsed after the start of negotiations until captive i was ransomed andX is a vector
of covariates. We estimate regression (1) separately for the Algiers and Tetuan samples.27
The results are presented in panel A of table 2. The first three columns limit the sample
to Algiers, whereas the last three columns estimate equation (1) using the sample of captives
rescued from Tetuan. The results in Algiers are consistent with the theoretical predictions.
In particular, an increase of one day in the time to ransom is associated with a roughly
four percent decrease in a captive’s ransom price in columns (1) and (2). This result is
statistically significant and is robust to the introduction of controls in column (2). Controls
include a captive’s time in captivity, age at captivity, a gender dummy, a dummy equal to
one if the captive was younger than 16 when ransomed and profession dummies.
In column (3) we restrict the Algerian sample to captives that were rescued in the first
(observed) ransoming trip to arrive in Algiers after they had been taken captive. This is
the cleanest estimation of within trip delay, comparing only individuals for whom previous
ransom trips likely did not generate any extra information on their value for the captors
(at least to the extent that we can measure this quantity). Here the point estimate of the
coefficient of the days elapsed variable decreases in absolute value, but remains statistically
significant.
In columns (4)-(6) of panel A of table 2 we present the results for Tetuan. Here, the
estimated β is close to zero and insignificant for every specification.
The next set of regressions examines the effect of earmarked money for an individual cap-
tive, a component of the value of the captive for the rescuing team that we directly observe,
on the release price of the captive, and on the time of negotiations. Note that qualitative
evidence suggests that conditional on covariates earmarked money can be considered exoge-
nous. If this assumption is true, then the results in panels B and C can be interpreted as
causal.27We omit all captives ransomed in two extremely long negotation trips that took over 270 days because
these outliers distort the results.
20
In panel B of table 2 we estimate regression (1) replacing the variable dayselapsedij with
the indicator variable earmarkedij which is equal to one if captive i in ransoming trip j had
money sent for his rescue from Spain. Our estimates show that earmarked has a large and
significant effect on a captive’s release price, increasing it by roughly 35% in Algiers, and by
a little bit less in Tetuan.
In Panel C of table 2 we again estimate regression (1) replacing the dependent variable
ln(priceij) with dayselapsedij and dayselapsedij with earmarkedij. In other words, we are
interested in examining whether earmarked captives were rescued earlier than other captives.
Results in columns (1) and (2) show that earmarked captives were rescued roughly 0.40 days
earlier in the negotiations, although the point estimate decreases in absolute value and
becomes statistically insignificant when we limit the sample to captives rescued in the first
ransoming trip to arrive in Algiers after they had been taken captive. In Tetuan the point
estimates are negative but insignificant, for all specifications.
3.2.2 Uncertainty and Delay
Theoretical considerations suggest that all else equal greater uncertainty should lead to
longer negotiations. In order to measure uncertainty, we use the coefficient of variation (the
standard deviation divided by the mean) of earmarked money by profession and by corsair
center (that is we calculate the coefficient of variation of earmarked money by profession in
both Tetuan and Algiers).28 Although this metric is imperfect at best, results in panel D of
table 2 provide some support for the theoretical prediction.
We estimate a regression of the form
dayselapsedipj = αj + β1Ep + β2σpEp+ γ X+ εij (2)
where dayselapsedipj is the days elapsed after the start of negotiations until captive i was
28Here we use disaggregated professions as given in the original sources instead of the aggregated categoriesused throughout the rest of the paper to increase statistical power.
21
ransomed which varies by captive i in profession p in ransoming trip j. Ep denotes the mean
amount of money sent for captives in profession p, σpEprepresents the coefficient of variation
of this quantity and the vector X contains controls (which includes a dummy indicating how
a captive was captured, the captive’s age when captured and dummy variables equal to one
if the captive was a child or female). The theory predicts that the coefficient β2 will be
greater than zero.
Results in columns (1)-(3) estimate equation (2) in the Algerian sample. The point
estimate show that a unit increase in the coefficient of variation of a professions was associated
with captives from that profession being ransomed between 0.36 and 0.57 days later in the
negotiations. Results for Tetuan, however, are of the “wrong” sign and are not statistically
significant.
3.2.3 Summary of the reduced form analysis
We find that in Algiers, where qualitative evidence suggests higher impatience and transac-
tion costs, the theoretical predictions regarding relationships among release price, length of
negotiations, and amount of uncertainty are generally supported by the data. At the same
time in Tetuan, where qualitative evidence points to lower discount rates and transaction
costs, we do not find empirical support for the predicted relationships. This may be because
the effects we are trying to estimate are small, and we have a relatively small number of
observations (our Tetuan sample is roughly one-half of the size of the Algerian sample), or
because for low amounts of impatience the qualitative predictions of dynamic bargaining
theory are not valid for real world negotiations.
4 Structural analysis
Although the large number of captives has allowed us to investigate the empirical relevance
of some general predictions of bargaining models in a reduced-form manner, such techniques
22
do not allow for a detailed comparison between negotiations at different locations, given that
there are only two of them, and they can differ in many important dimensions (distributions
of buyer evaluations, discount factors, and transaction costs). In this section we investigate
differences in bargaining outcomes between Algiers and Tetuan through structural analysis,
using the model in Cramton (1991) as our theoretical framework.
Our paper is one of the first conducting structural estimation of a dynamic bargaining
model in a buyer-seller setting with asymmetric information. In the context of negotiations
for out of court settlements in medical malpractice cases, Waldfogel (1995) and Sieg (2000)
estimate bargaining models with one round of negotiations, while Watanabe (2004) considers
a finite horizon dynamic bargaining model.29 Merlo et al. (2008) consider the problem of a
homeowner wanting to sell their house, and estimates a model which for simplicity assumes
that potential buyers are bidding automatons, instead of strategic players. Most recently,
Keniston (2011) estimates a model of bargaining with two-sided asymmetric information,
using data generated by a field experiment in the context of bargaining for the fare between
riksha drivers and passengers in India.30 There are important differences between both the
data analyzed and the methodology used in Keniston’s paper and ours. First, while Keniston
(2011) investigates a context in which the stakes for bargaining are low, in our context the
agreed upon price for the release of a hostage was typically multiple of the annual income of
the person. Second, Keniston (2011) assumes uncertainty on both sides, and both regarding
valuations and time costs. Third, Keniston (2011) does not impose a unique equilibrium
selection and allows for the possibility of any equilibrium to be played, but for tractability
of the estimation procedure it only requires beliefs on the opponents’ actions to be correct
along the equilibrium path (along the lines of self-confirming equilibrium in Fudenberg and
Levine (1993)). Instead, we focus on a specific sequential equilibrium in the Cramton model,
29There is also a small earlier literature computing point estimates of parameters of dynamic bargainingmodels, based on US data on wage negotiations between unions and employers (see Fudenberg et al. (1985)and Kennan and Wilson (1989)).30For other field experiments in bargaining contexts, investigating unrelated issues to the current paper,
see Ayres and Sigelman (1995) and Castillo et al. (2011).
23
motivated by general features of our data (no pooling of buyer types after time 0).
Subsection 4.1 summarizes the theoretical model. Subsection 4.2 provides the econo-
metric model specification. Subsection 4.3 describes the estimation procedure. Subsection
4.4 presents the main findings. Lastly, in Subsection 4.5 we conduct welfare analysis and
counterfactual investigations.
4.1 Theoretical model
The model we selected for the structural analysis is the one featured in Cramton (1991),
because it introduces a transaction cost for bargaining (on top of discounting), that is a
fixed cost per unit of time that a player has to incur until the negotiations are over. This
further facilitates the possibility of terminating negotiations in equilibrium, if a party foresees
that they would have to incur too high transaction costs before reaching an agreement.31
Qualitative evidence suggests that both of these features are potentially important in our
context.
Formally, we consider the following specification of the model. We treat each captive as
a unique commodity, initially owned by a monopolist seller. If the seller and the buyer agree
on price p after negotiations with length t, the seller and buyer get the following utilities:
Us = (p− S) · e−rt
Ub = (B − p) · e−rt − cr(1− e−rt)
where S represents the seller’s evaluation (outside option) for the captive, B is the buyer’s
evaluation, r is the discount rate for the parties, and the c is the transaction costs per unit
of time. All of the above parameters are commonly known for the bargainers, besides B.
31As Fudenberg et al. (1985) points out, introducing transaction costs and the possibility of exit to a purescreening model such as in Fudenberg and Tirole (1983) results in only trivial equilibria existing, in whichevery buyer type quits the negotiations immediately. As Perry (1986) shows, negotiations end immediatelyeven if exit is not a possibility, and the party with the lower transaction cost gets all the surplus. Thereforescreening models with transaction costs cannot explain delay in negotiations.
24
Only F (B), the cumulative distribution function of the buyer’s valuation, is known by the
seller.
The above specification entails various simplifying assumptions. One is that the trans-
action cost of the seller is zero. Here we are motivated by the fact that the negotiations
were in corsair territory, where the sellers were not exposed to any kind of danger during the
negotiations. Another is that the we assume the same discount factor for the seller and the
buyer. We apply this restriction mainly for the tractability of the model and the resulting
estimations.
Negotiations involve alternating-move bargaining, with the seller making the first offer.
A party whose turn it is to make a move can delay making the offer, for any amount of time.
Moreover, any party can decide to terminate negotiations and quit, at any point during the
game.
The game has typically many equilibria. We focus on the equilibrium featured in Cramton
(1991). This is the only equilibrium in which buyer types who stay in the game after time 0
fully separate. This is consistent with our observation that the empirical density function of
the times of agreement is fairly smooth, with no obvious candidate for a point of time where
a significant fraction of buyer types are pooled together (see Section 2 of the supplementary
appendix on observed settlement rates).
The equilibrium exhibits the following intuitive structure: The seller makes an offer
immediately, without waiting. Buyer types with high enough evaluations immediately accept
the offer. Buyer types with low enough evaluation turn down the offer and quit negotiations,
to save transaction costs. Finally, buyer types in an intermediate range turn down the initial
offer of the seller, and then delay making the counter-offer, for an amount of time that is
specific to the given type, thereby fully revealing the type of the buyer. Correspondingly, the
counter-offer that the buyer makes after the delay is the unique subgame-perfect equilibrium
price that would prevail in a game where the type of the buyer was commonly known.
Solving for this equilibrium backwards, for a given valuation B in the intermediate region,
25
equilibrium price in the continuation game after the buyer’s type is revealed, hence the
counter-offer of the buyer is:
p(B) =1
2(B + S) +
λ
2
where it is convenient to use λ = cr.
Assume now that the seller’s initial offer is p(b). As shown in Cramton (1991), the optimal
strategy for the buyer given this first offer is:
1. if B b, the buyer accepts without delay;
2. if b0(b) < B < b, the buyer waitsΔ(B, b) = −1rlog(λ+B−S
λ+b−S ) and then make counter-offer
p(B) which is immediately accepted, where b0(b) = λ(−1 + 2(1 + (b− S)/λ));
3. for values B b0 negotiation is terminated.
Lastly, the seller makes his first offer maximizing his expected payoff in the continuation
game:
b ∈ argmaxB(1− F (B))xB + u(B) (3)
s.t. u(b) =b
b0(b)
xB · e−rΔ(B,b)dF (B).
4.2 Econometric specification
The primary outcomes we observe are the duration of negotiations (Δ) and agreed upon price
(p). We do not observe the total value of a captive for the ransoming team (B). The first
observation we make is that we cannot fit our data perfectly by adjusting valuation B for
each observation. Any specification of private values would predict a negative relationship
between price and duration, which does not hold perfectly for the data. Hence, we assume
that we observe the duration and price data with errors. Denote Ω the set of parameters
which characterize equilibrium outcome for a given B. In our main specification we assume
26
that data is generated by the following process involving multiplicative errors:
p = p(B,Ω)eεp
Δ = Δ(B,Ω)eεd
where (εp, εd) are distributed normally with parameters N(0,Σ), independently from the
buyer’s valuation.
While assuming multiplicative errors is standard, and gives an interpretation of the er-
ror term that is independent of the unit of measurement of prices and duration, it has a
disadvantage in our setting: it introduces a sharp distinction between negotiations of zero
vs strictly positive equilibrium duration. In order to smooth out predictions, we assume a
minimal necessary time for negotiations, Δ0, equal to a small exogenous constant (0.5 days).
We change predicted equilibrium durations less than Δ0 (including negotiations predicted
to end with immediate agreement) to be equal to Δ0. In one of our alternative specifica-
tions, presented in the Supplementary Appendix, we assume additive error terms, instead of
multiplicative ones, and do not require a positive minimal necessary time for negotiations.
We also assume that B = v + f , where v is the unobserved base value of the captive for
the negotiating team, and f is the amount of earmarked money that we observe for released
captives. We allow these components to be correlated, for multiple reasons. First, there
could be positive correlation between v and f if the negotiating team had a preference to
release captives with characteristics that were positively correlated with the wealth of the
captive’s family. On the other hand, there could be a substitution effect between v and f ,
with families anticipating a lower v sending more of their own money to release loved ones.
We allow the mean of unobserved component to depend linearly on a set of personal
characteristics X, assumed to be observed by the seller.
We assume that the distribution of earmarked money is normal conditional on some
money having been sent. To sum up, our assumptions about total value distribution are the
27
following:
B =
⎧⎪⎨⎪⎩ υ + f
υ
pr
1− pr
(v
f) ∼ N((
μv
μf
),Λ)
μv = γX , Λ = (σv ρ
ρ σf
)
where pr is probability that some money were sent, μv and σv are the mean and standard
deviation of the general funds distribution, μf and σf are parameters of the family money
distribution and ρ is a covariance coefficient. We exogenously set pr = 0.075 in the analysis,
and estimate the rest of the parameters from the data. The data does not allow us a reliable
way to estimate pr, given that we only observe the fraction of captives with earmarked
money among rescued individuals, not among all captives that were part of the negotiations.
However, our parameter estimates presented below are not sensitive to the value assigned to
pr, as long it is from a reasonable range (0.025− 0.15).In the main body of the paper we only report our basic specification, which only includes
a constant in X, pools observations from all rescue trips to the same location, and includes
rescued captives who based on the data might have been rescued as part of bundled nego-
tiations. In the appendix, as robustness checks for our results, we report estimates from
alternative specifications. In the first one, we include age and region of origin in X.32 Al-
lowing for such differentiation slows down the computations significantly, because for each
vector of observed characteristics a separate equilibrium strategy has to be calculated. In the
32The region of origin could be revealed to the captors by various ways, for example the accent and clothingof the captive, as well as where he was captured.
28
second alternative specification we derive the time period of our data into two subperiods,
and conduct the estimation separately for those. In the third one we exclude captives who
might have been released through bundled negotiations. We also report the results from a
specification in which instead of normal, we assume that the distribution of the buyer’s eval-
uation is lognormal, thereby ruling out negative valuations.33 Finally, we report the results
from a specification in which instead of multiplicative error terms we assume additive error
terms in the data generating process. The qualitative conclusions we get from all the above
alternative specifications are very similar to those from the basic specification.
As a consistency check, we also estimate our basic specification for the Algiers data
during periods of political instability, and show that at least for the largest profession group,
estimated transaction costs and discount rates are both significantly higher during these
periods, consistent with our expectations.
4.3 Estimation procedure
The set of parameters to be identified consists of interest rate r, buyer transaction cost λ,
parameters for the buyer’s value distribution (γ,σv,μf ,σf ), and the variance of error terms
Σ. Since we observe earmarked money, parameters μf and σf can be identified from there.
Identifying the rest of the parameters is less trivial, because the model is highly non-linear
and all parameters influence all dimensions of the observed data. However, some intuition
can be provided.
As described in subsection 4.1, r−1 is a multiplier for duration of negotiations. Intuitively,
the higher the interest rate, the more costly it is for the redemptors to wait, and thus the
stronger is the signal. When interest rate is high, equilibrium duration is low. Therefore
observed negotiation durations help identify r. Short observed durations can also result from
a small variation of values. However, this also leads to small variation in observed prices,
33Note that negative evaluations do not change the qualitative features of the equilibrium, as the buyerterminates negotiations immediately both when having a negative evaluation and when having a smallpositive evaluation.
29
helping to separate the effects of the interest rate and the variation in values. The trans-
action cost parameter is identified from prices, because λ/2 enters as an additive constant
in threshold levels b and b0. Higher transaction costs lead to higher prices and a smaller
signalling region, thus to smaller variation in duration and prices. Comparing it with the
effect of interest rate, the latter influences only variation in duration, while transaction costs
influence variations both in duration and price. The mean of the unobserved component also
increases prices, but it extends the signalling region [b0, b], as opposed to shrinking it.
Formally, we use a standard maximum likelihood approach for model estimation. How-
ever, additional complexity arises because the solution for the equilibrium strategy cannot be
derived in closed form. We construct a numerical likelihood function through the following
steps.
1. First, we estimate the distribution of earmarked money using the subsample of ob-
servations with positive earmarked money. Here we deviate from the pure theory of
maximum likelihood, because we ignore information about how these parameters influ-
ence optimal strategies. However, we have enough observations to estimate it precisely,
and since only less than 10% of captives had earmarked money, its influence on the
seller’s strategy is small.
2. Having set values for other parameters, we numerically solve for the seller’s optimal
first offer b using equation (3). For the normal distribution we are able to obtain closed
form solution for the integral, so optimization on this level is fast and precise.
3. We then construct a grid of unobserved valuations in the relevant region (about 1000
bins with flexible step lengths). For each B and each observation we calculate the
predicted price and duration, and compute errors.
4. After constructing the likelihood function for each value, we take expectation with
respect to B. This is a one dimensional integral, so we use Simpson’s approximation
rule rather than simulation.
30
5. Finally, we sum the likelihoods of all observations to obtain the total likelihood func-
tion.
An important feature of this problem is that the parameters of the error terms do not
influence equilibrium outcome, and thus one may optimize the objective function with respect
to Σ without reestimating the equilibrium each time. Using this inner circle makes the
optimization slower, but more robust and precise. We use this technique for all reported
tables. We calculate standard errors by a bootstrap procedure, using repeated estimation of
100 different draws from the original sample.
We conclude this subsection by formally describing the construction of the likelihood
function.
There are two types of captives: with and without earmarked money. For observations
without earmarked money probability of given realization is:
log(P (fm = 0)P (rescued = 1, pi,Δi)) =
log(1− pr) + log(∞
b0
f(εp(p,Δ), εd(p,Δ))dF (B))
,where f is pdf of N(0,Σ) and F (B) is cdf of unobserved value distribution with parame-
ters μv and σv. Here εp and εd are differences between logarithms of predicted and realized
outcomes:
εp = log(p)− log(p(B))
εd = log(Δ)− log(Δ(B, b) +Δ0)
The Jacobian of this transformation is
|∂(ξp, ξd)∂(p,Δ)
| = |1p0
0 1Δ
| = 1
pΔ
31
This term doesn’t depend on parameter values and unobserved value, so we can ignore
it in maximization. The corresponding probability for observations with earmarked money
is similar:
log(P (fm > 0)P (fmi|fm > 0)P (rescued = 1, pi,Δi)) =
log(pr) + log(ϕ(fmi − μf
σf)) + log(
∞
b0
f(εp(p,Δ), εd(p,Δ))dF (B))
,where ϕ is the pdf of the standard normal distribution and F (B) is the cdf of total
value, which is normal with parameters μv + fmi and σv. To complete the construction of
likelihood function we take into account that we observe only successful negotiation, thus to
make conditional probability we divide each probability by probability of being rescued:
log(P (observed = 1)) = log(1− Φ(b0 − μ1 − fmi
σ1))
The total log-likelihood function has the usual additive form, because we assume that ob-
servations are independent:
logL =i
logLi
4.4 Basic results
We estimate the equilibrium separately for each of the three largest recorded professions
among captives, separately for the two corsair strongholds. These professions - fishermen,
soldiers and sailors, - together constitute roughly one third of the whole sample. We exclude
all government owned captives (about 25% of rescued captives). This leaves about 1500
observations, and the average sample size for our estimations is roughly 250.
Estimation results are presented in table 3. For most professions the estimated interest
rate is twice as high in Algiers as in Tetuan. This mainly results from the fact that nego-
tiations in Algiers were shorter. However, estimated transaction costs are about the same
32
across territories and professions. The above suggest that in Algiers negotiators were more
concerned about the danger the captives faced in captivity, and about negotiations breaking
down for exogenous reasons, rather than about their own personal safety.
Estimated correlation coefficients between general fund’s money and earmarked money
for the majority of the regressions is negative and over 0.5 in absolute value. This suggests
that in most cases earmarked funds partially substituted other sources of funding. However,
the finding should be evaluated taking into account that the standard errors are very high.
Figure 1 shows equilibrium price and time to negotiation trajectories based on our esti-
mations for each profession. The first thing to notice is that equilibrium prices are higher in
Tetuan, and signaling regions are shifted to higher values.
For all four professions, on the equilibrium path an additional day of delay in Algiers
resulted in higher price discounts, because higher interest rates made signaling faster. These
graphs also reflect that observed negotiations in Tetuan were longer on average. Our esti-
mates predict a maximum duration of 20 to 35 days in Tetuan depending on profession, and
of 15 to 20 days in Algiers.
Finally, table 3 also reports the probability a negotiation was terminated. For all profes-
sions but fisherman this probability is larger in Algiers, which is in line with the available
qualitative and empirical evidence.34 Our estimates for the probability of termination are
very high, for sailors and soldiers for Algiers the estimated model predicting 70% of the
negotiations to be terminated. However, our data is not suitable for reliable estimation here.
The main reason is that we do not observe failed negotiations, and we cannot identify non-
parametrically the value distribution below the threshold b0. Thus, we are left to rely purely
34To analyze this question in a reduced-form manner, we created a data set of captives for whom at leastone previous negotiation failed. The main idea here is that if the same individual appears as earmarked inseparate trips, then we know that he is alive and that negotiations failed in the previous trip. After usingextremely conservative matching criteria (detailed in the appendix) that appears to have done a good job ofminimizing type-II error (that is, only keeping individuals for whom there is a high certainty that (at least)one previous negotiation failed), we were left with 63 individuals. Of these individuals (for whom at least onenegotiation failed) 54 (85.71%) were captive in Algiers. Formal statistical tests comparing this proportionto the percentage of total rescued captives from Algiers (73%), the proportion of earmarked captives fromAlgiers (74%) or the proportion of rescued earmarked captives from Algiers (70%) all reject equality at the1% significance level. This result is consistent with a larger number of failed negotiations in Algiers.
33
on our parametric specification.
Besides parameter values estimates, it is interesting to know what part of variation in
the data we can explain with this model, a measure similar to R2 in linear regression. If the
outcome is y and its predicted value is y, this exercise would involve comparing (y− y)2 and(y − y)2. However, in our case y is not a constant, but a random variable, since underlying
value B is unknown. Thus, we use the mean of conditional distribution E[y|y] instead ofy in the formula above. For a given outcome and a prior distribution of valuations one
may calculate the posterior distribution of B given y using Bayes’ rule. The graphical
interpretation of this would be finding the closest point from the outcome (p,Δ) to the line
(p(B,Ω),Δ(B,Ω)), and taking the corresponding B as posterior estimate of B. From it
it we calculate posterior distribution for equilibrium outcome y, and compute its mean to
compare variances.
The estimated share of predicted variation in outcomes is presented in the last two rows
of the tables before the number of observations as EVduration and EVprice, which stands for
estimated variance in duration and price outcomes. These numbers are quite different for
different professions, but the average for duration is around 0.65 and for price it is 0.30.
The difference is noteworthy in the light of the error components being very similar. The
interpretation is that for plausible parameter values signaling does not give as much variation
in prices as it gives in duration of negotiations. Overall, we explain variation in the duration
of negotiations quite well, but only satisfactorily for prices.
4.5 Welfare analysis and counterfactuals
Given the estimated parameter values of the underlying model, we can calculate the equi-
librium expected division of surplus between the bargaining parties, and various sources of
losses. The total expected surplus is simply the expected buyer valuation:
Total_surplus =∞
0
BdF (B)
34
The seller surplus is already featured in the definition of equilibrium strategy:
Seller_surplus =b
b0
x(B)Be−rΔ(B,b)dF (B) + (1− F (b))x(b)b
The buyer gets the present value of his realized surplus at the time of agreement, minus
total transaction costs:
Buyer_surplus =b
b0
(1− x(B))Be−rΔ(B,b)dF (B) +∞
b
(B − x(b)b)dF (B)−
−b
b0
λ(1− e−rΔ(B,b))dF (B)
Finally, the expected social costs from various sources are:
Termination_costs =b0
0
BdF (B)
Delay_costs =b
b0
B(1− e−rΔ(B,b))dF (B)
Transaction_costs =b
b0
λ(1− e−rΔ(B,b))dF (B).
Calculation of surpluses using estimated parameters for each profession is presented in
table 4. In most cases, total surplus is much higher in Tetuan, reflecting that the estimated
mean buyer valuations tend to be higher in Tetuan. Comparison of surpluses extracted by
sellers and buyers shows that in most cases they obtained approximately equal shares of total
surplus, about 40% each. Delay costs range from 3%-5% for Sailors to 6%-7% for Fishermen
and Soldiers. Transaction costs were relatively small for all professions and all territories
and constituted about 2% of total surplus. Termination costs were also quite high: about
10% in most cases. Overall, this calculations suggest that negotiations were quite efficient,
since total costs were only about 20% of total surplus.
35
Next, we address the question that how much sellers would have gained by selling captives
in bundles, focusing only on captives for whom we can confidently rule out that they were
rescued as part of a bundle. Here we assume a bundle of 10 captives. Columns 5-8 in table
4 present our estimates. We find that selling in bundles would have mostly benefited the
seller, increasing his share from 40% to 50%-60% of total surplus. Buyers would have lost,
but not much: their average share would have fallen from 40% to 35%. One can see that all
types of costs are lower for trading in bundles, hence trading in bundles would have increased
realized social surplus.
We also examine the possibility that the seller could give a take it or leave it offer to the
buyer, or commit to leave negotiations if the initial offer is declined. The resulting estimated
distribution of surplus is presented in the last four columns (9)-(12) of table 4. As expected,
in all cases this commitment power would have benefited the seller. Since all negotiations
end immediately in this scenario, transaction and delay costs are zero. However, termination
costs are very high: 20% to 30%, leading to lower realized social surplus than in the basic
model. In all cases buyer would have been much worse off, and would have received only
25%-30% of the surplus.
5 Conclusion
While the Barbary Corsairs have long ceased to roam the seas, the practice of ransoming
captives continues today. The empirical results presented in this paper suggest that existing
theoretical work on bargaining under asymmetric information can inform our understanding
of such ransom negotiations.
Qualitative evidence suggests the relevance of the theory by showing that the negotiating
teams’ behavior was consistent with many of the theoretical predictions. These teams worked
to conceal the identities of the captives they wanted most, and delayed the ransom of many
captives to obtain lower prices.
36
Reduced-form empirical analysis using data on ransom prices and time to rescue for
captives ransomed between 1575 and 1739 bolster this qualitative evidence in Algiers. In
particular, delay within ransoming trips led to a substantial decrease in prices, reducing
the price by an estimated 4% per each day. In Tetuan, however, we find considerably less
support for the empirical predictions.
To further investigate the differences between Algiers and Tetuan, we structurally es-
timate the model in Cramton (1991). Our estimates suggest that higher interest rates in
Algiers drive most of the observed differences in bargaining outcomes between Tetuan and
Algiers. This result, in turn, emphasizes the importance of the rate of physical decay in cap-
tivity, as well as the possibility of exogenously terminated negotiations. We use our structural
model for welfare analysis, and find that the bargaining institution was remarkably efficient,
allowing parties to realize more than 80% of total social surplus.
Our investigation also suggest avenues for future research, such as investigating the issue
of bundling in dynamic bargaining.
Finally, the historical response of many European powers to the Corsairs may provide
insights into negotiating with Somali pirates (and possibly other criminal groups). For
example, the historical preference for centralized ransoming organizations suggests that such
institutions might aid negotiations with pirates today by both enabling negotiations for
multiple cargoes at once and by reducing transaction costs (which, besides saving costs
directly, improves the bargaining power of the negotiating team).
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(a) Equilibrium prices for differentprivate values
0 5 10 15 20 25 30 35 40800
1000
1200
1400
1600
1800
2000
2200
equilib
rium
pri
ce
equilibrium duration
Sailors
Algiers
non-Algiers
0 5 10 15 20 25 30 35 40600
800
1000
1200
1400
1600
1800
2000
equilib
rium
pri
ce
equilibrium duration
Fishermen
Algiers
non-Algiers
0 5 10 15 20 25 30 35 40600
800
1000
1200
1400
1600
1800
2000
equ
ilib
rium
pri
ce
equilibrium duration
Soldiers
Algiers
non-Algiers
(b) Equilibrium price-delay schedule
Figure 1: Private values, delay and pricesGraph details the equilibrium price and equilibrium time to negotiationtrajectories by profession.
Tab
le1:
Summary
Sta
tistics(forre
scued
individuals
unless
oth
erw
isenoted)
Algiers
Tetu
an
Variable
Description
Mean
St.dev.
Min
Max
NM
ean
St.dev.
Min
Max
N(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
Genera
l
YearsCaptive
Years
captive
6.10
6.45
0.02
60
7227
4.00
3.98
037
2335
Price
(SilverReales)
1658
2257
081458
7317
1973
1400
125
21439
2703
Earm
ark
ed∗
=1ifearm
arked
0.09
0.29
01
7398
0.11
0.31
01
2716
Moneysent(w
/NotRescued)
(Silver
Reales)
1224
4298
19
126440
1676
1083
1372
2515672
586
Moneysent(O
nly
Rescued)
(Silver
Reales)
1400
5299
19
126440
677
1103
1474
3415672
297
Bundled
=1ifbundled
0.43
0.50
01
7317
0.62
0.49
01
2703
Castile∗
=1iffrom
desired
areas
0.49
0.50
01
3824
0.68
0.46
01
1524
DaysE
lapsed
Daysto
agreem
ent
8.31
20.08
0272
4337
11.30
20.00
0284
2044
Govt
Captive
owned
byGov
t0.17
0.38
01
7398
0.25
0.43
01
2716
Fem
ale
∗0.05
0.22
01
7398
0.05
0.23
01
2716
Child∗
0.05
0.22
01
7253
0.08
0.27
01
2358
Age
Years
35.88
14.06
0.11
100
7253
33.68
14.21
0.83
94
2358
AgeC
apt
Agewhen
captured(Y
ears)
29.79
13.07
099.33
7185
29.97
13.66
091
2286
Pro
fession
Fisherman
0.12
0.33
01
7398
0.11
0.31
01
2716
Soldier∗
0.18
0.38
01
7398
0.17
0.37
01
2716
Majesty∗
InKing’s
Service
0.06
0.23
01
7398
0.06
0.23
01
2716
Shepherd
0.01
0.08
01
7398
0.01
0.12
01
2716
Sailor
0.06
0.23
01
7398
0.06
0.23
01
2716
Peasant
0.01
0.08
01
7398
0.02
0.14
01
2716
Indias∗
Enroute
toAmericas
0.01
0.09
01
7398
0.01
0.07
01
2716
Cleric∗
0.02
0.14
01
7398
0.01
0.12
01
2716
Other
Other
Identified
0.01
0.09
01
7398
0.02
0.12
01
2716
NI
NotIdentified
0.54
0.50
01
7398
0.54
0.50
01
2716
Notes:
∗den
otesacategory
thatwasfavoredbytheransomingmissions.
Theva
riable
Day
sElapsedis
equalto
0thefirstday
an
agreem
entis
reach
ed.See
textfordetails.
Tab
le2:
Reduced-F
orm
Estim
atesbyLocation
Algiers
Algiers
Algiers
Tetu
an
Tetu
an
Tetu
an
(1)
(2)
(3)
(4)
(5)
(6)
PanelA:Pricesand
Delay(dep
endentvariable
is100*logarithm
ofransom
price)
Delay(D
ays)
-3.93
-3.77
-1.98
0.01
-0.07
0.03
(0.85)
(0.84)
(0.60)
(0.10)
(0.07)
(0.14)
N308
130
2893
015
0512
9852
6
PanelB:Pricesand
Private
Inform
ation(dep
endentvariable
is100*logarithm
ofransom
price)
Earm
ark
ed35.04
36.26
33.05
33.96
32.44
28.42
(4.83)
(4.38)
(5.22)
(4.84)
(4.27)
(6.37)
N308
130
2893
015
0512
9852
6
PanelC:Delayand
Private
Inform
ation(dep
endentvariable
isdelay
inday
s)Earm
ark
ed-0.38
-0.39
-0.13
-0.59
-1.19
-2.04
(0.17)
(0.19)
(0.10)
(1.19)
(1.27)
(1.31)
N314
330
8693
915
1413
0352
8
PanelD:Delayand
Uncertainty
(dep
endentvariable
isdelay
inday
s)S.D
./M
ean
0.36
0.42
0.57
-0.20
-0.05
-0.18
(0.13)
(0.17)
(0.28)
(1.95)
(2.08)
(2.16)
Mean/100
0.00
0.01
-0.03
0.23
0.28
1.85
(0.00)
(0.01)
(0.02)
(0.14)
(0.15)
(1.69)
N306
030
0590
114
8212
7351
7
Con
trols
No
Yes
Yes
No
Yes
Yes
Sample
All
All
Non-D
elayed
All
All
Non-D
elayed
Notes:
Delay
measuresthenumber
ofday
selapsedsince
thestart
ofnegotiations.
Earm
arked
isanindicatorvariable
equalto
oneifthecaptivehadmoney
sentfrom
Spain
thatwasearm
arked
forhis
rescue.
S.D
./M
eanis
thecoeffi
cien
tofva
riationofthe
amountofearm
arked
money
bycalculated
byprofession.
Mea
n/100is
themean
ofthesamequantity
divided
by100(and
isgiven
inconstantsilver
reales).N
isthenumber
ofobservations.
Controls
includeacaptive’stimein
captivity,
ageatcaptivity,
agen
der
dummy,
adummyeq
ualto
oneifthecaptivewasyounger
than16when
ransomed
andprofessiondummies(thesedummies
are
omittedin
panel
D).
The“Non-D
elay
ed”sample
consistsofallcaptives
whowereransomed
onthefirstobserved
ransoming
exped
itionfollow
ingtheircapture.Standard
errors
clustered
byransomingtrip
are
inparentheses.
Tab
le3:
Structu
ralEstim
atesbyPro
fession,Location
Sailors,
Sailors,
Sailors,
Fisherm
en,
Fisherm
en,
Fisherm
en,
Soldiers,
Soldiers,
Soldiers,
Tetu
an
Algiers
Diff
erence
Tetu
an
Algiers
Diff
erence
Tetu
an
Algiers
Diff
erence
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
λ1128
471
657
681
365
315
669
425
244
(120)
(44)
(128)
(234)
(62)
(242)
(88)
(58)
(105)
μv
1803
118
168
519
6014
8847
211
9662
1134
(710)
(439)
(834)
(914)
(264)
(951)
(830)
(593)
(1020)
σv
2280
1634
646
1970
944
1026
2295
1989
306
(638)
(247)
(687)
(610)
(149)
(628)
(477)
(294)
(560)
μf
985
925
6098
592
560
985
925
60(173)
(114)
(207)
(173)
(114)
(207)
(173)
(114)
(207)
σf
763
658
105
763
658
105
763
658
105
(102)
(80)
(130)
(102)
(80)
(130)
(102)
(80)
(130)
ρ0.54
-0.79
1.33
-0.65
-0.48
-0.17
-0.78
-0.85
0.07
(0.62)
(0.44)
(1.19)
(0.65)
(0.22)
(0.67)
(0.62)
(0.39)
(0.73)
r0.015
0.028
-0.013
0.027
0.042
-0.015
0.017
0.031
-0.014
(0.004)
(0.004)
(0.005)
(0.013)
(0.006)
(0.014)
(0.002)
(0.007)
(0.007)
c17.12
13.40
3.72
18.28
15.24
3.04
11.67
13.36
-1.69
(3.87)
(1.21)
(4.04)
(4.89)
(1.02)
(5.00)
(1.38)
(2.46)
(2.82)
pr t
erm
ination
0.53
0.70
0.43
0.28
0.57
0.70
EVduration
0.74
0.82
0.53
0.85
0.75
0.53
EVprice
0.26
0.19
0.38
0.26
0.30
0.18
N93
146
149
359
211
454
Notes:
λisthemonetary
equivalentoftransactioncostsper
day
(given
inconstantsilver
reales).μvisthemeanofthedistribution
ofmoney
whichcould
bepaid
from
thegen
eralfundsforacaptive(given
inconstantsilver
reales).σvis
thestandard
dev
iation
ofthis
distribution(given
inconstantsilver
reales).μfis
themeanofthedistributionofearm
arked
fundssentforaparticular
captive(given
inconstantsilver
reales).σfis
thestandard
dev
iationofthis
distribution(given
inconstantsilver
reales).ρis
the
covariance
betweenearm
arked
andgen
eralfunds.
ris
theinterest
rate.cis
themonetary
equivalentoftransactioncostsper
day
(given
inconstantsilver
reales).pr t
erm
inationis
theprobabilitythenegotiationwasterm
inated.EVdurationgives
thepercentof
thevariationin
durationsthatissuccessfullyex
plained
bythemodel
andEVpric
egives
thepercentofvariationin
pricesex
plained
.N
gives
thenumber
ofobservationsusedin
estimation.Standard
errors
inparentheses.
Tab
le4:
Distribution
ofSurp
lusbetw
een
theCorsairsand
theRanso
mingTeam
Estim
ated
Ifbundled
in10
Takeit
orleaveit
Tetu
an
Algiers
Tetu
an
Algiers
Tetu
an
Algiers
Values
Share
Values
Share
Values
Share
Values
Share
Values
Share
Values
Share
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
PanelA:Sailors
Sellersurplus
931
0.43
299
0.40
1295
0.59
422
0.54
992
0.46
326
0.44
Buyersurplus
821
0.38
282
0.38
651
0.30
243
0.31
595
0.27
205
0.27
Delay
costs
67
0.03
37
0.05
33
0.01
60.01
00.00
00.00
Transactioncosts
31
0.01
14
0.02
21
0.01
40.01
00.00
00.00
Terminationcosts
328
0.15
117
0.16
183
0.08
110
0.14
590
0.27
217
0.29
Totalcosts
425
0.20
168
0.22
237
0.11
120
0.15
590
0.27
217
0.29
Total
217
71
749
121
821
785
121
771
749
1PanelB:Fisherm
en
Sellersurplus
919
0.42
684
0.44
1218
0.56
861
0.55
1023
0.47
779
0.50
Buyersurplus
846
0.39
620
0.40
768
0.35
595
0.38
615
0.28
410
0.26
Delay
costs
140
0.06
115
0.07
106
0.05
85
0.05
00.00
00.00
Transactioncosts
44
0.02
32
0.02
42
0.02
25
0.02
00.00
00.00
Terminationcosts
234
0.11
117
0.07
59
0.03
50.00
545
0.25
380
0.24
Totalcosts
418
0.19
265
0.17
207
0.09
114
0.07
545
0.25
380
0.24
Total
2183
115
681
2193
115
701
2183
115
681
PanelC:Soldiers
Sellersurplus
682
0.41
329
0.39
931
0.54
465
0.52
755
0.45
372
0.44
Buyersurplus
650
0.39
333
0.39
574
0.34
298
0.33
484
0.29
245
0.29
Delay
costs
104
0.06
60
0.07
56
0.03
18
0.02
00.00
00.00
Transactioncosts
34
0.02
18
0.02
28
0.02
10
0.01
00.00
00.00
Terminationcosts
214
0.13
114
0.13
120
0.07
104
0.12
443
0.26
237
0.28
Totalcosts
351
0.21
192
0.22
204
0.12
132
0.15
443
0.26
237
0.28
Total
168
31
854
117
091
896
116
831
854
1
Notes:
Sellersu
rplusis
expectedvalueofthetransactionto
seller.Buyer
surp
lusis
expectedva
lueofthetransactionto
buyer.
Delaycostsis
expecteddep
reciationofthecaptiveduringnegotiationprocess.Tra
nsa
ction
costsare
expectedcostsofwaiting
fornegotiationsto
finishfortherescueteam.Termination
costsis
expectedva
lueofcaptives
forwhichnegotiationsterm
inated
withouttransactionto
takeplace.Totalcostsis
thesum
ofDelaycosts,
Tra
nsa
ctioncostsandTermination
costs.
Allva
lues
are
given
inconstantsilver
reales.