Learning using Incentives: Evidence from a Drug
Selling Gang in Singapore∗
Kaiwen Leong†, Huailu Li‡and Haibo Xu§
Abstract
In a Singaporean drug-selling gang, an assistant decides how much to
charge a pusher for drugs. Then, the pusher whose drug selling ability is
unknown to anyone decides whether to call potential customers to estimate
demand for the drug. The pusher purchases drugs from the assistant and
payoffs are realized. This process repeats itself. We build a model to capture these dynamics. Using data about the gang’s historical transactions, we
find that both assistant and pusher engage in continuous learning about the
pusher’s ability to sell drugs. The assistant’s price offer is affected by what is
learnt about the pusher’s ability and exhibits a non-monotonic pattern.
∗
We have numerous people to thank for the invaluable roles they played in our academic journey. In no particular order, we would like to express our sincerest gratitude to the many who have
helped bring the insights in this paper to light. To begin with, we are very grateful to Jimmy
Chan, Kevin Lang, Barton Lipman, Stephen Holland, Erik Snowberg and seminar participants at
Fudan University, Nanyang Technological University, Singapore Management University, National
University of Singapore, Central Narcotics Bureau of Singapore for their helpful comments and suggestions. We also wish to thank the selfless individuals at various authorities and organizations for
valuable insights from the countless hours they spent working to give ex-offenders a second chance.
These include Philip Lim (Christian Counseling Services), Elvis Overee (Industrial & Services Cooperative Society Ltd), Teo Tze Fang (Singapore Corporation of Rehabilitative Enterprises), Chan
Ying Lock (Social Enterprise Hub), Walter Teo (Micro Credit Business Scheme), Chia (The Helping
Hand), Peh Beng Seng (PERTAPIS Halfway House), Serene Goh (Tanah Merah Prison), and Kochitty Abraham (Singapore Anti-Narcotics Association). Leong also thanks Nanyang Technological
University for start-up funding and its support for this research. The usual caveat applies.
†
Division of Economics, Nanyang Technological University. Email: kleong@ntu.edu.sg
‡
Department of Economics, Fudan University. Email: huailuli@fudan.edu.cn
§
Department of Economics, Fudan University. Email haiboxu@fudan.edu.cn
1
Introduction
There has been debate over whether criminals are rational individuals. Many studies
in criminological literature suggest that individuals involved in crimes are irrational,
since they act on their impulses (e.g., Taft and England, 1964; Gottfredson and
Hirschi, 1990; Kahneman and Tversky, 1997). On the other hand, in his seminal
paper, Becker (1968) argues that criminals respond to incentives and punishments
and therefore a framework of cost-benefit analysis applies to criminal activities.
Thereafter, many studies have found evidence of Becker’s claims that criminals are
rational individuals (e.g., Stigler 1970; Ehrlich, 1973). In this paper, we closely examine a dataset of illegal drug-selling transactions and investigate whether criminals
involved in these transactions make decisions rationally. We show that the behavior
of the individuals in our dataset matches the predictions of our theoretical model
which assumes that individuals are rational.
The dataset of transactions is obtained from a drug-selling gang in Singapore.
The gang, which was in operation during the 1990s, primarily consisted of three
layers: a boss (who was the leader of the gang), some assistants (who obtained drugs
from the boss and then sold to individual pushers), and many pushers (who bought
drugs from assistants and then sold to retailers and customers)1 . The transactions
recorded in our dataset happened between the assistants and pushers. Due to the
roles these participants played in the supply side of the market, as well as the
illegality of transactions they made, a pusher typically transacted with a single
assistant repeatedly. These repeated interactions provided a natural environment
for the investigation of whether there is learning by the participants; if so, how
a assistant’s pricing strategy reflect the learning and how a pushers’ selling effort
adjusts to the learning process accordingly.
Interviews with market insiders show that pushers differ in their abilities to sell
drugs in the market. Some pushers can sell consistently large quantities of drugs and
thus generate considerable profits for both the pusher and assistant to share, while
others may fail to sell so well and their selling revenues may not be enough to cover
the costs that the pushers and assistants need to bear. Building upon these aspects,
we consider a learning model where an assistant repeatedly trades with a pusher.
Within each period, the assistant decides to set a selling price (high or low) for a
drug. The pusher, after observing the price, decides whether to work or shirk, where
working means calling potential clients to see if they are interested to purchase the
drug. A transaction between the players is then made and payoffs are realized. The
pusher’s ability to sell drugs is either good or bad. If the pusher works, a good
pusher always generates a positive payoff for both players, and each one’s payoff is
higher when the price is lower; a bad pusher has some probability he can sell just
well as a good pusher, with the remaining probability he cannot. Specifically, when
the bad pusher does not sell well, if the price is high then the assistant and pusher’s
payoffs are positive and negative, respectively. Conversely, if the price is low then
their payoffs are negative and positive, respectively. On the other hand, if the pusher
shirks, the outcome in the period is independent of ability and price, and each player
obtains a payoff zero. At the outset, neither player knows whether the pusher is good
1
In local terms, the boss is called “LongTouLaoDa”, and each assistant is called “ZuoYouShou”.
1
or bad. Instead, they hold a common prior regarding the pusher’s ability. Over time,
the players update their beliefs based on the decisions and outcomes they observe,
and adjust their future decisions according to their learning process.
This payoff structure indicates that if the pusher is good, then the players have
aligned preferences in the sense that both of them prefer the pusher to work at the
low price; but if the pusher is bad, the players have conflict of preferences in the
sense that the assistant prefers the pusher to work at a high price while the pusher
prefers to work at a low price. These preferences give rise to a unique Markov perfect
equilibrium of the dynamic game. In equilibrium, the pusher is induced to work if
and only if the players hold a belief that is larger than a cutoff value. Intuitively, if
the players’ belief is so low that they expect that the pusher is more likely to fail
to sell well when he works, then the benefit of learning about the pusher’s ability
generated from working is outweighed by the current loss in payoffs, and therefore
it is optimal for the players to stop learning. Our result also predicts that for the
range of beliefs that the pusher works, the assistant sets prices non-monotonically
in the beliefs.
Specifically, if the belief that the pusher is good is very high, then we observe the
same behavior as we would get if the pusher is definitely good. Namely, the assistant
sets a low price, and the pusher works. Of course, if the transaction fails, then the
payoffs are 0 thereafter. If not, the belief that the pusher is good goes up and the
trend continues as such. Thus, there exists a largest cutoff value, and the assistant
will set a low price for beliefs above this value. Below this largest cutoff value, the
assistant is skeptical of the pusher’s abilities. Thus, the assistant prefers a set a high
price so as to avoid incurring a loss. However, the assistant cannot set a high price for
all beliefs below the largest cutoff. There exists a second belief cutoff value whereby
should the assistant set a high price for any belief value below this cutoff, the pusher
is skeptical of his ability to earn profits if he chooses to work. If the assistant wants
the pusher to work, she has no choice but to share the burden of the costs borne by
the pusher by setting a low price. As the beliefs starts to decrease from the second
cutoff value, there exists a third cutoff value whereby below this value, the assistant
will set a high price. The assistant does so because she knows that setting a high
price will not cause the pusher to shirk. The pusher is willing to work hard because if
he succeeds, the continuation payoffs for the pusher are positive. Since the assistant
also prefers to set a high price, she will do so. Following this logic, the assistant
will then set a low price below the next cutoff followed by a high price. Thus, the
assistant sets prices non-monotonically.Our model also predicts that if the players’
initial belief about the pusher’s ability is drawn from the range that the pusher is
induced to work, then there is full revelation of ability over time. This is because if
the pusher is induced to work at the very beginning, then should the pusher fail to
sell well once over time, this immediately reveals that the pusher is bad. Similarly, if
the pusher sells well all the time, this gradually reveals that the pusher is good. At
the end of section 4, we also explain why the non-monotonic pricing strategy used
by the assistant is not an artifact of the assumption that there are only two possible
prices. Even if we allowed the assistant to choose from a continuum of prices, the
results remain unchanged.
We have transaction data between assistants and pushers in the gang’s formative
2
years. It consists of 2956 trading transactions2 between 354 pushers and the assistants. Since at least two orders of different types of drugs occurred in each trade, we
have a total of 9132 orders. In addition, we have also obtained detailed individual
characteristics of the pushers before they started trading with the assistant. The
key advantage of our data is that we have the assistants’ assessment of the pushers’
ability after each trade is completed and the level of effort exerted by the pusher
in each trade, which is proxied by the number of phone calls he made to the end
consumers.
With this rich dataset, we are able to verify that both the assistants and pushers
engage in learning about the pushers’ abilities, and the assistants’ pricing strategy
reflects their learning. They applied a non-monotonic pricing tactic based on a
periodically updated ability assessment. Assistants learn about pusher’s ability using
both the pusher’s individual traits and trading outcomes. In particular, the sales in
the most recent trade is a key determinant of the assistant’s evaluation of pusher’s
ability, which helps to verify that the MPE concept used in the theory model is
plausible. On the other hand, we find that as long as the pushers stay in the trade, he
learns. The trading outcome is observable to both the assistant and pushers, and the
trading prices we observed are the equilibrium prices that the assistant and pusher
both agree upon. We find the pusher’s effort is associated with the assistant’s ability
rating and the effort level responds to the assistant’s price offers. This evidence
indirectly supports the pusher’s learning about his own ability. Specifically, high
prices discourage effort.
An assistant’s assessment of every pusher’s ability is measured on a scale of 1
to 10, and it is used as good proxy for the belief held by the assistant that the
pusher is a good type as defined in our theoretical model. We take pushers with
a rating of 5 and 6 as average ability pushers, and use it as the reference group
for comparison. We have the following findings: compared with pushers with a
rating below 5, namely those below average, these average pushers receive a 3-5%
price discount; top pushers (those with a rating of 9 and 10) receive a 10-13% price
discount compared with average pushers; most interestingly, we observe assistant
requesting a 2% price premium on pushers with an ability rating of 7 and 8 when
compared to average pushers. The assistant’s non-monotonic pricing strategy is
consistent with our theoretical predictions. The rationale behind non-monotonic
pricing is as follows: the assistant considers pushers with a rating of 7 and 8 as good
but not top pushers. The assistant also knows that she will not cause these pushers
to shirk just because of the price premium she is asking for, because the pushers
with this rating have lower expected loss compared to the pushers with a rating of
5. In fact, the assistant knows that if these pushers are indeed of high ability, they
will eventually become top pushers down the road, and it will not be too late to give
them lower prices when they prove themselves to be top performers. By doing this,
the assistant can protect herself from losing money (i.e with a price discount) due to
an incorrect assessment of the pusher’s type in the current period. The risky nature
of the market also motivates the assistant to care more about the short run gain
2
A trade refers to the trading interaction between an assistant and a pusher occurred on a
particular day; Pushers may buy different types of drugs in a trade. For example, if purchase of
Ice and Ketamine take place in a trade, then we say there are 2 orders in a trade.
3
rather than potential long run loss. In a competitive labor market, employers set
wages equal to the marginal productivity of the worker. That implies that assistant
should assign a monotonic price to the pushers with different ability. Because of the
monopoly power of the drug gang, the pushers face high search and switching cost.
It helps us to understand why the non-monotonic pricing can be sustained in the
drug market.
The dataset from the drug markets have unique advantages for the study of
learning and price offers in trading interactions. In many formal markets, the change
of prices may be caused simply by bargaining between the trading parties, especially
when some party can learn to bargain or accumulate bargaining skills in a better
way. With the existence of bargaining, it is challenging to identify how sellers or
employers price the product or make wage offers based on learned productivity of
the buyers or employees. In the drug market in Singapore, the monopoly power of
the drug gang grants almost no bargaining power to the pushers, which allows us to
better identify how assistants make price offers based on the learned ability of the
pushers. More than this, we have fairly precise assessments of the pushers’ selling
abilities and selling effort for transactions, which enable us to better disentangle the
effect of ability from the effect of effort on the pushers’ selling performances. Lastly,
unlike transactions in many formal markets in which prices are governed by predetermined long-term contracts or governed by laws and regulations, transactions
occurring in the illegal drugs market are not subject to such restrictions and therefore
prices can capture the trading parities’ learning processes more accurately.
Our model is related to the literature on the bandit problem and experimentation.
Rothschild (1974) studies a firm’s dynamic pricing problem with experimentation.
The firm is initially uncertain about its true market demand and so has to learn
about it. Prices differ in their current returns to the firm as well as in their informativeness regarding the firm’s true market demand. So the firm’s problem is
to optimally experiment prices to learn more about the true demand while maximizing its expected discounted profits. Since then, studies extend this analytical
framework to various economic settings, ranging from learning and pricing in markets (Bergemann and Valimaki (1996), Keller and Rady (1999), Bar-Isaac (2003),
etc) to financing and compensation in organizations (Bergemann and Hege (2005),
Manso (2011), Horner and Samuelson (2013), etc), and many others in between. In
this literature, the key point for examination is to determine the optimal trade-off
between maximizing for the period versus experimenting for information to use in
the future. When multiple parties with conflicts of interests are involved, incentive
problems may arise and optimal experimentation is further affected by the ways
that parties interact with others. Our model embeds the essence of experimentation
in the literature. In terms of the bandit problem, the pusher’s shirking is a safe
arm which yields fixed returns and does not generate any information regarding the
pusher’s selling ability; while working is a risky arm which yields uncertain returns
and generates information for the learning of the pusher’s ability. The novelty of
adding the assistant’s price decisions into the model then allows us to study the
impact of the players’ incentives and interactions on the process of experimentation,
and derive testable empirical implications.
Our paper is also related to the broad literature on employer learning in the
4
labor market. Holmstrom (1999) studies a model in which an agent’s ability is
revealed over time through observations of performance, and examines how this
agent’s career concerns may affect his effort decisions. Since this seminal work,
studies, both theoretical and empirical, have examined various settings to investigate
how an agent’s ability is learned by employers over time and how the agent’s wage
compensation is affected by employer learning, e.g., Farber and Gibbons (1996),
Altonji and Pierret (2001), Gibbons and Waldman (1999a, 2006), Pinkston (2009),
and Kahn and Lange (2014). However, unlike our paper in which the assistant sets
different trading prices to actively learn about the pusher’s ability, in these papers as
a simplification the agents’ wage compensation is set to their marginal productivity
and therefore learning is passive. The difference is also reflected with respect to
the models’ results. In our paper, the assistant’s price is non- monotonic in the
assessment of the pusher’s ability, while in these papers compensation is always
monotonic in the employer’s assessment of the worker’s ability. Bose and Lang
(2015) consider an employer learning model in which the employer can actively
monitor the agent’s performance to evaluate the agent’s ability and determine task
assignments accordingly. Monitoring is costly, and so it is most valuable when the
employer’s assessment regarding the agent’s ability is not too high or too low. As a
result, the employer’s optimal monitoring strategy exhibits non-monotonicity in the
evaluation of the agent’s ability. In contrast to Bose and Lang’s setting in which
only the employer makes decisions, we consider a game-theoretical environment in
which the pusher also plays a strategic role. If in our model the pusher is simply
assumed to always work and thus is passive, then the assistant’s pricing strategy is
necessarily monotonic-she sets a low price if and only if the belief about the pusher’s
ability is larger than a cutoff value. Therefore, the strategic role of the pusher is
crucial to our result of non-monotonicity.
The paper is organized as follows. In Section 2 we summarize the information
we have collected via interviews with people involved in the underground economy
related to the illegal drug market in Singapore and the gang from which we obtained
the dataset. In section 3 and 4 we build the theoretical model and derive the optimal
non-monotonic pricing strategy of the assistants. Section 5 and 6 contain data
description and empirical analysis. We conclude the paper with Section 7.
2
2.1
Drug Market in Singapore
Background about a Drug Selling Gang
For the past five years, the first author volunteered and worked closely with ex-drug
offenders. He was able to help some of them turn their lives around, for which he
earned their gratitude and trust. In return, one of the ex-drug offenders was willing
to help him interview 98 other ex-drug offenders. The information collected from
these interviews form the bedrock of the background information in this section.
During the time period covered by our dataset, there were five major gangs
dominating and controlling the illegal drug-selling market in Singapore, with approximately 70% to 80% market share. One of the gangs was slightly larger in size
whereas the other four were almost equally powerful. Our data was drawn from
5
one of these five gangs. The gang covered by our data had three main layers in the
hierarchy: the boss, assistants, and pushers. The boss was the mastermind behind
the gang’s drug business. He was in charge of importing the drugs from oversea suppliers and oversaw the drug trade of the entire gang. Assistants were high-ranking
members of the gang who worked directly under the boss. After acquiring drugs
from the boss, each assistant sold drugs to the pushers. Pushers then sold drugs
to individual drug abusers or to small retailers. At any point in time, there were
about 20 assistants in the gang and each assistant had about approximately 10 to
15 pushers.
Everyone in the gang was well informed of the high risks, costs and high expected returns of selling drugs. Indeed, the drug trade was extremely lucrative. On
average, an assistant could earn approximately 40 to 70 thousand Singapore dollars
a month.3 Assistants typically marked up the drugs by at least 50% before selling them to the pushers. A pusher’s earnings averaged between 5 to 30 thousand
Singapore dollars a month. On average, a pusher’s markup on drugs was around
60% to 100% for Ice and Ketamine, 80% to 100% for Heroin and 50% to 80% for
Ecstasy. Most pushers were also drug abusers. Working as pushers allowed these
drug abusers access to a stable supply of high quality drugs at lower prices. Each
assistant recruited pushers independently. When making recruitment, the assistants
were extremely cautious and never took in strangers. Pushers consisted of individuals whom the assistants already knew or were introduced by people they trusted.
Each pusher had to undergo a probation period before commencing formal trading
relations. Sometimes assistants competed for talented candidates by promising price
discounts and other forms of preferential treatment in future transactions. However,
once the trading relationship between the assistant and pusher was formalized, there
was no more competition between the assistants for any pushers. Should an assistant try to poach another assistant’s pusher, the boss would step in to mediate.
The rules of the gang would be enforced and the penalties were severe enough to
deter such poaching behavior. Although most pushers were formal gang members,
the role they played in the gang was clear and: they only purchased drugs from
assistants, and were not involved in any other gang-related activities. Thus, the
relationship between these two layers was very much like a seller-buyer relationship
in a market. If a pusher attempted to quit the gang, he was free to leave and would
not be punished.4
Meanwhile an assistant also had the discretion to fire a pusher at any point
of time. Assistants typically stayed in the gang for many years and rarely quit
voluntarily, so the group of assistants in the gang was very stable. On the contrary,
the average tenure of a pusher was short–approximately 6 months. The majority
of pushers exited the market either voluntarily or as a result of being caught by
the authorities. Promotion to higher levels in the gang’s hierarchy was not among
the chief considerations of a pusher when choosing to enter and stay in the market.
Drug transactions between assistants and pushers were typically as follows. After
3
1 US dollar was equivalent to approximately 1.60 Singapore Dollar during the 1990s.
If a pusher intended to buy drugs from another assistant or another gang instead of quitting
the market, it was a rule that he had to terminate the trading relationship with his current assistant
first.
4
6
knowing that the boss possessed large quantities of drugs or after having obtained
these drugs from the boss, each assistant called his pushers to inform them about
the fixed prices of the drugs. To make a transaction, the pusher then called his
customers to promote the drugs and estimate the quantity that he needed to buy
from the assistant. After these phone calls, the pusher placed an order with the
assistant. The pusher either came to the assistant to pick up the drugs by himself
or the assistant sent someone to deliver the drugs to the pusher. Payment was in
cash upon delivery. Lastly the pusher distributed the drugs to his customers.
Normally, a pusher’s estimation on the amount that he could sell was reasonably
precise. Due to the high costs and risks involved in possessing and selling drugs,
pushers would only buy enough stock to satisfy their customers’ demands. The
amounts that a pusher asked for were mostly large enough for the pusher and assistant to earn profits. But, occasionally, the quantities purchased were unexpectedly
low, and the earnings generated in such situations could be outweighed by the pusher
and assistant’s costs. To maximize profits, the assistants paid close attention to the
pushers’ drug-selling performances. To safeguard the business, assistants regularly
sent scouts to do checks on the pushers and frequently questioned them to ensure
that the number of customers the pusher claimed was indeed real. Scrutinizing a
pusher’s phone log was among the methods an assistant would use to check the
validity of the claim that the pusher had numerous customers. Moreover, assistants
strongly preferred pushers who had the ability to purchase large amounts of drugs
on a consistent basis. The primary way to evaluate a pusher’s selling ability was
to examine his connections to managers in entertainment venues (like nightclubs,
discos, etc.), as these venues were places where large numbers of customers would
congregate and form a large potential market for drugs.5 Assistants normally made
clear records on their assessments regarding the pushers’ abilities to form a more
accurate expectation of future drug-selling. Besides, since pushers were frequently
caught and imprisoned, another reason the assistants kept detailed evaluations was
to know which pushers to pull back in the market again after they were released.
2.2
Assistants and Pushers
It may be a concern that the pushers and assistants were not as rational as their
counterparts in formal economic sectors, because they may have self-control issues
like addiction to alcohol and drugs, or they may be inexperienced in selling drugs.
However, as indicated in the interviews, pushers and assistants in this gang earned a
significant sum and were well aware of the high risks of being in this market, which
shows their rationality. Evidence may also be seen from other studies. Desroches
(2007) states that “the upper levels (including assistants) are rational actors who
enter the drug business consciously and deliberately after considering risks versus
potential rewards.” Lien (2014) makes the similar claim that “the majority at the
higher levels are not drug dependent. They are rational and usually intelligent,
5
The pusher had to bribe managers at these entertainment venues with money, drugs or a
combination of both to ensure that he could sell the product in these areas. Sometimes the
managers would be asked to help sell the drugs, or the managers would purchase drugs from the
pusher to sell by themselves.
7
and they may be weaned from narcotics”. Though not at the higher levels of the
gang’s hierarchy, the pushers’ behavior also reflected clear rationality. Most of the
pushers were addicts and many chose to enter the drug market due to access to
cheaper or free drugs. Some pushers chose to be part-time drug sellers with the
concern that working full-time would require them to actively seek for customers
and thus increase the risk of getting caught. The pushers and assistants’ trading
behavior also revealed their rationality. For instance, prices varied among different
drugs. Specifically, agricultural substances, such as Cocaine, Heroin and Marijuana,
were less expensive than synthetic drugs that require costly chemical processing in
laboratories. When selling drugs, pushers set prices optimally and frequently gave
customers discounts. Some interviewees stated that “instead of you not buying from
me, I’m going to give you $10 off because I can afford to and still make a profit”,
and “In order to compete, I have to keep my prices pretty steady” (N.A.,2012).
Bargaining rarely took place between an assistant and a pusher. The strong
bargaining power wielded by the assistants was firstly due to the pushers’ lack of
outside options. Most pushers were either jobless or low-income earners, and were
desperate for money. Further, many of them were gamblers and were saddled with
debt. Thus, pushers were in weak position to bargain, as the assistants knew about
their dire financial situation. Secondly, the majority of the pushers were addicted
to drugs and a considerable proportion of them were heavy abusers. The assistants
were fully aware of the pushers’ addictions. Most importantly, there were limited
alternative sources supplying the drugs the pushers needed at a lower price. Thirdly,
it was also very difficult for pushers to build up bargaining power given their expected
tenure in selling drugs was relatively short. Therefore, the asking price was typically
the final price.
People may also wonder whether pushers can search for and purchase from different assistants, as there were several gangs each with multiple assistants in the
market. However, a pusher typically bought drugs repeatedly from an assistant because of the high switching cost of finding a new supplier. The switching cost was
high from several perspectives. First, finding a new assistant was challenging. These
assistants often stayed under the radar to ensure the safety of the boss, and this does
not just include evading official enforcement but also lower-level members within the
gang (Lien, 2014). Second, a pusher needed to go through a time-consuming probation period before being able to purchase drugs from an assistant.6 During the
probation period, the pusher would be unable to continuously provide drugs to his
existing customers.
This links to the third perspective of switching cost; namely, pushers would lose
customers during the probationary period since they would be unable to provide
drugs during that time. Customers had varying demand for drugs, at different
frequencies.7 Thus, a pusher who could not meet customers’ needs due to a supply
shortage faced a high risk of losing them. In addition to the threat of losing existing
6
For example, a drug pusher, Yogaras, was commanded “to perform three drug deliveries before
he could become a permanent daily runner” (Ho, 2015).
7
For instance, a drug dealer, Marshall, recalled “One client was smoking nine joints a day”. In
another interview, the interviewee mentioned that his customers demanded for drugs every day or
every other day (N.A., 2012).
8
customers, a pusher also found it risky and difficult to rebuild his customer base,
since looking for new customers involved significant risk.
3
Model
We model the trading relationship between an assistant (A or she) and a pusher (P
or he) as a repeated game. Time is denoted as t = 0, 1, ..., and goes to infinity.
Both players are risk neutral and share the same discount factor δ ∈ (0, 1).8
In each period of the game, the assistant first decides the unit price, high (H)
or low (L), of a drug that she will sell to the pusher. After knowing the price of the
drug, the pusher then decides how much effort to exert, working (W ) or shirking
(S), in selling drugs to the market. Finally, the transaction between the assistant
and pusher is made and payoffs are realized.
This summarizes the process of transactions occurring in the drug-selling market.
An assistant sets the price of the drug. After knowing the price, the pusher decides
whether he wants to work or shirk. If he works, he calls up potential clients to
estimate demand. The number of calls he makes is a proxy for his effort level. As a
final step, the pusher returns to the assistant and purchases the amount of drugs he
is able to sell based on his interactions with potential clients. Payments are made
and revenues are realized.
The pusher’s type, his ability to sell drugs to potential clients, is either good (G) or
bad (B). Let a ∈ {L, H}, e ∈ {W, S} and θ ∈ {G, B} denote the assistant’s price,
the pusher’s effort and the pusher’s type. In any period, if the pusher shirks, each
player gets a payoff normalized to 0. If the pusher works, both players’ payoffs are
price and type dependent. In the drug-selling market, a pusher’s selling ability is
related to the market demand that he has access to. A good pusher faces a demand
function for drugs which is elastic. If the assistant sets a low price and a good pusher
works, both players get the same payoff x. If the assistant sets a high price and a
good pusher works, both players get the same payoff y. When the assistant sets a
low price, this enables the pusher to sell a significant amount of drugs to the market,
and so the transaction generates large profits for the players to share; on the other
hand, a high price set by the assistant lowers the amount of drugs that the pusher
can sell, and the transaction generates less profits for the players to share. Thus,
x > y > 0.
Payoffs:
a=L
a=H
θ=G
e=W e=S
x, x
0, 0
y, y
0, 0
A bad pusher’s demand function is unstable–sometimes it is as good as a good
pusher’s, but sometimes it is inelastic. Specifically, with probability p ∈ (0, 1) a bad
pusher can sell well and the players’ payoffs are the same as depicted above, with
8
Alternatively, the discount factor can be interpreted as the probability that a trading relationship is maintained to the next period.
9
probability 1 − p that a bad pusher sells poorly. In the latter case, if the assistant
sets a low price and a bad pusher works, the assistant and pusher get payoffs z
and w respectively; if the assistant sets a high price and a bad pusher works, the
assistant and pusher get payoffs w and z respectively. When the demand function
turns out to be inelastic, the pusher can only sell a small amount of drugs and the
revenues from the transaction can not cover the total costs that the players have to
bear, such as delivery costs. In this case, if the assistant sets a low price, then the
negative loss is mainly absorbed by the assistant; if the assistant sets a high price,
then the negative loss is mainly absorbed by the pusher. The payoff outcomes (z, w)
and (w, z) with z < 0 depict these situations, respectively.
Payoffs:
θ=B
e=W e=S
a=L
z, w
0, 0
a=H
w, z
0, 0
With probability 1 − p
Payoffs:
θ=B
e=W e=S
a=L
x, x
0, 0
a=H
y, y
0, 0
With probability p
The payoffs are summarized in the tables. We make the following assumption
on the players’ payoffs.
Assumption 1. px + (1 − p)z < 0 < py + (1 − p)w.
This assumption says that if the pusher is of low ability and chooses to work,
one player’s expected payoff from the transaction is negative and the other player’s
expected payoff is positive. With this payoff structure, the assistant and a good
pusher have aligned interests in the sense that to motivate the pusher to work,
both prefer a low price set by the assistant; alternatively, the assistant and a bad
pusher have conflict of interests in the sense that to motivate the pusher to work,
the assistant prefers a high price while the pusher prefers a low price.9
Initially, none of the players knows the pusher’s type. Instead, they hold a
common prior that with probability µ0 ∈ (0, 1) the pusher is of the good type. The
players’ decisions and the outcomes are publicly observed, and all aspects of the
period game are common knowledge.
In this model, no player has private information, therefore, they always share
the same beliefs and update these beliefs according to the Bayes’ rule identically.
The law of motion of beliefs is as follows. Let µt denote the players’ belief at the
beginning of period t. If the pusher shirks in period t, the players’ belief in period
t + 1 satisfies
µt+1 = µt
so there is no change of beliefs. If the pusher works in period t, the players’ belief
in period t + 1 satisfies
µt
µt+1 =
µt + (1 − µt )p
9
The assumption of symmetric payoffs for the assistant and pusher is inconsequential for the
model. If the conflict of interests for both players remains unchanged, the results remain unchanged.
The proof for this claim is available upon request.
10
after observing a payoff outcome (x, x) or (y, y), and satisfies
µt+1 = 0
after observing a payoff outcome (z, w) or (w, z). Only a bad pusher might cause
an outcome (z, w) or (w, z), so after observing an outcome (z, w) or (w, z), the
players are certain that the pusher is bad. On the other hand, when the pusher
works, a good pusher generates an outcome (x, x) or (y, y) with a higher probability
compared to a bad pusher. Thus, after observing an outcome (x, x) or (y, y), the
players believe that the pusher is more likely to be good.
A public history mt summarizes all actions and outcomes up to period t. Let
Mt denote the set of all possible histories. A pure strategy of the assistant σ A is a
function that specifies the price of the drug she sets for any history mt ; formally,
σtA : Mt → {L, H}
Similarly, a pure strategy σ A of the pusher is a function that specifies the pusher’s
effort level for any history mt and the price set by the assistant at ; formally,
σtA : Mt × {L, H} → {W, S}
Denote ΣA and ΣP as the sets of all pure strategies for the assistant and the pusher,
respectively.
We study Markov perfect equilibrium where strategies only depend on the payoffrelevant part of the histories. In our model, this is represented by the common belief
0
µt . Specifically, for two histories mt and mt0 such that the players hold the same
belief, a Markov strategy requires the players to take the same actions after these
histories; formally,
0
µt (mt ) = µt0 (mt0 )
⇒
0
0
σtP (mt ) = σtP0 (mt0 ) and σtA (mt , at ) = σtA0 (mt0 , at0 )
for at = at0 . A Markov perfect equilibrium is then a sub-game perfect equilibrium
where the players only use Markov strategies. We restrict our attention to pure
strategy Markov perfect equilibrium hereafter.
4
Equilibrium Analysis
In this section we analyze the model by investigating the joint dynamics of the players’ beliefs and decisions. In a one-shot game where the players only interact once,
if m = 1, the pusher is definitely good. The unique sub-game perfect equilibrium
of this game involves the assistants setting a low price and the pushers working.
Both players receive a payoff of x. If m = 0, the pusher is definitely bad, and the
unique sub-game perfect equilibrium involves the assistants setting a high price and
the pushers shirking. Both players receive a payoff of 0.10 Intuitively, the assistant
10
Consider µ = 0. If the assistant sets a low price, given px + (1 − p)w > 0, the pusher’s optimal
response is to work. If the assistant sets a high price, given py + (1 − p)z < 0, the pusher’s optimal
response is to shirk. By backward induction, when px + (1 − p)z < 0, the assistant’s optimal
action is to set a low price. In this case, the unique sub-game perfect equilibrium consists of the
assistant’s setting a high price and the pusher’s shirking.
11
prefers a good pusher to work and a bad pusher to shirk. The difference between
these two equilibrium results in the one-shot game, which captures the relative importance of the players’ conflict of interests. If µ = 0, the payoff outcome (z, w)
or (w, z) happens with probability 1 − p when the pusher works. In this instance,
the large conflict of interests between both players’ preferences results in the noncooperative outcome of the assistants setting a high price and the pushers shirking.
If µ = 1, the negative payoff z never occurs and the players’ preferences are perfectly
aligned. This gives rise to the cooperative outcome with the assistants setting a low
price and the pushers working.
Now consider the dynamic game. A good pusher that works certainly generates
the payoff outcome (x, x) or (y, y). Therefore, after observing the payoff outcome
(z, w) or (w, z), the players are certain that the pusher is bad and thus their belief
drops to 0. We derive the equilibrium result for this event.
Lemma 1. (degenerate beliefs)
(1) When µ = 0, the Markov perfect equilibrium is unique. In equilibrium, the
assistant always sets a high price and the pusher always shirks.
(2) When µ = 1, the Markov perfect equilibrium is unique. In equilibrium, the
assistant always sets a low price and the pusher always works.
When µ = 0, this belief will remain forever unchanged. A Markov perfect equilibrium requires the players’ decisions to be stationary over time and consists of a
sub-game perfect equilibrium in each period. In each period, a low price will induce
the pusher to work while a high price will cause him to shirk, so it is optimal for
the assistant to set a high price all the time. Then, the pusher always shirks.
For comparison, we will also derive the equilibrium result for the case µ = 1.
In this case, in each period, the pusher chooses to work regardless of the price set
by the assistant. It is optimal for the assistant to set a low price. In this unique
equilibrium, the assistant sets a low price and the pusher works.
Proposition 1. The Markov perfect equilibrium in this game is unique and takes
the form of a partition equilibrium; there is a number k and a sequence of cutoff
values, µ∗1 , µ∗2 , ..., µ∗2k , where µ∗1 < 1, µ∗2k = 0 and µ∗i > µ∗j for i < j, such that the
assistant sets a low price if µ ∈ [µ∗2i−1 , µ∗2i−2 ) and sets a high price if µ ∈ [µ∗2i , µ∗2i−1 ).
The pusher works if and only if µ ≥ µ∗2k−1 .
Let µ∗ be given by
µ∗ =
(1 − p)(w − z) − p(x − y)
(1 − p)(x + w − y − z)
Then,
[µ + (1 − µ)p]x + (1 − µ)(1 − p)z ≥ [µ + (1 − µ)p]y + (1 − µ)(1 − p)w
if and only if for µ ≥ µ∗ , where µ∗ ∈ (0, 1) by Assumption 1. This inequality implies
that in the one-shot game, if the pusher will work, then the assistant prefers to
set a low price if and only if µ ≥ µ∗ , where the terms on the left and right hand
12
sides are the assistant’s expected payoffs from setting a low price and a high price,
respectively.
In this game, the assistant prefers to induce a good pusher to work and conversely
induce a bad pusher to shirk. So long as the pusher’s ability is still unclear, learning
about it is valuable since it enables the players to make better decisions. However,
given that the players only learn about the pusher’s ability through working, and
that working may also cause the payoff outcome (z, w) or (w, z), there is a tradeoff
between learning and incurring the negative payoff z. Specifically, in the unique
equilibrium of the game, the pusher is induced to work if and only if the players
hold a belief that is larger than a cutoff value, denoted by µ∗2k−1 . Intuitively, if both
players’ belief is very low, they would expect a high probability of the payoff outcome
(z, w) or (w, z) when the pusher works. In this case, the benefit from learning about
the pusher’s ability generated by working will be less than the current loss caused
by the negative payoff z, and therefore it is optimal for the players to stop learning
by inducing the pusher to shirk.
This proposition indicates that for the range of beliefs that the pusher is induced
to work, there will be a non-monotonic relationship between the price set by the
assistant and her belief that the pusher is good. When the belief is very high, the
players’ interests are well-aligned and the assistant prefers to set a low price to
maximize payoffs. However, when the belief lies in the moderate range, the players’
conflict of interests comes into play. Specifically, to induce the pusher to work in this
moderate range of beliefs, the assistant may have to set a low price when the belief
is relatively low. Conversely, she may set a high price when the belief is relatively
high. For this range of beliefs, the assistant prefers to set a high price as long as the
pusher can still be induced to work. However, she may not choose to do so when
the belief is relatively low as the pusher’s expected loss from the negative payoff z
would discourage him from choosing to work.11
In any equilibrium, if µ ≥ µ∗ , then the assistant sets a low price and the pusher
works. To illustrate this, if the assistant sets a low price, it is certain that the pusher
will work as he does not need to bear the negative payoff z. In some period t, if
there is a putative equilibrium where the assistant sets a high price given some belief
µ ≥ µ∗ , then the pusher may choose to either work or shirk. For either case, when
µ ≥ µ∗ , if the assistant deviates by setting a low price in period t and switches back
to the equilibrium strategy starting from the next period, she can obtain a larger
payoff in the current period without reducing her future payoffs. Hence it is optimal
for the assistant to set a low price whenever µ ≥ µ∗ .
Now we will explain why the equilibrium is unique. For any belief µ ∈ (0, µ∗ ),
there is a finite number l and a sequence of beliefs, µ1 , µ2 , ..., µl , satisfying µ1 = µ,
µl−1 < µ∗ ≤ µl and µi = µsi−1 (so if the payoff outcome (x, x) or (y, y) is observed
at belief µi−1 , the updated belief increases to µi ).
When the players’ belief is at µl , the assistant sets a low price and the pusher
works. Now consider the belief µl−1 . Since the equilibrium play in the continuation
game is uniquely determined after any outcome, the players’ decisions at belief µl−1
11
If we assume that the pusher always works, this game is reduced to a decision problem of the
assistant. Then the assistant’s pricing strategy is monotonic. She sets a low price if and only if
µ ≥ µ∗ .
13
are also uniquely determined: firstly, for any price offered by the assistant, the
pusher works if and only if his total payoff (the payoff in the current period and the
payoff in the continuation game) is non-negative; secondly, if the pusher will work
given a high price, then the assistant sets the high price. However, if the pusher
will shirk given a high price, then the assistant sets the low price unless her total
payoff becomes negative. Applying this argument inductively, we can show that at
any belief µ, the players’ decisions are unique.
The argument in the previous paragraph is extended in the proof to inductively
construct the sequence of cutoff values, µ∗1 , µ∗2 , ..., µ∗2k . For the general case with
k > 1, the reasoning goes as follows. For any µ ∈ [µ∗ , 1), the assistant sets a low
price and the pusher works, and each obtains a positive payoff. Denote µ∗1 = µ∗ .
Then for beliefs µ that are smaller than but close enough to µ∗1 , the pusher can
be induced to work even when the price is high, and it is therefore optimal for the
assistant to set a high price. Inductively, there is a cutoff value µ∗2 such that for
beliefs µ that are smaller than but close enough to µ∗2 , the pusher would shirk if the
price is high. In this case, the assistant has to set a low price. Whenever there is a
cutoff µ∗3 such that for µ that are smaller than but close enough to it that the pusher
is again incentivized to work when the price is high, the assistant would set a high
price until the cutoff µ∗4 , at which the high price becomes unacceptable to the pusher
again. The cutoff values µ∗2k−1 and µ∗2k can be constructed repeatedly. Intuitively,
for beliefs µ < µ∗ , the players have to bear the negative payoff z if the pusher should
be induced to work. However, for sufficiently low beliefs, no one would like to bear
this payoff z, resulting in the outcome that the assistant always sets a high price
and the pusher always shirks.
In this model, both players learn about the pusher’s type when the pusher works.
At any point in time, if the players are still uncertain about the pusher’s type after
observing the payoff outcome (z, w) or (w, z), their belief drops to 0 and stays there
forever. On the other hand, after observing the payoff outcome (x, x) or (y, y) their
belief rises gradually. Regardless of the prior and how the players initially interact
with one another, so long as the players’ belief is sufficiently high, µ ≥ µ∗ , the
pattern of interactions in the continuation game is deterministic: a series of payoff
outcome (x, x) or (y, y) without any observation of negative payoff z helps the players
to maintain a cooperative relationship in which the assistant sets a low price and
the pusher works, while an observation of payoff outcome (z, w) or (w, z) causes the
players to interact non-cooperatively with the assistant always setting a high price
and the pusher shirking.
Only a bad pusher can cause a negative payoff z when he works. Our previous
results together with the law of large numbers implies that in some period t, if the
players hold a common belief that µ ≥ µ∗2k−1 , the pusher’s type will eventually be
revealed . We summarize this result in the following corollary.
Corollary 2. In some period t, if the players have a common belief that the pusher
is good µ ≥ µ∗2k−1 , then the pusher’s type will eventually be revealed.
The non-monotonic pricing strategy used by the assistant is not due to the assumption that there are only two possible prices. Intuitively, it would seem that
when there is a continuum of prices, the assistant would always set a price that
14
makes the pusher indifferent between working and shirking. This is not the case.
The equilibrium analysis can be extended to allow more levels of prices or even a continuum of prices, and our main results would remain unchanged. In the drug-selling
market, a transaction between the assistant and pusher is a variation of stackelberg
competition; that is, the assistant sets her price to the pusher first, and then conditional upon participating in the transaction, the pusher learns about current market
demand and sets his price for the market. The market demand is determined by the
pusher’s selling ability. The higher the assistant’s assessment regarding the pusher’s
ability, the lower the assistant’s preferred price. However, the assistant may not able
to set the price she prefers. The players bear fixed costs for each transaction such as
delivery costs. When both players hold a low assessment of the pusher’s ability, setting a relatively high price may discourage the pusher to participate. As a result, the
assistant has to lower the price for low-ability pushers. As the players’ assessment
of the pusher increases over time, the pusher’s participation constraint loosens and
the assistant can therefore set a higher price closer to her preferred price. Moreover,
when the players’ assessment crosses some cutoff point, the pusher’s participation
constraint is no longer binding even if the assistant sets her preferred price. This
argument implies that a non-monotonic pricing strategy will also arise when the
assistant can set a price from a continuum: as the assessment of the pusher’s ability
increases, the price will first increase before eventually decreasing.
5
Data Description
The data set we obtained contained information about the activities of a drugselling gang that operated in Singapore during the 1990s12 . The entire data set was
retrieved from the gang’s books13 . Hence, it is not susceptible to any memory-related
issues like imperfect recall. Our data set is also representative in the sense that it
contains every single pusher’s (referred to with the pronoun “he”) transactions with
his respective assistant (referred to with the pronoun “she”) in the gang’s formative
years. Details about each pusher’s characteristics prior to joining the gang are
included as well.
The full sample comprises 354 pushers with a total of 2,956 trades and 9,132
orders. We have the following information for each order that took place between
a pusher and an assistant: the type of drug traded, quantity, quality, price, cost,
whether the assistant gave any drugs as a gift to the pusher, delivery method,
whether the pusher had proposed a counter offer and whether consignment is given.
Market factors (i.e tight police monitoring etc) that influenced the trading price and
quantity are also captured in this data set. This information allows us to control
12
See Appendix A about how we obtained this data.
The boss of this gang kept these records for several reasons. First, the gang wanted to evade
enforcement. Each assistant thoroughly vetted a pusher’s background before they transacted.
Scouts were sent regularly to monitor each pusher. This was to ensure that none of the pushers
were undercover officers. Furthermore, the boss wanted to ensure that the pushers’ daily activities
were not attracting the attention of the authorities. Second, the goal of the gang was to make
money. If a pusher was arrested and released from prison, the assistant wanted to know whether
it was worth re-inviting him back into the gang.
13
15
for possible market conditions that may have affected the transactions between a
pusher and the assistant. Moreover, we have the assistant’s assessment of each
pusher’s ability, as well as data on each pusher’s effort. Effort is represented by
the number of phone calls made by the pusher to sell the drugs. Apart from trade
history, we also have each pusher’s demographic information, arrest history, reasons
why he entered into this business, income, debts, whether he had a drinking habit,
and details of his business connections with entertainment establishments before
they started trading relationship with assistant. Trade occurred at different points
in time. In each trade, the pushers may purchase different types of drugs from
the assistants. For example, at trade N, a pusher bought drug B and C from his
assistant. We define two orders to have taken place because two different types of
drugs were purchased by the pusher at trade N. We will define trades and orders
in this way throughout the remainder of this paper. The date of Xth trade for
the pusher X is likely to be different from that of the pusher Y. We have date of
each trade, but it was already artificially adjusted in a systematic way to avoid the
direct disclose of the true dates. Nevertheless, the chronological order of the events
remains. The majority of trading incidences involve purchase of at least two types
of drugs; only 6% of them involves a single type.
This data set has an unbalanced panel structure. We have almost every pusher’s
trading records in the first 5 trading incidences, but only three quarter of the pushers’
records up to the 6th trade and one third of the pushers’ records up to the 9th trade.
Though it is counter-intuitive to expect every pusher exit the market after the same
number of trades, it is still important to discuss why we may have an unbalanced
data structure. In fact, one major reason is that pushers exit at different times
(voluntarily or involuntarily). This is a natural result of this highly risky and volatile
market and the characteristics of its participants. We have information on each
pusher’s tenure with this gang and found that 37% of the pushers exited within 2
months, half of them left after 3 months, only one quarter remained after 8 months.
A pusher could exit involuntarily due to unexpected policing by the authorities.
He could be arrested during a police raid against illegal drug trade or caught for
drug abuse during a regular police check-up. A pusher’s exit also depended on his
own personal characteristics. A less able pusher may drop out earlier than a more
capable pusher. However, a more able pusher may also exit early because they
have made enough money. Moreover, a pusher with heavy debt may stay longer
even though they are not able to sell very well, because they are more desperate
for money. In a nutshell, the special environment in the drug-selling market and
the traits of its participants dictate unbalanced nature of the data in this market.
Another possible reason is that the missing later trading history for some pushers
is simply due to the fact that we only have access to one of the books. This is not
a significant concern because we have no valid reason to believe the records in this
book is a selectively biased sub-sample of full sample, especially given the fact this
book contains all the early transactions of every pusher in the gang. Furthermore,
on average approximately 50% of the total number of transactions of the pushers
are covered in our sample, and we therefore have sufficient information to make
inferences from the assistants’ pricing strategy.
16
5.1
Pusher’s Assessed Ability
An essential advantage of this data set is that it contained the assistant’s assessment
of each pusher’s ability over time. Assistants were required to keep good track of
the pushers’ ability for the boss. According to an ex-assistant, an assistant was particularly interested in pushers who consistently purchased a large quantity of drugs.
These type of big customers are considered as high ability pushers. Frequent purchases of small quantities is not preferred to a single large purchase of an equivalent
amount, because repeated transactions increase the assistant’s risk of being exposed.
The assistants were not directly concerned about whether the pusher sold all the
drugs in the retail market or had some leftover stock. They only cared about it to
the extent that it could affect pusher’s next purchase. A pusher’s selling ability was
not completely observable to an assistant at the outset, but was gradually revealed
as the pusher and assistant interacted. Based on the assistant’s observation, she
created proxy to indicate the pusher’s ability to sell. The assistants in the gang
used a uniform rating system and assigned a quantitative measure of ability for each
pusher ranging from 1 to 10, with 1 being the lowest ability and 10 being the highest.
We have an ability rating for each pusher on each trading date. The ability
measure for the Nth trade reflects the assistant’s assessment of the pusher after
the (N-1)th trade is completed. Figure 1 displays the distribution of the pushers
’abilities across trades. We intentionally omit the histogram after the 11th trade,
because the dataset only captured 16% of the pushers’ population after that point.
The changes in the histograms reveal two facts. Firstly, the distribution of pusher’s
ability is approximately a normal distribution with light tails in early trades. Most
of the weights fall on the middle ratings, which means the assistants may have had
a hard time assessing the ability of the pusher at the outset. From the assistant’s
point of view, most of the pushers did not differ much from each other and they may
become better or worse. Secondly, the distribution starts to skew to the right after
the 4th trade, indicating that the majority of those who stayed in the business for
longer are capable pushers.
A pusher’s assessed ability is not a static measure in the drug-selling market.
This market is highly dynamic and risky. The quantity of drugs a pusher ordered
from the assistant is not only dependent on his sales network or salesmanship but
is also influenced by external factors such as raids conducted by the authorities.
For example, the pusher could suddenly lose his clients if they were caught. Hence,
the pusher’s assessed ability was periodically updated after each trade to reflect the
pusher’s purchase potential up to date. The table 2 -11 presents the transition matrix
of ability between different trades. Between any 2 trading periods, the movement of
the ability assessment was never drastic. The rating moved up or down maximally
by only two units. A rating of 1 & 2 is the absorbing state, meaning pushers with
these two lowest ratings never move up the performance scale. During the 1st to
4th trade, pushers with a rating from 3 to 8 have at least a 50% of chance of moving
up by at least one unit. This trend continued to the 7th trade, but with a more
moderate probability of moving up. The first 4 trading periods can be regarded
as the screening period. After the 4th trade, 10% of the pushers dropped out.
Assistants were prudent in assigning the ratings; they never assigned any pusher
17
with rating of 10 before trading with him at least 4 times. There were very few
top performers. After the 5th trade, only 10-15% of pushers with rating of 9 had
a chance to be rated 10. After the 7th trade, pushers with a rating below 5 never
improved. It implies that drug-selling ability is not type of skill one can acquire
through experience. If a pusher cannot sell well after several attempts, he is expected
to not sell well thereafter.
To learn whether the assistant’s ability assessment of a pusher exhibited a converging trend over time, we also plot the evolution of each pusher’s ability in figures
2 and 3. For analysis, we break these pushers into two groups: those who started
with an ability rating below 5 (see figure 2) and those started with an ability rating
equal to or more than 5 (see figure 3). In each figure, from left to right, the sub-graph
represents the group who ended below 5 and the group who ended above 5, respectively. It is important to point out that each line in the plot could represent different
pushers who exhibited the same ability pattern. The evidence collectively suggests
the following facts: as long as a pusher did not begin as an ultra low performer (with
rating below 3), he had the chance to improve his selling performance. Pushers who
started out as with an above average ability rating could also perform very poorly
down the road. This further proves that the initial evaluation of the pusher’s ability
was noisy, hence less accurate. Unfortunately, we are not able to identify which
assistant traded with which pusher. Otherwise, it would be interesting to examine
whether different assistants had different learning rates.
Across all pushers and trades, the average assessed ability of a pusher was 6.2
with standard deviation of 1.8. The within standard deviation of ability is 1.2
and between standard deviation is 1.3. There are no pushers whose rating stayed
constant. 96 pushers had ratings which varied by 2 units. There are 118 and 84
pushers whose rating moved 3 and 4 units, respectively.
5.2
Pusher’s Effort
Another key variable of interest in this paper is the pusher’s effort. The pusher’s
effort could be multidimensional, making it impossible for the assistant to fully
observe his effort despite having her scouts paying frequent and random trips to
check on pushers. However, we identified a viable proxy for effort, which is the total
number of phone calls the pusher made to sell drugs. A typical trade occurred in
the following sequence: first, the assistant informed the pusher of the prices for each
type of drug. Second, the pusher made phone calls to his potential customers to
try and sell the drugs. Third, once the pusher was able to estimate the amount of
drugs he was able to sell, he approached the assistant and purchased the drugs. A
pusher had several untraceable prepaid phones cards and could make up to 10 calls
to each customer when he tried to make a sale. However, before finalizing the trade,
the assistant scrutinized the pusher’s phone logs. Checking the pusher’s phone logs
was a tactic the assistant employed to verify the pusher’s order and to ensure the
pusher was not an undercover agent for the authorities. As a security measure, the
assistant sometimes randomly dialled the number in the phone log and asked the
pusher to confirm the order in front of her. The average number of phone calls the
pushers made to sell the drugs purchased from the assistant is 44 with a standard
18
deviation of 22. The within variation is 13 and the between variance is 17. The
drug retail market was a competitive marketplace for pushers at that point in time.
A pusher needed to make multiple calls to close the sale. Thereafter, the pusher
had to make additional calls to finalize the trading venue with his buyer. Intense
monitoring by enforcement officials complicated this process further and required
the pusher to constantly call his customer to change the transaction venue whenever
necessary.
Notice that the ability and effort measurements only vary at the trade level,
not at the order level. We recognize that the pushers with larger networks would
have needed to make more calls. However, a large network did not guarantee that a
pusher would be able to generate high sales if he did not work hard, because pushers
faced stiff competition from one another. The correlation between ability and effort
was calculated to be 0.92. We believe that hardworking pushers made more phone
calls and therefore were more likely to secure larger sales. The correlation between
the ability and effort is hard to disentangle completely in any context. However,
it is not a key concern in this paper. We take advantage of these two measures to
examine how an assistant learned about a pusher’s ability and how a pusher adjusted
his effort to influence the assistant’s learning.
5.3
Pushers’ Characteristics
Table 13 - Table 14 presents the pushers’ characteristics. Pushers were mostly
male and ethnic Chinese-only a handful were of Indian ethnicity. The average age
of a pusher was 32, with the youngest being 19 and oldest being 52 years old.
Pushers had low educational attainment-half of them did not complete primary
school. Approximately three quarters of the pushers were single. 42% were jobless
and 5% had no regular place to stay. The average monthly income was S$1490 for
162 pushers who had full time jobs, whereas the pushers with part time jobs earned
S$500 less per month than those with full time jobs. 66% of the pushers were gang
members. 60% of the pushers had been arrested prior to becoming pushers, and the
average prison sentence for those who had been arrested previously was about one
year.
This pool of pushers also exhibited certain behavioral patterns. 69% of the
pushers were addicted to drugs before becoming a pusher and half of these addicts
had previously undergone drug rehabilitation. Of those who had undergone drug
rehabilitation, 90% had spent at least one year in rehabilitation. A substantial
number of pushers were saddled with debt.
Table 14 shows drug trade related characteristics of the pushers. The top reasons
these pushers decided to sell drugs were either because they needed money for drugs
or debt repayment. Some pushers were doing it because their friends were doing it.
Each pusher distributed drugs in clearly demarcated regions, with pushers evenly
distributed across various regions in Singapore. Having business connections was the
key to success as a pusher. 58% of the pushers had certain types of connections with
entertainment establishments, which included KTVs, clubs, discos and brothels. If
the pusher knew the staff or a manager in a KTV, he was expected to distribute drugs
at a much faster pace and in larger quantities. The average number of business ties
19
the pushers had before selling drugs was 6. From industry insiders, we also learned
the two main reasons pushers exited the market: they were caught by the authorities
or they were sent to rehabilitation.
5.4
Order Level Characteristics
Generally speaking, trade occurred on a weekly basis. The methamphetamine-known
colloquially as Ice-sales comprises 40% of all 9134 orders, followed by Ketamine
(21%), Ecstasy (20%) and Erimin (13%) . The remaining 5% sales were attributed
to Heroin and other drugs14 (see table 16). In each trade, the pusher bought at least
2 types of drugs from the assistant. We have 35%, 27%, 30% of trades pertaining to
the purchase of 2, 3, and 4-5 types of drugs, respectively. The drugs sold were mostly
of high quality, with a third of the drugs sold being of top quality. There were no
prevalent modes of delivery for drugs: pushers picked the drugs up in person half of
the time, while the assistants delivered it to the pushers the other half of the time.
When a new batch of drugs arrived, an assistant had an incentive to give a pusher
new drugs for free to reward them. The incidence of giving a gift occurred 35% of
the time. Bargaining was not common in the drug market. Only 13% of the orders
entailed bargaining whereby pushers proposed counter offers. A simple regression
exercise confirms this fact. The occurrence of bargaining was independent of the
pusher’s order level characteristics. Bargaining was more likely to happen when the
asking price of the assistant was high. However, the attempted bargaining failed
most of the time. Pushers were price takers-similar to buyers in a monopolistic
market.
The types of drugs sold were all addictive, but they differed in potency. In
table 17, we summarized the order level characteristics by drug types. Distinct from
other drugs, Ice was the most expensive drug in both cost and market price. One
gram of Ice cost around S$83, which was 4 times higher than most drugs. The
average amount of Ice sold in a single order was 11 grams, which was also about
4 times smaller than the quantity of other drugs sold. Heroin and other drugs
were traded at higher quantities. 68% and 94% of Heroin and other drugs were of
medium quality, respectively, suggesting the high trading quantity may be due to
lower concentration/purity. Ice was the staple of the gang. They made S$75 profit
per gram, which was at least 5 times higher than those of other types of drugs. The
average revenue from each drug order was between S$1300-S$1620 for the 4 major
drugs, including Ice and Erimin.
6
Estimation Strategys
Empirically, we explore three main questions. First, does the assistant learn about
pusher’s ability, and if so, how does she learn? Second, how did the assistants
determine which prices to set for pushers of different abilities? Third, what was the
14
The small share of Heroin trade does not mean that Heroin was not popular at the time. In
fact, Heroin was one of the most popular drug in the market at the time, but that the gang we
have studied specialized in Ice
20
effect of the assistant’s incentive scheme on the pusher’s effort level? We answer
these questions by estimating following equations:
Abilityij = ΦXi + δHij 0 + εij
(1)
P riceijt = Σg β1g Abilityij + ΦXi + Σj Ωj Tj + ΓZijt + ε1ijt
(2)
Ef f ortijt = Σg β2g P riceijt + ΦXi + Σj Ωj Tj + ΓZijt + ε2ijt
(3)
, where i is the pusher, j is the trade and t is the order. Abilityij is the assistant’s
assessment of the pusher’s ability after the (j-1)th trade is completed. Xi is the set
of the pusher’s characteristics which can be divided into two separate parts. The
first part of the Xi comprises the pusher’s characteristics that are indicative of his
ability, denoted as Si : whether the pusher had business connections, whether he was
a gang member, whether he had a full time job, part time job or was jobless, whether
he was addicted to drugs, whether he had undergone rehabilitation and whether he
was heavily in debt. According to industry insiders, these ability indicators can
either improve or reduce each pusher’s sales performance. We will discuss this in
further detail in the empirical analysis section. The second part of the Xi includes
the pusher’s individual information that is not covered in Si . It includes the pusher’s
age, the area where the pusher conducted his business and how he was introduced
to the assistant. Hij 0 comprises the pusher’s trading history, which includes pusher’s
effort level in the previous trade, total log sales in the previous trade, accumulative
log sales up to the current trade, the accumulated number of times the pusher
proposed counter offers, the accumulated number of times gifts were given to the
pushers. Tj is a dummy variable capturing any variable trend that occurred during
the time period that the trades were captured. Zijt represents the set of each order’s
characteristics, comprising drug type, drug quality, unit cost of the drug, delivery
mode, whether the pusher proposed a counter offer, whether the assistant gave the
pusher a free gift, the trades no.15 . and order-specific circumstances such as market
conditions. We have information about whether the drug was sold at market price or
not, and the reasons behind it. This information helps us to capture supply shocks
due to enforcement or demand shocks due to seasonal factors.
In equation (1), we study how the assistant formed her beliefs and whether the
assistant’s subsequent updates reflected a learning process. The construction of the
equation (1) is influenced by our knowledge acquired from market insiders. This
method of estimating the formation of ability also allows us to test the validity of
the assumptions we made about the belief updating process in the model. Notice
that in our model, we assume only that the factors summarize the history of trading
would affect the assistant’s ability assessment formation.
To estimate how the assistant’s learning about the pusher’s ability influences her
own price offer strategy, we use only equation (2). More specifically, this is to explore
how the assistant’s price offer varies according to different levels of ability rating.
Throughout the analysis, we will use the unit price of the drug type as a proxy for
15
Trades are organized in chronological order. Both ability and effort do not vary at the trade
level. Adding this variable into the regression allows us to control for the assistant’s change in offer
price due to the increased trading tenure.
21
the terms an assistant offers to the pusher. Unit price is a summary statistic for
equilibrium price and quantity. The price we use is the final price for the quantity
purchased by the pusher. Since price bargaining rarely happens, it also reflects the
assistant’s asking price for the quantity demanded. Note that the total quantity of
the drugs the pusher purchased from the assistant alone cannot indicate whether
the assistant offered the pusher preferential terms or not. For example, drugs such
as Ketamine and Ecstasy are consumed in large quantities on averagee, due to their
chemical composition, and are purchased in large amounts. Moreover, the trading
quantity is indicative of both the pusher’s demand and the assistant’s supply, and
perhaps more heavily influenced by the former. The key coefficient of interest is
β1g , Abilityij is a categorical variables with g + 1 groups. For better comparison,
we set an ability rating of 5 as the benchmark group. Rating 5 indicates average
ability and 30% of all 354 pushers were rated 5 at least once. We use both OLS
and fixed effects in our estimations. Fixed effects regression would better allow us
to understand how assistants adjusted their pricing strategy as they learned about
pusher’s ability over time.
Lastly, we use equation (3) to study how the pusher adjusts his effort level given
the assistant’s offer both OLS and IV estimation,β2g is the variables of interest.
7
7.1
Empirical Analysis
Formation of Ability Assessment
There are two possible channels an assistant can learn about the pusher’s ability.
One way is to use pusher’s individual characteristics, which include the pusher’s
age, gang membership, job status, drug addiction status, the way in which he was
introduced to assistant. Some market insiders claimed that the pusher’s personal
traits helped the assistant to form expectations about the pusher’s ability. It is
worth mentioning here that these variables in our data set are a static measure and
only reflect the initial profile of the pushers before any trade occurred. Even though
they may change over time, we expect them to have negligible or no changes in
the short run. The second channel is to use trading interactions, which is more
dynamic. In particular, we examine how the assistant’s ability rating reflects the
pusher’s effort she has observed, total sales in the previous trade, as well as the past
trading outcomes. The effect of individual traits is presented in column (4), while
the effect of trading outcomes is shown in column (1)-(3) on table 18. We examine
all factors collectively in column (5).
We start the analysis with effort16 . We used two measures of effort: the observed
effort in the latest trade and the average historical effort. They explain 64% and
50% of the variation in assessed ability in the current order, respectively. This result
suggests that when the assistant evaluates a pusher’s ability, she takes the pusher’s
effort into consideration and cares more about the most recent effort than the average
historical effort. Given the high volatility in the drug trade, the immediate past
16
Notice that the ability measure in a given period is derived after the prior trade was completed,
whereas the effort measure reflects the pusher’s effort in that given period.
22
effort is more informative than the average historical effort. In addition, in our
theory model, a pusher whose ability belongs to the top tier would always receive a
low price from the assistant. A pusher whose ability belongs to the second tier has an
incentive to exert effort after receiving a high price from an assistant in the current
period. By doing so, the pusher can influence the assistant’s belief that the pusher is
of a high ability, thereby possibly getting a low price from the assistant in the next
period. The fact that the assistant updates her belief based on the most recent effort
of the pusher is compatible with our theory’s prediction that the pusher would try
to manipulate the assistant’s belief by adjusting his effort level. The coefficient of
the estimation when using the prior period’s effort is 0.07 (see column (1) in Table
18), meaning every incremental 10 unit increase in phone calls is associated with
an approximate 0.7 unit increase in assessed ability. Similar coefficients are found
when using average historical effort.
The latest trade is another important determinant of ability rating. The total
sales in the previous trade explains 20% of differences in ability (column (2) in
table 18). By using the Markov Perfect Equilibrium (MPE) concept in our theory
model, we assume that the assistant’s ability assessment was updated using the
trading outcome in the previous period. This regression tests the validity of the
MPE assumption, and shows that assistants use latest trading outcomes to make
assess the pushers’ ability. Holding other factors constant, every 2.5% increase in
sales in the last trade will inflate the ability rating by 1 unit. Similar to discussion
on effort, ability rating better reflects the sales in the previous trade rather than
average of historical sales. Furthermore, the more often pushers proposed an counter
offer, the less likely he is considered as capable. Though significant, the magnitude
of this effect is very small.
In column (4), we see that fixed individual characteristics alone explain 13% of
difference in the assistant’s ability assessment, which implies that a pusher’s potential ability is not completely random. Rather, an experienced assistant can predict
a pusher’s potential ability by using these personal traits as signals. Pushers who
started out with business ties, had jobs, were not severely addicted to drugs and
were invited to be a pusher by the assistant were considered more capable. The
impact of the these initial ability indicators on assessed ability diminished when dynamic factors like trading outcomes are added. However, the significant association
between these individual characteristics and ability assessment remains.
Brief Discussion on Assistant’s Learning
By studying the formation of ability rating, we can conclude that learning took
place and that the assistants learned about the pusher’s ability mainly by observing
the pusher’s effort and purchase potential revealed in the latest trade. The assistants
employed this method of updating both in early periods and later periods. Moreover,
certain fixed individual traits persistently influence the assistant’s assessment. From
the previous discussions on the evolution of ability, we learn that a pusher may turn
out to be a good pusher or a bad pusher regardless of his rating in the early periods
as long as it is above 2. Any experienced assistant is expected to know this before
the commencement of her trade with the pushers. Hence, she should be aware that
it takes time to learn the pusher’s true ability.
23
7.2
Assistant’s Pricing Strategy
In this subsection, we explore the assistant’s pricing strategy as she learns about
the pusher’s ability using the estimation equation (2). All the variables described
in the equation (2) are controlled in the regression. Due to space limitations, only
the key estimates are presented in the table. Results using absolute unit price and
log unit price as dependent variables are presented in column (1)-(3) and column
(4)-(6), respectively. We deal with unobserved heterogenity across pushers by using
robust error in all OLS specification.
Non-Monotonic Pricing Strategy
The key variable of interest is Ability. It corresponds to the µt in the theory
model and can be thought of as the assistant’s belief that the pusher is of high
ability. Recall that one third of the pushers in the gang have been rated 5 at least
once throughout their career. Hence, we refer to the ability rating of 5 as the base
group. The most interesting findings in price regression from table 18 is that the
assistant’s price offer exhibits a non-monotonic pattern.
In our model, when an assistant believes a pusher is of very low ability (i.e
pushers with rating of 1 or 2), it is optimal for the assistant to offer a high price.
However, when the assistants believe that the pusher is less capable but is not at
the bottom rung of the ability ladder (i.e pushers with a rating of 3 or 4), the
assistants may have the incentive to give him a price discount. Our theoretical
rationale is that the assistant knows these pushers will not attempt to work hard
without their help. With the assistant’s help, the pushers are more likely to continue
to trade in the next period rather than drop out. Thus, the assistant is willing to
sacrifice her profits today and give them a price discount to incentivize them to
work hard, so the assistant can continue to profit from this group of pushers in
future trades. Empirically, we did not observe such non-monotonic pricing to the
second lowest ability group. In fact, pushers with an ability rating below 5 are all
offered a significantly higher price. If the pushers with a 5 rating are of average
ability, then the less capable the assistant believed the pushers are compared to
an average ability pusher, the higher the offer price. The least able pushers were
asked for a 3-5% price premium (see column (4)). Though this does not support a
possible theoretical outcome, this is not an entirely surprising result. In this highly
profit-driven and risky market, it is expected that any assistant would not want to
give away short time profits for long term gain, because the pusher or the assistant
herself may be caught by police any time. Assistants view pushers with a rating of 5
and 6 as equal in selling ability, and grant them equal treatment in pricing. However,
if the assistant sees the pusher as a top performer (with a rating of 9 or 10), she
will give him a significant discount of 10-13%. These exceptionally capable pushers
typically demanded quantities 3 to 4 times larger than average ability pushers. Price
discounts helped the assistant keep these big customers.
The most intriguing result is that assistants also ask for a 2% price premium from
pushers with a rating of 8 and 9 compared to average ability pushers. This finding is
consistent with the assistant’s non-monotonic pricing predicted in our model. In our
model, when it comes to the pushers with an ability rating of 7 or 8, the assistant
24
consider them as capable but not yet a top pusher. The assistant also knows that
she will not cause these pushers to shirk just because of the price premium she is
asking for, because the pushers with this rating have lower expected loss compared
to the pushers with a rating of 5. In fact, the assistant knows that if these pushers
are indeed of high ability, they will eventually perform equally well as those with
ability assessments of 9 and 10, and it will not be too late to give them lower prices
when they prove themselves to be top performers. By doing this, the assistant can
protect herself from losing money (i.e with a price discount) due to an incorrect
assessment of the pusher’s type in the current period. Again, the risky nature of the
market motivates the assistant to care more about the short term gain rather than
potential long run loss.
We know that distributions of ability exhibit very similar patterns from the 1st
trade to the 4th trade with 75% of the pushers rated between 4-6. One third of
the pushers dropped out of the market after the 4th trade. It was only after that
we see the distribution of the ability converging to a lightly right-skewed normal
distribution with a small left tail. Likewise, the transition dynamics of the ability
also reveal a consistent trend in the first 4 trades. Therefore, we repeat this analysis
using two sub groups: the sub-sample consists only trades from 1-4 (see column
(2) and (5)) and the sub-sample consists all the transactions after the 4th trade
(see column (3) and (6)). A similar non-monotonic pricing pattern emerged, but is
less pronounced in the early trades. In the initial stage, both assistants and pushers
learn with more noise, hence the assistant plays a more conservative pricing strategy
with less exaggerated pricing variation.
Unit prices vary greatly by drug type due to the different chemistry composition and cost of each drug. The average unit price of Ice is high at S$150, whereas
other drugs are priced between S$20-30. A magnitude increase of S$4 in the price
of Ketamine had a very different implication from the same increase in the price
of Ice. The observed results in table 19 are mainly driven by Ice transactions, not
only because 40% of the transactions are Ice-related but also because the price response to the pusher’s ability rating in Ice was more exaggerated. Hence, we further
analyze the assistant pricing strategy by drug type in table 20. The assistant’s nonmonotonic pricing strategy is clearly observed in transactions with most of the drug
types. Due to the small sample of Ecstasy and Erimin transactions, some coefficients
on ability rating are less precisely estimated in columns (3) and (4). In comparison
with an average pusher, pushers with a rating of 7 and 8 receive a S$2-4 discount,
the lowest ability pusher receives price premium of S§10-15, and top ability pushers
receive a S$12 price discount when purchasing Ice. The general pricing patterns are
the same for other types of drugs, but with smaller magnitude. In short, top pushers
receive a S$2-3 discount and lousy pushers receive a S$2-3 premium, while pushers
with a rating of 7 and 8 were asked for a S$1 (equivalent of 3%) premium.
The pooled OLS captures both within and between differences. To understand
how an assistant’s learning about a pusher’s ability influences her pricing strategy
over time, we conduct fixed effects analysis using table 21. Based on previous results,
we simplify the ability measure by grouping it into 4 categories: low ability with
rating 1 and 2; second lower ability with rating 3 and 4; average ability with 5
and 6; second top ability with 7 and 8; and top ability with rating 9 and 10. The
25
findings are very much consistent with what we observed from OLS regressions,
indicating that assistants learn about a pusher’s ability as they transact and adopt
non-monotonic pricing strategy as she learns.
Discussion on other Explanatory Variables
The assistant’s pricing strategy is not solely influenced by his belief of the
pusher’s ability. The pusher’s other individual characteristics and the characteristics of each order also matter. The unit price and drug type are the key factors
that influence the assistants’ pricing. Together, they explain 96% of price differences.
Pushers who started out with business ties on average received a 1.4% discount on
purchases of a particular type of drug. This implies that pushers with business ties
tend to perform better and receive more preferential treatment prices. Similarly,
pushers who are gang members tend to perform better with easier access to the
end-user market. Whether a pusher has a formal job is irrelevant in the assistant’s
pricing decision. A more interesting result is that the assistant requested a 1% price
premium from pushers who are also drug abusers themselves. It could be because
the assistant believed drug-addicted pushers are less productive and are more likely
to be sent to rehabilitation or arrested. It could also be the case that the assistant
knew drug addicted pushers need drugs or money to feed their addiction. A higher
price would not discourage purchases given the assistant’s monopoly power. Moreover, the assistant consistently requested a 2% price premium from the pushers who
are not picked by herself or people she trusted.
Lastly, a mixture of different types of drugs was sold in most of the trades.
Considering the quantity and the types of drugs the pusher buys, the assistant may
make a choice to give discounts on only one type of drug and give more free drugs
as gifts. We will further explore it in later drafts.
7.3
Pusher’s Effort Response
Our theory predicts that the pusher will always shirk if the common belief that he
is a high performer is below a cut-off value and he will always work if it is above
this cut-off.For tractability, we assumed that the pusher’s action was binary-work
or shirk. In reality, the pusher may exert different levels of effort when he works. In
this subsection, we briefly discuss how a pusher adjusts his effort in response to the
assistant’s price offer. Therefore, we create a new measure of price: price quintile.
Price quintile organizes the price distribution for each drug type into 5 equal groups.
The reference group is the first quintile, which captures the transactions where prices
fall into the bottom 20% of price distribution for the drug traded. This way, we do
not need to be concerned that the changes in effort are a response to the differences
in drug type.
We use the equation (3) to perform estimation, and controlled for past trading
outcomes and the pusher’s individual characteristics in all specifications. Log effort
is used as the dependent variable. A summary of results is present in table 22.
The basic OLS results in Column (1) and (2) tell us that high prices discouraged
effort. The assessed ability of pushers is not controlled in column (1) but controlled
in column (2). Due to the strong correlation between effort and ability assessment,
26
the effect of price on the pusher’s effort is reduced by half when the ability measure
is omitted in the first specification. However, there may be variables that influence
both price and the pusher’s effort that we cannot control for. To mitigate the
estimation bias, we use instrument variable estimation. We use cost of drug, drug
type and quality as instruments for price. For ease of interpretation, we now use log
unit price instead of the price quintile. As mentioned, over 95% of variation in price
is explained by these three factors. We also show in the summary table that effort is
independent of these characteristics of drugs. Hence, they make decent candidates of
IV. The OLS output is shown in column (3) and IV output is in column (4). These
two estimation results illustrate that high offer prices discourages effort. IV estimate
is smaller than the OLS estimate on log unit price. A 10% decrease price correlates
to a 30% increase in effort. Notice that the prices we observed were the equilibrium
prices at which pushers agree to trade. Only when the pushers believed he were
offered with fair value that reflects his productivity (ability), he would continue to
trade with the assistants. Therefore, as long as the pusher stay in the business with
assistants, we can infer that he learns.
8
Conclusion
This paper provides an analysis of the transaction data that we obtained from a
drug selling gang in Singapore. We find that when selling drugs to the pushers, the
assistants learned about the pushers’ selling abilities by observing their performance,
and adjusted the prices of drugs according to the learning processes. Interestingly,
the pricing strategies employed by the assistants were typically non-monotonic in
their assessments of the pushers’ abilities. We also observe that pushers reacted to
the assistants’ learning processes and price variations by changing their effort levels.
These findings indicate that in an underground economy like the drug-selling market
in Singapore, agents are forward-looking and act rationally. Economic models with
profit maximizing agents fit these activities well and thus can generate insights both
positively and normatively in future research.
Our analysis differs from previous research on drug-selling activities by focusing
on the transactions happened between two layers of members in the gang. Levitt
and Venkatesh (2000) study the organizational structure of a drug selling gang in a
Chicago neighborhood by closely investigating the gang’s financial activities. They
find that compensation in the gang is highly skewed, with bottom members only
earning roughly the minimum wage while top leaders earning far more than their
formal sector counterparts. They interpret these findings as showing that potential
promotion along the organizational hierarchy and the prospects of future returns
are the primary motivations for bottom members to work for the gang. Galenianos,
Pacula and Persico (2012) study the market structure of drug-selling activities by
examining the trading interactions between drug customers and retailers. In the
search model with the presence of a moral hazard problem that they develop, a
customer searches for new retailers by incurring some costs, and can observe the
quality of drugs only after the trade is made. The market equilibrium of the model
shows the existence of both short-term motivated rip-offs and long-term relation-
27
ships between customers and retailers, as well as the presence of considerable price
dispersion. Our study bridges views between organization and market structure,
and thus complements these previous studies.
Pushers and assistants were interested in maximizing profits. There were several
gangs in the drug-selling market in Singapore and each gang had many assistants,
so the total amount of assistants was considerable. If assistants could compete
for talented pushers, then pushers of higher selling abilities should get lower prices
when purchasing drugs. However, competition among assistants did not arise in this
market. According to industry insiders, this was because current gangs inherited
knowledge from gangs of the past. Historically, the Ghee Hin Kongsi, one of the
most powerful secret societies of the time, monopolized the opium industry in Singapore in the 1800s through the exclusive rights to purchase raw opium from the
colonial government. To corner the market, the secret society formed specialized
gang divisions to nab opium smugglers who were trying to sell opium themselves
(Lim, 1999). As such, the gang was able to control opium prices in Singapore and
make large profits. Even after the authorities dismantled the Ghee Hin Kongsi, its
market structure in selling drugs continued to strongly influence subsequent gangs
throughout the recent decades. Drug-selling gangs, regardless of whether they had
historical bonds to Ghee Hin Kongsi, realized that huge profits could be made if they
had monopoly power. Thus, the gangs began to divide Singapore up geographically
and to monopolize drug selling in their own territories. According to an ex-gang
leader, gangs would prevent rival gangs from operating in their “turf” by threatening violence, which sometimes resulted in gang wars. As such, a gang had complete
monopoly of drug-selling within its own territory and the assistants of the gang
could determine prices as they desired (Singapore History Museum, 2002).
According to industry insiders who operated in Singapore and other countries,
many other underground markets such as loansharking, smuggling, contraband tobacco and illicit wildlife trading in East Asia and Southeast Asia share crucial similarities with the drug-selling market that we studied. Activities in these markets
are illegal and are often controlled or overseen by gangs or gang-like organizations.
Buyers and sellers typically transact exclusively with each other to overcome shortterm opportunism or holdup problems. Moreover, due to their advantages in the
organizational hierarchy, sellers have significantly more bargaining power compared
to buyers. All these aspects indicate that learning about the characteristics of the
buyers is important since it may help the sellers to make better decisions in the
future, and the sellers can use prices as incentive devices to influence the buyers’
behavior. For instance, in the loansharking market in Singapore, the loan sharks
(lenders) are concerned about the borrowers’ repayment abilities, and their treatments to the borrowers greatly depend on the borrowers’ repayment performance.
When the loan is cleared fully and punctually, the loan shark often tries to keep
the borrower by offering more loans or discounts17 . However, punishment would
17
An interviewee said that “I made my payments. No one ever bothered me, and once in a while
they would ask if I needed more money” (Hill & Kozup, 2007.) Another borrower borrowed $400
with the condition of paying $150 weekly for a month. She receives a re-loan offer with a favorable
discount after her last instalment (Hubba, 2013).
28
be executed given any late installments18 . The various treatments under different
conditions may be explained by the updating of the borrowers’ repaying abilities in
the view of loan sharks. Therefore, our analysis also has the potential to generate
insights for the analysis of these markets.
Our study can also shed light on how firms in the formal sector choose to pay
their employees. Companies that are monopolies may also set employees’ compensations in a similar non-monotonic pattern. For example, the bonus that a wrestler
obtains from WWE (World Wrestling Entertainment) is typically a large proportion
of his/her total income. We find that the bonuses given do not match the wrestlers’
ranking meaning that some more highly ranked wrestlers obtained low bonuses and
vice versa. Thus, WWE sets its bonus scheme non-monotonically. Our theoretical
results could explain this.
In our model we assume that both the pusher and assistant do not know the
pusher’s selling ability, hence information is symmetric between the players. This
assumption is plausible in markets such as the drug-selling market in question, where
an agent’s ability is task-specific and can be evaluated by measurable criteria. However, in other markets, an employee may have private information at the outset or
gain private information during his work. In this scenario, learning will still occur,
but may go through a more complicated process. Specifically, the employee may manipulate the employer’s belief about the employee’s ability by making unexpected
decisions, and benefit from such belief manipulation. Taking this into account, the
employer may have to alter the compensation scheme. It would be interesting to
build a model for empirical testing in future research.
18
A borrower said that “If I was a little late with my payment they would call me and the tone
would change”, and another one claimed that “They would call two or three times a day” (Hill &
Kozup, 2007).
29
Table 1: Summary at Trade Level
Trade No.
No. of Pushers
No. of Orders
354
354
351
351
320
276
202
185
153
128
96
58
33
24
22
2-6
1219
1220
1231
1233
1089
885
560
478
381
309
221
108
58
42
40
2-8
1st
2nd
3rd
4th
5th
6th
7th
8th
9th
10th
11th
12th
13th
14th
15th
16th and Above
Figure 1. Distribution of Ability
30
Table 2: Transition Matrix (Between 1st and 2nd Trade, N=354 )
Ability
Ability
2
3
4
5
6
7
8
9
100.0 0.0
0.0 52.2
0.0
1.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.3
3.7
0.0
47.8
65.0
2.8
0.0
0.0
0.0
22.3
0.0
0.0
34.0
61.7
7.7
0.0
0.0
29.7
0.0
0.0
0.0
35.5
64.6
6.8
0.0
23.4
0.0
0.0
0.0
0.0
27.7
68.2
7.1
13.8
0.0
0.0
0.0
0.0
0.0
25.0
78.6
6.2
0.0
0.0
0.0
0.0
0.0
0.0
14.3
0.6
(T =1)
2
3
4
5
6
7
8
Total
(T =2)
Table 3: Transition
Total
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
Matrix (Between 2nd and 3rd Trade,
N=354 )
Ability
Ability
(T =2)
2
3
4
5
6
7
8
9
Total
1
2
3
4
5
(T =3)
6
7
8
50.0 50.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 41.7 55.6 2.8 0.0 0.0 0.0
0.0 0.6 1.7 52.0 44.1 1.7 0.0 0.0
0.0 0.0 0.9 5.2 48.1 44.8 0.5 0.5
0.0 0.0 0.0 0.0 6.8 43.9 48.6 0.7
0.0 0.0 0.0 0.0 2.2 11.8 46.2 39.8
0.0 0.0 0.0 0.0 0.0 2.8 19.4 58.3
0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0
0.1 0.3 2.8 17.5 27.4 24.7 17.4 8.8
9
0.0
0.0
0.0
0.0
0.0
0.0
19.4
0.0
1.0
Total
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
Table 4: Transition Matrix (Between 3rd and 4th Trade, N=351 )
Ability
Ability
1
2
3
4
5
6
7
8
9
Total
(T =3)
1
100.0
100.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.4
2
3
4
5
(T =4)
6
7
8
9
Total
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0
15.0 25.0 55.0 5.0 0.0 0.0 0.0 0.0 100.0
0.8 4.8 30.6 61.3 2.4 0.0 0.0 0.0 100.0
0.0 3.1 7.7 34.5 51.5 2.1 1.0 0.0 100.0
0.0 0.0 1.1 6.9 33.7 55.4 2.9 0.0 100.0
0.0 0.0 0.0 7.5 12.5 35.0 43.3 1.7 100.0
0.0 0.0 0.0 0.0 8.1 19.4 43.5 29.0 100.0
0.0 0.0 0.0 0.0 0.0 14.3 57.1 28.6 100.0
0.6 2.4 9.4 23.4 25.8 22.1 12.8 3.1 100.0
31
Table 5: Transition Matrix (Between 4th and 5th Trade, N=320 )
Ability
Ability
(T =4)
1
2
3
4
5
6
7
8
9
Total
1
100.0
100.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.4
2
3
4
5
(T =5)
6
7
8
9
10
Total
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
64.3 14.3 21.4 0.0 0.0 0.0 0.0 0.0
0.0 20.3 21.9 56.2 1.6 0.0 0.0 0.0
0.0 3.8 7.0 39.9 46.2 2.5 0.6 0.0
0.0 0.0 2.9 8.6 37.7 47.4 3.4 0.0
0.0 0.0 0.0 7.3 8.7 38.0 44.7 1.3
0.0 0.0 0.0 0.0 5.8 14.0 45.3 34.9
0.0 0.0 0.0 0.0 0.0 4.8 9.5 76.2
1.3 3.1 4.9 18.6 23.5 23.4 17.1 7.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
9.5
0.3
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
Table 6: Transition Matrix (Between 5th and 6th Trade, N=276 )
Ability
Ability
2
3
4
5
6
7
8
9
10
Total
(T =5)
1
100.0
7.7
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.8
2
3
4
5
(T =6)
6
7
8
9
10
Total
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0
69.2 7.7 15.4 0.0 0.0 0.0 0.0 0.0 0.0 100.0
0.0 37.0 33.3 25.9 3.7 0.0 0.0 0.0 0.0 100.0
0.0 3.4 10.3 40.5 44.0 1.7 0.0 0.0 0.0 100.0
0.0 0.7 4.1 7.6 40.0 45.5 2.1 0.0 0.0 100.0
0.0 0.0 0.7 2.7 8.8 40.8 46.9 0.0 0.0 100.0
0.0 0.0 0.0 0.0 3.0 9.0 45.0 42.0 1.0 100.0
0.0 0.0 0.0 0.0 0.0 0.0 9.8 78.0 12.2 100.0
0.0 0.0 0.0 0.0 0.0 0.0 100.0 0.0 0.0 100.0
1.5 2.7 5.1 11.6 21.2 23.1 20.5 12.5 1.0 100.0
32
Table 7: Transition Matrix (Between 6th and 7th Trade, N=202 )
Ability
Ability
(T =6)
2
3
4
5
6
7
8
9
10
Total
1
100.0
12.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.1
2
3
4
5
(T =7)
6
7
8
9
10
Total
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0
50.0 25.0 12.5 0.0 0.0 0.0 0.0 0.0 0.0 100.0
0.0 20.0 55.0 25.0 0.0 0.0 0.0 0.0 0.0 100.0
0.0 3.0 17.9 29.9 47.8 1.5 0.0 0.0 0.0 100.0
0.0 0.9 2.6 10.5 43.9 41.2 0.9 0.0 0.0 100.0
0.0 0.0 0.8 0.0 9.3 54.2 34.7 0.8 0.0 100.0
0.0 0.0 0.0 0.0 2.2 8.8 42.9 45.1 1.1 100.0
0.0 0.0 0.0 0.0 0.0 0.0 14.0 72.0 14.0 100.0
0.0 0.0 0.0 0.0 0.0 0.0 25.0 0.0 75.0 100.0
0.8 1.9 5.9 7.8 20.0 25.2 18.7 16.4 2.3 100.0
Table 8: Transition Matrix (Between 7th and 8th Trade, N=185 )
Ability
Ability
2
3
4
5
6
7
8
9
10
Total
(T =7)
1
100.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.5
2
3
4
5
(T =8)
6
7
8
9
10
Total
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0
60.0 40.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0
0.0 26.3 42.1 31.6 0.0 0.0 0.0 0.0 0.0 100.0
0.0 0.0 34.3 22.9 40.0 2.9 0.0 0.0 0.0 100.0
0.0 0.0 1.1 17.0 43.2 37.5 1.1 0.0 0.0 100.0
0.0 0.0 1.0 2.9 6.8 59.2 29.1 1.0 0.0 100.0
0.0 0.0 0.0 0.0 1.5 16.2 51.5 30.9 0.0 100.0
0.0 0.0 0.0 0.0 0.0 1.8 14.3 69.6 14.3 100.0
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 100.0
0.8 1.8 5.7 8.3 15.5 27.7 19.2 15.8 4.7 100.0
33
Table 9: Transition Matrix (Between 8th and 9th Trade, N=153 )
Ability
Ability
(T =8)
1
3
4
5
6
7
8
9
10
1
4
5
6
7
8
9
10
100.0 0.0
0.0
0.0 100.0 0.0
0.0 81.8 18.2
0.0
0.0 50.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
50.0
15.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
55.0
7.1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
30.0
61.9
2.8
10.5
0.0
0.0
0.0
0.0
0.0
0.0
31.0
75.0
15.8
0.0
0.0
0.0
0.0
0.0
0.0
0.0
22.2
57.9
14.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
15.8
85.7
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
6.5
9.2
22.9
28.1
13.1
5.9
100.0
0.7
Total
3
(T =9)
7.8
5.9
Total
Table 10: Transition Matrix (Between 9th and 10th Trade, N=128 )
Ability
Ability
1
3
4
5
6
7
8
9
10
Total
(T =9)
1
100.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.7
2
3
4
5
(T =10)
6
7
8
9
10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
45.5 54.5 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 85.0 15.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 54.2 41.7 4.2 0.0 0.0 0.0 0.0
0.0 0.0 0.0 20.0 46.7 33.3 0.0 0.0 0.0
0.0 0.0 0.0 0.0 7.6 60.6 31.8 0.0 0.0
0.0 0.0 0.0 0.0 1.3 7.9 71.1 19.7 0.0
0.0 0.0 0.0 0.0 0.0 5.4 10.8 70.3 13.5
0.0 0.0 0.0 0.0 0.0 0.0 0.0 13.3 86.7
1.8
8.2
5.7
5.7
34
7.5
20.6 28.1 15.3
6.4
Total
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
Table 11: Transition Matrix (Between 10th and 11th Trade, N=96 )
Ability
Ability
(T =10)
1
2
3
4
5
6
7
8
9
10
1
100.0
50.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
1.8
Total
2
3
4
(T =11)
5
6
7
8
9
10
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
50.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
64.3 35.7 0.0 0.0 0.0 0.0 0.0 0.0 0.0
0.0 64.3 35.7 0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 40.0 46.7 13.3 0.0 0.0 0.0 0.0
0.0 0.0 0.0 17.6 58.8 23.5 0.0 0.0 0.0
0.0 0.0 0.0 0.0 12.8 61.5 25.6 0.0 0.0
0.0 0.0 0.0 0.0 1.4 14.3 61.4 22.9 0.0
0.0 0.0 0.0 0.0 0.0 0.0 11.8 79.4 8.8
0.0 0.0 0.0 0.0 0.0 0.0 0.0 6.7 93.3
4.9
6.2
4.9
4.5
8.0
17.0 25.4 19.6
7.6
Total
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
Table 12: Transition Matrix (Between 11th and 12th Trade, N=57 )
Abilityt+1
Abilityt
1
2
3
4
5
6
7
8
9
10
Total
1
2
3
4
5
6
7
8
100.0 0.0
71.4 28.6
0.0 71.4
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
28.6
37.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
62.5
22.2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
55.6
20.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
22.2
73.3
12.5
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
6.7
66.7
11.9
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
16.7
61.9
14.3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
4.2
0.0
26.2
0.0
82.1
3.6
0.0 100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
100.0
3.3
4.6
5.2
10.5
14.4
22.2
22.9
100.0
4.6
4.6
35
9
10
7.8
Total
Figure 2. Movement of Assessed Ability (Pushers Start with Ability Rating below 5)
Figure 3. Movement of Assessed Ability (Pushers Start with Ability Rating At Least 5)
36
Table 13: Pusher’s Characteristics (1)
%
Education level
Illiterate
Primary
Secondary
High School
Advanced Diploma or College
5.6
38.4
47.5
7.1
1.4
Marital status
Single or In relationship
Married
Divorced
70.4
12.4
17.2
Full-time or part-time?
Jobless
Full-time
Part-time
42.1
45.8
12.1
Gang member
No
Yes
33.6
66.4
Arrested before
No
Yes
41.2
58.8
Ever sent to Rehab
Not a Drug Addicted
Yes
No
31.4
30.0
38.7
The scale of frequency borrow money
Never
Seldom
Sometimes
Usually
Always
2.5
18.1
21.5
33.1
24.9
N
354
37
Table 14: Pusher’s Characteristics (2)
%
Why be pusher
He needs money for drugs
He needs money to repay his debt
He needs money for lavish
His friends are doing it
56
39
19
18
How the Pusher was Introduced
Pusher approached me directly
By non-gang Friends who I trust
By a member in my gang
I pick pusher myself
By my pusher
43.2
25.7
10.7
5.4
0.6
Type of business partner Collaborating
No Business Partner
KTV
KTV Club
Club
Disco
Brothel
Club Disco
Pub
KTV Disco
KTV Pub
Brothel KTV
Brothel Club
Club Pub
Disco Pub
KTV Club Pub
42.4
25.1
6.5
3.4
3.4
2.3
1.7
1.7
0.8
0.8
0.8
0.3
0.3
N
354
Table 15: Summary Statistics of Pushers
Variable
Mean
Std. Dev.
Min.
Max.
N
Age
Time known him till first trade (Months)
Number of business ties
Month income from job
32.1
10.9
6.6
1490.6
8.7
11.2
4.1
528.5
19
1
2
700
52
120
26
3500
354
353
203
203
38
Table 16: Order Level Characteristics
%
Drug Type
Ice
Ketamine
Ecstasy
Erimin
Other
Heroin
40.0
21.2
20.0
13.7
4.5
0.7
Drug quality
Very Good
Good
Average
34.1
64.7
1.1
Pusher Picks Up
Yes
No
55.1
44.9
Bargain Occurs
No
Yes
87.0
13.0
Have some Fraction as Gift
No
Yes
65.3
34.7
Market factors
Market price
Consignment
No reason specified
Tighter police monitoring
High demand period
Short of supply
My supply reason
He buy small amount
He buys a lot
Competition
46.4
28.5
7.8
5.4
4.7
4.3
0.7
0.7
0.4
0.1
N
9,103 - 9,134
Table 17: Order Level Characteristics by Drug Type
Ice
Ketamine
Ecstasy
Erimin
Heroin
Other
All
Unit Cost
83.16
17.44
15.65
20.19
14.17
10.87
43.43
Unit Price
158.27
26.89
24.58
35.22
17.89
16.83
79.61
Unit Profit
75.11
9.46
8.93
15.04
3.72
5.96
36.18
Quantity sold(gram)
10.86
53.01
70.60
42.93
390.25
162.54
45.50
Sales (Order Level)
1621.47
1330.79
1598.43
1387.42
5377.50
2514.64
1588.89
Ability
6.17
6.27
6.33
6.16
7.63
6.27
6.24
Effort
43.54
43.43
45.78
42.79
70.41
50.75
44.50
39
Table 18: Determinants of Ability
(1)
ability
Effort (In Previous Trade)
(2)
ability
(3)
ability
(4)
ability
0.069∗∗∗
(0.001)
0.065∗∗∗
(0.001)
1.364∗∗∗
(0.033)
Log Sales in Previous Trade
(5)
ability
0.405∗∗∗
(0.024)
0.123∗∗∗
(0.006)
Accumulative Log Sales upto Previous Trade
Accumulated Times Bargained
Accumulative Times Gift was Given
-0.018
(0.019)
-0.075∗∗∗
(0.012)
0.161∗∗∗
(0.014)
0.014
(0.009)
Gang Member
0.496∗∗∗
(0.040)
0.019
(0.027)
Have Business Connection
0.520∗∗∗
(0.038)
0.111∗∗∗
(0.026)
Job Status (Base=Jobless)
ref.
ref.
0.078∗∗
(0.039)
0.080∗∗∗
(0.027)
0.377∗∗∗
(0.063)
0.140∗∗∗
(0.040)
ref.
ref.
-0.662∗∗∗
(0.045)
-0.133∗∗∗
(0.031)
-0.806∗∗∗
(0.046)
0.023
(0.032)
-0.014
(0.039)
0.139∗∗∗
(0.026)
Full Time
Part Time
Drug Addition (Base=Not Addicted)
Addicted, been to Rehab
Addicted, not yet Rehabed
Heavy Borrower
Introduced By(Base=I pick myself)
ref.
ref.
By non-gang Friends who I trust
-1.244∗∗∗
(0.076)
-0.118∗∗
(0.058)
By a member in my gang
-0.908∗∗∗
(0.092)
-0.344∗∗∗
(0.065)
Pusher approached me directly
-1.250∗∗∗
(0.072)
-0.075
(0.055)
Pusher was my client
-0.794∗∗∗
(0.083)
0.030
(0.063)
-0.099
(0.190)
-0.805∗∗∗
(0.133)
9102
0.125
7888
0.668
By my pusher
N
R2
7913
0.644
Standard errors in parentheses
∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
40
7913
0.197
9132
0.134
Table 19: Assistant’s Non-Monotonic Pricing Strategy
Unit Price
(1)
Full
Ability (Base=5)
(2)
Before Trade 5
Log Price Unit
(3)
After 4th Trade
(4)
Full
(5)
Before Trade 5
(6)
Before Trade 5
ref.
ref.
ref.
ref.
ref.
ref.
1
12.032∗∗∗
(2.370)
-1.529
(2.512)
15.073∗∗∗
(2.805)
0.100∗∗∗
(0.019)
0.051∗∗
(0.022)
0.117∗∗∗
(0.025)
2
6.595∗∗∗
(1.991)
0.622
(1.581)
7.883∗∗∗
(2.666)
0.058∗∗∗
(0.016)
0.026∗
(0.016)
0.071∗∗∗
(0.021)
3
1.769∗
(0.956)
-0.725
(1.221)
3.561∗∗
(1.496)
0.027∗∗∗
(0.008)
0.005
(0.011)
0.051∗∗∗
(0.014)
4
1.157∗∗
(0.529)
0.464
(0.590)
2.740∗∗
(1.201)
0.009
(0.006)
-0.001
(0.007)
0.039∗∗∗
(0.011)
6
-0.060
(0.396)
0.118
(0.454)
-0.749
(0.806)
-0.006
(0.004)
-0.005
(0.005)
-0.003
(0.007)
7
1.963∗∗∗
(0.399)
1.461∗∗∗
(0.498)
2.041∗∗∗
(0.755)
0.027∗∗∗
(0.004)
0.023∗∗∗
(0.005)
0.035∗∗∗
(0.007)
8
1.334∗∗∗
(0.483)
1.058
(0.729)
1.321∗
(0.796)
0.014∗∗∗
(0.005)
0.009
(0.009)
0.025∗∗∗
(0.007)
9
-5.583∗∗∗
(0.732)
-2.422
(1.655)
-5.926∗∗∗
(0.974)
-0.092∗∗∗
(0.008)
-0.082∗∗∗
(0.021)
-0.084∗∗∗
(0.009)
10
-6.608∗∗∗
(1.149)
0.000
(.)
-6.066∗∗∗
(1.341)
-0.133∗∗∗
(0.012)
0.000
(.)
-0.122∗∗∗
(0.014)
Have Business Connection
-1.843∗∗∗
(0.309)
-0.689∗
(0.398)
-3.463∗∗∗
(0.499)
-0.020∗∗∗
(0.004)
-0.014∗∗∗
(0.005)
-0.032∗∗∗
(0.005)
Gang Member
-0.958∗∗∗
(0.309)
-0.791∗∗
(0.400)
-1.113∗∗
(0.488)
-0.015∗∗∗
(0.003)
-0.011∗∗
(0.005)
-0.017∗∗∗
(0.005)
Job Status (Base=Jobless)
ref.
ref.
ref.
ref.
ref.
ref.
Full Time
-0.806∗∗∗
(0.313)
-0.953∗∗
(0.405)
-1.123∗∗
(0.485)
0.003
(0.004)
0.007
(0.005)
-0.005
(0.005)
Part Time
-1.297∗∗∗
(0.456)
-0.863∗
(0.505)
-1.397∗
(0.784)
-0.005
(0.005)
0.006
(0.006)
-0.014∗
(0.008)
Drug Addition (Base=Not Addicted)
ref.
ref.
ref.
ref.
ref.
ref.
Addicted, been to Rehab
-0.117
(0.417)
-0.210
(0.589)
-0.618
(0.598)
0.012∗∗∗
(0.004)
0.010∗
(0.006)
0.008
(0.005)
Addicted, not yet Rehabed
-0.351
(0.362)
-0.160
(0.504)
-0.302
(0.528)
0.015∗∗∗
(0.004)
0.015∗∗∗
(0.005)
0.014∗∗
(0.006)
Heavy Borrower
0.050
(0.302)
-0.143
(0.356)
0.113
(0.504)
0.001
(0.003)
0.003
(0.004)
-0.002
(0.005)
Introduced By(Base=I pick myself)
ref.
ref.
ref.
ref.
ref.
ref.
By non-gang Friends who I trust
-0.784
(0.599)
-1.183
(0.885)
-0.914
(0.817)
0.010
(0.007)
0.009
(0.009)
0.011
(0.010)
By a member in my gang
-0.060
(0.672)
0.248
(0.907)
-0.449
(1.016)
0.024∗∗∗
(0.007)
0.023∗∗
(0.009)
0.027∗∗
(0.012)
Pusher approached me directly
-0.198
(0.568)
-0.563
(0.842)
-0.236
(0.768)
0.022∗∗∗
(0.007)
0.023∗∗∗
(0.009)
0.021∗∗
(0.010)
Pusher was my client
-0.949
(0.682)
-0.328
(0.921)
-2.191∗∗
(0.985)
0.015∗∗
(0.007)
0.024∗∗
(0.009)
0.003
(0.011)
-5.013∗∗
(2.409)
-5.144
(3.388)
-5.365
(3.426)
0.025
(0.024)
0.021
(0.026)
0.017
(0.037)
9024
0.967
4834
0.971
4190
0.963
9024
0.978
4834
0.978
4190
0.980
By my pusher
N
R2
Standard errors in parentheses
∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
41
Table 20: Assistant’s Non-Monotonic Pricing Strategy (By
Drugtype)
(1)
Ice
(2)
Ketamine
(3)
Ecstasy
(4)
Erimin
ref.
ref.
ref.
ref.
15.599∗∗∗
(3.510)
5.839
(4.841)
3.208
(2.303)
3.811∗∗
(1.732)
2
10.487∗∗∗
(3.675)
2.534
(2.113)
1.735
(1.159)
2.796∗∗
(1.174)
3
1.466
(2.167)
1.126∗∗
(0.529)
1.349∗∗∗
(0.359)
0.646
(1.169)
4
1.776
(1.224)
0.117
(0.230)
0.418∗∗
(0.187)
1.442∗
(0.839)
6
0.636
(0.909)
-0.236
(0.283)
-0.102
(0.196)
-0.305
(0.563)
7
4.071∗∗∗
(0.927)
0.740∗∗∗
(0.176)
1.074∗∗∗
(0.211)
1.563∗∗∗
(0.520)
8
2.738∗∗
(1.135)
1.085∗∗∗
(0.230)
0.363
(0.273)
1.101
(0.718)
9
-11.803∗∗∗
(1.719)
-2.307∗∗∗
(0.643)
-1.861∗∗∗
(0.330)
-1.080
(1.083)
10
-12.444∗∗∗
(2.837)
-3.746∗∗∗
(0.579)
-2.744∗∗∗
(0.451)
-4.590∗∗∗
(1.207)
Unit Cost
1.450∗∗∗
(0.025)
0.955∗∗∗
(0.065)
0.843∗∗∗
(0.091)
0.329∗∗
(0.139)
Have Business Connection
-4.315∗∗∗
(0.706)
-0.181
(0.225)
-0.433∗∗∗
(0.165)
-0.580
(0.460)
Gang Member
-2.058∗∗∗
(0.708)
-0.137
(0.223)
-0.176
(0.152)
-0.273
(0.518)
ref.
ref.
ref.
ref.
-1.949∗∗∗
0.708∗∗∗
0.409∗∗∗
(0.705)
(0.222)
(0.150)
-0.125
(0.412)
-2.227∗∗
(1.089)
0.579
(0.480)
0.114
(0.212)
0.185
(0.871)
Ability (Base=5)
1
Job Status (Base=Jobless)
Full Time
Part Time
ref.
ref.
ref.
ref.
Addicted, been to Rehab
Drug Addition (Base=Not Addicted)
-0.273
(0.934)
0.642∗∗
(0.259)
0.499∗∗
(0.206)
1.474∗∗
(0.577)
Addicted, not yet Rehabed
-1.904∗∗
(0.904)
0.854∗∗∗
(0.211)
1.280∗∗∗
(0.181)
0.708
(0.493)
Heavy Borrower
-0.457
(0.697)
0.083
(0.215)
0.264∗
(0.154)
-1.019∗∗
(0.436)
ref.
ref.
ref.
ref.
-1.057
(1.293)
1.137∗∗∗
1.200∗∗∗
(0.363)
(0.291)
-0.705
(0.707)
By a member in my gang
0.326
(1.539)
1.537∗∗∗
(0.363)
1.390∗∗∗
(0.300)
-0.281
(0.703)
Pusher approached me directly
0.509
(1.269)
1.579∗∗∗
(0.413)
0.927∗∗∗
(0.258)
0.673
(0.688)
Pusher was my client
-1.036
(1.449)
1.422∗∗∗
(0.406)
0.476
(0.331)
0.554
(0.706)
-18.860∗∗∗
(7.047)
0.000
(.)
0.000
(.)
2.384∗∗
(1.076)
3608
1911
0.292
1799
0.321
1232
0.201
Introduced By(Base=I pick myself)
By non-gang Friends who I trust
By my pusher
N
R2
Standard errors in parentheses
∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
42
0.758
Table 21: Assistant’s Non-Monotonic Pricing Strategy
By Drugtype (Fixed Effects)
(1)
Ice
(2)
Ketamine
(3)
Ecstasy
(4)
Erimin
ref.
ref.
ref.
ref.
13.524∗∗∗
4.722∗∗∗
3.344∗∗∗
(1.894)
(1.045)
(0.637)
2.110
(1.934)
3-4
2.517∗∗∗
(0.877)
0.080
(0.318)
0.364∗
(0.199)
2.343∗∗∗
(0.536)
7-8
3.914∗∗∗
(0.718)
1.712∗∗∗
(0.251)
1.152∗∗∗
(0.166)
1.130∗∗
(0.455)
9-10
-10.202∗∗∗
(1.267)
-0.842∗
(0.456)
-1.195∗∗∗
(0.285)
-1.374∗
(0.812)
1.281∗∗∗
(0.034)
1.160∗∗∗
(0.107)
1.386∗∗∗
(0.089)
0.469∗∗∗
(0.109)
ref.
ref.
ref.
ref.
3.867
(5.788)
-0.961∗∗∗
(0.331)
-0.656
(1.053)
2.206∗
(1.302)
14.385∗∗
(5.802)
0.000
(.)
0.053
(1.134)
1.060
(2.393)
Pusher comes to picks up
0.393
(0.534)
-0.044
(0.182)
0.273∗∗
(0.118)
0.629∗∗
(0.312)
Have Counter Offer
1.220∗
(0.721)
0.068
(0.271)
-0.101
(0.164)
-0.095
(0.512)
Have Some Given as Gift
-0.288
(0.539)
-0.301
(0.193)
-0.040
(0.127)
0.030
(0.389)
My supply reason
9.022∗
(5.068)
0.234
(1.535)
0.096
(0.781)
-1.666
(1.376)
He buys a lot
-13.230∗∗∗
(4.820)
0.549
(3.157)
-0.026
(0.846)
0.267
(2.919)
Competition
0.000
(.)
0.000
(.)
0.005
(1.845)
0.000
(.)
Tighter police monitoring
9.450∗∗∗
(1.272)
0.713
(0.436)
1.632∗∗∗
(0.308)
-0.030
(0.966)
High demand trade
9.852∗∗∗
(1.304)
0.812
(0.547)
0.729∗
(0.377)
-1.369
(1.034)
3.187
(8.193)
0.990
(0.876)
1.285∗∗
(0.637)
3.594∗
(1.909)
Short of supply
15.155∗∗∗
(1.503)
-0.067
(0.594)
1.216∗∗
(0.584)
2.774∗∗
(1.284)
Consignment
9.289∗∗∗
(0.786)
0.171
(0.273)
0.451∗∗∗
(0.169)
0.938∗
(0.498)
4.805
(6.083)
0.514
(3.853)
0.372
(2.445)
3.631
(4.460)
12.914∗∗∗
(1.211)
-0.520
(0.491)
0.579∗
(0.337)
1.234
(0.796)
-2.865
(3.883)
2.798∗∗
(1.211)
0.309
(0.742)
-0.066
(2.383)
34.382∗∗∗
(6.555)
6.542∗∗∗
(1.897)
2.806
(1.734)
21.633∗∗∗
(2.091)
3461
0.652
1848
0.229
1738
0.264
1192
0.120
Ability (Base=5)
1-2
Unit Cost
Drug Quality (Base=Average)
Good
Very Good
He buy small amount
Pusher has no other supply
No reason specified
Others
Constant
N
R2
Standard errors in parentheses
∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
43
Table 22: Pusher’s Effort Response (Log Effort) (1/2)
(1)
OLS
(2)
OLS
(3)
OLS
(4)
IV
Price (Base= 1st Quintile)
2nd Quintile
ref.
-0.127∗∗∗
(0.017)
ref.
-0.079∗∗∗
(0.013)
ref.
ref.
3rd Quintile
-0.230∗∗∗
(0.018)
-0.117∗∗∗
(0.014)
4th Quintile
-0.322∗∗∗
(0.022)
-0.165∗∗∗
(0.017)
5th Quintile
-0.436∗∗∗
(0.025)
-0.223∗∗∗
(0.019)
0.922∗∗∗
(0.016)
0.923∗∗∗
(0.012)
-0.043∗∗∗
(0.006)
-0.034∗∗∗
(0.006)
0.897∗∗∗
(0.016)
Ability Above 5
Log Unit Price
Log Accumulative Sales
0.251∗∗∗
(0.015)
0.093∗∗∗
(0.012)
0.100∗∗∗
(0.012)
0.101∗∗∗
(0.012)
Accumulative Times Bargained
-0.049∗∗∗
(0.007)
-0.025∗∗∗
(0.005)
-0.031∗∗∗
(0.005)
-0.031∗∗∗
(0.006)
Have Counter Offer
0.030∗
(0.017)
0.028∗∗
(0.014)
0.027∗
(0.014)
0.026∗
(0.015)
Have given Gift
0.026∗
(0.014)
0.007
(0.011)
-0.001
(0.011)
-0.000
(0.011)
Age
-0.001
(0.001)
-0.003∗∗∗
(0.001)
-0.002∗∗
(0.001)
-0.002∗∗
(0.001)
Gang Member
0.102∗∗∗
(0.016)
0.068∗∗∗
(0.013)
0.077∗∗∗
(0.013)
0.078∗∗∗
(0.012)
Have Business Connection
0.060∗∗∗
(0.014)
0.006
(0.011)
0.006
(0.011)
0.008
(0.012)
Job Status (Base=Jobless)
ref.
ref.
ref.
ref.
Full Time
∗∗
0.030
(0.015)
-0.017
(0.012)
-0.017
(0.012)
-0.017
(0.012)
Part Time
0.054∗∗
(0.024)
0.009
(0.018)
0.013
(0.019)
0.014
(0.018)
ref.
ref.
ref.
ref.
Addicted, been to Rehab
-0.138∗∗∗
(0.018)
-0.047∗∗∗
(0.015)
-0.056∗∗∗
(0.015)
-0.057∗∗∗
(0.015)
Addicted, not yet Rehabed
-0.215∗∗∗
(0.018)
-0.125∗∗∗
(0.015)
-0.144∗∗∗
(0.015)
-0.145∗∗∗
(0.014)
-0.015
(0.015)
-0.030∗∗
(0.012)
-0.026∗∗
(0.012)
-0.028∗∗
(0.011)
Drug Addition (Base=Not Addicted)
Heavy Borrower
Standard errors in parentheses
∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
44
Table 23: Pusher’s Effort Response (Log Effort) (2/2)
Introduced By(Base=I pick myself)
(1)
OLS
(2)
OLS
(3)
OLS
(4)
IV
ref.
ref.
ref.
ref.
By non-gang Friends who I trust
-0.233
(0.025)
-0.111
(0.019)
-0.124
(0.020)
-0.126∗∗∗
(0.026)
By a member in my gang
-0.177∗∗∗
(0.032)
-0.078∗∗∗
(0.025)
-0.098∗∗∗
(0.025)
-0.099∗∗∗
(0.029)
Pusher approached me directly
-0.234∗∗∗
(0.023)
-0.110∗∗∗
(0.018)
-0.128∗∗∗
(0.018)
-0.129∗∗∗
(0.025)
Pusher was my client
-0.105∗∗∗
(0.025)
-0.055∗∗∗
(0.019)
-0.062∗∗∗
(0.019)
-0.063∗∗
(0.028)
By my pusher
0.232∗∗∗
(0.062)
ref.
0.184∗∗∗
(0.046)
ref.
0.162∗∗∗
(0.048)
ref.
0.164∗∗
(0.074)
ref.
My supply reason
0.160∗∗∗
(0.043)
0.140∗∗∗
(0.028)
0.154∗∗∗
(0.026)
0.158∗∗
(0.065)
He buys a lot
0.258∗∗∗
(0.041)
0.216∗∗∗
(0.033)
0.220∗∗∗
(0.035)
0.223∗∗∗
(0.083)
Competition
0.087
(0.054)
0.013
(0.058)
0.045
(0.056)
0.050
(0.221)
Tighter police monitoring
0.029
(0.027)
-0.016
(0.022)
-0.013
(0.021)
-0.014
(0.024)
High demand trade
0.085∗∗∗
(0.029)
0.069∗∗∗
(0.021)
0.079∗∗∗
(0.021)
0.079∗∗∗
(0.025)
He buy small amount
0.243∗∗∗
(0.075)
0.048
(0.050)
0.022
(0.050)
0.026
(0.061)
Short of supply
0.110∗∗∗
(0.031)
0.081∗∗∗
(0.024)
0.012
(0.024)
0.011
(0.027)
Consignment
-0.097∗∗∗
(0.017)
-0.037∗∗∗
(0.013)
-0.052∗∗∗
(0.014)
-0.051∗∗∗
(0.012)
Pusher has no other supply
0.226∗∗∗
(0.077)
0.110
(0.078)
0.114
(0.080)
0.116
(0.113)
No reason specified
-0.004
(0.027)
0.016
(0.022)
-0.020
(0.022)
-0.022
(0.020)
Yes
Yes
Yes
Yes
Yes
Yes
Yes
Yes
9045
0.269
9045
0.536
9046
0.531
9045
0.530
Market Condition
Trade Region
Trade No.
N
R2
∗∗∗
Standard errors in parentheses
∗
p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01
45
∗∗∗
∗∗∗
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48
Appendix A: Data Collection
Over the past five years, the first author, Kaiwen Leong has spent more than ten
thousand hours working with government, non-government organizations and
private sectors firms to help rehabilitate ex-offenders. He started out as an unpaid
social worker giving motivational talks to the inmates in the Singapore prison and
individuals that have been released from prison. Eventually,he has helped many
ex-offenders to become successful entrepreneurs. His charitable works were
repeatedly reported by the media(i.e Strait Times (2013), National Council for
Social Service (2013)). He also helped their family members with their medical
expenses and any other issues they needed help with. Eventually, the first author
earned their trust and was allowed access to some confidential data these
individuals kept. All of the industry information in this paper were based on
interviews the first author conducted with drug addicts over the last five years.
The first author also hired an ex-drug offender to interview ninety- eight other
convicted drug offenders to collect supplementary and corroborating information
about the drug industry in Singapore.
Appendix B: Proofs.
Proof of Lemma 1.
Proof. When µ = 0, the players’ are certain that the pusher is bad. Bayes’ law
implies that after observing any outcome j ∈ {m, s, f }, both players belief does not
change. Markov perfect equilibrium requires every player’s strategy to be
stationary in equilibrium; that is, a player chooses the same action in all periods.
Thus, a strategy profile is a Markov perfect equilibrium if and only if the strategy
profile in any period t consists of a sub-game perfect equilibrium.
In any one-shot game (an extensive form game), if the assistant sets a low price,
then the pusher’s optimal response is to work, which yields payoffs px + (1 − p)z
and px + (1 − p)w to the assistant and the pusher respectively. If the assistant sets
a high price, then the pusher’s optimal response is to shirk. Both players obtain a
payoff of 0. Since px + (1 − p)z < 0 by Assumption 1, the unique sub-game perfect
equilibrium in this one-shot game is the one where the assistant sets a high price
and the pusher shirks.
Hence, when µ = 0, the Markov perfect equilibrium is unique. In this equilibrium,
the assistant always sets a high price and the pusher always shirks.
A similar argument applies to the case when µ = 1. In the one-shot game, when
µ = 1, the unique sub-game perfect equilibrium is one where the assistant sets a
low price and the pusher works. Hence, the unique Markov perfect equilibrium in
the dynamic game is the one where the assistant always sets a low price and the
pusher always works.
Proof of Proposition 2.
49
Proof. Let µ− = µ − ε, where ε > 0 is sufficiently small. After observing an
outcome (x, x) or (y, y), the updated belief (µ− )s satisfies (µ− )s > µ.
Claim 1. In any equilibrium, if the assistant sets a low price at the belief µ > 0,
then the pusher works. We will do a proof by contradiction. Suppose on the
contrary, there is an equilibrium in which the assistant sets a low price at a belief
µ > 0 in period t. But, the pusher shirks. In the next period, the players have the
same belief µ and face an identical decision problem as in period t. Recursive
induction implies that starting from period t, the assistant sets a low price and the
pusher always shirks. This results in the pusher getting a discounted payoff of 0.
However, if the pusher deviates, working in period t and shirking in any other
period t0 > t, his payoff is [µ + (1 − µ)p]x + (1 − µ)(1 − p)w > 0 which is positive.
This represents a contradiction.
End of Claim 1.
Let µ1 be given by
(1 − p)(w − z) − p(x − y)
µ1 =
(1 − p)(x + w − y − z)
In a one-shot game, if the pusher definitely works, the assistant prefers to set a low
price if and only if µ ≥ µ1 , where µ1 ∈ (0, 1) by Assumption 1.
Claim 2. In any equilibrium, if µ ≥ µ1 , the assistant sets a low price.
We will do a proof by contradiction. Suppose on the contrary there is an
equilibrium in which the assistant sets a high price at a belief µ ≥ µ1 in period t.
First, suppose the pusher’s equilibrium response is to shirk. In this case both
players obtain a discounted payoff of 0. If the assistant deviates and sets a low price
in period t and sets a high price in any other period t0 > t, the pusher responds to
this deviation by working in period t. Hence the assistant can obtain a payoff of
[µ + (1 − µ)p]x + (1 − µ)(1 − p)z > 0
in the current period (by Assumption 1) and is guaranteed a non-negative
discounted future payoff. So this is a profitable deviation.
Second, suppose the pusher’s equilibrium response to this high price is to work.
Then the assistant obtains a payoff [µ + (1 − µ)p]y + (1 − µ)(1 − p)w in period t and
a discounted future payoff of 0 or ∆ > 0. The value of the discounted future payoff
is contingent on what outcome is observed in period t. If the assistant deviates,
sets a low price in period t and then switches back to the equilibrium strategy
starting from period t + 1, she obtains a payoff of [µ + (1 − µ)p]x + (1 − µ)(1 − p)z
in period t, and her payoffs in the continuation game will not change. Since
[µ + (1 − µ)p]x + (1 − µ)(1 − p)z ≥ [µ + (1 − µ)p]y + (1 − µ)(1 − p)w
when µ ≥ µ1 , this deviation is profitable. In summary, in any equilibrium, the
assistant sets a low price whenever µ ≥ µ1 .
End of Claim 2.
50
Together, these claims show that in any equilibrium, if µ ≥ µ1 , the assistant sets a
low price and the pusher works. Denote function U1 (µ) as
U1 (µ) = [µ + (1 − µ)p]x + (1 − µ)(1 − p)w + δ[µ + (1 − µ)p]U1 (µs )
and function V1 (µ) as
V1 (µ) = [µ + (1 − µ)p]x + (1 − µ)(1 − p)z + δ[µ + (1 − µ)p]V1 (µs )
When µ ≥ µ1 the assistant and the pusher’s payoffs are captured by V1 (µ) and
U1 (µ), respectively. Using standard arguments, it can be shown that both
functions V1 (µ) and U1 (µ) are continuous and strictly increasing on [0, 1].
Now we derive the players’ equilibrium behavior for µ < µ1 . Let Π be
Π = [µ1 + (1 − µ1 )p]y + (1 − µ1 )(1 − p)z + δ[µ1 + (1 − µ1 )p]U1 (µs1 )
Denote function U2 (µ) as
(
U2 (µ) =
U1 (µ)
if µ ≥ µ1
s
[µ + (1 − µ)p]y + (1 − µ)(1 − p)z + δ[µ + (1 − µ)p]U2 (µ ) if µ < µ1
)
and function V2 (µ) as
(
V2 (µ) =
V1 (µ)
if µ ≥ µ1
[µ + (1 − µ)p]y + (1 − µ)(1 − p)w + δ[µ + (1 − µ)p]V2 (µs ) if µ < µ1
)
We have two cases, Π ≤ 0 and Π > 0, to discuss.
Claim 3. Suppose Π ≤ 0. The Markov perfect equilibrium is unique. In this
equilibrium, there is a cutoff value µP such that, if µ ≥ µP the assistant sets a low
price and the pusher works. If µ < µP , the assistant sets a high price and the
pusher shirks. Note that µP is given by V1 (µP ) = 0 and satisfies µP ∈ (0, 1).
We show that in any equilibrium if the assistant sets a low price at belief µ < µ1 ,
then the pusher shirks. Notice that if the pusher works, his payoff is bounded
above by
[µ + (1 − µ)p]y + (1 − µ)(1 − p)z + δ[µ + (1 − µ)p]U1 (µs )
His largest possible payoff in the continuation game is U1 (µs ), which is obtained
when an outcome (x, x) or (y, y) in the current period causes the assistant to set a
low price in all future periods (as long as the belief is non-zero). However, for
µ < µ1 , the above payoff is strictly less than Π (it is negative). On the other hand,
if the pusher shirks in all periods, he can guarantee himself a payoff 0. Therefore,
when Π ≤ 0, for any µ < µ1 , the pusher works if and only if the assistant sets a low
price.
Now consider the assistant’s optimal decision for µ < µ1 . Notice that there is a
cutoff value µP ∈ (0, µ1 ) such that V1 (µP ) = 0. Standard arguments show that the
assistant should set a low price if and only if µ ≥ µP .
51
End of Claim 3.
Hereafter, assume Π > 0. There exists a cutoff value, denoted as µ2 , such that
µ2 ∈ (0, µ1 ) and U2 (µ2 ) = 0. Define a sequence as (µ1,1 , µ1,2 , ..., µ1,k , ...) such that
µ1,1 = µ1 and µ1,k = µs1,k+1 , such that if an outcome (x, x) or (y, y) is observed at
belief µ1,k+1 , the updated belief becomes µ1,k . Let µ2 be drawn from the set
[µ1,l+1 , µ1,l ).
Claim 4. Suppose Π > 0. In any equilibrium, if µ ∈ [µ2 , µ1 ), the assistant sets a
high price and the pusher works.
To illustrate this, first consider the case that µ ∈ [max{µ2 , µ1,2 }, µ1,1 ). For any
price the assistant sets, in equilibrium, if the pusher shirks, his payoff is zero.
However, if he deviates and works, then his payoff is either U1 (µ) or U2 (µ) (since
the equilibrium play in the continuation game starting from the next period is
uniquely determined). Because
U1 (µ) > U2 (µ) ≥ 0
for µ ∈ [max{µ2 , µ1,2 }, µ1,1 ), it is optimal for the pusher to work. On the other
hand, given that the pusher chooses to work for any price set by the assistant, her
payoff is V1 (µ) or V2 (µ) if she sets a low price and high price respectively. Since
V2 (µ) > V1 (µ)
for µ ∈ [max{µ2 , µ1,2 }, µ1,1 ), it is optimal for the assistant to set a high price.
This argument can be applied recursively on sets [max{µ2 , µ1,k+1 }, µ1,k ) to show
that for µ ∈ [µ2 , µ1 ), the assistant always sets a high price and the pusher always
works. In addition, for µ ≥ µ2 , the assistant and pusher’s equilibrium payoffs are
captured by V2 (µ) and U2 (µ), respectively.
End of Claim 4.
Denote function U3 (µ) as
(
U3 (µ) =
)
U2 (µ)
if µ ≥ µ2
s
[µ + (1 − µ)p]x + (1 − µ)(1 − p)w + δ[µ + (1 − µ)p]U3 (µ ) if µ < µ2
and function V3 (µ) as
(
V3 (µ) =
V2 (µ)
if µ ≥ µ2
s
[µ + (1 − µ)p]x + (1 − p)(1 − µ)z + δ[µ + (1 − µ)p]V3 (µ ) if µ < µ2
)
Define a sequence as (µ2,1 , µ2,2 , ..., µ2,k , ...) such that µ2,1 = µ2 and µ2,k = µs2,k+1 .
Notice that by construction
U3 (µ2,k ) < U3 (µ−
2,k )
and for any µ ∈ [µ2,k+1 , µ2,k ), U3 (µ) strictly increases.
Consider µ ∈ [µ2,2 , µ2,1 ). Since U2 (µ) < U2 (µ2 ) = 0, if the assistant sets a high
price, the pusher will shirk. Thus, to induce the pusher to work, it is necessary for
52
the assistant to set a low price. On the other hand, if V3 (µ) > 0 and
µ ∈ [µ2,2 , µ2,1 ), the assistant sets a low price. If there is a cutoff value
µP ∈ [µ2,2 , µ2,1 ) such that V3 (µP ) = 0, the assistant sets a low price if and only if
µ ∈ [µP , µ2,1 ). For the latter case, it can be further verified that, for any µ < µP ,
the assistant sets a high price and the pusher shirks.
Now suppose V3 (µ2,2 ) > 0 and consider µ ∈ [µ2,3 , µ2,2 ). (a) If
−
−
−
−
− s
[µ−
2,2 + (1 − µ2,2 )p]y + (1 − µ2,2 )(1 − p)z + δ[µ2,2 + (1 − µ2,2 )p]U3 ((µ22 ) ) > 0
this implies that if the assistant sets a high price at belief µ−
2,2 , the pusher’s payoff
from working is strictly positive, then there is a cutoff value, denoted as µ4 , such
that U4 (µ4 ) = 0, where function U4 (µ4 ) is defined as
(
U4 (µ) =
U3 (µ)
if µ ≥ µ2,2
s
[µ + (1 − µ)p]y + (1 − µ)(1 − p)z + δ[µ + (1 − µ)p]U4 (µ ) if µ < µ2,2
)
In addition, denote µ3 = µ2,2 . Then, for µ ∈ [µ3 , µ2 ) the assistant always sets a low
price. For µ ∈ [µ4 , µ3 ) she sets a high price. On the other hand, the pusher always
works when µ ≥ µ4 .
(b) If
−
−
−
−
− s
[µ−
2,2 + (1 − µ2,2 )p]y + (1 − µ2,2 )(1 − p)z + δ[µ2,2 + (1 − µ2,2 )p]U3 ((µ22 ) ) < 0
then similar to the argument shown previously, it is impossible for the assistant to
induce the pusher to work by setting a high price for µ ∈ [µ2,3 , µ2,2 ). We will do
this analysis by induction. For µ ∈ [µ2,k , µ2,1 ), it is necessary to set a low price if
the assistant wants to induce the pusher to work.
But
−
−
−
−
− s
[µ−
2,k + (1 − µ2,k )p]y + (1 − µ2,k )(1 − p)z + δ[µ2,k + (1 − µ2,k )p]U3 ((µ2k ) ) > 0
If there is a cutoff value µP ∈ [µ2,k , µ2,1 ) such that V3 (µP ) = 0, then sets a low
price if and only if µ ∈ [µP , µ2,1 ). In addition, it can be verified that for any
µ < µP , the assistant sets a high price and the pusher shirks.
If V3 (µ) > 0 for any µ ∈ [µ2,k , µ2,1 ), denote µ3 = µ2,k and let µ4 be the largest value
satisfying U4 (µ4 ) = 0, where function U4 (µ4 ) is defined as
(
U4 (µ) =
U3 (µ)
if µ ≥ µ2,k
s
[µ + (1 − µ)p]y + (1 − µ)(1 − p)z + δ[µ + (1 − µ)p]U4 (µ ) if µ < µ2,k
)
Then, for µ ∈ [µ3 , µ2 ) the assistant always sets a low price. For µ ∈ [µ4 , µ3 ), the
assistant sets a high price. The pusher always works if µ ≥ µ4 .
In this case, we define function V4 (µ) as
(
V4 (µ) =
V3 (µ)
if µ ≥ µ3
s
[µ + (1 − µ)p]y + (1 − µ)(1 − p)w + δ[µ + (1 − µ)p]V4 (µ ) if µ < µ4
For µ ≥ µ4 , the assistant and the pusher’s equilibrium payoffs are captured by
V4 (µ) and U4 (µ), respectively.
53
)
(c) If
−
−
−
−
− s
[µ−
2,k + (1 − µ2,k )p]y + (1 − µ2,k )(1 − p)z + δ[µ2,k + (1 − µ2,k )p]U3 ((µ2k ) ) < 0
for any k, then for any µ < µ2 it is impossible to induce the pusher to work when
the assistant sets a high price. On the other hand, there is a unique cutoff value
µP ∈ (0, µ2 ) satisfying V3 (µp ) = 0. Therefore, in equilibrium the assistant sets a
low price and the pusher works if µ ≥ µP . The assistant sets a high price and the
pusher shirks if µ < µP .
If there exists a µ4 > 0 constructed in the same way as previously (and therefore a
µ3 > µ4 ), then the argument can be recursively applied to construct cutoff values
µ5 , µ6 , ..., µ2i−1 , µ2i . Finally, there exists a number i∗ with µ2i∗ = 0 such that, the
assistant sets a low price for µ ∈ [µ2i−1 , µ2i−2 ) and the high price for
µ ∈ [µ2i , µ2i−1 ), where i ≤ i∗ , and the pusher works if and only if µ ≥ µ2i∗ −1 .
54