Gradualism in Repeated Games with Hidden
Information: An Experimental Study
Melis Kartaly
Wieland Müllerz
University of Vienna & VCEE
University of Vienna & VCEE
Tilburg University, CentER & TILEC
James Tremewanx
University of Vienna & VCEE
May 4, 2015
Abstract
Two players are matched to play in a trust-game like setting in which the sender
does not fully know the trustworthiness of the receiver. We show that equilibrium is
characterized by “gradualism” in trust; i.e., the sender starts by choosing small trust
levels and increases the trust level gradually as long as the receiver returns. Next, we
conduct a series of experiments to empirically evaluate the role gradualism plays in
games ridden with incomplete information. We …nd strong evidence that senders use
gradualist strategies as we predicted. Moreover, we show that using such strategies is
optimal given the observed receiver behavior. JEL classi…cation numbers: C73, C91,
C92, D82, D83.
We would like to thank Guillaume Frechette, Karl Schlag and Andy Schotter for helpful comments.
Financial support from Hardegg Foundation via Heinrich Graf Hardegg’sche Stiftung is gratefully acknowledged.
y
Department of Economics, University of Vienna, Email: [email protected].
z
Department of Economics, University of Vienna, Email: [email protected].
x
Department of Economics, University of Vienna, Email: [email protected].
1
1
Introduction
How do trading partners establish trust and long-run cooperation in the face of hidden
information? Is it better to build the relationship cautiously using small, testing levels
of trade or could this type of strategy back…re and damage relationships (because such
cautiousness may be perceived as o¤ensive)? These are the questions which our paper
addresses. Our theoretical and experimental analysis sheds light on the best way to build
relationships in the shadow of hidden information.
Economic relationships may involve considerable uncertainty especially at the beginning. In particular, partners in a new relationship may not immediately observe each other’s
true motives and degree of commitment to the relationship. Indeed, one major concern at the
beginning of relationships is the prospect of facing an opportunistic partner who is around
for short term gains. Thus, an important question follows: How to build cooperation in the
face of such uncertainty?
To investigate this question theoretically and experimentally, we develop a simple
model in which two players are matched to play a general version of the so-called trust
game in…nitely many times. The trust game features a sender and a receiver. Our game
incorporates hidden information about the receiver. The receiver can be one of two types:
high or low. The high type has a higher discount factor than the low type. We interpret
discount factor as a proxy for trustworthiness.1 It is well known from the theory of repeated
games that a high discount factor is associated with cooperative behavior, whereas a low
discount factor typically results in myopia and opportunism. Indeed, the discount factors of
the two types in our model are such that the sender can achieve full cooperation with the
high type receiver whereas the low type receiver would never return the sender’s trust in a
symmetric information setting. Thus, the high type is “cooperative” whereas the low type
1
Trustworthy behavior is, without doubt, a complex, multi-faceted phenomenon. Here, we take
a very simplistic approach and focus on one particularly materialistic aspect of trustworthiness.
Laboratory experiments show that prosocial behavior has a strongly materialistic component. For
example, it has been observed in various experiments that varying the monetary stakes in games
alters prosocial incentives signi…cantly (see, for example, Slonim and Roth (1998), Gneezy (2005),
Johansson-Stenman et al. (2005), and List et al., (2011)).
2
is “opportunistic”.
In the hidden information game, the sender may prefer lower trust levels initially,
as this is less risky (i.e., the payo¤ loss of the sender from betrayal is limited at low trust
levels). However, this may result in a problem of imitation: Although the low type is not
cooperative, he may sometimes …nd returning the sender’s trust in his best interest in order
to mimic the high type and behave opportunistically afterwards. Therefore, the sender must
devise an appropriate strategy in order to screen the receiver and test his incentives.
Our general theoretical message is simple and intuitive: Equilibrium is “gradualist”
under reasonable assumptions. Gradualism implies that the relationship starts o¤ with small,
testing levels of trust, and the sender increases the trust level gradually as long as the receiver
chooses to return the sender’s trust. The theme of gradualism has appeared in a large gametheoretic literature with hidden information, such as Sobel (1985), Ghosh and Ray (1996),
Kranton (1996), Watson (1999, 2002), Halac (2012) and Kartal (2014). These papers study in
distinct environments how cooperation is established in the presence of hidden information
and point out that gradualism is an e¢ cient solution against problems that uncertainty
poses.2 Unlike the previous literature, we design a model that not only generates gradualism
in equilibrium but also is straightforward to implement in the laboratory because our ultimate
aim is providing a rigorous empirical evaluation of the gradualism hypothesis. This is the
most important distinction between the current work and the previous theoretical literature.
We denote the main experimental game in our design as the “large game.” In the
large game, the sender chooses either no trust or one of three positive trust levels: low
trust, medium trust or high trust. If the sender chooses a positive trust level, then the
receiver decides whether to return or default. If high trust is chosen and followed by return,
this generates the most e¢ cient outcome (i.e., the highest joint payo¤). In this game, the
sender can use smaller trust levels in order to test the receiver’s type. Indeed, we show that
equilibrium play is characterized by gradualism and, further, pin down equilibria precisely.
We distinguish between non-equilibrium and equilibrium gradualist strategies in the data;
this is important because not every gradualist strategy is a good strategy. We also have
2
Our work is closest to Kartal (2014), which studies informal contracts with hidden information
about the discount factor of the principal.
3
a control treatment in which the sender chooses between only no trust and high trust— we
denote this as the “small game.” The small game di¤ers from the large game only in that
low and medium trust levels are absent; all other parameters are held constant and, thus,
the e¢ cient outcome is the same across the two games. In the unique equilibrium of the
small game, the sender never trusts because choosing high trust is simply too risky for the
sender. In the large game, however, the equilibrium always entails positive levels of trust
unless the receiver defaults. More speci…cally, the sender starts by choosing low trust and
increases the trust level gradually as long as the receiver returns.
The two experimental games, the large game and the small game, relate to di¤erent
institutional lending mechanisms. Indeed, the distinction between the large game and the
small game is analogous to the contrast between micro…nance institutions and traditional
banking. Commercial banks have usually refrained from lending small amounts to the poor
mainly due to hidden information problems and associated costs.3 However, following the
succesful expansion of micro…nance in recent decades, especially, in developing countries,
many commercial banks shifted their operations to the large game setting. The comparison
of the small game and the large game enables us to put into perspective the decision of large
banks to shift their attention to micro…nance.
Our main experimental results are as follows. First of all, we …nd strong support
for our gradualism hypothesis. Most senders choose to start with low trust and gradually
increase the trust level as long as the receiver returns, consistent with our equilibrium prediction. Furthermore, we classify a large majority of the observed behavior as belonging
to one of …ve simple categories, the largest category comprising the equilibrium behavior.
We also …nd that equilibrium strategies make senders better o¤ given the empirical receiver
behavior. For example, starting the relationship with medium or high trust are dominated
choices.
Next, we investigate how observed payo¤s compare with the theory. In the large
3
Hidden information problem arises due to the fact that borrowers who need small loans are
typically poor and lack the credit history and/or the collateral needed to borrow from commercial
banks. Moreover, operational costs are higher if the bank handles multiple small accounts rather
than one large account.
4
game, match payo¤s are very close to but also slightly higher than the predicted payo¤.
This is not surprising given the fact that players conform to equilibrium predictions to a
signi…cant extent in the large game. To be more speci…c, senders receive payo¤s similar
to those predicted by the equilibrium whereas receivers do slightly better than what the
equilibrium predicts. This is mainly due to the following. On the one hand, a nonnegligible
proportion of senders choose medium and high trust levels, earlier than that predicted by the
theory. On the other hand, low type receivers do not fully exploit this; i.e., they return at
elevated rates— relative to our prediction. Since choosing higher trust levels earlier results in
mixed success for the senders, their payo¤s are not signi…cantly di¤erent than the equilibrium
prediction whereas receivers do better than the equilibrium prediction— but to a limited
extent because they do not fully exploit the senders’goodwill.
Although payo¤s in the large game are higher than our equilibrium predictions, the
extent of this improvement is limited, as we noted above. In the small game, however, payo¤s
are remarkably higher than that predicted by the theory. This is because a substantial
proportion of senders choose to trust and many receivers choose to return, in stark contrast
with the equilibrium prediction. In relation with this, we also observe that welfare is not
lower in the small game in comparison to the large game, in contrast with our prediction.
In fact, the small game generates higher joint payo¤s than the large game, which is strongly
signi…cant.
At …rst blush, the observation that the small game is more e¢ cient than the large
game— because many senders choose to trust— may seem puzzling. However, just as in
Camerer and Weigelt (1988), subjects seem to have a “homemade” belief about the receivers’type, in addition to the prior probability that we generated in the experiment. The
senders’homemade belief may represent, for example, the percentage of receivers who are
altruistic so that they will always return rather than default. If senders have a small, positive homemade prior in addition to the controlled probability that the receiver will pay back,
then this changes the theoretical prediction in the small game but not in the large game.
Put di¤erently, the small game prediction is— theoretically and experimentally— not robust
to small changes in the prior whereas the large game is robust in both dimensions.
In addition, when we separate sender and receiver payo¤s, we see that the increased
5
e¢ ciency in the small game stems from the fact that the small game boosts receiver payo¤s.
Sender payo¤s do not di¤er across the two games. Moreover, the large game provides senders
with “insurance”: The variance of sender payo¤s is signi…cantly lower in the large game than
in the small game whereas its expected value does not di¤er across the two treatments. Thus,
taking risk-aversion into account (which is commonly observed in laboratory experiments)
would imply that the large game makes senders better o¤ whereas the small game makes
receivers better o¤.4
Finally, joint match payo¤s exhibit a (weakly signi…cant) downward trend in both
games. In the large game, this is due to a downward trend in the receiver payo¤s (there is
no signi…cant change in sender payo¤s) and, in the small game, due to a downward trend
in the sender payo¤s (there is no signi…cant change in receiver payo¤s). Most of the trends
observed in the data disappear if we discard the …rst 10 matches during which subjects
presumably learn the game and adapt to it. Due to this and due to the fact that we would
like to focus on experienced play rather than learning, we discard the …rst 10 matches in the
data analysis of the main text. The analysis with the full data can be found in Appendix ?.
2
Related Literature
There is an extensive reputation literature dating back to the seminal works by Kreps and
Wilson (1982), Milgrom and Roberts (1982) and Kreps et al. (1982). While Kreps and
Wilson (1982) and Milgrom and Roberts (1982) analyze an entry-deterrence game, Kreps
et al. (1982) shows that incomplete information about players’ preferences can sustain
cooperation in equilibrium, even in a …nitely repeated prisoners’ dilemma game. Suppose
that each player is a type that is “committed” to a tit-for-tat strategy with some positive
probability. Then, reputation concerns can motivate self-interested players to imitate a
committed type and generate cooperative behavior even with self-interested players.
Experimenters have built on the theoretical work by Kreps et al. (1982) in order
to test the sequential equilibrium predictions when there is incomplete information about
4
Although we assume risk-neutrality in the model, moderate levels of risk-aversion does not
a¤ect any of our theoretical predictions.
6
players’ preferences. Camerer and Weigelt (1988) analyze a binary trust game whereas
Andreoni and Miller (1993) study a prisoners’dilemma game. Camerer and Weigelt (1988),
and the subsequent studies by Neral and Ochs (1992), Anderhub et al. (2002), Brandts
and Figueras (2003), and Grosskopf and Sarin (2010) study whether sequential equilibrium
is successful in predicting behavior, varying certain dimensions of the game such as the
probability with which the receiver is a speci…c type or the observability of past choices of
the receiver to the sender.
All of the experimental papers discussed above as well as Kreps et al. (1982) assume
that there are only two choices available to players and that the game is …nitely repeated.
Given this structure, it is not possible for the players to build up a relationship gradually.
What if the sender has more than two choices in an in…nitely repeated trust game with
incomplete information? We show that, in this case, equilibrium exhibits gradualism—
under reasonable conditions. This relates to a large branch of the reputation literature that
comes up with a similar conclusion in diverse settings (see, for example, Sobel (1985), Ghosh
and Ray (1996), Kranton (1996), Watson (1999, 2002, 2003), Halac (2012), Kartal (2014)).
Halac (2012) and Kartal (2014) study relational contracts with hidden information about the
principal. Kartal (2014) is the closest to our study since a relational contract is analogous
to a trust game, and she assumes that there is hidden information about the discount factor
of the principal. In relational contracts, the discount factor of the principal matters: It is
the value of the future relationship that sustains relational contracts, and this value depends
precisely on the discount factor of the principal. In Ghosh and Ray (1996), Kranton (1996),
and Watson (1999, 2002), two partners simultenously choose the level of cooperation in
every period in a prisoners’dilemma-like setting and decide individually whether or not to
behave opportunistically. There is two-sided hidden information: “High”type players prefer
to cooperate as long as their partners also cooperate whereas “low”types have an incentive
to take advantage of the other player’s trust. There is an initial testing phase in which
partners of high type build trust through their actions, and the stakes in the relationship
rise as players trust each other more.5 Also, the paper by Andreoni and Samuelson. Finitely
5
There is a separate strand of the literature in which gradualist strategies improve relation-
ships ridden with moral hazard (see, for example, Lazear (1981), Harris and Holmstrom (1982),
7
repeated and exogeneously imposed gradualism.
Finally, our paper relates to a burgeoning experimental literature on in…nitely repeated games (see, for example, Palfrey and Rosenthal (1994), Engle-Warnick and Slonim
(2004), Dal Bó (2005), Aoyagi and Fréchette (2008), and Dal Bó and Fréchette (2011)).6
To our knowledge, our paper is the …rst experimental study that incorporates incomplete
information in an in…nitely repeated setting. Moreover, most of the experimental literature on in…nitely repeated games focuses on simple settings with only two possible actions.
Therefore, previous studies do not address the issues that we study in the current paper.
3
A General Model
Two players, a sender and a receiver, interact repeatedly in periods t = 0; 1; : : : : At the
beginning of period t
0, the sender chooses a trust level, denoted by mt . We assume that
mt 2 f0g[[m; m], where m
0. That is, the sender can choose from a continuum of positive
trust levels and also has the option not to trust.
If the sender chooses mt > 0, then the receiver chooses between “return” and “default.” If mt > 0 and the receiver chooses to return, then rS (mt ) and rR (mt ) denote the
respective period-t payo¤ to the sender and the receiver, whereas dS (mt ) and dR (mt ) denote
the respective period-t payo¤ to the sender and receiver if the receiver chooses to default.
If the sender chooses mt = 0, then the receiver has no choice to make, and both players
Thomas and Worrall (1994) Albuquerque and Hopenhayn (1997), Ray (2002)). There is no hidden
information in these papers unlike in our setting.
6
Engle-Warnick and Slonim (2004) show in a trust game with complete information that senders
are signi…cantly more likely to send in the inde…nitely repeated game than in the …nitely repeated
game. However, some other studies present weaker experimental evidence regarding the positive
e¤ects of repetition on cooperation. For instance, Palfrey and Rosenthal (1994) …nd that repetition
indeed leads to higher cooperation, but the magnitude of this increase is small even though they use
a very high discount rate. Dal Bó and Fréchette (2011) analyze two repeated prisoners’dilemma
games: one in which perpetual cooperation is an equilibrium and another one in which cooperation
is not an equilibrium. They …nd that cooperation is not necessarily higher in the game in which
cooperation can be supported in equilibrium. Put di¤erently, even when cooperation is supported
in equilibrium, the actual cooperation rate may be low.
8
end up with their outside options, denoted by rS and rR for the sender and the receiver,
respectively. We assume that all the payo¤ functions are continuous in the trust level, mt .
Payo¤s are in line with the typical trust game à la Berg et al. (1996). The key
assumptions on payo¤s are as follows.
If m0 > m, then rS (m0 ) > rS (m) and rR (m0 ) > rR (m); i.e., the payo¤ from mutual
cooperation increases in the trust level for both parties.
If m0 > m, then dS (m) > dS (m0 ); i.e., the higher the trust level, the lower the payo¤
of the sender if the receiver defaults.
If m0 > m, then dR (m0 ) > dR (m); i.e., the higher the trust level, the higher the payo¤
of the receiver from default.
For every m 2 [m; m], rS (m) > dS (m) and dR (m) > rR (m).
Given these assumptions, our model accommodates the payo¤ structure of the typical
trust game.
The game involves one-sided hidden information. The sender’s discount factor is ,
which is …xed and known, whereas the receiver’s discount factor is
the receiver’s private information, and
l
<
h.
; where
2 fl; hg is
Let p0 denote the prior probability that the
receiver is high type. The receiver learns her type at the beginning of the initial period,
and this type remains the same in all subsequent periods. We assume that the high type
receiver is cooperative in the following sense. In the complete information game, there exists
a trusting equilibrium for every 0 < m 2 [m; m] such that the sender always chooses m and
the high type receiver returns; i.e.,
rR (m)
> dR (m) +
1
1
h
h
rR
(1)
h
for every m 2 [m; m]. This implies that the highest level of trust m is supported in the
equilibrium with a high type receiver. However, (1) never holds in the complete information
game with the low type receiver; i.e.,
rR (m)
< dR (m) +
1
1
l
9
l
rR
l
(2)
for every m 2 [m; m]. Therefore, the low type’s discount factor
l
is such that trust is never
an equilibrium outcome in the complete information game. Despite the fact that (2) holds,
the low type may sometimes return in the incomplete information game in order to imitate
the high type. This is the case if, for example, the sender’s strategy prescribes m 2 (0; m)
at t = 0, m at t = 1 if the receiver returns at t = 0, and
rR (m) +
l
dR (m) +
l
1
rR
l
dR (m) +
l
1
rR
l
holds.
We study the perfect Bayesian Nash equilibrium of this game assuming that the
sender follows a Markovian trust strategy. In particular, we assume that the trust schedule
depends only on the posterior belief of the sender at the beginning of period t, which we
denote by pt . In any period t such that mt > 0, the sender updates his posterior belief
according to Bayes rule, wherever possible. Otherwise, mt = 0, and pt = pt+1 . Finally, we
assume that once the posterior belief is degenerate, then it does not change no matter what
happens thereafter.7 Our main result is as follows:
Proposition 1 If p0 is at an intermediate value, then every equilibrium under the abovestated assumptions exhibits gradualism and fully reveals the receiver’s type in …nite time; i.e.,
there exists an interval (p; p)
(0; 1) such that if p0 2 [p; p] then there is a …nite T > 0 such
that mt < mt+1 for every t 2 f0; :::; T
If p0 < p, then mt = 0 for every t
1g until mT = m, as long as default is not observed.
0, whereas m0 = m if p0 > p.
The proof of Proposition 1 is relegated to the Appendix. The intuition for this result is
simple. If the prior probability that the receiver is high type is su¢ ciently high, then the
sender experiments with the highest trust level m because the default risk is low. If, however,
the prior probability is too low, then the unique equilibrium prescribes zero trust because
trusting is too risky— for any m > 0. Finally, if p0 2 [p; p], then the equilibrium prescribes
gradualism. Since the initial prior p0 is not high enough, starting o¤ with a su¢ ciently low
7
This assumption is without loss of generality. Proposition 1 still holds if we allow degenerate
beliefs to change. The proof is available upon request (the interested reader is referred to Kartal
(2014), Section 5).
10
trust level limits the potential loss of the sender from default. If the receiver returns at t
0,
then the trust level at t + 1 increases due to the following. On-the-equilibrium path, the
high type always returns, whereas the low type strictly randomizes between returning and
defaulting in every period. This randomization by the low type implies that the posterior
increases as long as the receiver keeps returning.8 Higher posterior, in turn, makes higher
trust feasible and desirable in the next period.
4
Experimental Design
4.1
The Experimental Game
The experimental game is designed on the basis of our general model, which we modi…ed
with the following consideration in mind. From a logistical standpoint, we cannot use a game
in which the sender’s strategy space is a continuum. As a result, we needed to discretize our
game. In the experimental game, the sender has, in total, four choices: no trust and three
positive trust levels. We chose a trust game with four possible levels of trust to satisfy two
main concerns. On the one hand, we wanted the number of trust levels to be su¢ ciently
high so that gradualism is a viable equilibrium outcome. On the other hand, we wanted the
number to be su¢ ciently low so that the game would be easy to explain to our subjects.
Our choice of experimental parameters for the game serves our primary purpose of
testing for the gradualism hypothesis. As we explain in more detail in the upcoming section,
we chose parameters such that all equilibria entail gradualist strategies.
To reiterate, in each period, the sender chooses either no trust (N ) or one of three
positive trust levels: low trust (L), medium trust (M ) and high trust (H). Thus, the sender
can use smaller trust levels in order to test the receiver. If a positive trust level is chosen,
then the receiver chooses between return (R) and default (D). Figure 2 displays the game
and the payo¤ parameters that we use in the experiment. The experimental payo¤s are
consistent with the assumptions that we imposed in the description of the general model.
8
If the receiver defaults, then the sender believes that the receiver is low type with probability
one, and, thus, chooses m = 0, thereafter.
11
Sender
..u.
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N................
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24
....D
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15
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32
20
44
12
56
0
22
42
36
72
42
80
Figure 1: Experimental game with actual payo¤ parameters
In the experiment, we simplify our theoretical setting and computerize the high type:
The computerized high type always returns if the sender chooses a positive trust level. We
use a computerized high type in order to have more control over the predictions of the model
as well as the subjects’behavior.9
In our design, the sender has a discount factor of 0.75 whereas the low type receiver
has a discount factor of 0.5. We explain how we implement di¤erent discount factors in
detail, in Section 4.3. Finally, the prior probability of meeting a high type player is 0.125.
4.2
Equilibrium Analysis
We analyze the perfect Bayesian Nash equilibrium (PBE) of the stage game given the parametrization discussed above and the payo¤s in Figure 2. Unlike in Proposition 1, we do not
restrain ourselves to Markovain trust strategies because a complete equilibrium analysis is
feasible.
Hereafter, the receiver refers to the low type receiver (unless otherwise stated) since
the high type is computerized and always returns. Every PBE is gradualist, and all equilibria
are nearly identical at t = 0 and t = 1. We focus, in particular, on one equilibrium, which we
denote by E. E has a number of attractive properties: it generates the highest possible payo¤
for every party, provides the fastest information revelation and involves the least amount of
9
Numerous experiments involved computer players in various strategic settings with hidden
information (see, for example, Andreoni and Miller (1993), Anderhub et al. (2002), Grosskopf and
Sarin (2010), and Embrey et al. (2014)).
12
mixing— mixing is usually problematic in experimental studies as it cannot be observed in
the data.10 After explaining this equilibrium, we brie‡y discuss brie‡y how other PBE di¤er.
E can be described as follows.
(i) At t = 0, the sender chooses low trust (L), and the receiver randomizes between return
(R) and default (D).
(ii) If the receiver defaults at t = 0; 1; :::, the sender always chooses no trust (N), thereafter.
(iii) At t = 1, the sender randomizes between low trust (L) and medium trust (M) if the
receiver returned in the …rst period.
(1) If L is chosen at t = 1, then the receiver returns with probability one.
(2) If M is chosen at t = 1, then the receiver defaults with probability one.
(iv) At t = 2, the sender chooses:
(1) M if low trust was chosen and the sender returned at t = 1; and
(2) H if M was chosen and the receiver returned at t = 1.
Finally, if M is chosen at t = 2, then the receiver defaults with probability one.
Thus, the sender learns the true type of the receiver with probability one, latest, at
the end of t = 2. A formal statement of equilibrium E and the proof of existence can be
found in Appendix A. Note that, in equilibrium, H is never reached with a low type receiver.
This is because the parameters are such that M must precede H in equilibrium, and if
M is chosen, then the receiver defaults with probability one no matter what the sender’s
strategy prescribes for any continuation game. In contrast, the receiver returns with positive
probability after L is chosen in order to build up a reputation and default later when the
trust level becomes higher.
Parts (i), (ii) and (iii) are common to all PBE, and the only distinction across equilibria at t = 0 and t = 1 is numerical. More speci…cally, the probability with which the receiver
returns at t = 0 and the probability with which the sender chooses low trust at t = 1 after
the low type returns vary across equilibria. In (iv), only (1) di¤ers across equilibria: if low
trust was chosen and the sender returned at t = 1, the sender could randomize between low
trust and medium trust rather than choosing medium trust with probability one.
10
All equilibria are identical for the sender and the low type receiver in terms of expected total
payo¤ but E is the unique e¢ cient equilibrium if the high type’s payo¤ is also taken into account.
13
Sender
.u
.........
.
.
.
.. .....
....
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.
.
.... H
N ....
....
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....D
..
R.....
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24
56
0
15
42
80
Figure 2: Small game
4.3
Experimental Protocol
We have two treatments. Our main treatment is what we call the “large game” treatment.
The large game treatment implements the experimental game discussed above. Note that the
presence of low and medium trust levels in our experimental game is crucial for the existence
of an equilibrium that involves trust. If the sender had only the two options N and H, then
he would always choose N in equilibrium because H would be too risky an option given the
parameters of the game. This theoretical observation underlies our secondary treatment,
namely, the “small game” treatment. The small game treatment is identical to the large
game treatment with the exception that low and medium trust levels are absent in the small
game (see Figure 3). All other parameters and design details are held constant across the
two treatments. To reiterate, the unique equilibrium in the small game is such that the
sender always chooses N .
Our design is between-subject: Each subject participates only once in the experiment
and play either in the large game treatment or the small game treatment. In either treatment, subjects participated anonymously in a sequence of in…nitely repeated games. At the
beginning of each session, each subject was randomly assigned to be a sender or a receiver.
Subjects remained in the same role throughout the session. Each session involved 15 “human”players: 8 senders and 7 receivers. Overall, there were 8 receivers in each session: The
eighth receiver was the “computer”player which was programmed to always return.
We refer to each repeated game as a “match.”At the beginning of each match, each
14
sender was randomly matched with a receiver, and played either the large game or the small
game depending on the treatment. Each sender knew that there was 1/8 chance of being
matched to a computerized receiver in each match but senders were never informed about
the true type of the receiver with whom they were interacting.11 Each sender was randomly
rematched with another receiver after the end of a match. Each session ended either after
25 matches were completed, or at the end of the …rst match that was completed after 75
minutes have passed, whichever lasted shorter.
We induced an in…nitely repeated game in the lab by having a random continuation
rule. The probability of continuation
was the same for all matches in both treatments and
was equal to the discount factor of the sender; i.e.,
= 0:75. The human receiver, on the
other hand, had a discount factor of 0:5, which was implemented as follows. Starting from
the second round of a match, the receiver payo¤s in Figure 2 were reduced by a factor of
1/3. For example, if the match continues to the second round and the sender chooses no
trust in the second round, then the receiver obtains 10 points in the second round, rather
than 15. Another example: If the match proceeds to a third round, the sender chooses M ,
and the receiver returns, then the receiver obtains 16 points in the third round, rather than
36.
The experiment was conducted at the experimental laboratory of Vienna Center for
Experimental Economics (VCEE) at University of Vienna. Subjects were recruited from
the general undergraduate population via e-mail solicitations. A total of 180 undergraduates participated in 12 experimental sessions. We had 6 sessions for each treatment, which
translate to 6 independent observations per treatment. All sessions are conducted through
computer terminals, and the computer program is written in Z-Tree. After subjects read the
instructions, they had to answer a set of control questions that were meant to test whether
they understood the instructions. At the end of each session, the total number of points
earned by each subject were converted to Euros at the exchange rate of 100 points =1 Euro
and paid privately in cash. The experiment lasted about two hours? and subjects received
an average of ? euros, which is the usual average payment for such a period of time in the
lab. The instructions for the treatments can be found in the Appendix.
11
Of course, senders can infer that a receiver who chooses to default is human (i.e., the low type).
15
5
Results
We focus on experienced play; therefore, we drop the data from the …rst 10 matches in all
sessions.12 Whenever possible, we report two p-values for statistical tests. The …rst comes
from a non-parametric test that uses session-level data as the independent unit of observation.
This is denoted by p. The second p-value, denoted by p0 , comes from a regression-based
approach that clusters standard errors at the session level.
5.1
5.1.1
Behavior
Is Behavior Gradualist in the Large Game?
We begin this subsection by reporting sender behavior in the …rst round of a match in the
large game. Table 1 summarizes the senders’choice and receivers’response in the …rst round
of a match. On average, low trust is by far the most common choice: 67.1% is the average
proportion of low trust (L) in the …rst round of a match. Moreover, the proportion of L
choice in the …rst round exceeds 57% in every session of the large game treatment whereas
random play would generate a proportion of 25%. Thus, the senders’preference for L in the
…rst round is far from being random according to a non-parametric binomial test (p < 0:001).
The second most popular sender choice in the …rst round is medium trust (M), which
is chosen 16.5% of the time. A Wilcoxon signed-rank test and a sign test both reject the
equality of proportions of L and M at p < 0:001. Of course, this also holds when we compare
L with high trust (H) or no trust (N). We conclude that there is strong support for our
prediction of L choice in the …rst round. Finally, we regress the share of L choices in the
…rst round of each match on the match number (i.e., trend). The coe¢ cient is insigni…cant,
demonstrating that there is no time trend if we focus on experienced play.
We, now, turn to the manner in which receivers respond to senders’trust choice in
the …rst round. Table 1 shows that the rate of return is at its highest with L and much lower
12
We observe that if we discard the data from the …rst 10 matches, then time trends disappear,
and the percentage of behavior consistent with equilibrium predictions increases without a¤ecting
other experimental results. Therefore, we conclude that subjects learn the game in the …rst 10
matches.
16
Low Trust (L)
Medium Trust (M)
High Trust (H)
Return
300(64:2%)
45(39:1%)
21(33:3%)
Default
167 (35:8%)
70(60:9%)
42(66:7%)
Total
467(67:1%)
115(16:5%)
63(9:1%)
No Trust (N)
51(7:3%)
Notes ...
Table 1: Actions in the First Round of a Match
with M or H. Thus, aggregate data is consistent with our predictions. As we discussed in
Section 4.2, the receiver has, in theory, no willingness to return after M or H is chosen but
if he returns after L, he can build up a reputation and default later at a higher trust level.
The di¤erence between the return rate after L and after M (or H) is statistically signi…cant
according to both a Wilcoxon signed-rank test and a sign test (p < 0:001).13
Next, we classify behavior based on the outcomes from the …rst 2 rounds of matches
that last at least 2 rounds. We, then, provide a classi…cation of behavior that is observed
in the …rst 3 rounds of matches, as well as, in the …rst 4 rounds. We restrict the data,
in the former classi…cation, to matches that last at least three rounds and, in the latter,
to maches that last at least 4 rounds. We do not classify behavior for a higher number of
rounds because restricting the analysis to longer matches results in substantial loss of data,
and, thus, noise.14
The classi…cation involves …ve categories that we identi…ed based on the experimental
data: gradualist, weakly gradualist, always no trust, always high trust, and lenient punishment. We, now, explain how each category is de…ned.
Gradualist: We de…ne behavior as gradualist if it is consistent with the equilibria of the
large game, which we discussed in Section 4.2. Roughly speaking, the sender starts the
relationship with low trust, (weakly) increases the trust level as long as the receiver
13
A confounding factor that we will have to account for is that the “price of being kind” for the
receiver is low if L is chosen.
14
Figure ? in Appendix ? shows the frequency of matches with respect to match length. To be
precise, classifying behavior in the …rst 5 rounds in matches that lasted at least 5 rounds requires
discarding more than 71% of our data in the large game treatment.
17
returns, and punishes default with perpetual no trust.15 The proportion of gradualist
behavior is our main variable of interest.
Weakly gradualist: We de…ne behavior as weakly gradualist if the sender starts the relationship with low or medium trust, (weakly) increases the trust level as long as the
receiver returns, and punishes default by perpetual no trust. Gradualist behavior is, of
course, weakly gradualist but not vice versa because weakly gradualist behavior need
not be consistent with equilibrium.
Always no trust: The sender always chooses no trust in which case the receiver has no
choice to make.
Always high trust: The sender starts by choosing high trust in the …rst round and chooses
high trust conditional on return in all previous rounds and no trust, otherwise.
Lenient punishment: We de…ne behavior as lenient punishment if the sender starts with
a positive trust level, (weakly) increases the trust level as long as the receiver returns,
and punishes default more leniently than perpetual no trust; i.e., strictly reducing the
trust level rather than choosing no trust or choosing no trust for only a limited period
and then reverting to a trusting action.
We classify any behavior that does not …t into any one of these categories as “other.”
In order to describe our classi…cation and …ndings in more detail, we introduce the following
notation.
XY: 2-round sender history consistent with weakly gradualist behavior, where X 2 fL; M; Hg
and Y 2 fN; L; M; Hg denote the sender action in the …rst and the second rounds,
respectively.
15
One caveat is that we will never truly observe whether a sender reverts to perpetual no trust
since all matches end after a …nite number of rounds.
18
19
3:1%
0:07%
47:7%
38:3%
n=3
n=4
55:7%
66:4%
1:9%
2:4%
2:1%
No Trust
Gradualist
75:2%
Always
Weakly
4:9%
5:2%
5:4%
High Trust
Always
16:3%
9:1%
3:7%
Punishment
Lenient
Table 2: Classi…cation of Behavior in the …rst n rounds in matches that lasted at least n rounds, where n = 2; 3; 4
10:7%
Random Play
% acc. to
55:8%
% in Data
n=2
at least n rounds
Matches that lasted
Gradualist
Recall that in equilibrium, the sender chooses L in the …rst round, and randomizes
between L and M if the receiver returns. Otherwise, the sender chooses N in the second
round. Given these, XY histories that are gradualist (i.e., consistent with equilibrium predictions) are LL, LM , and LN . Recall that if behavior is weakly gradualist, then default
must be punished by N . Thus, if XY 2 fLL; LM g, then it is understood that the receiver
returned in the …rst round whereas if XY = LN , then the receiver must have defaulted
in the …rst round by the de…nition of XY . Finally, behavior is weakly gradualist but not
gradualist if it results in a sender history XY 2 fLH; M M; M H; M N g.
Next, we de…ne the following.
XYZ: 3-round sender history consistent with weakly gradualist behavior. In addition to what
we de…ned above, Z 2 fN; L; M; Hg denotes the sender action in the third round.
Given our equilibrium analysis, XY Z histories that are gradualist are LLL, LLM ,
LM H, LN N , LLN , and LM N .16 The …rst three imply that the receiver returned in both
the …rst and the second rounds, whereas the latter three imply that a default has taken place
either in the …rst round or the second round.17
Since equilibrium consists of mixed strategies, which we cannot observe in the data,
we conduct the analysis of 2-round behavior on the basis of XY histories. As discussed
above, behavior is gradualist if XY 2 fLL; LM; LN g. The column labeled “gradualist” in
Table 2 presents the proportion of gradualist behavior in the …rst 2 rounds, as well as the
expected proportion of gradualist behavior that would be generated if both the sender and
the receiver played randomly. Overall, 55.8% of the sender behavior is gradualist, and the
three gradualist XY histories are the three most frequently observed outcomes. Moreover,
the proportion of gradualist behavior always exceeds 40% at the session level whereas random
16
Conditional on LL having been realized, and the receiver having returned in the second round,
there are two possible types of equilibrium play in the third round. In the …rst type, which we
denoted by E and explained in detail in Section 4.2, the sender chooses M in the third round
whereas in the other type of equilibrium, the sender randomizes between L and M in the third
round.
17
XY Z histories that are not gradualist are LLH , LM M , LHH , M M M , M M H , M HH , LHN ,
M N N , M M N , M HN .
20
play would generate a proportion of about 10.7%. Thus, the prevalence of gradualism is far
from being random according to a non-parametric binomial test (p < 0:001). Finally, the
proportion of weakly gradualist behavior (which includes gradualist behavior, as well) is
75:2%.
We, now, analyze behavior in the …rst 3 rounds of matches that lasted at least 3
rounds. Since we cannot observe mixing in the data we conduct the analysis of 3-round
behavior on the basis of XY Z histories similar to what we did above. As discussed above,
behavior is gradualist if XY Z 2 fLLL; LLM; LM H; LLN; LM N; LN N g. Table 2 shows
the proportion of gradualist behavior in the …rst 3 rounds of matches that lasted at least 3
rounds, as well as the expected proportion of gradualist outcomes that random play would
generate. Remarkably, the six gradualist XY Z histories are the most common six outcomes
in the data and comprise about 48% of the data. Moreover, the proportion of gradualist
behavior always exceeds 34% in every session whereas the percentage of gradualist behavior
generated by random play would be slightly above 3%. Thus, the prevalence of gradualism
in the data is, again, far from being random (p < 0:001). Finally, weakly gradualist behavior
is ubiquitous: On average, more than 66% of behavior is weakly gradualist.
When we investigate behavior in the …rst 4 rounds (of matches that last 4 rounds
or longer), we …nd that 38.3% of the observed behavior is gradualist whereas random play
would generate a minuscule proportion— just 0.7%. Moreover, 55.7% of behavior is weakly
gradualist.18
Table 2 demonstrates that a nontrivial proportion of behavior belongs to one of the
three categories, always no trust, always high trust and lenient punishment. In total, these
three categories describe about 11% of behavior in the 2-round analysis, about 17% in the
3-round analysis and about 23% in the 4-round analysis. Of particular importance is the
lenient punishment. Observe that this type of behavior becomes more and more prevalent
as the number of rounds that we focus on increases. This is natural because the longer
the match, the more likely default becomes. Thus, it also becomes more likely to observe
lenient punishment. Finally, observe that we are able to classify about 79% of the observed
outcomes in the 4-round analysis— the percentage is even higher in the 2-round or 3-round
18
A full list of observed outcomes and the associated frequencies can be found in the Appendix.
21
analysis.19
Next, we compute the expected payo¤ of the sender strategy that supports equilibrium
E— which we described in Section 4.2— given the observed receiver behavior, and compare
this with the expected payo¤ of various strategies that support the behavior listed in Table
2. In the …rst round, E prescribes L.20 We …nd that, in matches that last exactly one period,
L is the best choice given the observed receiver behavior and provides a payo¤ that is 13%
higher than that of the second best choice, M . H is the worst choice in matches that last
exactly one period. The expected payo¤ in matches that last up to two rounds as well as
three rounds, the expected return for the strategy that supports E is strictly greater than
the return to any strategy that gives rise to weakly gradualist behavior, as well as choosing
always no trust or high trust— until default is observed.
In our comparison, we use the expected payo¤ conditional on the match lasting at
most 3 rounds. Computing the expected payo¤ for matches that last longer than 3 periods
creates a drastic problem of noise in terms of comparison with other strategies. For example,
in a match that lasts four rounds, there are 10 distinct pure strategies that give rise to
behavior that is weakly gradualist but not gradualist. Moreover, about only 39% of matches
continue at least 4 rounds. As a result, there are too few observations for the various possible
histories of each individual strategy -it would be good to have an example about MM?. Due
to precisely the same reason, our payo¤ comparison is only descriptive. At the session
level, the data becomes even more sparse, and, thus, it is not possible to have a meaningful
statistical test.
19
A caveat to keep in mind is that behavior which we classify as being gradualist based on the
…rst three rounds may end up being classi…ed as weakly gradualist or lenient punishment based on
the …rst four rounds— provided that the match continues to the fourth round.
20
In order to compute the expected payo¤ of choosing L in matches that last 1 round, we used the
receiver choice in the …rst round of all matches in which the sender chose L— not only the matches
that actually lasted one round. This is intuitive because receivers can never know whether or not
the match stops after the …rst round so there is no reason to focus on a subset of the data. The
remaining analysis is based on an analogous logic.
22
5.2
5.2.1
Payo¤ Comparisons
Payo¤ comparison with the theory
In our experiment, the length of each match is random. Since the match length is an
important factor that a¤ects payo¤s, we would like to control for the number of rounds
played in each match in our comparison of observed earnings with theoretical predictions.
To that aim, we calculate the payo¤-index
data, and
eq
obs
eq
, where
obs
denotes the match payo¤ in the
denotes the expected match payo¤ in equilibrium, given the actual receiver
type and the realized number of rounds.21 We compute three types of indices: one for the
joint-payo¤s, one for the sender payo¤s and the last one for the receiver payo¤s. Table 3
reports the summary statistics of these indices by treatment.
TABLE 3: Summary of Payo¤ Indices by Treatment
obs
eq
Joint-payo¤ Index
Sender-payo¤ Index
Receiver-payo¤ Index
Large Game
1:105
;
(0:016)
1:038 (0:024)
1:180
;
(0:017)
Small Game
1:662
;
(0:044)
1:161; (0:067)
2:819
;
(0:147)
N ote : For xa;b , a represents the signi…cance level according to the nonparametric
test, and b denotes the signi…cance level according to the regression-based analysis.
First, we report our …ndings in the large game treatment. The average of the jointpayo¤ index in the large game treatment exceeds one by 10.5 percent. Thus, joint matchpayo¤s are slightly higher in the data than in theory. Both a nonparametric binomial test
and a regression that regresses the joint-payo¤ index on a constant (controlling for the
match number and the number of rounds in the match) con…rm that observed payo¤s are,
on average, larger than the predicted values (p; p0 < 0:01).22 However, calculating the
respective payo¤ indices for senders and receivers shows that this result is driven by receiver
earnings that are higher in the data than in theory. The average of the receiver-payo¤ index
exceeds one by 18 percent, and this is strongly signi…cant according to both a nonparametric
binomial test and regression-based analysis. However, there is no evidence that sender payo¤s
21
When the sender is matched to a computer player, the joint-payo¤ index is computed on the
basis of the sender payo¤ since the computer has no earnings.
22
See the regression analysis and the related statistics in Appendix ?.
23
are di¤erent from the predicted values: The average of the sender-payo¤ index exceeds
one by just 3.8 percent, and this is statistically insigni…cant using both parametric and
nonparametric methods (p; p0 > 0:1). Thus, we conclude that the observed joint-match
payo¤s are higher than the theoretical predictions because receiver payo¤s are higher in the
data than in theory.
Overall, theory seems to be quite succesful in predicting payo¤s in the large game
treatment. This is not surprising given the fact that players conform to equilibrium predictions to a signi…cant extent. Receivers are better o¤ in the data, to a certain degree, due
to the following. On the one hand, a nonnegligible proportion of senders choose medium
and high trust levels, earlier than that predicted by the theory— aside from a substantial
proportion that conforms to our equilibrium predictions. On the other hand, receivers do not
fully exploit this; i.e., the return rate is elevated in the data relative to the equilibrium prescription. Since choosing higher trust levels earlier results in mixed success for the senders,
their payo¤s are not signi…cantly di¤erent than the equilibrium prediction, whereas receivers
do better than the equilibrium prediction, albeit only to a limited extent because they do
not fully exploit the senders’goodwill.
Finally, inspecting the coe¢ cient of the match number in the regressions we see that
there is no signi…cant time trend in neither the aggregated match payo¤s nor the playerspeci…c match payo¤s.
Next, we report our …ndings in the small game treatment. As Table 3 shows, the
average of the joint-payo¤ index exceeds one by 60%, which is highly signi…cant (p; p0 < 0:01).
The same is true to an even greater extent when we disaggregate the data and calculate the
receiver-payo¤ index. Receivers do much better than the prediction in the small game; the
average of the receiver payo¤-index exceeds one by 182 percent, as Table ? shows. This
di¤erence from the theoretical prediction is due to the fact that a substantial proportion
of senders choose to trust, in stark contrast with the equilibrium prediction of perpetual
no trust. Senders, however, do merely slightly better than the theoretical prediction: The
average of the sender-payo¤ index exceeds one by about 16 percent, which is signi…cant only
according to our regression-based analysis and at the 10 percent level (p > 0:1; p0 = 0:064).
This is because senders’trust results in limited success: a substantial proportion of receivers
24
choose to default following trust (on average, more than 50 percent of human receivers default
in rounds 1-3 of a match).
Looking at the coe¢ cient of the match number in the regressions for the small game,
we see that there is no signi…cant time trend with the exception of a slight downward trend
in the sender payo¤s, which is barely signi…cant (p0 = 0:096). This is due to a very slight
trend towards more equilibrium play in the small game. For example, the frequency of no
trust choice in the …rst round of a match increases over time— this is statistically signi…cant
at the 10% level according to a regression-based analysis.
5.2.2
Payo¤ comparison across treatments
As discussed above, the length of a match is random in each session. Since the match length
is an important factor that a¤ects the comparison of payo¤s across the two treatments, we
would like to control for the number of rounds played in each match for our across-treatment
comparison. To that aim, we calculate the e¢ ciency-index
obs
C
where
obs
N
N
denotes the match payo¤ in the data as before,
N
denotes the payo¤ if the
static Nash equilibrium is played repeatedly (i.e., the sender always chooses no trust), and
C
denotes the payo¤ from repeated full cooperation (i.e., the sender always chooses trust
and the receiver always returns). Note that both full cooperation and the static Nash play
are identical across the two treatments.
We compute three types of e¢ ciency indices: one for joint-payo¤s, one for the sender
payo¤s and the last one for the receiver payo¤s. All three types of e¢ ciency indices equal one
if players coordinate on the cooperative outcome whereas they will be zero if the static Nash
equilibrium outcome prevails. Table 4 reports the summary statistics of the three indices
by treatment. The joint-e¢ ciency index is clearly below one for both treatments. A further
look at the respective indices for senders and receivers display a divergence: The indice is
quite close to 0 for senders and close to 1 for receivers. This indicates that while there is
indeed some cooperation, many receivers exploit senders’trust in both types of game.
The …rst column in Table 4 shows that the di¤erence in the joint-e¢ ciency index
25
across the two treatments is about 13%. This di¤erence is statistically signi…cant (p < 0:01
according to a one-tailed Mann-Whitney test, and p0 < 0:01 from a parametric approach
that regresses the e¢ ciency index on a binary treatment variable controlling for the match
number, the number of rounds in a match, as well as possible interaction e¤ects).
TABLE 4: Mean of E¢ ciency Indices by Treatment
obs
N
Joint-e¢ ciency Index
Sender-e¢ ciency Index
Receiver-e¢ ciency Index
Large Game
0:328 (0:021)
0:079 (0:019)
0:752 (0:032)
Small Game
0:455 (0:030)
0:121 (0:050)
1:014 (0:082)
C
N
When we calculate separate indices for senders and receivers, we see that the di¤erence
in the joint-e¢ ciency across the two treatments stems from the fact that receivers are better
o¤ in the small game. Both a Mann-Whitney test and a regression-based analysis show that
the e¢ ciency index di¤ers across the two treatments for receivers but not for senders— for
receivers, p < 0:01; and p0 < 0:05, whereas for senders, p; p0 > 0:1.23 Thus, the di¤erence
in the joint-e¢ ciency index across the two treatments arises because the receiver-e¢ ciency
index di¤ers across the two treatments signi…cantly. The latter result can be explained taking
into account the fact that a small, positive home-made prior switches the equilibrium in the
small game from perpetual no-trust to a trusting one, while the large game equilibrium is
una¤ected. In theory, receivers are better o¤ in the small game than in the large game if
senders choose to trust in the …rst round in the equilibrium of the small game.
Although the sender-e¢ ciency index does not di¤er across the two treatments, there
is a highly signi…cant di¤erence across the two treatments when we compare the standard
deviation of the sender-e…ciency index at the session level (p < 0:001 from a Mann-Whitney
test). To be more precise, the session-level standard deviation ranges from 48.3% to about
60.1% in the small game and from 29.7% to 39.8% in the large game. This is because a
substantial proportion of senders use equilibrium gradualist strategies in the large game, as
we discussed above. Such strategies provides senders with insurance against default; as a
23
To be more speci…c, the di¤erence in the sender-e¢ ciency index across the two treatments
translates to about 1.3 experimental points per round, which is not signi…cant. The di¤erence in
the receiver-e¢ ciency index across the two treatments translates to about 7.1 experimental points
per round, which is signi…cant.
26
result, the large game generates a lower variance (without reducing the expected return) and,
thus, bene…ts risk-averse individuals— although we have assumed risk-neutrality throughout
the model, moderate levels of risk-aversion does not a¤ect any of our theoretical predictions.
5.3
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