Melco Management Accounting Research
Discussion Paper Series
No.MDP2021-001

The Ratchet Effect in Teams: The Impacts of Learning and Inequity
Aversion
April 2021

Mami Koyama
Specially Appointed Assistant Professor, Graduate School of BusinessOsaka City University
3-3-138, Sugimoto, Sumiyoshi-ku, Osaka-shi, 558-8585, JAPANe-mail: mamikoyama.phd@gmail.com

Maki Makoto Foundation
Nagoya, Japan
Discussion Paper Series of the Melco Management Accounting Research, in order to promote
the study of management accounting, has published this unfinished paper onto the Web. Pleaseobtain the permission of the author when citing this paper.

The Ratchet Effect in Teams: The Impacts of Learning and Inequity
Aversion

Mami Koyama

ABSTRACTThis study theoretically and experimentally demonstrates that because of theinterdependent nature of work environments, the mechanism of the ratchet effect inteams qualitatively differs from that in independent work environments. First, I
theoretically proved that occurrence of the ratchet effect in teams depends on the agents’inequity aversion and that learning among teammates deters the ratchet effect. Then, I
ran a laboratory experiment of a team-production task that was designed according to
the theoretical model. The experiment results reveal that the ratchet effect occurs inteams. I confirm that participants’ inequity-aversion levels were sufficiently low to fallwithin a range for which the ratchet effect is theoretically expected to occur. The results
also reveal that the ratchet effect is significantly mitigated when learning is present thanwhen learning is absent. These results demonstrate that the mechanism of the ratchet
effect in teams differs from that in independent work environments. This also implies
that encouraging learning indirectly benefits firms.

Keywords: target setting; ratchet effect; learning; inequity aversion.
Data Availability:
Contact the author.

1. INTRODUCTION
This study theoretically and experimentally demonstrates that the mechanism of
the ratchet effect in teams differs qualitatively from that in independent work
environments, which most previous studies have focused on.
While targets are an important factor in incentive systems, they are often
1determined based on past performance, and such a target-setting method is susceptible
to an incentive problem, that is, the ratchet effect (Dekker et al., 2012; Murphy, 2000;
Weitzman, 1980). Employees expect that once good performance is recorded, the next
period’s targets will be revised into more difficult ones, and thus, they have an incentive
to limit their current performance to keep future targets easily attainable. If the ratchet
effect prevails, then the firm’s performance is lower than the performance level that
could be attained if every employee exerts their maximum capability. Thus, it isimportant to overcome this problem.
Prior studies have empirically investigated the ratchet effect in firms (e.g.,
Bouwens & Kroos, 2011; Chaudhuri, 1998; Leone & Rock, 2002; Murphy, 2000), in
addition to discussing how managers can overcome this problem (e.g., Aranda et al.,2014; Baron & Besanko, 1984; Cardella & Depew, 2018; Charness et al., 2011;
Indjejikian et al., 2014a; Indjejikian & Nanda, 2002). However, these studies haveinvestigated this incentive problem in an independent work environment.
In contrast to prior studies, this study investigates the ratchet effect in a teamenvironment. A team is a situation in which agents work interdependently, and aprincipal cannot observe the output per agent but the output per team (Holmstrom,1982). This interdependent nature of work environments, which is absent in
independent work environments, may impact the agents’ incentive to commit the ratchet
effect and the principal’s means to mitigate the ratchet effect. This study investigates
how the interdependent work environment impacts the mechanism of the ratchet effectby focusing on two factors: inequity aversion and learning among teammates. In thisstudy, I operationalized the notion of interdependence by employing a Leontief typeproduction function, that is, the team’s output is assumed to follow a minimum functionof two agents’ effort level.
First, this study investigates how the agents’ incentive to commit the ratchet effectis affected by interdependency, focusing on inequity aversion. Inequity aversion is asocial preference that represents individuals’ preference for disliking inequity among
members (Dhami, 2016). Since they work together and their rewards are determined
depending on each other’s behavior, employees are more likely to care about theircolleagues in teams than in independent work environments. Thus, inequity aversion is
expected to play more significant role in teams than in independent work environments.
This study theoretically proves that the occurrence of the ratchet effect in teams depends
on the agents’ inequity aversion. If the agents have a standard preference and do notcare about inequity at all, then the ratchet effect will occur. The more inequity-averse
the agents, the less likely is the occurrence of the ratchet effect.
2

This is brought about by either of the following two logics. First, when a skilled
agent misreports their type as unskilled, they will feel guilty from the dishonestly
enhanced payoff compared to their teammates. If this feeling of guilt outweighs thebenefit from easy future targets, then the agent refrains from committing the ratchet
effect. Second, when a skilled agent misreports their type as unskilled and enjoys easy
future targets, their teammates will envy that agent. If this feeling of envy is sufficiently
strong, then the teammates penalize that agent by deciding not to cooperate in the
team’s production activity, which drastically reduces the team’s performance andimpairs the agent’s benefits from the easy targets. Note here that the second logic arises
from the interdependent nature and inequity aversion. Because of their interdependent
nature, the agents can influence their teammates’ payoffs by changing the team’s output.
In addition, sufficiently inequity-averse agents have an incentive to prevent teammatesfrom committing the ratchet effect. Thus, teams with sufficiently inequity-averse agentscan mutually govern their behavior and deter the ratchet effect by themselves1.
Second, this study investigates how the principal’s means to mitigate the ratchet
effect is affected by interdependency, focusing on the impact of a phenomenon that iscaused by interdependency. Given the interdependent work environment, employees canimprove their skills by learning from their teammates. Berg et al. (1996) demonstrated
that workers in production teams often informally teach shortcuts, problem-solving, orother ways to improve their teammates’ work. Specifically, unskilled employees may
grow the most in heterogeneous teams that consist of skilled and unskilled employees.
This is because skilled employees pull up unskilled employees by teaching theirknowledge (Hamilton et al., 2003), or unskilled employees who have a larger room for
improvement spontaneously learn from a good role model. This study theoretically
proves that the more prominent such learning, the more easily the principal can preventthe ratchet effect2.
The logic is as follows3: The principal tries to adjust targets according to theagents’ skill levels. When there is no learning, each agent’s report provides information
The first logic could also hold in the case of independent work environments with multiple agents.
On the other hand, the second logic can not be observed in independent work environments, because
the agents cannot manipulate their colleagues’ performance, and they can not penalize each other.
2 The key concept of learning which this study deals with is that the agents’ growth amount differs
depending on the pair of their initial skill level and that of their teammates: unskilled agents whopaired with skilled agents grow more than others. If every agent grows the same amount regardlessof their initial skill level and that of their teammate, then the ratchet effect may not be deterred by
the presence of such learning as explained in Footnote 4.
3 Appendix A provides the detailed model analyses for the impact of learning on the ratchet effect.
1

3

about their own types only. If an agent reports their type as unskilled in the first period,
then the principal cannot ascertain whether this agent is lying. Thus, in the first and thesecond periods, the principal assigns a target corresponding to the unskilled agent’s skill
level, accounting for the possibility that the agent actually has low skill and gives up to
work under too demanding a target.
In contrast, in the presence of learning, the principal understands the correlation
between the agents’ skill levels in the second period, and then each agent’s reportbecomes informative about their teammates’ types. The principal can use this additional
information to resolve the ratchet effect. Specifically, when one agent reports their typeas unskilled and the other reports their type as skilled, then the principal can infer fromthe latter agent’s report that the former agent's skill level in the second period will be noless than an improved skill level of initially unskilled employees who are paired withskilled employees. Then, in the second period, the principal can assign the former agent
a target corresponding to the improved skill level of unskilled employees, and this target
is more demanding than that for unimproved unskilled employees. This means that even
if a skilled agent misreports their type, they will be assigned a target in the second
period that is more demanding than in the case when the learning is absent, and theirbenefit from misreporting their type decreases4.
This study experimentally investigates the above theoretical predictions using alaboratory experiment of a team-production task performed by undergraduate andgraduate students. Combining the theoretical analyses and the laboratory experiment
would be suitable for this study. Theoretical analyses contribute to clarify the logic of
the ratchet effect in teams which is unclear up to now, and clarification of the logic ofwhich we want to test is necessary for designing experiments properly. As theoreticalpredictions show, the theory expects multiple equilibria and it can not conclude which
equilibrium is likely to occur without any empirical investigation. In addition, it isuneasy to identify the occurrence of the ratchet effect using non-experimental data,

This logic of learning could be extended in the case of independent work environments as long as
the following two conditions are satisfied: (1) low-skill type may grow more than high-skilled typeat least on some occasion and (2) the principal knows the condition in which low-skill type growsignificantly. Suppose there is an agent who conducts an independent task, the agent’s possible typeare low skill or high skill, and the agent’s ability grows over time. In this case, if the low-skill typegrows more than the high-skill type, then learning may deter the ratchet effect likewise in teams. In
addition, if low-skill type grows more than high-skill type if and only if they attend some training or
pass some in-house exams and the principal understands this fact, then learning may also deter theratchet effect. If both types of agents grow the same amount, such learning may not contribute todeterring the ratchet effect.
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because it is difficult to know the employees’ true ability and discern whether theemployees strategically reducing their performance (Cardella & Depew, 2018; Charness
et al., 2011). I focused on reporting behaviors that the participants took in the task, andthe principal’s choices of targets given the reports were made by a computer program in
the experiment5.
The results of the experiment reveal that the ratchet effect occurs in teams. Skilledworkers misrepresented their types as unskilled more frequently in a two-periods
without learning treatment (NOLEARN) than in a one-period treatment (ONE). I
measured Fehr and Schmidt (1999) inequity aversion for each participant, andconfirmed that the participants’ inequity-aversion levels were low enough to fall withina range for which the ratchet effect is theoretically expected to occur.
The experimental results also reveal that the ratchet effect is significantly
mitigated when learning is most salient compared to when it is absent. Skilled workers
represented their type truthfully more frequently in a two-periods with the most salient
learning treatment (LEARN) than in a two-periods without learning treatment(NOLEARN).
The contributions of this study are as follows: First, this study sheds new light on
the literature on the ratchet effect by considering the agents’ incentive to commit theratchet effect and the principal’s means to mitigate the ratchet effect in teams. This
study theoretically explains that the ratchet effect is deterred in teams with strongly
inequity-averse agents because the agents mutually govern their behavior. This implies
that the ratchet effect is automatically deterred in teams with strongly inequity-averse
agents without any management activity by the principal. This can be a possible
explanation for an unanswered question, that is, why is the practice of setting targets
based on past performance prevalent? This study also theoretically and experimentallydemonstrates that learning among teammates works as a tool for the principal to deter
the ratchet effect by introducing the correlation between the agents’ skill levels, whichmakes each agent’s report on their own type informative about the other teammates’
type. This suggests that managers can cope with the ratchet effect in teams by
If the role of the principal is also played by real participants, then the participants who assume therole of the agents need to predict the strategic decision of the principal in addition to that of theirteammates to make their decisions. This would be too demanding for participants. I operationalized
the principal’s decision as follows: I assumed the principal's utility is determined as profit minus thesum of rewards paid to the agents and analyzed their on-the-pass-of-the-equilibria behaviors as in theequilibrium analyses in Section 4. The computer principal is designed to take thoseon-the-pass-of-the-equilibria behaviors. The detail of the principal’s decision rule is explained insubsection 3.1. This decision rule was informed to participants in the experimental instructions.
55determining targets considering the mechanism of learning: the managers can cope withthe ratchet effect problem by adjusting targets' level not only based on the past
performance but also the ability improvement that would arise as a result of learning.
Second, this study reveals the indirect benefit of employees’ learning for firms.
The results reveal that facilitating learning in teams may benefit firms by not only
inducing skill improvement in unskilled employees but also by preventing skilledemployees from committing the ratchet effect. In terms of human resource management
implications, this suggests that we need to account for this indirect role when discussing
the role of employees' learning and training. As a practical implication, this alsosuggests that facilitating knowledge sharing among team members and/or cultivating acooperative culture contributes to preventing skilled employees from shirking.
The remainder of this paper is organized as follows. Section 2 summarizes therelated literature. Section 3 explains the experimental design of this study, and Section 4
develops the hypotheses based on the equilibrium analyses. Section 5 presents theresults, and Section 6 concludes. Appendix A analyzes a theoretical model for the
impact of learning on the ratchet effect. Appendix B analyzes the equilibrium behaviors
in each experimental treatment. Appendix C explains the experimental procedure formeasuring the inequity aversion.

2. RELATED LITERATURE
Prior literature has found that targets are commonly determined based on past
performance (Dekker et al., 2012; Murphy, 2000). Such a target-setting method is called
target ratcheting (Indjejikian et al., 2014b).
Although prevalent in practice, target ratcheting is theoretically considered tocause an incentive problem called the ratchet effect (Indjejikian et al., 2014b; Weitzman,1980). Once good performance is recorded, the next period’s targets will be revised into
more difficult ones. Employees have an incentive to limit their current performance tokeep future targets easy. This problem is considered as a dynamic adverse selection
problem, namely, the principal and the agent contract with multiple periods, and there isinformation asymmetry on the agent’s type. The principal needs to provide additional
informational rent to resolve the information asymmetry (Laffont & Tirole, 1993). The
major research themes in the literature on the ratchet effect are (1) empiricalinvestigations on whether the ratchet effect occurs consistent with the theoreticalarguments and (2) theoretical and empirical investigations on overcoming the ratchet
effect problem. This study considers both these issues in teams.
62.1 The Existence of the Ratchet EffectPrior studies have undertaken laboratory experiments and statistical analyses ofactual firm data to empirically investigate the existence of the ratchet effect.
The classical experimental study was conducted by Chaudhuri (1998). Because of
the complicated experimental design, it was not easy for participants to understand thestrategic nature, and the ratchet effect was not observed. Subsequent studies modified
experimental designs to simplify it (Cardella & Depew, 2018; Charness et al., 2011;
Cooper et al., 1999). These studies introduced non-real players to reduce theparticipants’ roles, or reduce the number of alternatives that participants can select. Inline with the latter experiments’ approach, this study simplifies an experimental designby having the role of principals played by a computer.
In prior studies, the output is determined interdependently for each observation
unit; thus, it can be interpreted that prior studies investigate the ratchet effect problem inwork environments without interdependency. In laboratory experiments based on a
non-real-effort task, each participant individually selects their output level, and thus, theoutput is determined per person (Charness et al., 2011; Chaudhuri, 1998; Cooper et al.,1999). Another laboratory experiment based on a real-effort task also measures theoutput per person (Cardella & Depew, 2018). In Cardella and Depew’s (2018)
experimental task, each participant stuffs and seals mailing envelops. Each participant's
output is measured as a number of assembled envelops within a time limit, and thus,
there is no interdependency. The observation units of studies using data from actual
firms are the regional retail store managers (Bouwens & Kroos, 2011), the business unit
managers (Leone & Rock, 2002), and executives (Murphy, 2000). There seems to be nointerdependency in the output between each observation unit. In contrast to these priorstudies, in this study, the output is interdependently determined per team depending on
the two agents’ decisions.
2.2 Mitigating the Ratchet EffectPrior literature theoretically proposes and empirically verifies various ways tocope with the ratchet effect. However, the correlation between teammates’ private
information, which this study considers, has not been studied so far as a key tomitigating the problem.
The most straightforward way considered in the literature is the principal’scommitment power. If a principal can commit not to revise targets upward even after theagent records good performance, then revealing the true ability does not impair anagent's future utility (Aranda et al., 2014; Baron & Besanko, 1984; Bouwens et al.,7

2015; Indjejikian et al., 2014a; Indjejikian & Nanda, 2002). The logic here is theremoval of the target revision itself, and it is irrelevant whether work environments areindividual or group.
It is possible to reduce the ratchet effect without the principal’s commitment
power on some occasions. Existence of an outside option is one possibility (Charness etal., 2011). If a principal has an outside option and an incumbent agent records bad
performance, then the principal can get the agent fired and recruit a new agent. As aresult, this fear of being fired decreases an agent’s incentive to deceive. Charness et al.
(2011) experimentally demonstrated that an outside option reduces the ratchet effect.
The logic is the threat of replacement by an outside agent, and no interaction exists
between the incumbent agent and the outside agent. Thus, this approach is basicallytoward the ratchet effect in individual work environments.
Limiting an agent's strategic influence on their future contract is anotherpossibility (Cardella & Depew, 2018). Cardella and Depew (2018) considered asituation in which a principal contracts with multiple agents. Each agent conducts areal-effort task and their performance is measured per person. Thus, their experimental
task is an independent work environment. They found that the ratchet effect is lesslikely to occur when the second period’s contract, which is a piece rate amount per
output, is designed per group and everyone is paid at the same rate as the rate
determined per person. When the second period’s rate is designed per group, even if anagent lowers their output in the first period, as long as the majority of the others record
high output, the second rate will decrease. Thus, the agents cannot be better off by
restricting their output in the first period. The logic here is to eliminate the agent's
benefits from restricting output by reducing the degree to which the agent can influence
their future contracts.
Utilizing peer performance information and filtering out common exogenous
shocks is also an alternative possibility (Casas-Arce et al., 2018). They considered asituation in which the agent’s output is determined by their effort and two random
shocks, and the principal can only observe the output amount but cannot know the effort
level. One of the random shocks is common among agents, and the other one varies by
person. The common shock is serially correlated, and the agent-specific shock istransitory. Casas-Arce et al. (2018) theoretically and empirically demonstrated that if aprincipal determines a target based not only on past output but also on past peer output,
then the ratchet effect is reduced. The logic is that contrasting past peer outcomes withthose of the agent removes the impact of the common shock and reduces information
asymmetry between the principal and the agent. The rationale that the principal should8reflect the peer’s information is different from that of this study: the principal uses thepeer’s information because it contains information for guessing the agent's private
information, which is determined regardless of the peer's strategic decisions. In contrast,in this study, the information content of teammates’ behavior depends on teammates’
strategic decisions.
2.3 Inequity Aversion
Prior literature on the ratchet effect has assumed that agents have a standard
preference in that they only care about their own payoffs. Through experimental studies,
however, it is now well-recognized that some people may also care about others (Fehr &Schmidt, 2006; Cooper & Kagel, 2015). Such preferences are referred to as“other-regarding preferences (or social preferences)” (Dhami, 2016). Inequity aversion
is one type of other-regarding preference. Inequity-averse agents feel disutility when the
payoffs are not equal among agents (Fehr & Schmidt, 2006; Cooper & Kagel, 2015).
This study accounts for such inequity aversion when discussing the ratchet effect.
Various formalizations of inequity aversion have been proposed (Bolton &Ockenfels, 2000; Engelmann & Strobel, 2004). For the purpose of measurablity in alaboratory, I adopted the Fehr and Schmidt (1999) model of inequity aversion. They
considered two agents: agent 𝑖 and agent 𝑗. Let 𝑥𝑖 be the agent 𝑖’s monetary rewards.
Then, the agent 𝑖’s utility 𝑢𝑖 is
𝑢𝑖 = 𝑥𝑖 − 𝛼 max{𝑥𝑗 − 𝑥𝑖 , 0} − 𝛽 max{𝑥𝑖 − 𝑥𝑗 , 0}(1)where, 0 < 𝛽 < 𝛼, and 𝛽 < 1Parameter 𝛼 represents envy. Agent 𝑖 suffers from envy when the agent 𝑗 gets
more monetary rewards than the agent 𝑖. Parameter 𝛽 represents guilt. Agent 𝑖 feels
guilty when agent 𝑖 gets more monetary rewards than agent 𝑗.
Various methods have been applied for measuring parameters 𝛼 and 𝛽 inexperimental studies (Bellemare et al., 2008). I adopted Blanco et al.’s (2011) method
as it provides interval estimations for 𝛼 and 𝛽 for each subject, using a set of dictator
games with binary choices. These estimations allow for an empirical analysis of eachagent’s behavior in the team-production task with respect to their inequity aversion.

93. EXPERIMENTAL DESIGN6This is a laboratory experiment using z-Tree (Fischbacher, 2007). The experimenthas two parts. The first part is a team-production task. The second part is a measurementof the inequity aversion parameters 𝛼 and 𝛽 based on the BDM mechanism (Becker
et al., 1964). In the following, I mainly explain the team-production task7.
3.1 The Team-Production GameFirst, I explain the team-production game that underlies the team-production task.
The game is based on a two-period adverse selection model in which a principal
contracts with two agents who work as a team as same as the model in Appendix A8. In
this experiment, the role of the principal is played by a computer. Participants assume
the role of an agent. In each period, each agent puts in their effort with some effort cost.
The marginal cost differs depending on the agent type. There are two types: a low-cost
type (i.e., a skilled worker) and a high-cost type (i.e., a newcomer). The type is theagents' private information. Knowing their own type, each agent sends an opinion on the
production schedule. Based on the agents’ reports, the principal executes the production
schedule of the team that consists of the units of production and rewards for each agent.
The team’s output is determined based on the agents' efforts in a completely
complementary manner. Both agents must exert the same level of effort required toderive the output amount.
Insert Figure 1 here
Figure 1 shows the flow of decisions in this game. In the beginning, eachparticipant was privately informed of their own type. An agent is a skilled worker with

Experimental instructions are available upon request.
A procedure of the second part is explained in Appendix C.
8 We can say the model in Appendix A captures the ratchet effect problem because that accompanies
the following incentive problem. Since the principal cannot commit to any long-term contract, theprincipal revises the second period’s contract considering the first period’s results. If the good-typeagent reveals their high-capability in the first period, then in the second period, the principal willassign the first-best offer which is tuned to the good-type agent’s true capability. Such a first-bestoffer requires the agent to exert their maximum level of effort, while pays only the amount whichjust compensates their effort cost (it implies payment is minimum and the agents cannot enjoy rent).
On the other hand, if the good-type agent misreports their type as bad, then in the second period,
they will be assigned the first-best offer which is tuned to the bad-type agent’s true capability. Since
the good-type agent has higher capability than the bad-type agent, the good-type agent can achievethat effort requirement with lower cost and it renders some rent to the agent (i.e., if the good-typeagent misreports their type in the first period, then they can earn some benefit in the second period).
Thus, if the agents correctly expect what would happen in the second period, then they have an
incentive to misreport their type in the first period for the sake of the second periods’ rent.
6
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10

probability two thirds and an agent is a newcomer with probability one thirds.
After informing the type, the first period starts. First, participants were told that todetermine a team’s production schedule, a manager asked workers' opinions on how
much they could work in this period. Then, participants make the following decision
simultaneously. Each participant selects one of the three alternatives to represent theopinion: Much, Less, or Reject9. The principal interprets the opinion such that selecting
the alternative Much (Less) corresponds to reporting the type as a skilled worker(newcomer). Depending on the pair of participants’ choices, the principal determines theunits of production requirement for this team and rewards for each agent. Each agent
bears effort costs by conducting a production activity. Agents’ payoffs are calculated bysubtracting the cost of production effort from the rewards10. The units of production
follow the following order: “the amount when both select Much” > “the amount when
one selects Much and the other selects Less” > “the amount when both select Less” Itsuggests that the production requirement becomes more demanding when the agent
selects Much than when they selects Less. The cost of production effort depends on theunits of production and the agent’s type. The cost increases as the units of production
increase and the newcomer bears more cost than the skilled workers when they conduct
the same amount of production activity. As will be explained shortly, the agents’decisions in the first period impact their first and second period’s payoffs, thus we canconsider this experimental design captures the feature of the ratchet effect problem.
Insert Tables 1 and 2 here
After the first period’s decision, participants were informed of their teammates’
type.
Then, based on two participants’ decisions in the first period, the second period’s
schedule is determined by the principal according to Tables 3 and 4. The principal
assumes that the first period’s decisions correctly revealed the agents’ types as
explained in the above, then the principal believes there is no information asymmetry
with respect to the agents’ types and offers the first-best production schedule that
corresponds to the agents’ type pair in the second period. In other words, the second
period’s contract becomes more demanding if the agent selected Much in the first period
Four payoff charts (1st and 2nd period's payoff charts for each type of worker) are printed as a
one-pager. I hand them out to each participant. Participants can refer to the payoff charts at any time
during this task.
10 To facilitate better understanding of participants, I do not show the units of production or rewards.
Only payoffs are shown in the experimental payoff charts. The payoffs are calculated based on themodel shown in Appendix A.
911than if they selected Less. As will be explained shortly, the agents’ payoffs are impacted
by the principals’ schedule selection. Participants know how their first period's decisions
impact the second period’s schedule and their payoffs before they make the first
period’s decision. Thus, the agents’ decisions in the first period impact their second
period’s payoffs through the determination of the second period’s production schedule,
and the agents need to make their first period’s decision considering its impact on theirfirst and second period’s payoffs.
Insert Tables 3 and 4 here
In the second period, participants make the following decision simultaneously,
given the selected second period’s schedule. Each participant selects one of twoalternatives: Accept or Reject. If both participants select Accept, then both get non-zero
payoffs. Otherwise, both payoffs are zero. The concrete payoffs are shown in Tables 5, 6,
7, and 8.
Insert Tables 5, 6, 7, and 8 here
After the second period’s decision, all results were fed back to each participant.
Then the game ends.
3.2 Treatments
There are three treatments with the between-subject design: ONE, NOLEARN,
and LEARN. The base treatment is NOLEARN, in which participants play theteam-production game described above. ONE is a treatment manipulated such thatparticipants play the first period only in the team-production game. LEARN is atreatment manipulated such that participants can improve their ability by learning overperiods. The mechanism of learning here is consistent with the definition of learningthat I analyze in Appendix A. Specifically, when a newcomer is paired with a skilledworker, the newcomer learns from the skilled teammate in the first period and thenattains the same ability level as the skilled worker in the second period. However, theskilled workers never learn from any type of teammate, and their ability levels remain
the same in the second period11. In this LEARN treatment, participants were informed of
the outcome of learning after the end of the first period. More precisely, participants are
first informed of the type of their teammates that are given at the beginning of the first
period. Participants are then informed whether ability improvement occurs in their team,In the experiment, I normalize 𝜖 = 0 without loss of generality, and 𝛥 = 𝜃𝐵 − 𝜃𝐺 to highlightthe impact of learning.
1112

and if so, they are also told whose ability is improved.
In all three treatments, the team-production task had 20 rounds. In each round,
two participants were randomly paired without knowing each other. The participants’
types are assigned such that two-thirds of participants randomly selected in each session
are assigned the role of a skilled worker and the remaining one-third are assigned therole of a newcomer. Over 20 rounds, the team is randomly rematched by rounds12. In
addition, each participant’s type is randomly re-assigned by rounds.
3.3 Participants and Sessions
Participants were undergraduate and graduate students of Osaka University. They
are recruited via an online recruitment system for economic experiments (ORSEE) run
by the Institute of Social and Economic Research at Osaka University13.
I ran two sessions per treatment (six sessions in total). Each recruited student was
assigned to one of the six sessions. Each session was conducted with either 18 or 24
participants due to no shows. Participants are randomly assigned to a separate computer
terminal and conduct tasks independently. Each session took 90–120 minutes, including
ex-post questionnaires.
3.4 Payment Scheme
From this experiment, every participant receives a sum of the show-up fee andrewards from the two tasks. The show-up fee is 800 JPY. The rewards from theteam-production task are determined based on participants’ earned payoffs in thefollowing way. After the 20 rounds end, one round is randomly selected. Eachparticipant’s rewards from this task are determined according to their total payoffs inthis selected round by converting one payoff point to one JPY.
The method of determining rewards from the second task is explained inAppendix C.

4. EQUILIBRIUM PREDICTIONS AND HYPOTHESES
Using the team-production game, I derived hypotheses about the ratchet effect inteams and the impact of learning on the ratchet effect in teams. To derive these

The re-matching protocol that this study used allows the same participants to be matched as ateam again.
13 This study received approval from the Institute of Social and Economic Research at Osaka
University.
12

13

hypotheses, I used a game-theoretic equilibrium analysis.
Before conducting the analyses, I reconfirmed the situations wherein the ratchet
effect does and does not occur. As explained in Section 2, the ratchet effect is a problem
in that a high-ability agent misreports their type as a low-ability one, and then theprincipal cannot properly adjust the team’s production schedule based on the agents’ability. This problem is caused by the high-ability agent’s incentive to pursue theinformation rent from being assigned a relatively easy task in the future periods. In theexperiment, the ratchet effect occurs if both skilled workers and newcomers select thesame alternative in the first period, since the manager cannot obtain any additional
information to distinguish each worker’s type based on their first period’s decisions.
However, the ratchet effect does not occur if each type of worker selects a different
alternative in the first period, since the manager can distinguish each worker’s typebased on their first period’s decisions and assign the second period’s schedule based onthat ability.
For the equilibrium analysis, I focus on the following two strategy profiles as a
candidate for equilibrium: separation and pooling14. Separation is a strategy profile inwhich skilled workers select Much and newcomers select Less in the first period, andeveryone selects Accept in the second period. This separation corresponds to thesituation wherein the ratchet effect does not occur. In this case, the manager assigns thesecond period’s schedule corresponding to the pair of skills: (Skilled, Skilled), (New,
Skilled), and (New, New). Pooling is a strategy profile in which both types of workers
select Less in the first period, and everyone selects Accept in the second period. Thispooling corresponds to the situation wherein the ratchet effect occurs. In this case, themanager can only assign the second period’s schedule based on the prior information on
the type.
Table 9 summarizes the results of the equilibrium analyses for each treatment15. It
shows which strategy profile—separation or pooling—becomes an equilibrium in each
case.
Insert Table 9 here
Based on the above equilibrium analyses, I derived the following threehypotheses that investigate the existence of the ratchet effect in teams and themechanism of mitigating the ratchet effect in teams.

14
15

For simplicity, this study restricts its analysis to pure strategies.
All analyses are shown in Appendix B.
14

The first hypothesis intends to test the existence of the ratchet effect in teams, and
it is built on the comparison of ONE and NOLEARN in the case of agents with standard
preferences. The above equilibrium analysis shows that separation is the only
equilibrium in ONE. The logic is as follows: if skilled workers select Much in the first
period, then they will get either 560 or 615, depending on their teammates’ choices.
However, if skilled workers select Less, then they will receive either 500 or 427. Thus,
skilled workers can strictly better off by selecting Much in ONE.
However, pooling is the only equilibrium in NOLEARN. If skilled workers selectMuch in the first period, then they will get either 560 + 440 or 615 + 440 in total,
depending on their teammates’ choices. However, if skilled workers select Less, thenthey will get either 500 + 773 or 427 + 628 in total, depending on their teammates’
choices. Thus, skilled workers can be better off by selecting Less in NOLEARN.
The choice of Less by the skilled workers in NOLEARN is the ratchet effect. The
logic of this behavior is as follows: if the skilled workers select Less in the first period,
then the units of production in the first period become lower and their payoffs from thefirst period are lower—500 or 427 compared to 560 or 615. However, the choice of Lessbrings them higher rent in the second period: 773 or 628 compared to 440. This benefit
outweighs the loss in the first period. Thus, skilled workers have an incentive to selectLess in the first period. In contrast, in ONE, there is no such incentive for skilledworkers since they do not have an opportunity to earn any information rent in thesecond period.
From the above, I derive H1. H1 is tested by observing that separation occurs in
ONE and pooling occurs in NOLEARN, which confirms that pooling is driven by the
dynamics of the relationship.
H1: The ratchet effect occurs in a team without learning.
The second hypothesis tests that learning mitigates the ratchet effect in teams andis built on the comparison of NOLEARN and LEARN in the case of agents with standard
preferences. The equilibrium results reveal that there are two equilibria—separation andpooling—in LEARN.
The logic of separating equilibrium is as follows. The payoffs when they earn byselecting Much are the same as in the NOLEARN treatment. If skilled workers selectMuch in the first period, they will earn either 560 + 400 or 615 + 400 in total, dependingon their teammates’ decisions. However, if the skilled workers select Less in the first
period, then their payoffs may differ from NOLEARN, depending on their teammate’s
decision. Specifically, the skilled workers will earn either 500 + 440 or 427 + 628 intotal. In the second period, they will earn only 440 in LEARN, in contrast to 773 in
15

NOLEARN, when their teammate selects Much. This is because the manager knows thatthe agent who selected Less will improve the ability by learning from their skilled
teammate even if the agent initially has low skill, and the manager assigns the second
period’s schedule considering that both agents have high skill. Thus, as long as theirteammate selects Much in the first period, they will not get any additional rent in thesecond period. Hence, the skilled workers’ benefit from selecting Less does notcompensate for the loss in the first period and their incentive to select Less in the first
period vanishes; the ratchet effect does not occur in LEARN.
The logic of the pooling equilibrium in LEARN is similar to that in NOLEARN.
More precisely, in LEARN, the principal knows that the probability that both agents
have high skill in the second period is 8/9, much higher than that in NOLEARN.
However, the principal still assigns the production schedule for two low-skilled agentsin the second period fearing that both agents have low skill and reject the second
period’s schedule for high-skilled agents.
Given H1 is supported, if separation occurs, then it implies that the ratchet effectis mitigated by the existence of learning. From the above, I derive H2. H2 is tested byobserving that separation occurs in LEARN and pooling occurs in NOLEARN.
H2: Given H1 is supported, the ratchet effect is less likely to occur in a team withlearning than in cases without learning.
The third hypotheses investigates the impact of inequity aversion on the ratchet
effect in teams and is built on the comparative static analysis of equilibria in NOLEARN
with respect to the inequity aversion of the agents. For analyzing inequity-averse agents’behavior, I assume that agents have Fehr and Schmidt’s (1999) inequity-averse utility
function, as shown in Equation (1). Figure 2 shows how the equilibria differ dependingon the agents’ inequity-aversion parameters16.
Insert Figure 2 here
The orange region in Figure 2 shows the pair of 𝛼 and 𝛽 for which separation isan equilibrium, whereas pooling is not. In the yellow region, both separation andpooling are equilibria. In the green region, pooling is an equilibrium, whereas separation
is not. This result means that even without learning, agents prefer to tell their type
truthfully if they feel strong envy and guilt, and thus, the ratchet effect does not occur.
From the above, I derive H3. H3 is tested by observing that separation occurs in

For this equilibrium analysis, I put the following two assumptions. First, 𝛼 and 𝛽 are
symmetric among agents. Second, the value of 𝛼 and 𝛽 are common knowledge.
16

16

NOLEARN when participants’ inequity-aversion parameters belong to the yellow ororange region of Figure 2. To do so, I investigated each participant’s behavior pairedwith their inequity aversion. The inequity-aversion parameters are measured based on
the second part of this experiment, as explained in Appendix C.
H3: The ratchet effect does not arise for those who have strong inequity aversion.

5. RESULTS
The total number of participants was 126, of which 29.37% were female, and this
proportion of female students is not far away from the population of Osaka University
students (Osaka University, 2019). Table 10 shows the summary statistics. The
observation unit of Table 10 is the decision of each round. Since 126 participants
conduct 20 rounds of decisions, I have 2,520 observations in total. Recall that
two-thirds of the participants become skilled workers and one-third of participants
become newcomers in each round. For example, since NOLEARN has 48 participants intotal, there are 640 observations of skilled workers’ decisions in the first period17. The
average earnings of ONE, NOLEARN, and LEARN are 1,477.639 JPY, 1,936.292 JPY,
and 1,962.095 JPY, respectively.
Based on Table 10, more than 90% of newcomers select Less in ONE, NOLEARN,
and LEARN (90.8%, 96.9%, and 97.9%, respectively). However, skilled workers’
behavior differs between ONE and NOLEARN. More than 90% of skilled workers in
ONE select Much (98.3%), whereas skilled workers in NOLEARN select Less more than
Much (81.6% select Less). These observations are consistent with H1. This implies thatthe ratchet effect does occur.
Additionally, the ratio of skilled workers who select Much in LEARN is higher
than that in NOLEARN (67.5% vs. 18.4%). This observation is consistent with H2. Thisimplies that skilled workers are less likely to misrepresent their type in LEARN than in
NOLEARN, that is, the ratchet effect is less likely to occur in the presence of learning.
Table 10 also shows the average value of inequity aversion by treatment. Figure 3
shows the scatter plot of the pair of inequity-aversion parameters. Based on these results,
I confirm that most of the participants are weakly inequity-averse or do not care forinequity at all. Therefore, the inequity aversion does not relate to this participant’s
behavior.
In NOLEARN, one newcomer who is paired with a newcomer selected Reject in the first period
and that team did not experience the second period. Therefore, the number of observations of
newcomers’ decision in the second period is reduced to 318.
17

17

Insert Table 10 and Figure 3 here
5.1 Definitions of Variables
Since each participant repeatedly conducted 20 rounds of decisions, each decision
data might not be independent. To cope with this problem, I created the following two
variables and used them for testing:
Denote the participant i’s observed choice frequency that they select Much in thefirst period when they are a skilled worker as 𝑅𝑎𝑡𝑒_𝑀𝑢𝑐ℎ𝑆𝑘𝑖𝑙𝑙𝑒𝑑𝑖 .
𝑅𝑎𝑡𝑒_𝑀𝑢𝑐ℎ𝑆𝑘𝑖𝑙𝑙𝑒𝑑𝑖 =
Both

∑20
𝑡=1 𝕀(𝐷𝑒𝑐𝑖𝑠𝑖𝑜𝑛1𝑖,𝑡 = 𝑀𝑢𝑐ℎ, 𝑅𝑜𝑙𝑒𝑖,𝑡 = 𝑆𝑘𝑖𝑙𝑙𝑒𝑑)∑20
𝑡=1 𝕀(𝑅𝑜𝑙𝑒𝑖,𝑡 = 𝑆𝑘𝑖𝑙𝑙𝑒𝑑)

𝕀(𝐷𝑒𝑐𝑖𝑠𝑖𝑜𝑛1𝑖,𝑡 = 𝑀𝑢𝑐ℎ, 𝑅𝑜𝑙𝑒𝑖,𝑡 = 𝑆𝑘𝑖𝑙𝑙𝑒𝑑)

and

(6)

𝕀(𝑅𝑜𝑙𝑒𝑖,𝑡 = 𝑆𝑘𝑖𝑙𝑙𝑒𝑑)

are indicator functions that take 1 when the condition in each bracket is satisfied and 0
otherwise.
Likewise, I denote participant i’s observed choice frequency that they select Lessin the first period when they are a newcomer as 𝑅𝑎𝑡𝑒_𝐿𝑒𝑠𝑠𝑁𝑒𝑤𝑖 .
𝑅𝑎𝑡𝑒_𝐿𝑒𝑠𝑠𝑁𝑒𝑤𝑖 =∑20
𝑡=1 𝕀(𝐷𝑒𝑐𝑖𝑠𝑖𝑜𝑛1𝑖,𝑡 = 𝐿𝑒𝑠𝑠, 𝑅𝑜𝑙𝑒𝑖,𝑡 = 𝑁𝑒𝑤𝑐𝑜𝑚𝑒𝑟)∑20
𝑡=1 𝕀(𝑅𝑜𝑙𝑒𝑖,𝑡 = 𝑁𝑒𝑤𝑐𝑜𝑚𝑒𝑟)(7)

There were two sessions per treatment. I examine whether a difference exists inthe above two variables by session within the same treatment. Based on the
Kolmogorov–Smirnov test, I did not find any statistically significant difference between
sessions. Subsequently, I pooled the data by treatment.
5.2 Testing the Existence of the Ratchet Effect (H1 & H3)
I examined whether the ratchet effect is observed in teams comparing ONE andNOLEARN. Specifically, I examined (1) whether skilled workers are less likely to selectMuch in NOLEARN than in ONE and (2) whether newcomers are likely to select Less
regardless of treatments. To do so, I compared the distribution of the observed choicefrequency I defined above. Figures 4 and 5 show the empirical cumulative distributionfunctions of 𝑅𝑎𝑡𝑒_𝑀𝑢𝑐ℎ𝑆𝑘𝑖𝑙𝑙𝑒𝑑𝑖 and, 𝑅𝑎𝑡𝑒_𝐿𝑒𝑠𝑠𝑁𝑒𝑤𝑖 respectively.
Insert Figure 4 and Figure 5 here
First, I examined skilled workers’ behavior. Figure 4 shows that more than 80%
of participants in ONE always select Much. However, in NOLEARN, participants whoalways select the alternative Much are almost zero. Moreover, approximately 40% ofparticipants always select Less. The results of the Kolmogorov–Smirnov test revealed18

that the empirical cumulative distribution function 𝑅𝑎𝑡𝑒_𝑀𝑢𝑐ℎ𝑆𝑘𝑖𝑙𝑙𝑒𝑑𝑖 of NOLEARN
is statistically significantly greater than that of ONE (𝐷 = 0.93056, 𝑝 = 3.368𝑒 − 16,one-sided18). Table 11 shows that the Wilcoxon’s rank-sum test also provides consistentstatistically significant evidence (𝑊 = 2368, 𝑝 < 0.001, one-sided). Based on theseresults, I conclude that skilled workers in NOLEARN are less likely to select Muchcompared to those in ONE.
Insert Table 11 here
Second, I examined newcomers’ behavior. Figure 5 shows that newcomers arehighly likely to select Less in both ONE and NOLEARN. In both treatments, more than
60% of participants always selected Less. Consistent with the equilibrium predictions,the results of the Kolmogorov–Smirnov test do not reveal any statistical difference inthe empirical cumulative distribution function 𝑅𝑎𝑡𝑒_𝐿𝑒𝑠𝑠𝑁𝑒𝑤𝑖 of ONE andNOLEARN ( 𝐷 = 0.18056 , 𝑝 = 0.5137 , two-sided). Likewise, the results ofWilcoxon’s rank-sum test also reveal that these two distributions do not statisticallydiffer (𝑊 = 1411, 𝑝 = 0.0668, two-sided). Based on these results, I conclude that
newcomers are highly likely to select Less, and their behavior does not differ between
ONE and NOLEARN.
The above results demonstrate that the ratchet effect occurs; thus, H1 is
supported.
The analyses so far are based on the assumption that participants are
self-regarding individuals. I make sure whether inequity-averse participants exist, and if
so, how they behave. First, I divided participants of NOLEARN into two subsamples.
Recall that separation can be an equilibrium when individuals’ inequity-aversion
parameters belong to the orange and yellow regions of Figure 2. The downward-sloping
line in Figure 3 shows the boundary of the yellow and green regions in Figure 2. Figure
3 shows that almost all participants do not belong to the yellow region. Moreover, theonly participant who belongs to the yellow region is that of LEARN. Therefore, noparticipant of NOLEARN belongs to the yellow region. From the above, I concluded thatthe behavior predicted in H3—that individuals refrain from committing the ratchet
effect due to their strong inequity aversion—is not observed in this experiment.

Since the data include ties, all of the Kolmogorov–Smirnov test results rely on bootstrapping for
calculating the confidence interval. Likewise, for the Wilcoxon’s rank-sum test, I assigned eachobservation in a tie its average rank and deflated the variance of test statistics considering thenumber of ties.
18

19

5.3 Testing the Impact of Learning on the Ratchet Effect (H2)
I examined whether the ratchet effect is mitigated in the presence of learning bycomparing NOLEARN and LEARN. Specifically, I examined (1) whether skilled workers
are more likely to select Much in LEARN compared to NOLEARN and (2) whether
newcomers are likely to select Less regardless of treatments.
First, I investigated skilled workers’ behavior. Figure 4 shows that approximately
30% of participants in LEARN always select Much. In contrast, participants of
NOLEARN who always select Much are less than 5%. The results of the Kolmogorov–
Smirnov

test

revealed

that

the

empirical

cumulative

distributionfunction

𝑅𝑎𝑡𝑒_𝑀𝑢𝑐ℎ𝑆𝑘𝑖𝑙𝑙𝑒𝑑𝑖 of LEARN is statistically significantly less than that of NOLEARN(𝐷 = 0.66071, 𝑝 = 3.21𝑒 − 09, one-sided). In addition, the Wilcoxon’s rank-sum test
also provides consistent statistically significant evidence, as shown in Table 11 (𝑊 =1509.5, 𝑝 < 0.001, one-sided). Based on these results, I concluded that skilled workers
in LEARN are more likely to select Much compared to those in NOLEARN.
Second, I examined newcomers’ behavior. Figure 5 shows that newcomers arehighly likely to select Less in both treatments. In addition, the results of both the
Kolmogorov–Smirnov test (𝐷 = 0.047619, 𝑝 = 0.823, two-sided) and the Wilcoxon’s
rank-sum test ( 𝑊 = 2142.5 , 𝑝 = 0.2912 , two-sided) revealed that the empiricalcumulative distribution functions 𝑅𝑎𝑡𝑒_𝐿𝑒𝑠𝑠𝑁𝑒𝑤𝑖 of NOLEARN and LEARN are notstatistically significantly different. Based on these results, I concluded that newcomers
are highly likely to select Less, and their behavior does not differ between NOLEARN
and LEARN.
In addition, I examined whether the presence of learning completely resolved theratchet effect, comparing skilled workers’ behavior in LEARN and ONE. The results ofboth the Kolmogorov–Smirnov test (𝐷 = 0.57937, 𝑝 = 4.46𝑒 − 06, two-sided)
and the Wilcoxon’s rank-sum test (𝑊 = 1892, 𝑝 < 0.001, one-sided) revealed that theempirical cumulative distribution functions 𝑅𝑎𝑡𝑒_𝑀𝑢𝑐ℎ𝑆𝑘𝑖𝑙𝑙𝑒𝑑𝑖 of LEARN is
statistically significantly less than that of ONE. These results imply that the presence oflearning partially reduces the likelihood that the ratchet effect occrs.
The reason for why the ratchet effect is only partially reduced is asked for thefollowing discussions. As the equilibrium analyses in Section 3 proved, both separationand pooling are equilibria in LEARN. Figure 4 shows that the empirical cumulative
distiribution function of 𝑅𝑎𝑡𝑒_𝑀𝑢𝑐ℎ𝑆𝑘𝑖𝑙𝑙𝑒𝑑𝑖 of LEARN has two mass points. It
suggests that both equilibria occurs in the experiment. Although the difference is not20

statistically significant 19 , Figure 6 shows two empirical cumulative distributionfunctions of 𝑅𝑎𝑡𝑒_𝑀𝑢𝑐ℎ𝑆𝑘𝑖𝑙𝑙𝑒𝑑𝑖 of LEARN dividing them into the first-half rounds’
data and the latter-half rounds’ data. Figure 6 suggests that skilled workers in thelatter-half rounds are more likely to select Much compared to that in the first-halfrounds. This implies that separation is more likely to be selected over pooling as therounds proceed. From the above, the reason for the partial reduction of the ratchet effectcan be understood because of the multiple equilibria, and data is consistent with thetheoretical predictions.

6. CONCLUSION
This study theoretically and experimentally demonstrated that the mechanism of
the ratchet effect in teams differs qualitatively from that in independent work
environments. First, I theoretically proved that the occurrence of the ratchet effect inteams depends on the agents’ inequity aversion and that learning among teammates
deters the ratchet effect. Then, I ran a laboratory experiment to test these theoreticalpredictions. The results of the experiment demonstrate that the ratchet effect occurs inteams. I confirmed that participants’ inequity-aversion levels were sufficiently low tofall within a range for which the ratchet effect is theoretically expected to occur. The
results also reveal that the ratchet effect is significantly mitigated in the presence oflearning compared to when learning is absent.
The contributions of this study are as follows. First, this study extends theliterature on target ratcheting and the ratchet effect by demonstrating that themechanism of the ratchet effect in teams differs from that of independent work
environments. This study theoretically explains that the ratchet effect is deterred inteams with strongly inequity-averse agents because such agents mutually govern their
behavior. This study also theoretically and experimentally demonstrated that learning
among teammates works as a tool for the principal to deter the ratchet effect byintroducing a correlation between the agents’ skill levels, which makes each agent’s
report on their own type informative of the other teammates’ type.
Second, this study reveals the indirect benefit of employees’ learning for firms,
which is not discussed in the prior literature on learning. The results imply thatfacilitating knowledge sharing among team members and/or providing mentoring
opportunities for low-performing employees can contribute to mitigating the ratchet

19

The result of Kolmogorov–Smirnov test is as follows: 𝐷 = 0.19048, 𝑝 = 0.2179, one-sided.
21

effect. In other words, such management activities are beneficial to firms as a means of
not only improving the employees' abilities but also preventing skilled employees fromshirking.
The limitation of this study is that the experiment was not designed for directly
testing the impact of inequity aversion on the ratchet effect. Based on the results of thisexperiment, I cannot conclude whether strictly inequity-averse individuals refrain fromcommitting the ratchet effect. Future research can test this impact more directly byrecruiting participants with various inequity aversion and comparing the behavior basedon their inequity aversion.

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24

Figures
FIGURE 1
Timeline of a Round in Two-periods Treatments

NOTE: At the timing of 3, learning occurs only in LEARN treatment.

FIGURE 2
Equilibriums in NOLEARN depending on the Inequity Averseness

NOTE: [Green region] pooling is an equilibrium while separation is not, [Yellow region] both pooling and separation
is an equilibrium, [Orange region] separation is an equilibrium while pooling is not.

25

FIGURE 3
The Scatter Plot of Participants’ Inequity Averseness

NOTE: The values of 𝛼 and 𝛽 are calculated as shown in Appendix C. These two parameters can take a value from0 to 1 at an interval of 0.1. Two participants who represent more than two switching points are omitted; thus, the total
observation number is 124. The number of participants who have (𝛼, 𝛽) = (0,0) is 84. The number of participants
who have (𝛼, 𝛽) = (1,1) is 1 and that participant participates in the LEARN treatment.

FIGURE 4
The Empirical Cumulative Distribution Function of Skilled Workers’ Decisions in the First
Period26

FIGURE 5
The Empirical Cumulative Distribution Functions of Newcomers’ Decisions in the First Period

FIGURE 6
The Empirical Cumulative Distribution Functions of Skilled Workers’s First Period’s
Decisions in LEARN Treatment: Comparison of 1–10 Rounds’ Data and 11–20 Rounds’ Data

27

Tables
TABLE 1
A Payoff Chart for the First period’s Decision when a Decision-maker is a Skilled Worker

TABLE 2
A Payoff Chart for the First period’s Decision when a Decision-maker is a Newcomer

Table 3The Relationship between the First Period’s Decisions and the Second Schedule in NOLEARN

28

TABLE 4
The Relationship between the First Decisions and the Second Schedule in LEARNTABLE 5
A Payoff Chart for the Second period’s Decision when a Decision-maker is a Skilled Worker in
NOLEARNTABLE 6
A Payoff Chart for the Second period’s Decision when a Decision-maker is a Newcomer in
NOLEARN

29

TABLE 7
A Payoff Chart for the Second period’s Decision when a Decision-maker is a Skilled Worker in
LEARN

TABLE 8
A Payoff Chart for the Second period’s Decision when a Decision-maker is a Newcomer in
LEARN

Table 9Equilibriums in Each Treatment

30

TABLE 10
Summary Statistics

Table 11
The Results of the Wilcoxon Rank-Sum Tests for Decisions in the First Period31Appendix
A. The Model for the Impact of Learning on the Ratchet EffectA.1. Setting
Consider a principal that manages a team with two agents (𝑛 = 1, 2) to conduct a
productive activity for two periods. Suppose that every party is risk-neutral. Each agent
exerts their own effort with private cost, and the pair of two agents' efforts determines
the team's output. Agent 𝑛's effort in period 𝑡 is 𝑎𝑡𝑛 ∈ [0, 𝑎̅] ≡ Α. Let the output of theteam in period 𝑡 be 𝑥𝑡 = min{𝑎1𝑡 , 𝑎𝑡2 } ∈ Χ ⊆ ℝ. Agents are heterogeneous in terms of
their marginal cost of effort. Suppose there are two types of agents: good and bad. I
denote the type of agent 𝑛 in period 𝑡 as 𝜃𝑡𝑛 . Let 𝜃1𝑛 ∈ {𝜃𝐺 , 𝜃𝐵 } and 0 < 𝜃𝐺 < 𝜃𝐵 .
The cost of each agent is 𝐶(𝑎𝑡𝑛 , 𝜃𝑡𝑛 ) = 𝜃𝑡𝑛 𝑎𝑡𝑛 . Therefore, 𝜃𝐺 is a low-cost and efficient
type. Denote the pair of team members' types in period 1 as 𝜃 𝑖𝑗 ≡ (𝜃11 = 𝜃𝑖 , 𝜃12 = 𝜃𝑗 ).
Suppose 𝑃𝑟(𝜃 𝑖𝑗 ) = 𝑝𝑖𝑗 ≥ 0 , ∑𝑖𝑗 𝑝𝑖𝑗 = 1 , ∑𝑗 𝑝𝐺𝑗 = ∑𝑖 𝑝𝑖𝐺 = 𝑝 > 0 , and ∑𝑗 𝑝𝐵𝑗 =∑𝑖 𝑝𝑖𝐵 = 1 − 𝑝. Based on the team's output, income from the team's productive activity
is realized. Let the team's income when the team's output is 𝑥𝑡 be 𝑏(𝑥𝑡 ). Suppose 𝑏(⋅)
is a twice continuously differentiable function. Let 𝑏(0) = 0, 𝑏′(0) = +∞, 𝑏′(𝑥̅ ) = 0,
for any 𝑥 < 𝑥̅ , 𝑏′(𝑥) > 0, and for any 𝑥 ≥ 0, 𝑏 ′′ (𝑥) ≤ 0. Agent 𝑛 receives a reward
𝑤𝑡𝑛 based on the team's output. The agent’s utility in period 𝑡 is determined by 𝑤𝑡𝑛 −𝜃𝑡𝑛 𝑎𝑡𝑛 . The principal's utility in period 𝑡 is determined by 𝑏(𝑥𝑡 ) − 𝑤𝑡1 − 𝑤𝑡2 . Suppose
the principal cannot commit to any long-term contract. Therefore, the principal uses a
short-term contract that can refine the contracts for the subsequent periods at the
beginning of each period.
The timeline of this two-periods transaction is as follows.
Period 0
1. Nature determines the type of agents. Each agent privately observes their own type.
Period 1
2. The principal offers a contract for two periods for both agents. If at least one agent
rejects the contract, then everyone gets 0 and the transaction ends.
3. If both agents accept the contract, then each agent submits a report on their type.
4. Based on the reports, the amount of outputs and rewards for the first period isdetermined. The agents produce outputs as prescribed in the selected contract.
Thereafter, each party receives their payoff for the first period. Then, the first
period ends.
32Period 2
5. At the beginning of the second period, ability improvement by learning arises if
possible.
6. The principal offers a contract for the second period to both agents. If at least one
agent rejects the contract, then everyone receives zero and the transaction ends.
7. If both agents accept the contract, then each agent submits a report on their type.
8. Based on the reports, the contract to be executed is selected. Agents exert effort in
the second period, as prescribed in the selected contract, and the team's output isrealized. Thereafter, each party receives their payoff for the second period and theperiod ends.
Assumption 1 (Learning). As a result of learning in the first period, at the beginning of
the second period, the marginal cost of each agent improves as follows20.
𝐼𝑓 (𝜃11 , 𝜃12 ) = (𝜃𝐺 , 𝜃𝐺 ), 𝑡ℎ𝑒𝑛 (𝜃21 , 𝜃22 ) = (𝜃𝐺 − 𝜖, 𝜃𝐺 − 𝜖)𝐼𝑓 (𝜃11 , 𝜃12 ) = (𝜃𝐺 , 𝜃𝐵 ), 𝑡ℎ𝑒𝑛 (𝜃21 , 𝜃22 ) = (𝜃𝐺 − 𝜖, 𝜃𝐵 − 𝜖 − Δ)
𝐼𝑓 (𝜃11 , 𝜃12 ) = (𝜃𝐵 , 𝜃𝐺 ), 𝑡ℎ𝑒𝑛 (𝜃21 , 𝜃22 ) = (𝜃𝐵 − 𝜖 − Δ, 𝜃𝐺 − 𝜖)𝐼𝑓 (𝜃11 , 𝜃12 ) = (𝜃𝐵 , 𝜃𝐵 ), 𝑡ℎ𝑒𝑛 (𝜃21 , 𝜃22 ) = (𝜃𝐺 − 𝜖, 𝜃𝐺 − 𝜖)where, 𝜖 ≥ 0 and 0 ≤ 𝛥 ≤ 𝜃𝐵 − 𝜃𝐺 .
A.2. Benchmark
Suppose that the principal knows agents’ types perfectly. Then, the principal offers thefirst-best contract, which is a solution to the following problem.
max 𝑏(𝑥𝑡 ) − 𝑤𝑡1 − 𝑤𝑡2

for 𝑡 = 1,2𝑤,𝑥s. t. 𝑤𝑡𝑛 − 𝜃𝑡𝑛 𝑎𝑡𝑛 ≥ 0 for 𝑛 = 1,2
Let the first-best offer to the team with 𝜃

𝑖𝑗(𝐴. 1)(𝐴. 2)

be {𝑥𝑡𝐹𝐵 (𝜃 𝑖𝑗 ), 𝑎𝑡𝑛,𝐹𝐵 (𝜃 𝑖𝑗 ), 𝑤𝑡𝑛,𝐹𝐵 (𝜃 𝑖𝑗 )}for any t and any n. The first-best offer satisfies the following Lemma:
Lemma 1. The first-best offer satisfies the following two properties.
𝑥𝑡 (𝜃 𝑖𝑗 ) = 𝑎1𝑡 (𝜃 𝑖𝑗 ) = 𝑎𝑡2 (𝜃 𝑖𝑗 )𝑤𝑡𝑛 (𝜃 𝑖𝑗 ) = 𝜃𝑡𝑛 𝑎𝑡𝑛 (𝜃 𝑖𝑗 )for 𝑛 = 1,2(𝐴. 3)(𝐴. 4)For simplicity, 𝜖 is zero in the experiment. In NOLEARN, 𝛥 = 0. Whereas in LEARN, 𝛥 =𝜃𝐵 − 𝜃𝐺 .
20

33

The first property is driven by the characteristic of 𝑥𝑡 = min{𝑎1𝑡 , 𝑎𝑡2 }. The second
property is driven by the binding of participation constraints.
By Lemma 1, the first-best offer is characterized as follows.
𝑤𝑡𝑛,𝐹𝐵 (𝜃 𝑖𝑗 ) = 𝜃𝑡𝑛 𝑎𝑡𝑛,𝐹𝐵 (𝜃 𝑖𝑗 ) = 𝜃𝑡𝑛 𝑥𝑡𝐹𝐵 (𝜃 𝑖𝑗 )𝑏 ′ (𝑥𝑡𝐹𝐵 (𝜃 𝑖𝑗 )) = 𝜃𝑡1 + 𝜃𝑡2∀ 𝑛, 𝑡, 𝑖, 𝑗

∀ 𝑡, 𝑖, 𝑗(𝐴. 5)(𝐴. 6)

A.3. Short-term Contracts
In this section, I analyze a case in which the principal cannot observe the agents’ types.
Since this study assumes that the principal cannot commit to any long-term contract, theprincipal designs a short-term contract satisfying the incentive-compatibility
constraints21.
The next subsection shows the best short-term contract when the principal fully
distinguishes the agent’s type in the first period (fully separating). This means that theratchet effect does not occur under this contract. In other words, this contract allows a
principal to prevent the ratchet effect. In contrast, the subsection entitled fully poolingshows the best short-term contract when the principal gives up to distinguish the agent’s
type in the first period at all. Since distinguishing the agents’ type requires the principal
to owe informational rent, and such required rent may become too large, the principal
cannot bear such cost in some cases. Therefore, it is not always feasible to distinguishthe agents’ types. The condition for fully separating becomes feasible is shown in thesubsequent subsection.
As in the benchmark case, any optimal short-term contract satisfies Lemma 2.
Lemma 2. Any optimal short-term contract satisfies the following property.
𝑥𝑡 (𝜃 𝑖𝑗 ) = 𝑎1𝑡 (𝜃 𝑖𝑗 ) = 𝑎𝑡2 (𝜃 𝑖𝑗 )(𝐴. 7)

Fully Separating
I analyze the contract backward. By Lemma 2, let the best fully separating
short-term contract be 𝑆 𝑆 = {𝑥𝑡𝑆 (𝜃 𝑖𝑗 ), 𝑤𝑡1,𝑆 (𝜃 𝑖𝑗 ), 𝑤𝑡2,𝑆 (𝜃 𝑖𝑗 )}𝑖𝑗∈{𝐺,𝐵},𝑡=1,2.
Since the agents’ types are fully revealed as a result of the first period, the principal
knows the agent’s type and there is no information asymmetry at this point. That is, theFor simplicity, this study restricts its analysis to pure strategies. Thus, this study does not deny thepossibility that any partially separating contract outperforms the other two contracts.
21

34

principal’s problem in the second period is the same as in the benchmark case.
Therefore, the principal offers the first-best offer in the second period.
𝑤2𝑛,𝑆 (𝜃 𝑖𝑗 ) = 𝑤2𝑛,𝐹𝐵 (𝜃 𝑖𝑗 ) = 𝜃2𝑛 𝑥2𝐹𝐵 (𝜃 𝑖𝑗 )∀ 𝑛, 𝑖, 𝑗(𝐴. 8)𝑏 ′ (𝑥2𝑆 (𝜃 𝑖𝑗 )) = 𝑏 ′ (𝑥2𝐹𝐵 (𝜃 𝑖𝑗 )) = 𝜃21 + 𝜃22∀ 𝑖, 𝑗(𝐴. 9)

Then, the principal's problem in the first period is formalized as follows.
max ∑ 𝑝𝑖𝑗 [𝑏 (𝑥1 (𝜃 𝑖𝑗 )) − 𝑤11 (𝜃 𝑖𝑗 ) − 𝑤12 (𝜃 𝑖𝑗 )𝑤1 ,𝑥1𝑖𝑗+𝛿 {𝑏 (𝑥2𝐹𝐵 (𝜃 𝑖𝑗 )) − 𝑤21𝐹𝐵 (𝜃 𝑖𝑗 ) − 𝑤22,𝐹𝐵 (𝜃 𝑖𝑗 )}]
𝑠. 𝑡.

∑
𝑗

(𝐴. 10)𝑝𝐺𝑗 1 𝐺𝑗
[𝑤1 (𝜃 ) − 𝜃𝐺 𝑥1 (𝜃 𝐺𝑗 ) + 𝛿 × 0] ≥ 0
𝑝𝑝𝐵𝑗∑[𝑤 1 (𝜃𝐵𝑗 ) − 𝜃𝐵 𝑥1 (𝜃𝐵𝑗 ) + 𝛿 × 0] ≥ 0
1−𝑝 1
𝑗

∑
𝑖

∑
𝑖

∑
𝑗

𝑝𝑖𝐺 2 𝑖𝐺
[𝑤1 (𝜃 ) − 𝜃𝐺 𝑥1 (𝜃 𝑖𝐺 ) + 𝛿 × 0] ≥ 0𝑝𝑝𝑖𝐵[𝑤 2 (𝜃 𝑖𝐵 ) − 𝜃𝐵 𝑥1 (𝜃 𝑖𝐵 ) + 𝛿 × 0] ≥ 0
1−𝑝 1
𝑝𝐺𝑗 1 𝐺𝑗
[𝑤1 (𝜃 ) − 𝜃𝐺 𝑥1 (𝜃 𝐺𝑗 ) + 𝛿 × 0]
𝑝
≥∑
𝑗

∑
𝑗

𝑝𝐵𝑗
[𝑤 1 (𝜃𝐵𝑗 ) − 𝜃𝐵 𝑥1 (𝜃𝐵𝑗 ) + 𝛿 × 0]
1−𝑝 1
≥∑
𝑗

∑
𝑖

𝑝𝐺𝑗 1 𝐵𝑗
[𝑤1 (𝜃 ) − 𝜃𝐺 𝑥1 (𝜃𝐵𝑗 ) + 𝛿{𝑤21,𝐹𝐵 (𝜃𝐵𝑗 ) − 𝜃21 𝑥2𝐹𝐵 (𝜃𝐵𝑗 )}]𝑝

𝑝𝐵𝑗[𝑤 1 (𝜃 𝐺𝑗 ) − 𝜃𝐵 𝑥1 (𝜃 𝐺𝑗 ) + 𝛿{𝑤21,𝐹𝐵 (𝜃 𝐺𝑗 ) − 𝜃21 𝑥2𝐹𝐵 (𝜃 𝐺𝑗 )}]1−𝑝 1

𝑝𝑖𝐺 2 𝑖𝐺
[𝑤1 (𝜃 ) − 𝜃𝐺 𝑥1 (𝜃 𝑖𝐺 ) + 𝛿 × 0]𝑝𝑝𝑖𝐺 2 𝑖𝐵≥∑
[𝑤1 (𝜃 ) − 𝜃𝐺 𝑥1 (𝜃 𝑖𝐵 ) + 𝛿{𝑤22,𝐹𝐵 (𝜃 𝑖𝐵 ) − 𝜃22 𝑥2𝐹𝐵 (𝜃 𝑖𝐵 )}]
𝑝
𝑖

∑
𝑖

𝑝𝑖𝐵
[𝑤 2 (𝜃 𝑖𝐵 ) − 𝜃𝐵 𝑥1 (𝜃 𝑖𝐵 ) + 𝛿 × 0]
1−𝑝 1
𝑝𝑖𝐵
≥∑
[𝑤 2 (𝜃 𝑖𝐺 ) − 𝜃𝐵 𝑥1 (𝜃 𝑖𝐺 ) + 𝛿{𝑤22,𝐹𝐵 (𝜃 𝑖𝐺 ) − 𝜃22 𝑥2𝐹𝐵 (𝜃 𝑖𝐺 )}]1−𝑝 1𝑖Solving the above problem, the first period’s offer is characterized as follows.
35(𝐴. 12)𝑏 ′ (𝑥1𝑆 (𝜃 𝐺𝐺 )) = 𝜃𝐺 + 𝜃𝐺𝑏 ′ (𝑥1𝑆 (𝜃 𝐺𝐵 )) = 𝑏 ′ (𝑥1𝑆 (𝜃 𝐵𝐺 )) = 𝜃𝐺 + 𝜃𝐵 +𝑏 ′ (𝑥1𝑆 (𝜃 𝐵𝐵 )) = 𝜃𝐵 + 𝜃𝐵 +𝑝(𝜃 − 𝜃𝐺 )1−𝑝 𝐵2𝑝(𝜃 − 𝜃𝐺 )1−𝑝 𝐵

𝑤11,𝑆 (𝜃 𝐺𝐺 ) = 𝜃𝐺 𝑥1𝑆 (𝜃 𝐺𝐺 ) + (𝜃𝐵 − 𝜃𝐺 )𝑥1𝑆 (𝜃 𝐵𝐺 ) + 𝛿(𝜃𝐵 − 𝜃𝐺 − Δ)𝑥2𝐹𝐵 (𝜃 𝐵𝐺 )𝑤11,𝑆 (𝜃 𝐺𝐵 ) = 𝜃𝐺 𝑥1𝑆 (𝜃 𝐺𝐵 ) + (𝜃𝐵 − 𝜃𝐺 )𝑥1𝑆 (𝜃 𝐵𝐵 ) + 𝛿(𝜃𝐵 − 𝜃𝐺 )𝑥2𝐹𝐵 (𝜃 𝐵𝐵 )𝑤11,𝑆 (𝜃𝐵𝑗 ) = 𝜃𝐵 𝑥1𝑆 (𝜃𝐵𝑗 )𝑗 = 𝐺, 𝐵
2,𝑆
𝑆
𝐺𝐺
𝐺𝐺
𝑤1 (𝜃 ) = 𝜃𝐺 𝑥1 (𝜃 ) + (𝜃𝐵 − 𝜃𝐺 )𝑥1𝑆 (𝜃 𝐺𝐵 ) + 𝛿(𝜃𝐵 − 𝜃𝐺 − Δ)𝑥2𝐹𝐵 (𝜃 𝐺𝐵 )𝑤12,𝑆 (𝜃 𝐵𝐺 ) = 𝜃𝐺 𝑥1𝑆 (𝜃 𝐵𝐺 ) + (𝜃𝐵 − 𝜃𝐺 )𝑥1𝑆 (𝜃 𝐵𝐵 ) + 𝛿(𝜃𝐵 − 𝜃𝐺 )𝑥2𝐹𝐵 (𝜃 𝐵𝐵 )𝑤12𝑆 (𝜃 𝑖𝐵 ) = 𝜃𝐵 𝑥1𝑆 (𝜃 𝑖𝐵 )𝑖 = 𝐺, 𝐵(𝐴. 22)(𝐴. 23)(𝐴. 25)(𝐴. 26)

The above results reveal that first, the source of the ratchet effect is confirmed bycomparing the above results with the second-best contract for a case in which aprincipal–agent relationship is a one-shot relationship. The major difference between
these two contracts is the amount of rent the good-type agent can earn. In the fully
separating contract, the good type receives more rent than in the one-shot case. To
induce agents to report their type truthfully, the principal needs to provide some rent
that is at least as much as the benefit that agents can accrue by misreporting their types.
Moreover, in the two-period relationship, misreporting their type in the first period
benefits the good-type agent not only in the first period but also in the second period.
Therefore, in the two-period case, the principal needs to provide rent that covers theagent's deceiving benefit for the entire two periods. In other words, to prevent thisrent-seeking behavior caused by the dynamics of the relationship (i.e., the ratchet effect),the principal needs to provide additional rent in the first period compared to a one-shot
relationship case—at least as much as the agent's total deceiving benefit from two
periods.
Second, as a bad type paired with a good type grows significantly (i.e., as Δ
becomes large), the benefit that the good-type agent accrues by misereporting their typedecreases. Suppose both agents are of the good type. In this case, the misreporting
benefit that agent 1 can enjoy in the second period is 𝛿(𝜃𝐵 − 𝜃𝐺 − Δ)𝑥2𝐹𝐵 (𝜃 𝐵𝐺 ). This
term decreases as Δ increases. Thus, the ratchet effect is less likely to occur as Δ
increases. This implies that firms can mitigate the ratchet effect by facilitatingknowledge sharing among teammates and/or investing in employees’ training systems toincrease Δ.
36

Fully Pooling
Let 𝑆 𝑃 = {𝑥1𝑃 , 𝑤1𝑃 , 𝑥2𝑃 (𝜃 𝑖𝑗 ), 𝑤21,𝑃 (𝜃 𝑖𝑗 ), 𝑤22,𝑃 (𝜃 𝑖𝑗 )}𝑖𝑗∈{𝐺,𝐵} denotes the best fully
pooling short-term contract.
Since the first period is fully pooling, the principal designs the optimal contract forthe second period based on the common prior p. The principal's problem is written as
follows22:
max ∑ 𝑝𝑖𝑗 [𝑏 (𝑥2 (𝜃 𝑖𝑗 )) − 𝑤21 (𝜃 𝑖𝑗 ) − 𝑤22 (𝜃 𝑖𝑗 )]𝑤2 ,𝑥2𝑠. 𝑡.

𝑖𝑗

𝑤21 (𝜃 𝐺𝑗 ) − (𝜃𝐺 − 𝜖)𝑥2 (𝜃 𝐺𝑗 ) ≥ 0

𝑗 = 𝐺, 𝐵

𝑤21 (𝜃 𝐵𝐺 ) − (𝜃𝐵 − 𝜖 − Δ)𝑥2 (𝜃 𝐵𝐺 ) ≥ 0
𝑤2𝑛 (𝜃 𝐵𝐵 ) − (𝜃𝐵 − 𝜖)𝑥2 (𝜃 𝐵𝐵 ) ≥ 0

𝑛 = 1,2

𝑤22 (𝜃 𝑖𝐺 ) − (𝜃𝐺 − 𝜖)𝑥2 (𝜃 𝑖𝐺 ) ≥ 0

𝑖 = 𝐺, 𝐵

𝑤22 (𝜃 𝐺𝐵 ) − (𝜃𝐵 − 𝜖 − Δ)𝑥2 (𝜃 𝐺𝐵 ) ≥ 0
𝑤21 (𝜃 𝐺𝑗 ) − (𝜃𝐺 − 𝜖)𝑥2 (𝜃 𝐺𝑗 ) ≥ 𝑤21 (𝜃𝐵𝑗 ) − (𝜃𝐺 − 𝜖)𝑥2 (𝜃𝐵𝑗 )𝑗 = 𝐺, 𝐵(𝐴. 34)𝑤21 (𝜃 𝐵𝐺 ) − (𝜃𝐵 − 𝜖 − Δ)𝑥2 (𝜃 𝐵𝐺 ) ≥ 𝑤21 (𝜃 𝐺𝐺 ) − (𝜃𝐵 − 𝜖 − Δ)𝑥2 (𝜃 𝐺𝐺 )(𝐴. 35)𝑤21 (𝜃 𝐵𝐵 ) − (𝜃𝐵 − 𝜖)𝑥2 (𝜃 𝐵𝐵 ) ≥ 𝑤21 (𝜃 𝐺𝐵 ) − (𝜃𝐵 − 𝜖)𝑥2 (𝜃 𝐺𝐵 )(𝐴. 36)

𝑤22 (𝜃 𝑖𝐺 ) − (𝜃𝐺 − 𝜖)𝑥2 (𝜃 𝑖𝐺 ) ≥ 𝑤22 (𝜃 𝑖𝐵 ) − (𝜃𝐺 − 𝜖)𝑥2 (𝜃 𝑖𝐵 )𝑖 = 𝐺, 𝐵
𝑤22 (𝜃 𝐺𝐵 ) − (𝜃𝐵 − 𝜖 − Δ)𝑥2 (𝜃 𝐺𝐵 ) ≥ 𝑤22 (𝜃 𝐺𝐺 ) − (𝜃𝐵 − 𝜖 − Δ)𝑥1 (𝜃 𝐺𝐺 )𝑤22 (𝜃 𝐵𝐵 ) − (𝜃𝐵 − 𝜖)𝑥1 (𝜃 𝐵𝐵 ) ≥ 𝑤21 (𝜃 𝐵𝐺 ) − (𝜃𝐵 − 𝜖)𝑥2 (𝜃 𝐵𝐺 )(𝐴. 37)(𝐴. 38)(𝐴. 39)Solving the above problem, the second period's offer is characterized as follows.
𝑏 ′ (𝑥2𝑃 (𝜃 𝐺𝐺 )) = 𝜃𝐺 − 𝜖 + 𝜃𝐺 − 𝜖𝑏 ′ (𝑥2𝑃 (𝜃 𝐺𝐵 )) = 𝑏 ′ (𝑥2𝑃 (𝜃 𝐵𝐺 )) = 𝜃𝐺 − 𝜖 + 𝜃𝐵 − 𝜖 − Δ +𝑏 ′ (𝑥2𝑃 (𝜃 𝐵𝐵 )) = 𝜃𝐵 − 𝜖 + 𝜃𝐵 − 𝜖 +𝑝(𝜃 − 𝜃𝐺 − Δ) (𝐴. 41)
1−𝑝 𝐵

2𝑝
(𝜃 − 𝜃𝐺 )1−𝑝 𝐵

𝑤21,𝑃 (𝜃 𝐺𝐺 ) = (𝜃𝐺 − 𝜖)𝑥2𝑃 (𝜃 𝐺𝐺 ) + (𝜃𝐵 − 𝜃𝐺 − Δ)𝑥2𝑃 (𝜃 𝐵𝐺 )𝑤21,𝑃 (𝜃 𝐺𝐵 ) = (𝜃𝐺 − 𝜖)𝑥2𝑃 (𝜃 𝐺𝐵 ) + (𝜃𝐵 − 𝜃𝐺 )𝑥2𝑃 (𝜃 𝐵𝐵 )𝑤21,𝑃 (𝜃 𝐵𝐺 ) = (𝜃𝐵 − 𝜖 − Δ)𝑥2𝑃 (𝜃 𝐵𝐺 )𝑤2𝑛,𝑃 (𝜃 𝐵𝐵 ) = (𝜃𝐵 − 𝜖)𝑥2𝑃 (𝜃 𝐵𝐵 )𝑤22,𝑃 (𝜃 𝐺𝐺 ) = (𝜃𝐺 − 𝜖)𝑥2𝑃 (𝜃 𝐺𝐺 ) + (𝜃𝐵 − 𝜃𝐺 − Δ)𝑥2𝑃 (𝜃 𝐺𝐵 )(𝐴. 43)(𝐴. 44)𝑛 = 1,2

Each constraint for the second-period analysis is written as an ex-post constraint. This assumption
could be acceptable because agents may understand each other's ability by working together in thefirst period.
22

37(𝐴. 47)

𝑤22,𝑃 (𝜃 𝐵𝐺 ) = (𝜃𝐺 − 𝜖)𝑥2𝑃 (𝜃 𝐵𝐺 ) + (𝜃𝐵 − 𝜃𝐺 )𝑥2𝑃 (𝜃 𝐵𝐵 )(𝐴. 48)

𝑤22,𝑃 (𝜃 𝐺𝐵 ) = (𝜃𝐵 − 𝜖 − Δ)𝑥2𝑃 (𝜃 𝐺𝐵 )Since the principal does not distinguish each agent's type at all in the first period,
the principal's problem in the first period is formalized as follows.
max ∑ 𝑝𝑖𝑗 [𝑏(𝑥1 ) − 𝑤1 − 𝑤1 + 𝛿 {𝑏 (𝑥2𝑃 (𝜃 𝑖𝑗 )) − 𝑤21,𝑃 (𝜃 𝑖𝑗 ) − 𝑤22,𝑃 (𝜃 𝑖𝑗 )}] (𝐴. 50)

𝑤1 ,𝑥1

s. t.

𝑖𝑗

∑
𝑗

𝑝𝐺𝑗
[𝑤1 − 𝜃𝐺 𝑥1 + 𝛿{𝑤21,𝑃 (𝜃 𝐺𝑗 ) − (𝜃𝐺 − 𝜖)𝑥2𝑃 (𝜃 𝐺𝑗 )}] ≥ 0𝑝𝑝𝐵𝑗∑[𝑤 − 𝜃𝐵 𝑥1 + 𝛿{𝑤21,𝑃 (𝜃𝐵𝑗 ) − 𝜃21 𝑥2𝑃 (𝜃𝐵𝑗 )}] ≥ 0
1−𝑝 1(𝐴. 51)(𝐴. 52)

𝑗

∑
𝑖

∑
𝑖

𝑝𝑖𝐺
[𝑤1 − 𝜃𝐺 𝑥1 + 𝛿{𝑤22,𝑃 (𝜃 𝑖𝐺 ) − 𝜃𝐺 𝑥2𝑃 (𝜃 𝑖𝐺 )}] ≥ 0𝑝𝑝𝑖𝐵[𝑤 − 𝜃𝐵 𝑥1 + 𝛿{𝑤22,𝑃 (𝜃 𝑖𝐵 ) − 𝜃22 𝑥2𝑃 (𝜃 𝑖𝐵 )}] ≥ 0
1−𝑝 1(𝐴. 53)(𝐴. 54)

Solving the above, the first period’s offer is characterized as follows.
𝑏 ′ (𝑥1𝑃 ) = 𝜃𝐵 + 𝜃𝐵

(𝐴. 55)

𝑤1𝑃 = 𝜃𝐵 𝑥1𝑃(𝐴. 56)

Feasibility of Fully Separating
Fully separating is feasible as long as the incentive-compatibility constraint for abad type is satisfied. From (A.16),
𝑝𝐵𝐺[(𝜃 − 𝜃𝐺 ){𝑥1𝑆 (𝜃 𝐵𝐺 ) − 𝑥1𝑆 (𝜃 𝐺𝐺 )} + 𝛿(𝜃𝐵 − 𝜃𝐺 − Δ){𝑥2𝐹𝐵 (𝜃 𝐵𝐺 ) − 𝑥2𝐹𝐵 (𝜃 𝐺𝐺 )}]
1−𝑝 𝐵
𝑝𝐵𝐵
[(𝜃 − 𝜃𝐺 ){𝑥1𝑆 (𝜃 𝐵𝐵 ) − 𝑥1𝑆 (𝜃 𝐺𝐵 )} + 𝛿(𝜃𝐵 − 𝜃𝐺 )𝑥2𝐹𝐵 (𝜃 𝐵𝐵 )] ≤ 0(𝐴. 57)+
1−𝑝 𝐵
Remember that 𝛿(𝜃𝐵 − 𝜃𝐺 )𝑥2𝐹𝐵 (𝜃 𝐵𝐵 ) is a part of a rent for a good-type agent in
the first period when his teammate is bad, and this rent is non-negative. The above
inequality (A.57) suggests that if a rent for a good-type agent becomes sufficiently large,
then fully separating might not be feasible. The intuition is as follows: If the principal
provides sufficiently large rent to the agent who chooses to report a good type in thefirst period, then not only a good-type agent but also a bad-type agent prefers to report agood type in the first period.
Since the main focus of this model analysis is the impact of learning, I investigate
how does learning (the parameter Δ) impact the feasibility of the full-separation
38

contract using numerical examples. Specifically, I use the following two sets ofparameters.
The first set of parameters is (𝑝 = 0.3, 𝛿 = 1, 𝜃𝐺 = 0.1, 𝜃𝐵 = 1, 𝜖 = 0.01).
Figure 7 shows the left-hand side’s value of inequation (A.57) depending on the valueof Δ. It shows that inequation (A.57) is always satisfied for any possible value of Δ
under these parameters. Figure 7 also shows that the more Δ increases the more theleft-hand side’s value gets close to the boundary of inequation (A.57).
Figure 7
Feasibility of Full-separation when 𝒑 = 𝟎. 𝟑, 𝜹 = 𝟏, 𝜽𝑮 = 𝟎. 𝟏, 𝜽𝑩 = 𝟏, 𝝐 = 𝟎. 𝟎𝟏

The second set of parameters is (𝑝 = 0.3, 𝛿 = 1, 𝜃𝐺 = 0.7, 𝜃𝐵 = 1, 𝜖 = 0.01).
Under these parameters, inequation (A.57) is violated for some possible values of Δ asshown in Figure 8. Figure 8 shows that the more Δ increases the less the feasibility
constraint is likely to be satisfied under these parameters.
Figure 8
Feasibility of Full-separation when 𝒑 = 𝟎. 𝟑, 𝜹 = 𝟏, 𝜽𝑮 = 𝟎. 𝟕, 𝜽𝑩 = 𝟏, 𝝐 = 𝟎. 𝟎𝟏

Comparing Contracts
I denote the present value of the principal's expected payoff for two periods undereach contract as, Π𝐹𝐵 , Π𝑆𝐸𝑃 , and Π𝑃𝑂𝑂𝐿 . Then, in the region where full separation is
feasible, Π𝑆𝐸𝑃 always outperforms Π𝑃𝑂𝑂𝐿 . Thus, as long as it is feasible, the fully
separating short-term contract is the second-best contract.
Π𝐹𝐵 = ∑ 𝑝𝑖𝑗 [ 𝑏 (𝑥1𝐹𝐵 (𝜃 𝑖𝑗 )) − 𝑤11,𝐹𝐵 (𝜃 𝑖𝑗 ) − 𝑤12,𝐹𝐵 (𝜃 𝑖𝑗 )𝑖𝑗+𝛿 {𝑏 (𝑥2𝐹𝐵 (𝜃 𝑖𝑗 )) − 𝑤21,𝐹𝐵 (𝜃 𝑖𝑗 ) − 𝑤22,𝐹𝐵 (𝜃 𝑖𝑗 )}]39(𝐴. 58)Π𝑆𝐸𝑃 = ∑ 𝑝𝑖𝑗 [𝑏 (𝑥1𝑆 (𝜃 𝑖𝑗 )) − 𝑤11,𝑆 (𝜃 𝑖𝑗 ) − 𝑤12,𝑆 (𝜃 𝑖𝑗 )𝑖𝑗+𝛿 {𝑏 (𝑥2𝐹𝐵 (𝜃 𝑖𝑗 )) − 𝑤21,𝐹𝐵 (𝜃 𝑖𝑗 ) − 𝑤22,𝐹𝐵 (𝜃 𝑖𝑗 )}(𝐴. 59)

Π𝑃𝑂𝑂𝐿 = 𝑏(𝑥1𝑃 ) − 𝑤1𝑃 − 𝑤1𝑃 + 𝛿 ∑ 𝑝𝑖𝑗 [𝑏 (𝑥2𝑃 (𝜃 𝑖𝑗 )) − 𝑤21,𝑃 (𝜃 𝑖𝑗 ) − 𝑤22,𝑃 (𝜃 𝑖𝑗 )] (𝐴. 60)
𝑖𝑗

In the following, I numerically compare each contract. Specifically, I use two setsof parameters which I also used in the above.
Let me start with the first set of parameters (𝑝 = 0.3, 𝛿 = 1, 𝜃𝐺 = 0.1, 𝜃𝐵 = 1,
𝜖 = 0.01). As shown in Figure 7, the full-separation is always feasible under theseparameters. Figure 9 shows the value of Π𝑆𝐸𝑃 and Π𝑃𝑂𝑂𝐿 depending on the value of
Δ. It shows that the full-separation contract outperforms the full-pooling contract if Δ
is lower than a certain value. If 𝛥 becomes larger than that certain value, thefull-pooling contract outperforms the full-separation contract.
In the case of the second set of parameters (𝑝 = 0.3, 𝛿 = 1, 𝜃𝐺 = 0.7, 𝜃𝐵 = 1,
𝜖 = 0.01), the full-separation contract is not feasible in a certain region. Within theregion where the full-separation contract is feasible, Figure 10 shows that the
full-separation contract outperforms the full-pooling contract.
Figure 9
𝜫𝑺𝑬𝑷 and 𝜫𝑷𝑶𝑶𝑳 when 𝒑 = 𝟎. 𝟑, 𝜹 = 𝟏, 𝜽𝑮 = 𝟎. 𝟏, 𝜽𝑩 = 𝟏, 𝝐 = 𝟎. 𝟎𝟏

Figure 10
𝜫𝑺𝑬𝑷 and 𝜫𝑷𝑶𝑶𝑳 when 𝒑 = 𝟎. 𝟑, 𝜹 = 𝟏, 𝜽𝑮 = 𝟎. 𝟕, 𝜽𝑩 = 𝟏, 𝝐 = 𝟎. 𝟎𝟏

40

B. Equilibria
B.1. ONE with Self-regarding Individuals
Suppose that the teammate reports their type truthfully. Let the skilled worker’s
expected utility when they select Much be 𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ). Then,𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ) =21
21× 560 + × 615 > × 500 + × 427 = 𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠) (𝐵. 1)
3
3
3
3

Thus, skilled workers strictly prefer Much. Likewise, newcomers strictly preferLess as shown in the following.
𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ) =2
1
× (−190) + × 495 < 380 = 𝑢1 (𝑁𝑒𝑤, 𝐿𝑒𝑠𝑠)
3
3(𝐵. 2)

Hence, separation is an equilibrium in this case.
B.2. NOLEARN with Self-regarding IndividualsFirst, I confirm that the separation is not an equilibrium in this case. The analyses are
conducted backward.
In the second period, suppose that the first period achieved the separation. In thissituation, all agents face the first-best offer for their true types. Then, the agents’
expected utility is the same regardless of their true type. The agents’ expected utility
when they select Accept is 440, and this is strictly larger than that in case they selectReject. Thus, every agent selects Accept in the second period.
In the first period, the expected utility of skilled workers evaluated at the time of
the first period23 is21
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ)𝑆𝐸𝑃 = (560 + 440) + (615 + 440)
3
3
<

2
1
(500 + 773) + (427 + 628) = 𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠)𝑆𝐸𝑃
3
3(𝐵. 3)

Thus, skilled workers strictly prefer Less, and separation cannot be an equilibrium.
Next, I confirm that the pooling is an equilibrium. Suppose that every agent selects
Less in the first period. In this situation, all agents face plan D in the second period.
Skilled workers’ expected utility when they select Accept is 628 and is strictly greaterthan that when they select Reject. Moreover, newcomers’ expected utility when they
select Accept is 440 and is also strictly greater than what they can obtain when they
select Reject. From the above, every agent strictly prefers Accept in the second period.

23

For simplicity, I assume the common discount factor as 1.
41

In the first period, the expected utility of skilled workers evaluated at the time of
the first period is
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ)𝑃𝑂𝑂𝐿 =2
1(615 + 440) + (615 + 440)332
1
= (427 + 628) + (427 + 628) = 𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠)𝑃𝑂𝑂𝐿
3
3(𝐵. 4)

Thus, skilled workers weakly prefer Less.
Whereas, newcomers strictly prefer Less as shown in the following.
21
𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ)𝑃𝑂𝑂𝐿 = (495 − 330) + (495 − 330)
3
3
<

2
1
(380 + 440) + (380 + 440) = 𝑢1 (𝑁𝑒𝑤, 𝐿𝑒𝑠𝑠)𝑃𝑂𝑂𝐿
3
3(𝐵. 5)

Altogether, the pooling is an equilibrium in this case.
B.3. LEARN with Self-regarding Individuals
Suppose that every agent reports their type truthfully in the first period. In this case, allagents face the first-best offer for their true type, and such a contract renders 440 when
they select Accept. Thus, all agents strictly prefer Accept in the second period.
In the first period, skilled workers’ expected utility is,21
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ)𝑆𝐸𝑃 = (560 + 440) + (615 + 440)
3
3
>

2
1
(500 + 440) + (427 + 628) = 𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠)𝑆𝐸𝑃
3
3(𝐵. 6)

Therefore, skilled workers strictly prefer Much. On the other hand, newcomers
strictly prefer Less.
21
𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ)𝑆𝐸𝑃 = (−190 + 440) + (495 + 0)
3
3
2
1
< (380 + 628) + (380 + 440) = 𝑢1 (𝑁𝑒𝑤, 𝐿𝑒𝑠𝑠)𝑆𝐸𝑃
3
3(𝐵. 7)

Hence, separation is an equilibrium in this case.
B.4. ONE with Inequity-averse Individuals
Suppose the agents have an inequity-averse utility function, as defined in equation (1).
Skilled workers’ expected utility when they select Much is,
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ) =21[560] + [615 − 235𝛽]3342(𝐵. 8)

Whereas, their expected utility when they select Less is,
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠) =2
1[500 − 115𝛼] + [427 − 47𝛽]33

(𝐵. 9)

Because 𝛽 < 1, the value of equation (B.8) is strictly greater than that of equation(B.9) regardless of the pair of 𝛼 and 𝛽. This implies that skilled workers always
strictly prefer Much regardless of the extent of their inequity aversion.
Likewise, newcomers’ expected utility when they select either Much or Less is,21
𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ) = [−190 − 750𝛼] + [495 − 115𝛽]33
𝑢1 (𝑁𝑒𝑤, 𝐿𝑒𝑠𝑠) =2
1[380 − 235𝛼] + [380]3
3(𝐵. 10)(𝐵. 11)Since both 𝛼 and 𝛽 are strictly greater than zero, the value of equation (B.11) isalways greater than that of equation (B.10). Thus, newcomers strictly prefer Less,
regardless of their inequity aversion.
B.5. NOLEARN with Inequity-averse IndividualsFirst, I examine the separation equilibrium.
As in the above, I start with the analysis on the second period’s
on-the-pass-of-the-equilibrium behavior. Suppose every agent reports their type
truthfully in the first period. Then, in the second period, all agents’ expected utility
when they select Accept is simply equal to 440. This is because everyone earns the same
payoff and there is no room for the parameters 𝛼 nor 𝛽 impacts on. Thus, everyone
strictly prefers Accept.
In the first period, skilled workers’ expected utility when they select Much is,
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ)𝑆𝐸𝑃 =2
1[560 + 440] + [615 − 235𝛽 + 440]
3
3

(𝐵. 12)

Whereas, their expected utility when they select Less differs depending on thesecond period’s off-the-pass-of-the-equilibrium behavior. In the following, I analyze thecondition when skilled workers prefer Much in each case.
[Case1: 𝛼 ≤

440

]

333

In this case, both types of agents select Accept in the second period. Thus, theskilled workers’ expected utility when they select Less is,
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠)𝑆𝐸𝑃 =2
1[500 − 115𝛼 + 773 − 333𝛽] + [427 − 47𝛽 + 628 − 188𝛽] (B. 13)
3
3Comparing (B.12) and (B.13), skilled workers weakly prefer Much as long as the43following equation (B.14) is satisfied. Otherwise, they strictly prefer Less.
𝛽≥−
440

115
91
𝛼+
333
111

(𝐵. 14)110[Case2: 333 < 𝛼 ≤ 47 ]In this case, newcomers select Accept in the second period likewise in Case 1. In
contrast, skilled workers who selected on-the-pass-of-the-equilibrium behavior, Much,in the first period refuse to work in the second period. They select Reject because of
their non-negligible envy. Since they will suffer sufficient envy over their teammate
who took off-the-pass-of-the-equilibrium behavior in the first period, they prefer Reject.
Thus, the skilled workers’ expected utility when they select Less is,
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠)𝑆𝐸𝑃 =2[500 − 115𝛼 + 0]
31
+ [427 − 47𝛽 + 628 − 188𝛽]3(𝐵. 15)Comparing (B.12) and (B.15), skilled workers weakly prefer Much as long as thefollowing equation (B.16) is satisfied. From the assumption, 𝛼 ≥ 0. Thus, (B.16) isalways satisfied.
𝛼≥−

100
23

(𝐵. 16)

110[Case3: 47 < 𝛼]
In this case, skilled workers who selected on-the-pass-of-the-equilibrium behavior,
Much, in the first period select Reject as in Case 2. In addition, newcomers who selected
on-the-pass-of-the-equilibrium also become to select Reject. This is because of they will
suffer non-negligible envy if they select Accept. Thus, the skilled workers’ expectedutility when they select Less is,
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠)𝑆𝐸𝑃 =2
1[500 − 115𝛼 + 0] + [427 − 47𝛽 + 0]
3
3

(𝐵. 17)Comparing (B.12) and (B.17), skilled workers weakly prefer Much as long as thefollowing equation (B.18) is satisfied.
𝛽≤

115
407
𝛼+
94
47

(𝐵. 18)Now we move on to the analyses of newcomers’ behavior. Newcomers’
on-the-pass-of-the-equilibrium expected utility is,44

𝑢1 (𝑁𝑒𝑤, 𝐿𝑒𝑠𝑠)𝑆𝐸𝑃 =2
1[380 − 235𝛼 + 440] + [380 + 440]
3
3

(𝐵. 19)Their off-the-pass-of-the-equilibrium expected utility differs depending on their
envy parameters. If the envy parameter 𝛼 becomes sufficiently large, then the
newcomers who took off-the-pass-of-the-equilibrium behavior in the first period preferto reject in the second period. Hence,2
1
107[−190 − 750𝛼 + 0] + [495 − 115𝛽 + 107 − 333𝛼] , if 𝛼 ≤3
3
333(𝐵. 20𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ)𝑆𝐸𝑃 {21= [−190 − 750𝛼 + 0] + [495 − 115𝛽], 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
3
3=

Since 𝛼, 𝛽 > 0, the value of the equation (B.19) is always greater than that of theequation (B.20). Therefore, newcomers strictly prefer Less regardless of their inequity
averseness.
From the above, separation is an equilibrium if at least either the equation (B.14)
or (B.18) is satisfied. Otherwise, separation is not an equilibrium.
Next, I confirm how the inequity aversion impacts on the pooling equilibrium.
Let me begin with the analysis of the second period. Suppose every agent selects
Less in the first period and plan D is offered in the second period.
Skilled workers’ expected utility when they select Accept differs depending on thetype of their teammate. Let a skilled worker’s expected utility when he selects Accept
and his teammate is a newcomer be 𝑢2 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐴𝑐𝑐𝑒𝑝𝑡, 𝑁𝑒𝑤). Then, each type of
agent’s on-the-pass-of-the-equilibrium payoff in the second period is as follows.
(𝐵. 21)𝑢2 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐴𝑐𝑐𝑒𝑝𝑡, 𝑆𝑘𝑖𝑙𝑙𝑒𝑑)𝑃𝑂𝑂𝐿 = 628
𝑢2 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐴𝑐𝑐𝑒𝑝𝑡, 𝑁𝑒𝑤)𝑃𝑂𝑂𝐿 = 628 − 188𝛽

(𝐵. 22)𝑢2 (𝑁𝑒𝑤, 𝐴𝑐𝑐𝑒𝑝𝑡, 𝑆𝑘𝑖𝑙𝑙𝑒𝑑)𝑃𝑂𝑂𝐿 = 440 − 188𝛼(𝐵. 23)𝑢2 (𝑁𝑒𝑤, 𝐴𝑐𝑐𝑒𝑝𝑡, 𝑁𝑒𝑤)𝑃𝑂𝑂𝐿 = 440(𝐵. 24)Since 𝛽 < 1, the value of equation (B.22) is always strictly greater than zero. The
value of the equation (B.23) becomes weakly greater than zero if,
𝛼≤

110
47

(𝐵. 25)

Based on (B.21) and (B.22), skilled workers always select Accept on the pass ofthe equilibrium. Based on (B.24), newcomers who paired with newcomers also always
select Accept on the pass of the equilibrium. On the other hand, newcomers who pairedwith skilled workers select Accept if the equation (B.25) is satisfied, otherwise, they
select Reject.
Beforemoving

to

the

first

period’sanalysis,I

examineoff-the-pass-of-the-equilibrium behavior at the second period. If skilled workers deviate
45

and select Much, then both agents select Accept as long as the following condition(B.26) is satisfied. (B.26) is a condition when the deviator who paired with a skilledworker attains utility which is weakly greater than zero. Since the deviator’s teammate
always prefer Accept and also the deviator himself prefer Accept as long as he is pairedwith a newcomer, (B.26) is the only condition we need to care about.
𝛼≤

440
333

(𝐵. 26)

Besides, if a newcomer selects Much and he is paired with a skilled worker, thenboth team members select Accept if,
𝛼≤

107
666

(𝐵. 27)

If a newcomer selects Much and he is paired with a newcomer, then both of teammembers select Accept if,
𝛼≤

107
333

(𝐵. 28)

Next I start the first period’s analysis. First, I derive conditions when skilledworkers prefer Less in each case. Then, I also derive conditions when newcomers preferLess in each case.
440[Case1: 0 < α ≤ 333]
In this case, skilled worker’s on-the-pass-of-the-equilibrium expected utility is,21
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠)𝑃𝑂𝑂𝐿 = [427 + 628] + [427 − 47𝛽 + 628 − 188𝛽]3
3(𝐵. 29)Their off-the-pass-of-the-equilibrium expected utility is,21
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ)𝑃𝑂𝑂𝐿 = [615 − 115𝛽 + 440 − 333𝛼] + [615 − 235𝛽 + 440] (𝐵. 30)
33Comparing (B.29) and (B.30), then skilled workers always strictly prefer Less inthis case.
440

110

[Case2: 333 < 𝛼 ≤ 47 ]In this case, skilled worker’s on-the-pass-of-the-equilibrium expected utility is thesame as (B.29).
Their off-the-pass-of-the-equilibrium expected utility is,21
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ)𝑃𝑂𝑂𝐿 = [615 − 115𝛽 + 0] + [615 − 235𝛽 + 440]3
3
46

(𝐵. 31)Comparing (B.29) and (B.31), then skilled workers always strictly prefer Less inthis case.
110[Case3: 47 < 𝛼]
In this case, skilled worker’s on-the-pass-of-the-equilibrium expected utility is,21
𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠)𝑃𝑂𝑂𝐿 = [427 + 628] + [427 − 47𝛽 + 0]
3
3

(𝐵. 32)

Their off-the-pass-of-the-equilibrium expected utility is the same as (B.31).
Comparing (B.32) and (B.31), then skilled workers prefer Less as long as thefollowing (B.33) is satisfied.
𝛽≥−

252
418

(𝐵. 33)

By the assumption, 𝛽 > 0. Thus, (B.33) is always satisfied.
From the above analyses for Case 1 to Case 3, skilled workers always prefer Less.
Now we move on to the analysis on the newcomers’ behavior.
107[Case1: 0 < α ≤ 666]
In this case, newcomer’s on-the-pass-of-the-equilibrium expected utility is,𝑢1 (𝑁𝑒𝑤, 𝐿𝑒𝑠𝑠)𝑃𝑂𝑂𝐿 =
Their

2
1
[380 − 47𝛼 + 440 − 188𝛼] + [380 + 440]3
3

off-the-pass-of-the-equilibrium

expected(𝐵. 34)
utilityis21𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ)𝑃𝑂𝑂𝐿 = 3 [495 − 5𝛼 + 107 − 666𝛼] + 3 [495 − 115𝛽 + 107 − 333𝛼] (𝐵. 35)
Comparing (B.34) and (B.35), then skilled workers always prefer Less.
107

107

[Case2: 666 < 𝛼 ≤ 333]
In this case, newcomer’s on-the-pass-of-the-equilibrium expected utility is thesame as (B.34).
Their off-the-pass-of-the-equilibrium expected utility is21𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ)𝑃𝑂𝑂𝐿 = [495 − 5𝛼] + [495 − 115𝛽 + 107 − 333𝛼]3
3
47(𝐵. 36)

Comparing (B.34) and (B.36), then skilled workers prefer Less as long as thefollowing (B.37) is satisfied.
𝛽≥
107

127
868
𝛼−
115
115

(𝐵. 37)107

Since 666 < 𝛼 ≤ 333 and 𝛽 > 0, (B.37) is always satisfied in this case.
107

110

[Case3: 333 < 𝛼 ≤ 47 ]In this case, newcomer’s on-the-pass-of-the-equilibrium expected utility is thesame as (B.34).
Their off-the-pass-of-the-equilibrium expected utility is,𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ)𝑃𝑂𝑂𝐿 =2
1[495 − 5𝛼] + [495 − 115𝛽]3
3(𝐵. 38)Comparing (B.34) and (B.38), then skilled workers prefer Less as long as thefollowing (B.39) is satisfied.
𝛽 ≥ 4𝛼 −[Case4:

110
47

195
23

(𝐵. 39)< 𝛼]

In this case, newcomer’s on-the-pass-of-the-equilibrium expected utility is,𝑢1 (𝑁𝑒𝑤, 𝐿𝑒𝑠𝑠)𝑃𝑂𝑂𝐿 =2
1[380 − 47𝛼 + 0] + [380 + 440]
3
3

(𝐵. 40)

Their off-the-pass-of-the-equilibrium expected utility is the same as (B.38).
Comparing (B.40) and (B.38), then skilled workers prefer Less as long as thefollowing (B.41) is satisfied.
𝛽≥

84
19
𝛼−
115
23

(𝐵. 41)

The above results are summarized in Figure 2.
B.6. LEARN with Inequity-averse Individuals
Suppose every agent reports their type truthfully in the first period. Then, as discussed
in section B.5, all agents’ expected utility when they select Accept is 440. Therefore,
everyone strictly prefers Accept in the second period.
48

Skilled workers’ expected utility depending on their choice in the first period is,
2
1[560 + 440] + [615 − 235𝛽 + 440]
3
3

(𝐵. 42)2
1[500 − 115𝛼 + 440] + [427 − 47𝛽 + 628]3
3

(𝐵. 43)𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝑀𝑢𝑐ℎ) =𝑢1 (𝑆𝑘𝑖𝑙𝑙𝑒𝑑, 𝐿𝑒𝑠𝑠) =Since 𝛼 ≥ 𝛽, the value of the equation (B.42) is always greater than that of theequation (B.43). Therefore, skilled workers strictly prefer Much regardless of theirinequity averseness.
Similarly, newcomers’ expected utility depending on their decision in the first
period is,
𝑢1 (𝑁𝑒𝑤, 𝑀𝑢𝑐ℎ) =2
1[−190 − 750𝛼 + 440] + [495 − 115𝛽 + 0]
3
3

(𝐵. 44)2
1[380 − 235𝛼 + 440] + [380 + 440]
3
3

(𝐵. 45)𝑢1 (𝑁𝑒𝑤, 𝐿𝑒𝑠𝑠) =Since 𝛼, 𝛽 > 0 , newcomers strictly prefer Less regardless of their inequity
averseness.

C. An Experimental Procedure for Measurement of Inequity AversenessI measured the inequity-aversion parameters using the BDM mechanism (Becker et al.,1964). In this experiment, participants were randomly matched as a pair. This
experiment consists of two parts. The first part is for the measurement of 𝛼, and thelatter one is for measurement of 𝛽.
In the first part, each participant answered the following 11 binary-choiceproblems.
FIGURE 11
Choice Problems for Measuring the Preference Parameter of Envy

49

I measured 𝛼 based on the following equation.
𝛼=150 − 𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔𝑃𝑜𝑖𝑛𝑡𝛼
100(𝐶. 1)

𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔𝑃𝑜𝑖𝑛𝑡𝛼 is identified as the minimum value where participants selectthe right-hand-side alternative. For participants who have more than two kinks in theirchoice behavior, I do not define 𝛼 and let it be a missing value.
In the latter part, I measure 𝛽 using a similar procedure. Figure 12 shows 11
binary-choice problems in this section. 𝛽 is measured using equation (C.2). The
definition of 𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔𝑃𝑜𝑖𝑛𝑡𝛽 is similar to that of 𝑆𝑤𝑖𝑡𝑐ℎ𝑖𝑛𝑔𝑃𝑜𝑖𝑛𝑡𝛼 .
𝛽=250 − 𝑆𝑤𝑖𝑡ℎ𝑐ℎ𝑖𝑛𝑔𝑃𝑜𝑖𝑛𝑡𝛽100(𝐶. 2)FIGURE 12
Choice Problems for Measuring the Preference Parameter of Guilt

When every participant finished answering the above choice problems, one out of
22 problems was randomly selected. In addition, within each pair, either one participant
was randomly selected. Based on the selected participant’s answer at the selected
problem, each participant’s rewards from this task were determined.

50