Disclosure of Belief–Dependent Preferences
in a Trust Game
Giuseppe Attanasi
BETA, University of Strasbourg
Pierpaolo Battigalli
Bocconi University and IGIER, Milan
Rosemarie Nagel
ICREA, Universitat Pompeu Fabra, Barcelona Graduate School of Economics
December 2013 (This version: March 2014)
Abstract
Experimental evidence suggests that agents in social dilemmas have belief-dependent preferences. But in experimental games such preferences cannot be common knowledge, because
subjects play with anonymous co-players. Therefore, subjects play games with incomplete
information, for which it is hard to provide theoretical predictions. We address this issue
in the context of a trust game, assuming that the truster is self-interested and the trustee
has belief-dependent preferences given by a combination of guilt aversion and intention-based
reciprocity. We elicit the trustee’s belief-dependent preferences through a structured questionnaire. In the main treatment, the …lled-in questionnaire is disclosed and made common
knowledge within the matched pair, whereas in the control treatment, the …lled-in questionnaire is not disclosed to the co-player (truster). According to our main auxiliary assumption,
the treatment with disclosure approximately implements a psychological game with complete
information. To organize the data, we derive precise predictions for the complete-information
model, and robust qualitative predictions for the incomplete-information model. We …nd
that, in this social dilemma, guilt aversion is the prevalent psychological motivation, and that
behavior and elicited beliefs move in the direction predicted by the theory: In particular, in
the treatment with disclosure there is a polarization of behavior and beliefs, with more trust
and sharing in matched pairs with an elicited high-guilt trustee; in the control treatment
high-guilt trustees are less cooperative and we …nd a higher frequency of intermediate beliefs.
JEL classi…cation: C72, C91, D03.
Keywords: Experiments, psychological games, trust game, guilt, reciprocity, incomplete
and complete information.
We thank Olivier Armantier for great support in the statistical analysis. We thank Stefania Bortolotti, Martin
Dufwenberg, Alejandro Francetich, Ariel Rubinstein and the seminar participants at CISEI 2010 in Capri, ESA
2010 in Copenhagen, Behavioral & Quantitative Game Theory 2010 Conference in Newport, IMEBE 2011 in
Barcelona, University of Namur, Universitat Pompeu Fabra, Bocconi University, New York University and Max
Planck Institute of Economics in Jena for useful discussions. Giuseppe Attanasi gratefully acknowledges …nancial
support by the ERC (grant DU 283953) and by “Attractivité” IDEX 2013 (University of Strasbourg). Pierpaolo
Battigalli gratefully acknowledges the hospitality of NYU-Stern and …nancial support by the ERC (grant 324219).
Rosemarie Nagel gratefully acknowledges …nancial support from ECO2008-01768, ECO2011-25295, the Generalitat
de Catalunya and the CREA program.
1
1
Introduction
In recent years economists have become increasingly aware that belief-dependent motivation is
important to human decision-making, and that this can have important economic consequences
(see, for example, Dufwenberg 2008, Battigalli & Dufwenberg 2009, and the references therein).
Beliefs may a¤ect motivation in more than one way. First, as argued by Adam Smith (1759),
human action is a¤ected by emotion and a concern for the emotions of others; since emotions
can be triggered by beliefs (Elster 1998), beliefs a¤ect choice in a non-instrumental way, that is,
they a¤ect preferences about …nal consequences, such as consumption allocations. Second, beliefs
a¤ect the cognitive appraisal of the pre-choice situation and the reaction to this situation, as in
angry retaliations to perceived o¤ences (Berkowitz & Harmon-Jones 2004).1
We study belief-dependent motivations in the Trust Game, a stylized social dilemma whereby
agent A (the truster) takes a costly action that generates a social return, and agent B (the trustee)
decides how to distribute the proceeds between himself and A (Berg et al. 1995, Buskens & Raub
2008). We focus on the simplest version of this game, called Trust Minigame: A can either
take a costly action or not, and B can either (equally) share the proceeds or take everything
for himself. The goal of this paper is to study experimentally how B subjects’preferences over
distributions of monetary payo¤s in the Trust Minigame depend on their beliefs, and how the
disclosure of such belief-dependent preferences a¤ects strategic behavior.2
Two kinds of belief-dependent motivation seem salient in this social dilemma. Guilt aversion
makes B more willing share if he thinks that A expects him to do so; thus, B ’s willingness to
share is increasing in his second-order belief (Dufwenberg 2002, Battigalli & Dufwenberg 2007).
According to intention-based reciprocity, instead, B ’s willingness to share depends on his
perception of A’s costly action as either kind or neutral toward him: The less A expects B to
share, the kinder her costly action; therefore, B ’s willingness to share is decreasing in his secondorder belief (Dufwenberg & Kirchsteiger 2004).3 Experimental studies of the Trust Game show a
positive correlation between elicited second-order beliefs and sharing, supporting the hypothesis
that, in this social dilemma, guilt aversion is the prevailing psychological motivation of B -subjects
(Charness & Dufwenberg 2006, Chang et al. 2011).4 However, for di¤erent social dilemmas, Falk
et al. (2008) …nd evidence in support of intention-based reciprocating behavior. Furthermore,
some evidence of intention-based reciprocating behavior is also found in the Trust Game (Cox
1
There are other important sources of belief-dependent motivation. In particular, agents may have a noninstrumental concern for esteem (e.g. Ellingsen & Johanneson 2008) and self-esteem (e.g. Kuhnen & Tymula
2012), which are determined by beliefs. Tadelis (2011) experimentally studies the e¤ect of exposure in the Trust
Game: His results are consistent with a theory of shame avoidance.
2
From now on we refer to A (B ) as a female (male).
3
The intellectual home and mathematical framework for models of interacting agents with belief-dependent
motivations is an extension of traditional game theory, introduced and labeled “psychological game theory” by
Geanakoplos et al. (1989) and further developed by Battigalli & Dufwenberg (2009). In a nutshell, utility is
assumed to depend not only on (the consequences of) choices, but also on hierarchical beliefs. The theory of
intention-based reciprocity was …rst put forward by Rabin (1993) for simultaneous-move games. See also Charness
& Rabin (2002), Falk & Fischbacher (2006) and Stanca et al. (2009).
4
See also Guerra & Zizzo (2004), Bacharach et al. (2007), Charness & Dufwenberg (2011) and Bracht & Regner
(2013) for the Trust Game, and Dufwenberg & Gneezy (2000), Rueben et al. (2009) and Bellemare et al. (2011)
for other social dilemmas. Vanberg (2008), Ellingsen et al. (2010) and Kawagoe & Narita (2011) instead, do not
…nd evidence corroborating the guilt aversion hypothesis.
2
2004, Bacharach et al. 2007, Stanca et al. 2009). Thus, it seems that both motivations are present
in human interaction, but guilt aversion prevails for a large majority of the non-sel…sh B -subjects
in the Trust Game. Our study disentangles these two types of belief-dependent motivation for
each individual subject and con…rms this intuition.
A common feature of most game experiments where non-sel…sh preferences are likely to be
important is that such preferences are not controlled by the experimenter, hence they cannot be
made common knowledge among the matched subjects. This means that the matched subjects
are anonymously interacting in a game with incomplete information.5 Theoretical predictions
are harder to derive for such games, because incomplete-information models are more complex,
and their speci…cation is more arbitrary. Indeed, on top of the distribution of preferences, the
analyst also has to specify the distribution of possible hierarchical beliefs about such preferences,
that is, the beliefs about the co-player preferences, beliefs about such beliefs, and so on (see
Harsanyi 1967-68). Theoretical analysis and introspection give some guidance on the speci…cation
of preferences, but little guidance on the speci…cation of belief hierarchies.6
Our study addresses this issue. We assume that the truster, A, is self-interested and the
trustee, B, has belief-dependent preferences given by a combination of guilt aversion and intentionbased reciprocity. We elicit the trustee’s belief-dependent preferences through a structured questionnaire.7 In the main treatment, the …lled-in questionnaire is disclosed and made common
knowledge within the matched pair, whereas in the control treatment, the …lled-in questionnaire
is not disclosed to the truster. The experimental design is such that B -subjects have no incentives
to misrepresent their preferences, and indeed we …nd no signi…cant di¤erence in the pattern of
answers across treatments. This supports our main auxiliary assumption: in the treatment with
disclosure B ’s psychological type is truthfully revealed and made common knowledge; therefore,
this treatment implements a psychological game with complete information. To organize the
data, we derive precise predictions for the complete-information model, and robust qualitative
predictions for the incomplete-information model. We …nd that guilt aversion is the prevalent
psychological motivation, and that behavior and elicited beliefs move in the direction predicted
by the theory: First, the trustee’s propensity to share is indeed increasing with guilt aversion.
Second, in the treatment with disclosure there is a polarization of behavior and beliefs, with more
trust and sharing in matched pairs with an elicited high-guilt trustee. Third, high-guilt trustees
are less cooperative in the control treatment, where we …nd a higher frequency of intermediate
beliefs.
The rest of the paper is structured as follows. In section 2 we describe our experimental design.
Section 3 presents our theoretical model. In section 4 we present and discuss our experimental
results in the light of the theoretical predictions. An appendix collects some proofs and a detailed
speci…cation of the incomplete-information model.
5
In a game with complete information there is common knowledge of (i) the rules of the game, which include
how each player is paid as a function of all players’ actions, and (ii) players’ preferences. If at least one of these
conditions fails, there is incomplete information.
6
For an incomplete information analysis of the Trust MiniGame with guilt aversion see Attanasi et al. (2013).
7
Bellemare et al. (2013) and Khalmetski et al. (2013) elicit the dictator’s belief-dependent preferences in a
dictator game through a structured questionnaire comparable to ours. Regner & Harth (2014) let subjects answer to
a non-structured post-experimental questionnaire (developed by psychologists) from which measures of sensitivity
to guilt, positive reciprocity, and negative reciprocity are derived: They use these measures to analyze the trustee’s
behavior in a Trust Minigame, …nding support for guilt and negative reciprocity.
3
2
Design of the experiment
2.1
The Trust Minigame
We consider a one-shot game representing the following situation of strategic interaction (Trust
Minigame). Player A (“she”) and B (“he”) are partners on a project that has thus far yielded
a total pro…t of e2. Player A has to decide whether to Dissolve or to Continue with the
partnership. If player A decides to Dissolve the partnership, the contract states that the players
split the pro…t …fty-…fty. If player A decides to Continue with the partnership, total pro…t doubles
(e4); however, in that case, player B has the right to share equally or keep the whole surplus.
In the simultaneous-moves game summarized in Table 1 (strategic form of the Trust Minigame),
player B – before knowing player A’s choice – has to state if he would Take or (equally) Share
the higher pro…ts.
B
A
Dissolve
Continue
T ake
1,1
0,4
Share
1,1
2,2
Table 1 Payo¤ matrix for the Trust Minigame.
2.2
The experimental design
Participants were 1st and 2nd year Bocconi undergraduate students in Economics, recruited
at the Bocconi Experimental Laboratory of Social Sciences (BELSS) in Milan. The sessions
were conducted in a computerized classroom and subjects were seated at spaced intervals. The
experiment was programmed and implemented using the z-Tree software (Fischbacher, 2007).
We held 16 sessions with 20 participants per session with 16 sessions, hence 320 subjects in
total. Each person could only participate in one of these sessions. Average earnings were e8.86,
including a e5 show-up fee (minimum and maximum earnings were respectively e5 and e17);
the average duration of a session was 50 minutes, including instructions and payment.
The experimental design is made of three phases and three treatments, explained in detail in
Table 2.8 The di¤erence between treatments is in phase 2, depending on whether subjects playing
in the role of player B are asked to …ll a questionnaire and whether such answers are disclosed to
subjects playing in the role of player A. We refer to these treatments, to be explained in detail
below, as No Questionnaire (NoQ), Questionnaire no Disclosure (QnoD) and Questionnaire
Disclosure (QD). We run 4 sessions for NoQ and for QnoD (80 subjects each) and 8 sessions for
QD (160 subjects).
At the beginning of the experiment, each participant is randomly chosen to play in role
A or role B of the Trust Minigame. He/She maintains the same role until the end of the
experiment. Subjects in role A (henceforth A-subjects) are randomly matched with subjects in
role B (henceforth B -subjects), thus creating 10 matched pairs in each session. Participants are
told that the experiment is made of three phases. Instructions of the new phase are given and
read aloud only prior to that phase. Phase 1 consists of two decision tasks: the Trust Minigame
8
The English translation of the instructions is provided as an electronic supplementary material of this paper
that can be found at http://www.igier.unibocconi.it/
4
in Table 1 and a belief elicitation described in detail below. Phase 2 consists of a new random
matching and, in QnoD and QD, also of the Questionnaire explained below. Phase 3 consists
of the same decision tasks of phase 1. After phase 3, there is another questionnaire, the same
one for all treatments and similar to the one in phase 2 of QnoD and QD. Only after this Final
Questionnaire are subjects told the results of phase 1 and phase 3 and paid the sum of the
earnings in these two phases.
Treatments
NoQ
QnoD
Role Assignment and Random Matching
Phase 1
Phase 2
Phase 3
QD
Trust Minigame and Beliefs Elicitation
No Questionnaire
New Random Matching
Questionnaire with no Disclosure
Questionnaire with Disclosure
Trust Minigame and Beliefs Elicitation
Final Questionnaire with no Disclosure
Results of Phase 1 and Phase 3
Table 2 Summary of the Experimental Design.
We now describe in detail the three phases of the experimental design.
In Phase 1, which is the same for all treatments, there are two decision tasks. First, a beliefs
elicitation about strategies and beliefs in the Trust Minigame in Table 1:9 Each A-subject is
asked to guess the percentage of B -subjects in her session who will choose Share (A’s initial …rstorder belief);10 each B -subject is asked to state the guess of his co-paired A about the percentage
of B -subjects who will choose Share (B ’s unconditional second-order belief),11 and to guess the
choice that his co-paired A will make (B ’s deterministic …rst-order belief).12 Then, within each
pair, player A and player B simultaneously make their choice in the Trust Minigame in Table
1.13 Subjects do not receive any information feedback at the end of phase 1. Indeed, at the
9
See Schotter & Trevino (2014) for a survey on …rst- and second-order beliefs elicitation in two-player games
with belief-dependent motivations.
10
We do not ask A to guess the likelihood that the paired B would choose Share, since we do not observe this
likelihood: the observed binary choice would make this simply a Yes or No guess. We instead ask A to indicate a
percentage %, with = 10 n and n 2 f0; 1; 2; :::; 10g, since it is public information that there are 10 B -subjects
in each session.
11
As we did for player A, we ask player B to indicate a percentage %, with = 10 n and n 2 f0; 1; 2; :::; 10g
being the number of B -subjects in the session. Notice that we elicit B ’s unconditional second-order belief, while
Charness and Dufwenberg (2006) elicit B ’s second-order beliefs conditional on A choosing Continue: they ask B
to state A’s average guess of the percentage of B -subjects choosing Share, by considering only those A-subjects
choosing Continue. The motivations behind this feature of our design should become clear when discussing the
structure of the questionnaire in phase 2 (see footnote 17).
12
Here we elicit deterministic …rst-order beliefs for the sake of simplicity: we ask each B to guess whether his
co-paired A will choose Dissolve or Continue.
13
As Charness and Dufwenberg (2006), we also use the strategy method. Eliciting beliefs should not change
5
beginning of the phase, subjects are informed that the resulting payo¤s of the Trust Minigame
and the gains for the correctness of the beliefs will both be told at the end of the experiment.14
In Phase 2 subjects are randomly re-matched to form other 10 pairs. In NoQ they proceed
directly to phase 3. In QnoD and QD, B -subjects are asked to …ll the questionnaire in Table
3.15 In particular, each B is asked to consider the following hypothetical situation: His new
A-co-player has chosen Continue and he, B, has chosen Take, thereby earning e4 and leaving
A with e0. Given this, B has the possibility – if he wishes – to give part of the e4 back to A.
He is allowed to condition his payback on the new A-co-player guess of percentage of B -subjects
choosing Share. Since there are 10 B -subjects, when choosing Continue, A had 11 possible guesses
about how many B -subjects could choose Share (0%, 10%, ..., 100%), as shown in Table 3.16
Hence, each B -subject is asked to insert a value (between e0.00 and e4.00) in each of the 11 rows
of Table 3.17 In treatments where B -subjects are asked to …ll the questionnaire, QnoD and QD,
the following features of phase 2 are made public information among the subjects in a session:
Neither the responding subject nor anyone else will receive any payment for the answers he/she
gives in the questionnaire of Table 3; A-subjects read and listen to the instructions of phase 2.18
Furthermore, in QnoD it is public information that B ’s …lled-in questionnaire will not be disclosed
to anyone, while in QD it is public information that B ’s …lled-in questionnaire will be disclosed
to a randomly chosen player A.19 In fact, in QD player A receives the …lled-in questionnaire of
behavior in the subsequent Trust Minigame: Guerra and Zizzo (2004) …nd no di¤erence in trust and cooperation
between comparable treatments with and without belief elicitation when a Trust Minigame similar to ours is played
with the strategy method.
14
There are two reasons for this initial one-shot interaction without information on results. First, we want to
know how subjects form their beliefs and make their choices without public information within each matched pair
about B’s answers to the questionnaire in phase 2 (compared to phase 3 in QD ). Second, we want to let them
understand the Trust Minigame and the beliefs elicitation procedure before B would …ll the questionnaire in phase
2 of QnoD and QD. Indeed, in each session there are 10 B -subjects, hence 11 possible values for the frequency of
Share choices in phase 1: each of these values corresponds to one of the 11 rows of the questionnaire in phase 2
(see Table 3). For NoQ, phase 1 has been mainly introduced to maintain the same structure as in QnoD and QD,
thereby making players’behavior in phase 3 comparable between NoQ and the other treatments.
15
In both QnoD and QD, B -subjects are invited to …ll the questionnaire …rst in paper form and then copy the
answers on the questionnaire in their computer screen.
16
To check for framing e¤ects, in half of the sessions of each treatment the …rst column of Table 3 is shown in
reserve order, with 100% on the …rst row and 0% on the last row.
17
Here comes the explanation for having elicited B ’s unconditional rather than conditional second-order belief
in phase 1. First of all, eliciting B ’s conditional second-order belief would require a hypothetical question –
conditional on A’s choosing Continue – that might interfere with B ’s interpretation of the questionnaire in phase
2. Furthermore, in order to keep a reasonable number of rows in the questionnaire, we only have 10 B subjects
in each session. This is a too small number for making a reliable inference about what A-subjects think, if one
considers only A-subjects who choose Continue. As an extreme case, suppose that B thinks that there is only
one player A choosing Continue, or none. Then, he will give a conditional second-order belief that is not an
average: in the former case, it will concide with what a speci…c player A thinks. In the latter, it will be based on a
hypothetical situation and a serious problem about how to pay this belief would arise. Indeed, in our experiment
there are sessions where these two cases (0/10 As or 1/10 As choosing Continue) occur: phase 1 of sessions QnoD 2, and phase 3 of sessions NoQ -4, QnoD -2 and QnoD -3 (raw experimental data are provided in the electronic
supplementary material of this paper that can be found at http://www.igier.unibocconi.it/). An additional reason
to elicit unconditional second-order beliefs is that our theory provides testable predictions about them.
18
The latter is for A-subjects to know what B -subjects are asked to do in phase 2.
19
In QD our elicitation method is motivated by the goal of helping B to truthfully answer the questionnaire.
In particular, we wanted to avoid that B -subjects would …ll in the questionnaire with the aim of strategically
6
her new co-player B, randomly chosen at the beginning of phase 2. More precisely, at the end of
phase 2 the paired B ’s …lled-in questionnaire appears on A’s screen and the latter is invited to
copy it on a previously distributed paper.
A’s possible assessments of Share
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Your payback (in e)
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
between 0.00 and 4.00
Table 3 Questionnaire (Hypothetical Payback Scheme) in Phase 2.
In Phase 3, the two decision tasks are the same as in phase 1. However, unlike in phase 1,
in QnoD and QD, B can keep his previously …lled-in questionnaire in paper form in front of him
during the phase. Additionally, in QD, A can keep the paired B ’s …lled-in questionnaire in paper
form in front of her during the phase.20 It is public information that, in each pair, B ’s …lled-in
questionnaire disclosed at the end of phase 2 corresponds to matched B -subject of phase 3. As
in phase 1, subjects do not receive any information feedback at the end of phase 3.
After Phase 3, there is a Final Questionnaire, which is the same for all treatments: we
ask B -subjects to …ll another questionnaire – similar to the one in phase 2 of QnoD and QD –
knowing that it will not be disclosed to anyone. In NoQ this is the …rst time B -subjects …ll the
questionnaire in Table 3; in QnoD and in QD –being the second time B -subjects face the same
questionnaire –they are allowed to give answers di¤erent from those given in phase 2.
Results of both phase 1 and phase 3 are communicated after the …nal questionnaire. In
particular, each subject learns the co-player’s choice in the Trust Minigame in phase 1 and in
manipulating the matched A’s …rst-order beliefs and behavior in phase 3. This is why in phase 2 we try to tell
subjects as little as possible about phase 3: although subjects know that there is a phase 3, they do not know how
the experiment will continue in phase 3, hence they do not know if and how their answers in the questionnaire
will be used later. Thus, strategic behavior – e.g. cheap talk – should not be expected in phase 2. More details
about questionnaire …lling and disclosure can be found in the experimental instructions provided in the electronic
supplementary material of this paper.
20
The reason for asking player B (A) to …ll (copy) the questionnaire in paper form in phase 2 is twofold. First, by
asking B -subjects in QnoD and QD to …ll the questionnaire twice, we make them think more carefully about their
answers, and control for the correspondance between answers in the paper and in the computerized questionnaire.
Similar considerations apply to A-subjects in QD : asking them to copy B ’s answers on a paper questionnaire
should make them focus on these answers. Second, we do not want B -subjects (in QnoD and QD ) and A-subjects
(in QD ) to have the …lled-in questionnaire on their screen during phase 3: This would force a subject to look at it
during all phase 3, while instead having it on paper form would just give him/her the option to look at it whenever
the he/she thinks it is worth doing it; furthermore, since the …lled-in questionnaire is made of 11 combinations of
beliefs and choices, this could create possible confounds with beliefs and choices that subjects have to insert in the
screen during phase 3.
7
phase 3, and whether her …rst-order belief (player A) or his …rst- and second-order beliefs (player
B ) in phase 1 and in phase 3 were correct. Each subject is paid the sum of the resulting payo¤s
in the Trust Minigame in phase 1 and in phase 3, and is also paid for correct guesses (elicited
beliefs). Speci…cally, e5 are added to the total payo¤ of A-subjects for each correct “…rst-order
guess” (in phase 1 and 3). Similarly, e5 are added to the total payo¤ each B -subject for every
time he guessed correctly both the choice and the “…rst-order guess”of the co-player (in phase 1
and 3).21
3
Model
3.1
Belief-dependence, guilt, and reciprocity
We analyze the interaction of two players, i and j, who obtain monetary payo¤s (mi ; mj ), and
whose preferences over payo¤ distributions depend on beliefs. As in Battigalli & Dufwenberg
(2007, 2009), we allow a player’s preferences over outcomes to depend on the beliefs of the coplayer, which yields a simpler representation. Higher-order beliefs appear in the expected utility
maximization problems embedded in solution concepts. Speci…cally, we represent a player’s
preferences with a psychological utility function that depends only on (mi ; mj ) and on the coplayer’s …rst-order beliefs (which include the co-player’s plan of action, a belief about what he/she
is going to do). At this level of generality, we do not have to spell out the details about such
beliefs. Let j denote j’s …rst-order belief about the strategy pair (sj ; si ), where the marginal on
Sj represents j’s plan. We obtain a utility function of the form ui (mi ; mj ; j ) by assuming that i
dislikes disappointing j (the “guilt”component), and cares about the monetary payo¤ distribution
that j expects to achieve (the “intention-based reciprocity” component); both variables depend
on j . Assuming that j has a deterministic plan, viz. strategy sj , j is determined by the pair
(sj ; ji ), where ji is j’s belief about i’s strategy, and it makes sense to write j = (sj ; ji ).
For example, if A in the Trust Minigame plans to continue and expects B to share with 60%
probability, then A = (Continue; AB (Share) = 0:6), and her expected monetary payo¤ is
EA [m
e A ; A ] = 2 0:6 = 1:2. The psychological utility of B depends on this expectation. Of
course, since B does not know A , his valuation of (mB ; mA ) is the subjective expectation
EB [uB (mB ; mA ; e B )] according to his second-order belief. Next we provide the details of our
speci…cation of the psychological utility function uB (mB ; mA ; e B ).
The disappointment of player j is the di¤erence, if positive, between j’s expected payo¤
and his/her actual payo¤:
Dj (
j ; mj )
= max f0; Ej [m
e j;
j]
mj g .
The kindness of player j is the di¤erence between the payo¤ that j expects to accrue to i
(what j “intends”to let i have, given her belief about i’s strategy) and the “equitable”payo¤ of
i, an average mei that depends on ji :
Kj (
21
j)
= Ej [m
e i;
j]
mei (
ji ).
The payment scheme of B ’s “second-order guesses” is coherent with the theoretical de…nition of second-order
belief as a joint distribution about the …rst-order beliefs and actions of the co-player.
8
Battigalli & Dufwenberg (2007) provide a theoretical analyses of these two belief-dependent
motivations separately in Trust Minigames. We assume that i’s preferences have an additively
separable form with three terms: the utility of i’s monetary payo¤, the disutility of disappointing
j, and the (dis)utility of increasing j’s payo¤ if j is (un)kind:
ui (mi ; mj ;
j)
= vi (mi )
gi (Dj (
j ; mj ))
+ ri (Kj (
j)
mj ), vi0 > 0; vi00
0; gi0 > 0; ri0 > 0:
(1)
Term gi ( ) captures i’s guilt aversion: i is willing to sacri…ce some monetary payo¤ to decrease
j’s disappointment. Term ri ( ) captures i’s reciprocity concerns: if j is kind (unkind), i is willing
to sacri…ce some monetary payo¤ to increase (decrease) the monetary payo¤ of j.
In the “simple guilt” model of Battigalli & Dufwenberg (2007), utility depends linearly on
both monetary payo¤ and disappointment, and the reciprocity term is zero:
ui (mi ; mj ;
j)
= mi
Gi Dj (
j ; mj ),
Gi
0.
(2)
Model (2) has been mostly used to analyze binary allocation choices, such as the choice of player
B (the trustee) in the Trust Minigame. Here, instead, we rely on belief-dependent preferences also
to analyze the payback scheme in Table 3 above, where B -subjects answer hypothetical questions
by choosing distributions in a …ne grid that approximates a continuum. With this goal in mind,
in both choices that B is asked to make in our experimental setting – the Trust Minigame and
the hypothetical payback scheme –, we use a parametric speci…cation of (1) where the utility of
monetary payo¤ is concave (with constant relative risk aversion equal to 1), the guilt term gi ( )
is quadratic, and the reciprocity term ri ( ) is linear:
ui (mi ; mj ;
j)
= ln(1 + mi )
Gi
[Dj (
4
2
j ; mj )]
+ Ri Kj (
j)
mj .
(3)
This achieves a good balance between tractability and ‡exibility. (We use di¤erent labels for
the guilt parameter in the two models because similar values of G and G may yield di¤erent
behavior.)
In the context of the Trust Minigame, we assume that A (the truster) has sel…sh risk-neutral
preferences. Since B is the only player who may be a¤ected by guilt and reciprocity, from now
on we drop the player index from the guilt and reciprocity parameters. In our experiment,
the subjects actually play the normal form of the Trust Minigame, a simultaneous-move game
(see Table 1 above). But we assume that B -subjects best respond as if they had observed the
trusting action Continue, as this is the only case where their decision is relevant. This is implied
by standard expected utility maximization, except for the case where B is certain that A chooses
Dissolve. The additional assumption is therefore that B has a belief conditional on Continue
even when he is certain of Dissolve, and he acts upon such belief. Furthermore, we assume
that Continue is regarded as fully intentional, i.e. as revealing the plan of the co-player A. The
latter assumption implies that the only relevant uncertainty for B (conditional on Continue) is
the initial belief of A about the strategy of B, AB . To simplify notation, from now on we let
= PA (Share) denote this variable, and = EB ( jCont:) denote B ’s expectation of , that is,
the conditional second-order belief of B.
9
3.2
Analysis of the hypothetical payback scheme
We start with a theoretical analysis of B ’s answers to the questionnaire. Our baseline assumption
is that B …lls the payback scheme in Table 3 as if the amount x that he hypothetically gives
back to A were really given to A, thus implementing the distribution (mA ; mB ) = (x; 4 x) with
x 2 [0; 4]. The expected payo¤ for A of Continue is 2 , hence, modeling disappointment as in
Battigalli and Dufwenberg (2007),
DA ( ; x) = max f0; x
2 g.
The kindness of action Continue as a function of is modeled as in Dufwenberg & Kirchsteiger
(2004), which implies that Continue is always a kind action, but less so the more A expects B to
share (the higher ). Indeed, the higher , the lower the increase in B ’s payo¤ that A expects
to induce by choosing Continue rather than Dissolve. Speci…cally, the equitable payo¤ of B in
A’s eyes is
1
1 + (4 2 )
5
meB ( ) = [EA (m
e B ; Diss:; ) + (EA (m
e B ; Cont:; )] =
=
2
2
2
;
hence, the kindness of Continue is
KA ( ) = (4
5
2
2 )
=
3
2
:
Plugging the disappointment and kindness functions in (3), we obtain the maximization
problem
G
3
max ln(5 x)
[max(0; 2
x)]2 + R
x :
4
2
x2[0;4]
However, there is a possible confound. Since we put the B responder in a hypothetical
situation in which he has “transgressed,” we have to allow for the possibility that B chooses a
higher x than implied by the above maximization problem. This is because the transgression
puts him in an ex post negative a¤ective state that can be alleviated by giving more than he
would ex ante. Such “moral cleansing” (Sachdeva et al. 2009) is consistent with experimental
…ndings by psychologists and economists (Ketelaar & Au 2003, Brañas-Garza & Bucheli 2011).
Therefore, we introduce in the maximization problem an ex-post feeling-mitigation parameter
p 2 [0; 1] that boosts the payback x adding to in the disappointment function and subtracting
from it in the kindness function. The modi…ed maximization problem is
max
x2[0;4]
ln(5
x)
G
[max(0; 2( + p)
4
x)]2 + R
3
2
(
p)
x :
(4)
By strict concavity, (4) has a unique solution x = ( ). We call ( ) the payback function.
The appendix contains a derivation of the payback function ( ) in closed form; here we provide
intuition. The …rst-order condition for an interior solution helps us understand how the payback
changes as a function of the …rst-order belief and of parameter shifts. It is useful to think in
terms of the “marginal cost” and “marginal bene…t” of the payback x:
M C(x)
1
5
x
=
G
max (0; 2p + 2
2
10
x) + R
p+
3
2
M B(x).
(5)
Figure 1.a shows a typical solution when G > R and is high. In this case, an increase in
increases the payback (Figure 1.c). Figure 1.b shows a typical solution when R > G and is
low. In this case, an increase in decreases the payback (Figure 1.d).
Fig. 1.a: Payback if guilt prevails and
Fig. 1.c: An increase in
is high.
Fig. 1.b: Payback if reciprocity prevails and
increases the payback.
Figure 1 How
Fig. 1.d: An increase in
is low.
decreases the payback.
a¤ects the payback
Next we describe the main features of the payback function ( ) and its dependence on guilt,
reciprocity and ex-post feeling mitigation components. Figure 2 provides some examples of the
graph of ( ) for di¤erent values of the parameters G, R and p.
11
Fig. 2.a: G = 1:90; R = 0; p = 0.
Fig. 2.b: G = 0:35; R = 0:10; p = 0:80:
Fig. 2.c: G = 0:05; R = 0:25; p = 0:25:
Fig. 2.d: G = 0:70; R = 0:20; p = 0:20:
Figure 2 The payback function ( ) for di¤erent values of G, R and p.
Proposition 1 shows how the slope of the payback function ( ) depends on the comparison
between guilt and reciprocity components.
Proposition 1 Consider the range of where an interior solution obtains (i.e. G (p + )+R(p+
3=2
) > 1=5, R(p + 3=2
) < 1). The payback function ( ) is (i) increasing if G > R and
R 1=[(5 2p)(3=2+p)], (ii) constant if G = R and R 1=[(5 2p)(3=2+p)], (iii) …rst decreasing
and then increasing ( U-shaped) if G > R and 1=[(5 2p)(3=2 + p)] < R < 1=[(3 2p)(1=2 + p)],
(iv) decreasing if either G < R or R 1=[(3 2p)(1=2 + p)]. Furthermore, ( ) is increasing in
a neighborhood of only if ( ) < 2 + 2p.
12
The results in Proposition 1 can be understood by looking at Figure 1. First note that an
interior solution obtains if maxx2[0;4] (M B(x) M C(x)) > 0 and minx2[0;4] (M B(x) M C(x)) <
0. This gives the condition on the range of . The payback function is increasing if and only
if the part of the M B schedule with negative slope shifts upward with an increase in and the
‡at part of the M B schedule is always below the M C schedule (see Figures 1.a, 1.c). The …rst
condition holds if and only if G > R; the second condition holds if and only if R( 32 + p
)
M C(2p + 2 ) 5 2p1 2 for every 2 [0; 1], that is, if and only if R 1=[(5 2p)(3=2 + p)]: the
reciprocity component is low enough that A’s disappointment matters to B. This explains result
(i); the intuition for result (ii) is similar. The payback function is decreasing if either the part
of M B with negative slope shifts downward with an increase in (i.e. G < R, see Figure 1.d),
or the M C schedule intersects the M B schedule in its ‡at part (Figure 1.b) for every . The
second condition is R( 23 + p
) 5 2p1 2 for all 2 [0; 1], i.e. R 1=[(3 2p)(1=2 + p)]: the
reciprocity component is high enough that A’s disappointment does not matter to B. This explains
result (iv). For the remaining set of parameter values, the payback function is not monotone. To
understand why it has to be U-shaped –result (iii) –, suppose that is very small and reciprocity
1
)
considerations induce B to pay back x
2p + 2 (condition R( 32 + p
5 2p 2 ); since
there is no disappointment/guilt at x, only the reciprocity matters for small changes in ; a
small increase in induces a decrease in kindness and in the payback determined by reciprocity;
further increases in bring the payback below the (increasing) 2p + 2 threshold, and then an
increase in makes x increase if G > R.
The foregoing analysis also explains why ( ) is locally increasing only if ( ) < 2 +2p. Thus,
within this model, an increasing payback above A’s expected payo¤ 2 can only be explained by
ex-post feeling mitigation: p > ( )=2
.
Proposition 1 implies that ( ) is locally increasing (hence it is an interior solution) if and
only if G > R and 0 < ( ) < 2 + 2p. This also follows from the implicit function theorem: an
interior solution x = ( ) 2 (0; 4) satis…es the …rst-order condition (5); di¤erentiating it, we get
(
R(5
( ))2
if ( ) 2p + 2 ,
0
2
( )=
2(5 ( ))
(G R) if ( ) < 2p + 2 .
G(5 ( ))2 +2
Proposition 2 establishes a link between the parametric speci…cation of (1) considered in
this paper and the “simple guilt” model of Battigalli & Dufwenberg (2007): when R is low and
G ! 1, the model with reciprocity and quadratic guilt (3) yields the same payback function as
the linear model of “simple guilt” (2) with G > 1.
Proposition 2 Suppose that R
1=[(5 2p)(3=2 + p)]. Then the payback function is approximately a¢ ne and increasing in
(i.e. ( ) = 2 + k) when G is large. In particular,
( ) ! 2 + 2p as G ! 1.
We have already explained that the M C schedule intersects the M B schedule where the
slope of the latter is negative (for each ) if and only if R 1=[(5 2p)(3=2 + p)]. As G ! 1,
the negative slope becomes approximately vertical and the intersection between M C and M B
converges to the point on M C with ascissa 2p + 2 . Notice that x = 2p + 2 is the solution
to the modi…ed maximization problem (4) for the “simple guilt” model (2) with G > 1. Indeed,
plugging the disappointment function DA ( ; x) and the kindness function KA ( ) in (2), and
13
considering the ex-post feeling-mitigation parameter p 2 [0; 1] that boosts the payback x adding
to in DA ( ; x), we obtain the modi…ed maximization problem
max
x2[0;4]
(4
x)
G max(0; 2p + 2
x)
with solution22
x =
3.3
0,
2p + 2 ,
G<1
G>1
Equilibrium analysis of the Trust Minigame
Since we assume that B -subjects choose as if they had observed the trusting action Continue,
we analyze the perfect Bayesian equilibria (PBE) of the sequential Trust Minigame, a game
with perfect information.23 We consider two situations: the complete-information benchmark of
common knowledge of the psychological utility function uB , which we claim we approximate in
the lab in the main treatment, and the incomplete-information case where uB is not common
knowledge, which is the standard situation in experiments.
It is well-known that psychological games have multiple PBEs even in situations where standard games have a unique PBE; the Trust Minigame is a case in point (Geanakoplos et al. 1989,
Battigalli & Dufwenberg 2007, 2009). Although our main theoretical model of belief-dependent
preferences is non-linear and allows for both guilt aversion and reciprocity concerns, it is easier
to convey the main intuitions by focusing …rst on the simpler linear guilt model without reciprocity concerns. The main qualitative predictions for the Trust Minigame carry over to the
more complex preference model (3).
Speci…cally, consider the linear guilt model (2) with complete information. Independently
of G, there is always an inferior PBE where A chooses Dissolve with probability one as the
best response to belief = 0, and B would choose Take after Continue because he would keep
believing that = 0 even after the zero-probability choice Continue by A: In this case = 0 and
B goes after the money. Dissolve is the unique equilibrium outcome for low values of G. But, for
higher values of G, (Continue,Share) is also an equilibrium. Furthermore, there is a third, mixed
equilibrium. A similar argument applies to the incomplete information case: there is always a
“non-cooperative” PBE in which A-types choose Dissolve independently of their beliefs about
B -types.
To obtain sharp predictions, we focus on a simple re…nement: We select the equilibrium with
higher monetary payo¤ s, which is the equilibrium with trust, whenever it exists.24
22
The knife-edge case of indi¤erence, G = 1, is omitted.
At the risk of being pedantic, let us remind the reader that “perfect information” means that players move in
sequence after they observe past choices, whereas “complete information” means that the rules of the game and
players’preferences are common knowledge.
24
This is also the equilibrium with higher psychological utility. Notice that forward-induction reasoning selects
the same equilibrium when the guilt component is high enough (Dufwenberg 2002; Battigalli & Dufwenberg 2009).
However, there is a range of intermediate values for which the “trusting equilibrium” exists, although it is not
selected by forward induction.
23
14
3.3.1
Complete information
In a PBE, initial beliefs are correct, A best responds to her initial …rst-order belief, and B best
responds to his conditional second-order belief about , which coincides with the initial secondorder belief when Continue has positive probability.25 Speci…cally, (i) , the …rst-order belief
of A, coincides with the probability of Share according to the (possibly mixed) strategy of B,
(ii) the unconditional belief of B assigns probability one to (sA ; ), the equilibrium strategy and
…rst-order belief of A, (iii) if the probability of Continue is positive, the conditional second-order
belief of B assigns probability one to the equilibrium value , hence := EB [e jCont:] = .
The equilibrium analysis of the linear guilt model (2) is well known:26 B chooses Share i¤ (if
and only if)27
4 2G
2.
(6)
Hence, (Continue,Share) is an equilibrium i¤ G
rium with lower expected payo¤s.28
1. When G > 1 there is also a mixed equilib-
Proposition 3 The Pareto-superior equilibrium of the linear guilt model (2) under complete
information is
(Diss:; T ake);
= 0,
if G < 1,
(Cont:; Share);
= = 1; if G 1:
The equilibrium analysis of the guilt-reciprocity model (3) is more complex, because, if R is
su¢ ciently high, B may prefer Take when = 1 and Share for lower values of . This creates
the possibility of a partially randomized equilibrium where A chooses Continue, B is indi¤erent,
and he chooses Share with probability
1=2 (cf. Battigalli & Dufwenberg 2009, section 4.3.3).
Player B chooses Share i¤
ln 3 + 2R
3
2
ln 5
G
EB [(2e )2 jCont:],
4
that is,
G
EB [(2e )2 jCont:] + 2R
4
3
2
ln
5
3
0.
(7)
25
Let
be the equilibrium …rst-order belief of A. In equilibrium, B ’s second-order beliefs are correct; hence,
PB [e = ] = 1. Since,
PB [e = ] = PB [e = jCont:] PB [Cont:] + PB [e = jDiss:] (1
PB [Cont:]) ,
if PB [e = ] = 1, then either PB [Cont:] = 0, or PB [e = jCont:] = 1 = PB [e = ].
26
See Dufwenberg (2002) and Attanasi et al. ( 2013).
27
In the spirit of our equilibrium selection criterion, we assume that B chooses Share when indi¤erent.
28
For a complete parametric analysis, see Attanasi & Nagel (2008). The weaker re…nement of forward induction
selects the “trusting equilibrium” if G > 2. See Battigalli & Dufwenberg (2009) for a general analysis of forward
induction reasoning in dynamic psychological games.
15
As explained above, in a complete-information equilibrium where A chooses Continue, the
conditional second-order belief of B assigns probability one to the true value . Therefore,
inequality (7) becomes
W S( ; G; R) := G
2
2R + 3R
ln
5
3
0.
Our equilibrium analysis depends on the shape of the “willingness to share”function W S( ; G; R)
implied by the utility type (G; R).
In particular, (Continue; Share; = = 1) is an equilibrium – hence the Pareto-superior
equilibrium – i¤ G + R
ln(5=3)
0:52. More generally, there is an equilibrium where A
chooses Continue with positive probability i¤ (7) holds when B assigns (ex ante, and conditional
on Continue) probability one to the true value of , which must be larger than 1=2 (otherwise A
would choose Dissolve). This holds i¤
1
W S( ; G; R) 0 for
.
2
The quadratic, convex function W S( ; G; R) (with unrestricted 2 R) as has a minimum at
= R=G, and attains value
W S(1; G; R) = G + R
ln
5
3
at = 1. If (Continue,Share) is not an equilibrium, that is, if W S(1; G; R) < 0, eq. W S( ; G; R) =
0 necessarily has two real-valued solutions, the largest of which must be larger than one. Therefore, only the smaller solution,
p
R
R2 G (3R ln (5=3))
(G; R) =
,
(8)
G
matters for the analysis. At =
(G; R), the graph of W S( ; G; R) cuts the horizontal axis
from above (see Figure 3).
Figure 3 B ’s equilibrium willingness to share for G + R small and R=G large.
16
If
(G; R)
1=2, then there is an equilibrium where A chooses Continue and correctly
believes that B –who is indi¤erent –chooses Share with probability =
(G; R). This is also
the Pareto-superior equilibrium.29
If
(G; R) < 1=2, A chooses Dissolve in equilibrium. If
(G; R)
0, the equilibrium
strategy of B is Take. If 0 <
(G; R) < 1=2, there are multiple, payo¤-equivalent equilibria
where the probability of Share (which is equal to the …rst-order belief ) is lower than 1=2.30
Replacing the expression for
(G; R) of eq. (8) in the above inequalities, we obtain
(G; R)
1=2 () R
(G; R)
0 () R
1
ln
3
1
G,
8
5
3
1
ln
2
5
3
.
Therefore, the Pareto-superior equilibrium depends on G and R as follows (see Figure 4):
Proposition 4 The Pareto-superior equilibrium of the guilt-reciprocity model (3) under complete
information is
(Continue; Share)
(Continue; )
(Dissolve; )
(Dissolve; T ake)
=
=
=
=
=1
(G; R)
(G; R)
=0
if
if
if
if
G + R ln
G + R < ln
G + R < ln
G + R < ln
5
3
5
3
5
3
5
3
,
1
and R 12 ln 53
8 G,
1
5
1
, 3 ln 3 < R < 2 ln 53
and R 13 ln 53 .
1
8 G,
Figure 4 The Pareto-superior equilibrium under complete information.
29
There is also a continuum of equivalent Pareto-inferior equilibria (Dissolve,Take) where the conditional secondorder belief of B (which cannot be derived from the correct, unconditonal one) is concentrated on some value above
(G; R). Since W S( ; G; R) < 0 for >
(G; R), Take is a conditional best response.
30
In the unique sequential equilibrium (Battigalli & Dufwenberg 2009), the probability of Share is precisely
(G; R).
17
3.3.2
Incomplete information
A detailed equilibrium analysis of the incomplete-information about psychological preferences
requires the speci…cation of players’hierarchies of beliefs about preferences by means of a type
structure, which yields a Bayesian psychological game. The analysis of a fully-‡edged incompleteinformation model is rather complex and beyond the scope of this paper. Here we only provide
a qualitative analysis based on intuition. In the appendix we study a model consistent with our
qualitative analysis; a more exhaustive analysis can be found in Attanasi et al. (2013).
We start with two observations. First, when subjects are matched at random and do not
observe anything, directly or indirectly, about the other subject they are matched with, the
behavior and beliefs of A-subjects must be independent of the behavior, beliefs and psychological
utility of B -subjects. Second, in an incomplete-information setting, it is plausible to assume that
A-subjects have heterogeneous and dispersed beliefs about the psychological utility function uB .
It is even more plausible that B -subjects have heterogeneous and dispersed (second-order) beliefs
about such beliefs of the A-subjects. An A-subject is characterized by her exogenous hierarchy
of beliefs, where the …rst-order belief is her belief about uB ; such hierarchy is summarized by A’s
type. A B -subject is characterized by his exogenous hierarchy of beliefs and by his psychological
utility uB ; these features are summarized by B ’s type, which comprises B ’s utility-type. Hence,
from now on, we describe the equilibrium behavior and beliefs of A-types and B-types. By the …rst
observation, the types of A and B must be independent; therefore, the beliefs and behavior of A
and B must also be independent. We further assume that the utility-type of B and his hierarchy
of beliefs are independent.
In our analysis of the complete-information model, we focused on the Pareto-superior equilibrium, which is the one with the highest degree of trust and cooperation. Similarly, here we
describe the main features of an incomplete-information equilibrium where a positive fraction of
A-types choose Continue, which implies that Continue has positive probability, so the conditional
belief PB ( jCont:) is derived from the initial belief of B.
Since an A-type chooses Continue only if she holds a …rst-order belief
1=2, the conditional
second-order belief of a B -type who observes Continue must assign probability one to
1=2.
With this, it is appropriate to look at the following “dominance regions” in the space of utilitytypes (G; R) (see Figure 4bis):
(
)
S :=
T :=
(G; R) 2 [0; L]2 : min W S( ; G; R) > 0 ,
2[ 21 ;1]
(
)
(G; R) 2 [0; L]2 : max W S( ; G; R) < 0 .
2[ 21 ;1]
To see the relevance of these regions, recall that W S( ; G; R) is the willingness to share of B
with utility-type (G; R) when he is certain that the …rst-order belief of A is . Here we take
into account that, even if B is not certain about the value of , he is certain – conditional on
Continue –that
1=2. Therefore, the B -types with (G; R) 2 S strictly prefer Share, and the
18
B -types with (G; R) 2 T strictly prefer Take.
Figure 4bis Dominance regions for Share and Take when B is certain that
1=2.
This allows us to derive a lower bound and an upper bound on . Let tA and PtA (S)
respectively denote the probability assigned by type tA to Share (an endogenous belief determined
in equilibrium) and to region S (an exogenously given belief). Then tA
PtA (S). Similarly,
1
PtA (T). Of course, since di¤erent A-types have di¤erent beliefs about B -types, they
tA
also hold di¤erent …rst-order beliefs about the strategy of B. But the previous observation allow
us to bound …rst-order beliefs: Let
: = inf PtA (S),
tA
: =1
inf PtA (T).
tA
Then, for every A-type tA ,
tA
.
Similarly, di¤erent B -types hold di¤erent beliefs about A-types, and therefore di¤erent initial
beliefs about the choice and …rst-order beliefs of A, and di¤erent conditional beliefs about the
…rst-order beliefs of A. But we can put bounds on these beliefs too: Since type tA choose Continue
(respectively, Dissolve) if tA > 1=2 (respectively, tA < 1=2), and tA
PtA (S) (respectively,
1
PtA (T)), we have
tA
PtB
tA : PtA (S) >
1
2
PtB (Cont:)
1
PtB
where PtB denotes the belief of a B -type tB . Furthermore,
EtB (e )
19
.
tA : PtA (T) >
tA
imply
1
2
,
The behavior of B -types tB with utility-type (G; R) out of the dominance regions depends
on their equilibrium conditional belief PtB ( jCont:).31 Since function W S( ; G; R) is increasing
on [1=2; 1] i¤ R=G
1=2, for every utility-type (G; R) with G
2R, a “higher” conditional
second-order belief (in the sense of stochastic dominance) implies a higher willingness to share
EtB [W S(e ; G; R)jCont:]:
Given our assumptions about the statistical distribution of A-types and B -types, our analysis
yields the following qualitative results:32
Proposition 5 Every equilibrium of the Trust Minigame with incomplete information where a
positive fraction of A-types choose Continue has the following features:
(1) A-types have heterogeneous, dispersed beliefs about B’s strategy, hence, a substantial fraction of types have well above 0 and well below 1. Furthermore, for every tA ,
tA
.
(2) B-types have heterogeneous, dispersed initial beliefs about A’s strategy and
second-order beliefs about are bounded by and : for every tB ,
EtB (e )
; the initial
:
Conditional second-order beliefs are also heterogeneous, but have support in [1=2; 1].
(3) The strategy and beliefs of A are independent of the strategy, utility-type, and beliefs of B;
(4) B-types with high values of G or R (i.e., with (G; R) 2 S) choose Share, B-types with low
values G and R (i.e., with (G; R) 2 T) choose Take, the choice of intermediate types tB depends
on the equilibrium conditional belief PtB ( jCont:). The proportion of B-types with G > 2R who
choose Share is positively correlated with the conditional second-order belief.
3.3.3
Link between theoretical predictions and experimental design
We can summarize the equilibrium predictions as follows. With complete information, there is a
strong polarization of behavior and beliefs because common knowledge of B ’s utility-type works
as a coordination device. In fact, if B is su¢ ciently sel…sh (low guilt and reciprocity parameters),
Dissolve is the unique equilibrium outcome and the probability of Share (which is equal to the
…rst-order belief ) is lower than 1=2. In the complementary region of the parameter space,
Continue is the Pareto-superior equilibrium strategy of A, and the probability of Share (which
is equal to the …rst-order belief ) is higher than 1=2 (see Proposition 4 and Figure 4).33
31
If the fraction of types tA such that
tA
= 1=2 has zero measure, then
PtB ( jCont:) = PtB ( j ftA :
32
tA
1=2g) .
Compare with Propositions 6 and 7 in the appendix, and the qualitative results derived from those propositions
given the assumptions about the distribution of types.
33
If we only relied on forward-induction reasoning to select among multiple equilibria, we would obtain an
intermediate parameter region where we may expect lack of coordination and intermediate beliefs. In this region
the guilt parameter is not su¢ ciently high for the weaker re…nement of forward induction to select the “trusting
equilibrium”.
20
With incomplete information, there is more heterogeneity of behavior and more dispersed
beliefs, including “intermediate” beliefs. This is quite obvious for the A-subjects (assuming
heterogeneous, dispersed beliefs about uB ). More interestingly, there is a parameter region with
intermediate values of G (and low values of R) where B -subjects would cooperate and hold high
second-order beliefs under complete information, while they exhibit less cooperative behavior
and intermediate second-order beliefs under incomplete information (see Proposition 5, compare
Figure 4bis with Figure 4).
Given these theoretical predictions, the main goal of the paper is to obtain a qualitative
comparison of the statistics of trust (A choosing Continue) and cooperation (B choosing Share)
across the two types of information in phase 3 of the experiment:
Treatment QD: In phase 3, the questionnaire …lled in by B is disclosed to the matched Asubject and made common knowledge within the matched pair. Assuming that the …lled-in
questionnaire identi…es B ’s utility-type, the matched subjects play a psychological game
with complete information.
Treatments NoQ, QnoD: In phase 3, A obtains no information about B. Therefore the
matched subjects play a psychological game with incomplete information.
We also compare behavior in phases 1 and 3 within the same treatment. For the “completeinformation treatment” QD, we should …nd di¤erences in the direction of a strong polarization
of behavior and beliefs in phase 3. On the other hand, we expect no di¤erences between phases
1 and 3 in the “incomplete-information treatments” NoQ and QnoD.
Furthermore, among the B -subjects for whom guilt aversion prevails, according to elicited
preferences, we should …nd a positive correlation between the elicited conditional second-order
belief and the propensity to share (cf. Charness and Dufwenberg 2006).
As speci…ed above, due to the multiplicity of equilibria, the comparative statics of the two
situations in part rely on our selection criterion. In the next section we discuss the data guided
by the theoretical predictions for the two di¤erent information settings.
4
Data analysis
Here we present and discuss our experimental data in the light of the theoretical model.34 Relying on the hypothetical payback function introduced in Section 3.2, in Section 4.1 we present the
classi…cation of B ’s belief-dependent preferences given by the questionnaire in Table 3. With this
classi…cation in mind, we analyze A’s and B ’s behavior in the Trust Minigame using the equilibrium predictions of Section 3.3. In particular, in Section 4.2 we use the complete-information
equilibrium to analyze subjects’behavior in phase 3 of the treatment with questionnaire disclosure (QD). In Section 4.3 we use our qualitative, incomplete-information predictions to analyze
behavior in phase 1 of QD and in the treatments without questionnaire disclosure (NoQ and
QnoD). There we also compare behavior in all these phase-treatment combinations with behavior in phase 3 of QD.
34
Raw experimental data are provided in an electronic supplementary material of this paper that can be found
at http://www.igier.unibocconi.it/
21
4.1
Experimental Elicitation of Belief-Dependent Preferences
As discussed in Section 3.1, the experimental elicitation of B ’s belief-dependent preferences in the
Trust Minigame relies on the questionnaire in Table 3. Here we analyze the answers of each B subject to the questionnaire. We call “payback pattern”the actual set of answers of a B -subject,
one payback value for each hypothesized (A’s belief about his strategy). Our aim is to estimate,
for each B -subject, the triple (G; R; p) that identi…es B ’s best response to the hypothesized ,
^ R
^ and p^ the estimated values of G,
i.e. his theoretical payback function ( ; G; R; p). We call G,
R and p respectively.
Recall that the payback pattern gives only 11 observations for B ’s payback function, i.e. one
^ R;
^ p^) of a given
for each 2 f0; 10%; :::; 100%g. The best-…t response function ^( ) := ( ; G;
B -subject minimizes the sum of the squared deviations (least square error) of the theoretical
payback function from the actual payback for the 11 rows of the …lled-in questionnaire. Given
that the maximization problem (4) is non-linear in one of the unknown parameters, G, we use
non-linear least square estimation, with bounds G; R 2 [0; 1000] and p 2 [0; 1]. In the electronic
^ R
^ and
supplementary material of the paper, we provide the non-linear least square estimates G,
p^ for each of the 160 B -subjects in our experiment and the standard deviation associated with
each estimated parameter. To account for the small size of the sample, standard deviations have
been estimated by a (non parametric) bootstrap approach of size 10,000.35
In Table 4, we report the distribution of B -subjects’estimated utility-types across the possible
shapes of the corresponding payback function ( ).
Categories of
elicited utility-types
^ > R)
^
Guilt prevails (G
^ = R)
^
Balanced (G
Guilt prevails for high
^ < R)
^
Reciprocity prevails (G
^=R
^ = 0)
Sel…sh (G
TOTAL
Estimated
Function
^0 ( ) > 0
NoQ
Treatment ( )
QnoD NoQ-QnoD
QD
23
20
43
45
^( )=0
3
2
5
9
^( ) U-shaped
^0 ( ) < 0
3
2
5
4
7
7
14
11
4
40
9
40
13
80
11
80
0
^( ) = 0
Table 4 Classi…cation of B -subjects according to the payback pattern.
( )
Column NoQ-QnoD pools the observations of NoQ and QnoD.
Recall that, according to Proposition 1 (see also Figure 2), the theoretical payback function
( ) can be (i) increasing (G > R and R low), (ii) constant (G = R and R low), (iii) …rst
decreasing and then increasing (G > R and R low: U-shaped), and (iv) decreasing (R > G, or R
high). We consider separately the case G = R = 0 of sel…sh preferences. Notice that p does not
play any role in the classi…cation of Table 4. The reason is that p was introduced to take into
account the possible confound of ex-post-feeling mitigation, which matters in the analysis of the
35
We also analyzed the actual payback function of each B-subject through a grid estimation algorithm (see Nagel
& Tang 1998) with step 0.05 for G; R 2 [0; 1000] and p 2 [0; 1], …nding similar estimates.
22
hypothetical payback and in the estimation of G and R for each subject, but does not matter in
the analysis of the Trust Minigame.
Since we …nd no signi…cant within-treatment di¤erence in the distribution of estimated utilitytypes,36 we pool the data within each treatment. (Note that for treatment NoQ we rely on the
…nal questionnaire –the only one …lled-in by B -subjects in this treatment –, while for treatments
QnoD and QD we refer to the questionnaire in phase 2 –the …rst one …lled-in in these treatments.)
Furthermore, we …nd no signi…cant di¤erence between NoQ and QnoD, which allows us to pool
the data of these two treatments (column NoQ-QNoD in Table 4), so as to have the same number
of observations with disclosure (QD) and without disclosure (NoQ-QnoD).37 We also …nd the
distribution of types in QD being not signi…cantly di¤erent from the one in NoQ-QNoD (last
two columns of Table 4: 2 test, P-value = 0:768). We claim that this last test is indirect
evidence of the fact that in the treatment with disclosure (QD) our elicitation method has not
been interpreted by B -subjects as an opportunity for manipulative cheap talk (see also footnote
17).
Table 4 shows that, independently of the treatment, the guilt component of the psychological
utility function uB is prevalent for more than half of B -subjects, while reciprocity prevails for
only 16% of them. There is, however, a non-negligible number of B-subjects (6%) for whom guilt
prevails when A’s …rst-order belief is high and reciprocity prevails otherwise (U-shaped payback
function). The remaining subjects have a ‡at estimated payback function (i.e., the payback is
independent of A’s …rst-order belief). The majority of them are sel…sh (0 payback whatever ,
15% of the sample). The estimated payback function of the others (only 9% of the sample) is
consistent with inequity aversion: these subjects aim at an interior distribution independent of
.
Considering the actual answers to the questionnaire (the payback pattern), rather than the
estimated payback function, we note that there are 138/160 (86%) B -subjects whose payback
pattern exactly mimics one of the theoretical shapes of ( ) implied by our model. The remaining
22/160 (14%) B -subjects provide a pattern of answers that does not …t any of these shapes. More
precisely, 16 of them (8 in NoQ-QnoD and 8 in QD) show a payback pattern with an inverted-U
shape and the other 6 payback patterns (4 in NoQ-QnoD and 2 in QD) have di¤erent shapes.38
However, our non-linear least square estimation procedure allows us to classify them within one
^ and R
^ (for a similar method, see Costa-Gomes et al.
of the …ve categories of Table 4, using G
2001). These 22 subjects are spread over the di¤erent categories of Table 4.39
The following statement summarizes the main experimental …ndings about the distribution
of B -subjects’payback patterns.
36
Recall that, to check for framing e¤ects, in half of the experimental sessions of each treatment the 11 lines
of the questionnaire in Table 3 have been shown in reverse order, starting with 100% instead of 0%: We do not
…nd any signi…cant di¤erence in the categorization of types in Table 4 caused by the order. Further, questionnaire
…lling by B -subjects shows no timing e¤ect, phase 2 vs …nal questionnaire (see the experimental design in Table
2). Only 4/40 (9/80) B -subjects provide a di¤erent payback pattern in the …nal questionnaire in QnoD (QD ), and
for only 1/4 (3/9) the classi…cation would change, moving from one category to another in Table 4.
37
No signi…cant di¤erence between the distribution of types in NoQ and the one in QnoD in Table 4 is con…rmed
by using a 2 test (P-value = 0:470).
38
Technically, they are neither quasi-convex, as our model would imply, nor quasi-concave (single-picked).
39
Of these subjects, 7 are classi…ed as “guilt prevails”, 6 as “balanced”, 8 as “reciprocity prevails”, and 1 as
sel…sh.
23
Result 1 The great majority (86%) of B -subjects’payback patterns are coherent with the theoretical shapes implied by our model. Across all B -subjects, the estimated payback functions
^( ) are mostly belief-dependent (76%); of these, the guilt component is prevalent for 72%,
while reciprocity prevails for only 20%. Similar results hold for the sub-populations of
subjects within the di¤erent treatments.
4.2
Choices and beliefs under disclosure of the …lled-in questionnaire
This section is split into two parts. First, we organize B -subjects and matched A-subjects
^ and R
^ obtained from the payback
according to the complete-information predictions using G
pattern (predicted behavior). Secondly, we compare actual behavior with predicted behavior.
Recall that the only phase in our experimental design that supposedly approximates a Trust
Minigame with complete information is phase 3 of QD. However, we consider subjects’behavior
under the complete-information predictions also for phase 1 of QD and for phases 1 and 3 of
NoQ-QnoD. The reason for this exercise is twofold.
First, we want to check whether the parameter constellations estimated in QD and in NoQQnoD lead to similar complete-information predicted behavior under the counterfactual assumption that subjects have complete information also in phase 3 of NoQ-QnoD. In the previous section
we showed that the distribution of B -subjects over the …ve categories of elicited utility-types is
similar across all treatments. However, it could be that given the speci…c category of utility-types
the estimated parameter constellations are di¤erent between two treatments, thereby leading to
di¤erent complete-information predictions across treatments. This virtual observability method
(see Camerer & Ho 1999) allows us to check for the presence of treatment bias. Indeed, the
portability of our methodology would be enhanced if: (i ) the number of “predictable”B -subjects
(Share vs Take), (ii ) the distribution of predictable B -subjects across the four di¤erent shaded
regions of Figure 4, and (iii ) same as (ii ) within each category of elicited utility-types in Table
4 (e.g. given that guilt prevails), were not signi…cantly di¤erent between QD and NoQ-QnoD.
Secondly, this would help us provide a clean comparison between the “complete-information
treatment”QD and the “incomplete-information treatment”NoQ-QnoD through within-treatment
and between-treatment di¤erences. In the next section, the former are analyzed by comparing
phase 3 with phase 1 of QD; the latter, by comparing phase 3 of QD with phase 3 of NoQ-QnoD.
Figure 5 reports the complete-information predictions for matched pairs in QD according to
Proposition 4. It refers to the four regions of predicted strategy pairs in the parameter space
(G; R) in Figure 4. In particular, it reports the number of B -subjects in QD for whom we do not
^ R
^
obtain a clear complete-information prediction (“unpredictable”B -subjects), i.e., disclosed G+
is not signi…cantly di¤erent than ln(5=3) at 10% level (P-values being estimated by bootstrap).
It also indicates the number of remaining subjects (“predictable” B -subjects) in each region of
prediction (in brackets, bold character) and the category of utility-types to which they belong,
taken from Table 4.
24
Figure 5 Complete-information predicted behavior of classi…ed types.
Results of virtual observability are encouraging for the portability of our method:40
(i ) Considering the whole subject pool (80 B -subjects in QD and 80 in NoQ-QnoD), we …nd
that we can make a prediction (Share vs Take) for about 84% (134/160) of B -subjects given
^ R).
^ This percentage is slightly higher in QD (90%, 72/80) than in NoQ-QnoD (78%, 62/80),
(G;
with this di¤erence being signi…cant at 5% level. However, we …nd that the distribution of
predictable/unpredictable B -subjects across the …ve categories of elicited utility-types in Table
4 is not signi…cantly di¤erent between QD and NoQ-QnoD ( 2 test, P-value = 0:494). In
particular, in both treatments all the B -subjects for whom guilt does not prevail (“balanced,”
“reciprocity prevails,” and sel…sh) are predictable, while a substantial proportion (29%, 26/72)
of B -subjects for whom guilt prevails are unpredictable. Hence, for all unpredictable B -subjects
guilt prevails. This is why in Figure 5 they are indicated as lying on the segment of line R =
ln(5=3) G where G > R.
(ii ) The frequency of predictable strategy pairs is not signi…cantly di¤erent between QD
and NoQ-QnoD ( 2 test, P-value = 0:666), with – independently of the treatment – almost all
predictable types (127/134) belonging to either the (Dissolve,Take) region (lightest color) or the
(Continue,Share) region (darkest color).
(iii ) The complete-information prediction for a pair with a sel…sh B -subject is straightforward: (Dissolve,Take) is the only equilibrium. Independently of the treatment, we …nd that:
(Dissolve,Take) is the predicted complete-information equilibrium behavior also for all pairs with
^ = R)
^ B -subject and for the great majority of pairs with a B -subject for whom
a “balanced” (G
40
Figure 5 for treatment NoQ-QnoD can be found in the electronic supplementary material of this paper available
at http://www.igier.unibocconi.it/
25
reciprocity prevails (lightest region); Dissolve (second-lightest region) is the predicted completeinformation equilibrium outcome (with the probability of Share being lower than 1=2) also for
^ > G);
^ conversely, for the great majority
all but one the remaining pairs of the latter group (R
of pairs with a B -subject for whom guilt prevails, (Continue,Share) is the complete-information
prediction (darkest region).
Predictable B -subjects for whom guilt prevails require a more thorough discussion. First,
we …nd that their distribution across regions (Continue,Share) and (Dissolve,Take) is not signi…cantly di¤erent between QD and NoQ-QnoD ( 2 test, P-value = 0:551). Further, and more
importantly, for all B -subjects being predicted to choose Share with probability 1 in the completeinformation equilibrium, the estimated guilt component is positive and prevails over reciprocity
^ > R).
^ In particular, for all but one41 of these subjects it is G
^ > ln(5=3), hence guilt aversion
(G
is, by itself, high enough to yield the cooperative equilibrium under complete information. For
these reasons, from now on we refer to the B -subjects in the (Continue,Share) region of Figure
5 as “high-guilt” types and to the B -subjects in the (Dissolve,Take) region of Figure 5 as
“low-guilt” types. Recall that in the latter group we have all sel…sh B -subjects, all those con^ = R),
^ the great majority of those for which reciprocity prevails
sistent with inequity aversion (G
^
^
^ > R).
^
(R > G), and a few of those for which guilt prevails (G
The following result summarizes the main experimental …ndings about B -subjects’predicted
behavior under complete information.
Result 2 Given the estimated guilt and reciprocity components, all B -subjects predicted to
chose Share with probability 1 under complete information are “high-guilt” types. Almost
all B -subjects for whom reciprocity prevails are predicted to choose Take with probability
1 under complete information. All this holds independently of the treatment.
Now we compare actual behavior with predicted behavior in phase 3 of QD. We use the
three phase-treatment combinations where the …lled-in questionnaire is not disclosed (phase 1 of
QD, phase 1 and phase 3 of NoQ-QnoD; from now on, “incomplete-information phases”)
as controls.42 In Figure 6, for each category of B ’s utility-types in a speci…c region of predicted
behavior, before the number of predictable pairs in QD (bold character, taken from Figure 5),
we report the number of predictable pairs that behave as predicted in phase 3 of QD (regular
character).43
^ ' 0:42, hence very close to the threshold value ln(5=3) ' 0:52.
For this B -subject it is G
Recall that we implemented a stranger-matching design: A-subjects and B -subjects are randomly re-matched
so as to have di¤erent pairs in phase 1 and in phase 3 and avoid repeated-game e¤ects. However, with the
goal of providing a clean check of within-treatment di¤erences, we analyze pairs’ behavior in phase 1 of each
treatment according to the random matching of phase 3 (where, in the main treatment, a feature of B – the
…lled-in questionnaire – is made public information within the pair). This can be done at no cost, since A’s (B ’s)
choice in phase 1 is told to the B (A) matched with her (him) only at the end of the experiment.
43
The corresponding …gures for the other three phase-treatment combinations can be found in the electronic
supplementary material of this paper available at http://www.igier.unibocconi.it/
41
42
26
Figure 6 Actual behavior vs complete-information prediction.
Figure 7 As and B s’choices and beliefs in phase 3 of QD, disentangled by B ’s type.
The controls work as they should. We …nd a similar, low rate of success of the completeinformation predictions in all incomplete-information phases (77/196, 39% on average), with no
signi…cant di¤erences among these three phases ( 2 test, P-value = 0:568). Conversely, we …nd
a signi…cantly (1% level) higher rate of success of complete-information predictions for phase
3 of QD (57%, 41/72) than for the incomplete-information phases considered together. Our
complete-information predictions are particularly successful for pairs being predicted to choose
(Dissolve,Take): 65% (24/37) in phase 3 of QD. However, the greatest di¤erence with respect to
27
the controls is found for pairs being predicted to choose (Continue,Share): 49% (17/35) in phase
3 of QD vs 14% (11/79) in the incomplete-information phases, again signi…cant at 1% level.
The following result summarizes the main experimental …ndings about matched pairs’actual
vs predicted behavior under complete information.
Result 3 Complete-information predictions are able to explain 57% of actual strategy pairs
after questionnaire disclosure (phase 3 of treatment QD). In particular, 65% of pairs in the
(Dissolve,Take) region and 49% of pairs in the (Continue,Share) region behave as predicted.
In Figure 7 we deepen the analysis presented in Figure 6: We present actual choices and
beliefs of matched pairs in phase 3 of QD, disentangled by role and by B ’s utility-type. Given
the few predictable utility-types in regions (Dissolve;
< 12 ) and (Continue;
> 21 ) of Figure
6 (4/72), here we focus only on the two regions of polarized predictions (Continue,Share) and
(Dissolve,Take). Recall the we de…ned B -subjects belonging to the (Continue,Share) region as
“high-guilt” types and those belonging to the (Dissolve,Take) region as “low-guilt” types:
For each utility-type of B -subjects – and respectively for the matched A-subjects – we use the
same color as the region they belong to in Figure 4. On the left side of the …gure, we report
the frequency of A-subjects’Continue choices and of matched B -subjects’Share choices. On the
right side of the …gure, we report the box plot (minimum, …rst quartile, median, third quartile
and maximum) of As’…rst-order belief ( ) and of B s’unconditional second-order belief (EtB (e ));
the average belief is indicated near to each box plot.
A clari…cation is needed about beliefs in Figure 7. Recall (see section 2.2) that A’s elicited
…rst-order belief is not only about the matched B,44 hence according to our complete-information
predictions we should get an elicited that is less polarized than the true one: e.g., an A who
faces a high-guilt B in phase 3 of QD is asked how many of the 10 B -subjects in the session
(the matched B and the other nine) will Share, and she can rationally presume – despite the
disclosed …lled-in questionnaire of the matched high-guilt B – that there are some low-guilt B subjects in the session. Furthermore, B knows that A did not state a belief about the choice
of the matched B only; hence, according to our complete-information predictions we should get
an elicited unconditional second-order belief, EtB (e ), that is less polarized than the true one.
Finally, recall that we elicit the unconditional, not the conditional second-order belief. The latter
is the relevant correlate of B ’s propensity to share, but the former shows how B thinks about
the game. For example, when B is a low-guilt type, knowing that A observes this, he should
expect that is very low. But B ’s conditional expectation of given Continue can reasonably
be larger than 1=2. With the same caveat explained above (B knows that A’s stated belief is not
just about him), the elicited unconditional belief EtB (e ) is a rough estimate of the conditional
belief for those B -subjects indicating Continue as the most likely choice of A.
First, we discuss experimental results about choices and …rst-order beliefs of A-subjects in
phase 3 of QD.
Coherently with the complete-information predictions, in phase 3 of QD, A-subjects matched
with a high-guilt B -subject show a signi…cantly greater frequency of Continue compared to Asubjects matched with a low-guilt B -subject (+47:19%; 2 test: P-value = 0:000). A similar
conclusion can be stated about the …rst-order beliefs (+26:65% on average, Mann-Whitney test
44
Charness & Dufwenberg (2006) have a similar issue.
28
rejecting the null hypothesis of equal population medians with P-value = 0:000). Furthermore,
in phase 3 of QD a strongly-signi…cant positive correlation is found between the Continue choice
and the …rst-order belief (Spearman’s = 0:55, P-value = 0:000): This also holds if we consider
A-subjects matched with high-guilt and low-guilt types separately.45
A further result coherent with the complete-information predictions is the induced correlation
between B ’s utility-type and A’s behavior when the former is disclosed: In phase 3 of QD we
^ R)
^ and both A’s choice of Continue and her …rst-order
…nd a positive correlation between (G;
^ and R
^ are statistically
belief. This point requires a thorough discussion. First, we checked that G
46
^
^ separately.
independent, so as to run the correlation analysis with As’ behavior for G and R
^
For G the prediction in phase 3 of QD is veri…ed: we …nd = 0:44 (P-value = 0:000) for the
correlation with the Continue choice and = 0:43 (P-value = 0:000) for the correlation with the
^ we …nd a non-signi…cant negative correlation, which is not consistent
…rst-order belief. For R
with our predictions: = 0:10 (P-value = 0:441) for the correlation with the Continue choice,
^ + R,
^ which
and = 0:18 (P-value = 0:147) for the …rst-order belief. However, if we consider G
is relevant for the complete-information equilibrium threshold (see Proposition 4), we …nd again
a signi…cant positive correlation with both the Continue choice ( = 0:45, P-value = 0:000) and
with the …rst-order belief ( = 0:42, P-value = 0:000).
Finally, if we disentangle the A-subjects in phase 3 of QD according to the matched utilitytype –high-guilt vs low-guilt –and we focus on each subgroup separately, we …nd no signi…cant
^ or R,
^ or G
^ + R)
^ and both
correlation between B ’s disclosed utility-type (relying on either G,
A-subjects’ choice of Continue and the …rst-order belief. Indeed, in line with the completeinformation predictions, all A-subjects in the former (latter) subgroup should choose Continue
^ R).
^
(Dissolve) independently of (G;
The following result summarizes the more salient experimental …ndings about A-subjects’
actual vs predicted behavior under complete information of B ’s utility-type.
Result 4 In line with the complete-information predictions, after questionnaire disclosure, both
the frequency of Continue choices and the …rst-order beliefs are signi…cantly greater for Asubjects matched with high-guilt B -subjects. More generally, both trust and A’s …rst-order
beliefs are positively correlated with the disclosed guilt-type of B.
Now we discuss experimental results about choices, …rst- and second-order beliefs of B subjects in phase 3 of QD.
Coherently with the complete-information predictions, compared to low guilt-types, disclosed
high-guilt B -subjects show a signi…cantly greater frequency of Share (+44:85%; 2 test: P-value
= 0:000) and signi…cantly greater (unconditional) second-order beliefs (+33:84% on average,
Mann-Whitney test: P-value = 0:000). Furthermore, in phase 3 of QD a strongly signi…cant
positive correlation is found between Share and the deterministic …rst-order belief ( = 0:44,
P-value = 0:000): This also holds if we only consider high-guilt types, but not for low-guilt
45
It is = 0:36 (P-value = 0:034) in the subgroup of A-subjects matched with a high-guilt type and = 0:40
(P-value = 0:020) in the other subgroup.
46
This holds both if we consider the whole subjects’ sample ( = 0:10, P-value = 0:191) and if we check for
correlation in the two subjects’subsamples separately ( = 0:15, P-value = 0:180 in treatment QD ; = 0:06,
P-value = 0:586 in treatment NoQ-QnoD ).
29
ones.47
Results about the correlation between B ’s disclosed utility-type and B ’s behavior con…rm the
^ and if we consider G
^ + R:
^ We …nd a signi…cant postheoretical predictions, both if we consider G
itive correlation of the utility-type with the Share choice, the deterministic …rst-order belief, and
the (unconditional) second-order belief.48 As it happens for A-subjects, also for B -subjects we
^ and both choice and beliefs. Disentangling
…nd a non-signi…cant negative correlation between R
by utility-type –high-guilt vs low-guilt –, we …nd no signi…cant correlation between B ’s disclosed
^ or G+
^ R)
^ and B -subjects’choices, …rst- and second-order beliefs,
utility-type (relying on either G,
in any of the two sub-groups considered separately: In line with the complete-information predictions, all B -subjects in the former (latter) subgroup should choose Share (Take) independently
^ R).
^
of (G;
The following result summarizes the more salient experimental …ndings about B -subjects’
actual vs predicted behavior under complete information of B ’s utility-type.
Result 5 In line with the complete-information predictions, after questionnaire disclosure, the
frequency of Share choices, the …rst- and the second-order beliefs are signi…cantly greater
for high-guilt than for low-guilt B -subjects. More generally, cooperation, B ’s …rst- and
second-order beliefs are positively correlated with the disclosed guilt-type of B.
4.3
Choices and beliefs without disclosure of the …lled-in questionnaire
In this section we focus on the “incomplete-information phases”, i.e. on the three phasetreatment combinations where the …lled-in questionnaire is not disclosed (phase 1 of QD, phases
1 and 3 in NoQ-QnoD): In these phases, subjects play a Trust Minigame with incomplete information of B ’s utility-type. First, we check the theoretical predictions about subjects’beliefs and
choices in the incomplete-information phases (see Proposition 5). Then, we provide a qualitative
comparison of behavior between the incomplete-information phases and phase 3 of QD.
We …nd that A-subjects’…rst-order beliefs are heterogeneous and dispersed in the incompleteinformation phases: Only 22% (1%) of A-subjects have
= 0 ( = 1), with no signi…cant
di¤erences across the three phases;49 the standard deviation is 83%, 90% and 95% of the average
belief, respectively in phase 1 of QD, in phase 1 and in phase 3 of NoQ-QnoD.50 Of course,
^ R),
^ and (…rst- and
A’s …rst-order belief is independent of the strategy, estimated utility-type (G;
second-order) beliefs of the matched B -subject.51 A similar result holds for A’s strategy. Finally,
47
It is = 0:47 (P-value = 0:005) in the subgroup of high-guilt types and = 0:04 (P-value = 0:817) in the
other subgroup.
48
^ it is = 0:43 (P-value = 0:000) with Share, = 0:53 (P-value = 0:000) with , and = 0:50 (P-value
For G,
^ + R.
^
= 0:000) with EtB (e ). We …nd similar results for G
49
Kruskal-Wallis equality-of-populations rank test over the three incomplete-information phases: P-value =
0:659; Mann-Whitney test (phase 1 of NoQ-QnoD vs phase 1 of QD ): P-value = 0:564.
50
The dispersion is slightly lower (Levene’s test of equality of variances: P-value = 0:082) in phase 1 of QD than
in phase 1 of NoQ-QnoD .
51
We tested both aggregate data and data disentangled by incomplete-information phase. We only …nd a spurious
negative correlation ( = 0:21, P-value = 0:061) of A’s …rst-order belief with the Share choice of matched B subjects in phase 3 of NoQ-QnoD. This is possibly due to the small number of B -subjects choosing Share (14/80)
in this incomplete-information phase, and to the fact that the average …rst-order belief of A-subjects matched with
them is, by chance, signi…cantly smaller than the average belief of the remaining A-subjects.
30
we …nd a signi…cant di¤erence in the frequency of Continue choices (85% vs 12%, signi…cant at
1% level) between A-subjects with
1=2 and A-subjects with < 1=2, with this di¤erence
being much higher in phase 1 (91% vs 7% in NoQ-QnoD, 100% vs 21% in QD), possibly due
to an end-game e¤ect in phase 3 of NoQ-QnoD (65% vs 6%, still signi…cant at 1% level). This
result corroborates the assumption that A has sel…sh risk-neutral preferences (hence she should
choose Continue if and only if
1=2).
As for B -subjects’…rst-order beliefs, recall that we only ask them whether they expect Continue or Dissolve: We …nd that these beliefs are not concentrated on one of the two A’s strategies,
with 40% of B -subjects indicating Continue with probability 1 and 60% of them indicating Dissolve with probability 1, and no signi…cant di¤erences across the three incomplete-information
^ R),
^
phases. B ’s …rst- and second-order beliefs are independent of the estimated utility-type (G;
corroborating our assumption that the epistemic component of B ’s type is independent of the
utility component (see the appendix). The unconditional second-order beliefs are heterogeneous
and dispersed: Only 26% (3%) of B -subjects have EtB (e ) = 0 (EtB (e ) = 1), with no signi…cant
di¤erences across the incomplete-information phases;52 the standard deviation is 86%, 89% and
93% of the average belief, respectively in phase 1 of QD, in phase 1 and in phase 3 of NoQ-QnoD,
with no signi…cant di¤erences. Recall that we only elicit the unconditional second-order belief,
but we obtain a rough estimate of the second-order belief conditional on Continue, , by focusing only on those B -subjects with Continue as …rst-order belief; we …nd that all of them have a
positive estimated conditional second-order belief and 46% of them have
1=2 (the standard
deviation being 55% of the average belief 0.68).
As for B -subjects’choices in the Trust Minigame with incomplete information, we organize
^ and R
^ obtained from the
them according to the incomplete-information predictions using G
payback pattern (predicted behavior). Figure 8 reports the incomplete-information predictions
for matched pairs in NoQ-QnoD according to Proposition 5. It refers to the three dominance
regions in the parameter space (G; R) in Figure 4bis. In particular, it reports the number of
B -subjects in NoQ-QnoD for whom we do not obtain a clear incomplete-information prediction
(“unpredictable” B -subjects) –similar procedure as for the complete-information predictions in
Figure 5. It also indicates the number of remaining subjects (“predictable” B -subjects) in each
region of prediction of Figure 4bis (in brackets, bold character) and the category of utility-types
to which they belong, taken from Table 4. This applies to phases 1 and 3 of NoQ-QnoD. Similar
results hold for phase 1 of QD, where predictions of behavior for B -subjects under incomplete
information can be easily inferred from Figure 5.53 Considering together the three incomplete
^ R)
^ 2 S, while it is
information phases, we …nd that Share is chosen by 44% of B -subjects with (G;
^
^
chosen by only 15% of B -subjects with (G; R) 2 T, the di¤erence in the frequency of Share being
^ R)
^ 2 (S [ T)c –
signi…cant at 1% level. The proportion of intermediate types – those with (G;
choosing Share is between the previous two (27%). Focusing only on those intermediate types
52
Kruskal-Wallis equality-of-populations rank test over the three incomplete-information phases: P-value =
0:971; Mann-Whitney test (phase 1 of NoQ-QnoD vs phase 1 of QD ): P-value = 0:931.
53
All “unpredictable”utility-types in Figure 5 are “unpredictable”under incomplete information too; all utility^ R)
^ 2 T in Figure 4bis (light-grey color
types in the two regions (Dissolve; ) of Figure 5 (light-grey colors) have (G;
1
^ R)
^ 2 (S [ T)c
too); the only utility-type in region (Continue;
> 2 ) of Figure 5 (second-darkest region) have (G;
in Figure 4bis (white-colored, intermediate region); also 7/35 of the utility-types in region (Continue,Share) of
^ R)
^ 2
Figure 5 (darkest region) belong to this intermediate region in Figure 4bis, while the remaining 28/35 have (G;
S in Figure 4bis (darkest region).
31
expecting Continue with probability 1, this proportion slightly increases (36%), and we …nd a
positive correlation between the Share choice and the conditional second-order belief ( = 0:49,
P-value = 0:130), not signi…cant because of the small number of observations (11). A signi…cant
positive correlation ( = 0:41, P-value = 0:003) is instead found if considering all predictable
^ > 2R.
^
B -subjects with G
Figure 8 Incomplete-information predicted behavior of classi…ed types.
The following result summarizes the more salient experimental …ndings about matched pairs’
actual vs predicted behavior under incomplete information.
Result 6 In line with the incomplete-information predictions, in each of the three phase-treatment
combinations where the questionnaire is not disclosed, we …nd heterogeneous and dispersed
beliefs about B ’s strategy, about A’s strategy, and about elicited (both unconditional
and conditional). Furthermore, B -subjects predicted to choose Share actually do it much
more frequently than those predicted to choose Take (44% vs 15%). For B -subjects with
^ > 2R,
^ the Share choice is positively correlated with the elicited , the conditional belief
G
about .
We conclude this section with a qualitative comparison of behavior under incomplete information (…lled-in questionnaire not disclosed) and under complete information (phase 3 of QD)
of B ’s utility-type. Figures 9 and 10 have been built by applying to the incomplete-information
phases the same separation between high-guilt and low-guilt types as in phase 3 of QD.
32
Figure 9 As’frequency of Continue and B s’freq. of Share, disentangled by B ’s type.
In particular, in Figure 9 we present, for all phase-treatment combinations, actual choices
of matched pairs disentangled by role and by B ’s utility-type, and in Figure 10 corresponding
As’ …rst-order belief ( ) and of B s’ unconditional second-order belief (EtB (e )) (box plot, with
average belief on the left side). For phase 3 of QD, histograms and box-plots coincide with
those in Figure 7, and same colors are used: dark-grey for high-guilt B -subjects (and matched
A-subjects), and light-grey for low-guilt B -subjects (and matched A-subjects). For low-guilt
B -subjects – (Dissolve,Take) region in Figure 4 – we use the same light-grey color also in the
^ R)
^ 2 T with
incomplete-information phases: this is because we …nd that all of them have (G;
1
^
R < 3 ln(5=3) in Figure 4bis. For high-guilt B -subjects –(Continue,Share) region in Figure 4 –
the dark-grey color is meshed with small white dots so as to indicate that many of them have
^ R)
^ 2 S in Figure 4bis and few of them (4/24 in NoQ-QnoD, 7/35 in QD) belong to the
(G;
intermediate white-colored region (S [ T)c in Figure 4bis. Recall that the latter is the parameter
region where B -subjects should cooperate and hold high second-order beliefs in phase 3 of QD,
while they should exhibit intermediate second-order beliefs and less cooperative behavior in the
incomplete-information phases. For A-subjects in the incomplete-information phases, we use a
mixture of the three colors in Figure 4bis (dark-grey, white, and light-grey) – independently of
the fact that the matched B is a high-guilt or a low-guilt type, which they actually do not know.
33
Figure 10 As’…rst- and B s’second-order beliefs, disentangled by B ’s type.
The controls for A-subjects work as they should: In each incomplete-information phase, we
…nd no signi…cant di¤erence in the frequency of Continue 54 and in the distribution of the …rstorder beliefs55 between A-subjects matched with a high-guilt type and A-subjects matched with
a low-guilt one.
Within-treatment and between-treatment comparisons work very well for A-subjects matched
with a high-guilt type: Within treatment, we …nd a signi…cantly greater frequency of Continue and signi…cantly greater …rst-order beliefs in phase 3 than in phase 1 of QD.56 Between
treatments, we …nd a similar result by comparing phase 3 of QD to phase 3 of NoQ-QnoD.57
Within-treatment and between-treatment controls do not work well for A-subjects matched with
a low-guilt B, possibly due to an end-game e¤ect –independent of the matched type –detected
in both treatments. This e¤ect indirectly ampli…es the signi…cance of the results for A-subjects
54 2
test: P-value = 0:667 in phase 1 of QD, P-value = 0:526 in phase 1 of NoQ-QnoD, P-value = 0:451 in
phase 3 of NoQ-QnoD.
55
Mann-Whitney test: P-value = 0:563 in phase 1 of QD, P-value = 0:186 in phase 1 of NoQ-QnoD, P-value
= 0:567 in phase 3 of NoQ-QnoD.
56
Respectively, +40% ( 2 test, P-value = 0:000) and +23:14% on average (Mann-Whitney test: P-value = 0:000).
57
Respectively, +58.93% ( 2 test, P-value = 0:000) and +26.12% on average (Mann-Whitney test: P-value
= 0:001).
34
matched with a high-guilt B -subject shown above.58
In line with Result 6 above, B -subjects behave di¤erently according to their utility-type:
High-guilt types have signi…cantly greater (at 5% level) frequency of Share choices than low-guilt
types in each of the three incomplete information phases; also the (unconditional) second-order
beliefs are greater for high-guilt than for low-guilt types, although this di¤erence is signi…cant (at
5% level) only if we pool data of the three incomplete-information phases. Furthermore, given
the utility-type (high-guilt or low-guilt), both frequency of Share choices and second-order beliefs
are not signi…cantly di¤erent across the incomplete-information phases.59
Within-treatment and between-treatment comparisons work quite well for high-guilt B -subjects:
Within treatment, we …nd a non-signi…cantly greater frequency of Share and a signi…cantly
greater second-order beliefs in phase 3 than in phase 1 of QD.60 The di¤erence in the frequency
of Share should be caused by the high-guilt subjects that do not belong to the dominance region
S, but there are few such subjects; therefore, we are not surprised that this di¤erence is not
signi…cant. Between treatments, we …nd similar di¤erences, both signi…cant, by comparing phase
3 of QD to phase 3 of NoQ-QnoD.61
Within-treatment and between-treatment controls work well also for low-guilt B -subjects:
The predicted behavior being the same under both complete and incomplete information, we
coherently …nd no signi…cant di¤erence about both the frequency of Share and the second-order
beliefs, either by comparing phase 3 to phase 1 within QD or by comparing phase 3 between QD
and NoQ-QnoD.62
The following result summarizes the experimental …ndings about pairs’actual behavior under
complete vs incomplete information.
Result 7 Comparing phase 3 of QD with the other phase-treatment combinations, we …nd
di¤erences in the direction of a polarization of subjects’ behavior and beliefs in phase 3,
both by taking phase 1 of QD and by taking phase 3 of NoQ-QnoD as controls. The most
signi…cant di¤erence is found for A-subjects matched with high-guilt B -subjects in phase 3
of QD. No signi…cant di¤erence is found between phases 1 and 3 in NoQ-QnoD.
58
We do not …nd a signi…cant decrease in either the frequency of Continue ( 12:12%; 2 test: P-value = 0:283)
and the …rst-order beliefs ( 2:73%; Mann-Whitney test: P-value = 0:570) within the same (main) treatment.
Same for both choices (+4:24%; 2 test: P-value = 0:673) and …rst-order beliefs ( 3:51%; Mann-Whitney test:
P-value = 0:658) between QD and NoQ-QnoD in phase 3.
59
For choices, 2 test: P-value = 0:838 for high-guilt types, and P-value = 0:706 for low-guilt types. For beliefs,
Kruskal-Wallis test: P-value = 0:997 for high-guilt types, and P-value = 0:823 for low-guilt types.
60
Respectively, +17:14% ( 2 test, P-value = 0:151) and +20:29% on average (Mann-Whitney test: P-value
= 0:005).
61
Respectively, +22.50% ( 2 test, P-value = 0:090) and +20.74% on average (Mann-Whitney test: P-value
= 0:012).
62
For choices, phase 3 vs phase 1 in QD : 2 test, P-value = 0:741; phase 3 of QD vs phase 3 of NoQ-QnoD : 2
test, P-value = 0:651. For beliefs, phase 3 vs phase 1 in QD : Mann-Whitney test, P-value = 0:426; phase 3 of QD
vs phase 3 of NoQ-QnoD : Mann-Whitney test, P-value = 0:293.
35
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38
Appendix
In this section, we derive B ’s hypothetical payback function (x) and we analyze the Trust
Minigame with incomplete information about B ’belief-dependent preferences.
Hypothetical payback function
Here we provide the details of the derivation of B ’s payback function as a best response to the
hypothetical questions in Table 3. We rewrite the objective function of the modi…ed maximization
problem (4) as:
U (x; ; G; R; p) :=
ln(5
G
[max f0; 2p + 2
4
x)
xg]2 + R p +
3
2
x:
As anticipated above, this functional form has the advantage of being strictly concave in x,
with a decreasing, continuous derivative
Ux =
1
5
x
+
G
max f0; 2p + 2
2
xg + R p +
3
2
.
The aim is to obtain B ’s best-response function to A’s initial belief about his strategy
( ; G; R; p) := arg max U (x; ; G; R; p),
x2[0;4]
both in the case B ’s payback depends on A’s disappointment and in the case it does not.
Let us …rst consider the case where A’s disappointment matters to B, because he gives back to
her less than (the sum of his ex-post feeling-mitigation payo¤ and) what A would have expected
to get after Continue, i.e. x 2 [0; 2p + 2 ). In this case Ux (x; ; G; R; p) = 0 yields:
1
5
x
+
G
(2p + 2
2
x) + R p +
3
2
=0
which, given that x < 5 by construction, can be rewritten as
G 2
x
2
G
5
+p+
2
+R p+
3
2
x + 5 G(p + ) + R p +
3
2
1 = 0:
The determinant of the second-order equation in x is
= G
5
2
p
3
R p+
2
2
+ 2G;
which is positive for G > 0. Therefore, the two roots of the second-order equation in x are
q
2
G 52 + p +
+ R p + 32
G 52 p
R p + 32
+ 2G
x1=2 =
G
39
Given that the greater of the two roots is never lower than 2p + 2 for any nonnegative vector
( ; G; R; p), the only acceptable solution is the smaller root. Therefore, the best-response function
when A’s disappointment matters, D ( ; G; R; p), is:
s
2
5
R 3
5
R 3
2
(
;
G;
R;
p)
=
+
p
+
+
+
p
p
+
p
+
D
2
G 2
2
G 2
G
h
3
1
for R 23 + p
2 51 G (p + ) ; 5 2(p+
2 0; 15 G (p + ) , then
) . If R p + 2
D ( ; G; R; p) = 0. Notice that lim
D ( ; G; R; p) = 2 + 2p, the same best-response function
G!+1
we would obtain by solving, for G > 1, the problem
max
x2[0;4]
(4
x)
G max(0; 2p + 2
x) ,
i.e. the linear model of “simple guilt” (2) with the introduction of the ex-post feeling-mitigation
parameter p 2 [0; 1].
Let us now consider the case where A’s disappointment does not matter to B, because x 2
(2p + 2 ; 4]. In this case, Ux (x; ; G; R; p) = 0 yields:
1
5
x
+R p+
3
2
= 0.
Thus, the best-response function when A’s disappointment does not matter is:
N oD (
; R; p) = 5
2
R (3 + 2(p
))
h
i
1
for R 32 + p
2 5 2(p+
2 [1; +1) it is N oD ( ; R; p) = 4.
;
1
. For R 32 + p
)
To sum up, B ’s payback function is given by the following formula:
8
0
if R p + 23
2 0; 15 G (p + ) ,
>
>
i
>
>
1
1
< D ( ; G; R; p) if R p + 3
2
G
(p
+
)
;
2
5
5 2(p+ ) ,
i
( ; G; R; p) =
3
1
>
2 5 2(p+
>
N oD ( ; R; p) if R p + 2
>
); 1 ,
>
:
3
4
if R p + 2
2 (1; +1).
A simple incomplete information model of the Trust Minigame
We …rst consider the “simple guilt” model of belief-dependent preferences (2) with incomplete
information about G. The model we present is simple enough to fully understand the logic of
the Bayesian equilibrium with the maximal degree of trust. Next we move to an incomplete
information analysis of the guilt-reciprocity model .
40
4.3.1
Incomplete information about linear guilt
It is common knowledge that A is sel…sh and risk neutral, and that the belief-dependent utility
function of B has the form
uB (mB ; mA ; ) = mB
G DA ( ; mA ), G
0.
To simplify the analysis, we assume that there are only two possible values of the guilt parameter
for player B, a low value GL such that B wants to take for every , and a high value GH such
that B wants to share whenever
1=2:
GB 2 fGL ; GH g, with 0
GL < 1, 2 < GH .
Recall that player A is assumed to be a sel…sh expected-monetary-payo¤ maximizer and that this
is assumed to be common knowledge.
In order to obtain a Bayesian game, we have to append to this parametrized Trust Minigame
a type structure (see Harsanyi, 1967-68), which yields an implicit speci…cation of the possible
hierarchies of beliefs of A and B about the preference parameter GB (A’s belief about GB , B ’s
belief about this belief, and so on). This implicit speci…cation is self-referential. We assume
that, for each player i 2 fA; Bg, there is a compact metrizable space Ti of possible types and a
continuous belief map i : Ti ! (T i ),63 with
TA = [0; 1], TB = fGL ; GH g
[0; 1]:
A type of A is associated with the subjective probability that GB = GH , i.e. that B ’s guilt
type is high. A type of B is a pair tB = (GB ; eB ) given by B ’s preference parameter and B ’s
epistemic type, the latter being a parameter that determines B ’s belief about the type of A.
Speci…cally, belief map A ( ) is determined by a strictly increasing and continuous cumulative
distribution function (CDF) F : [0; 1] ! [0; 1] (F (0) = 0, F (1) = 1) according to following
equation:64 For all tA and y,
A (tA )[GB
= G H ; eB
y] = tA F (y).
(9)
This means that the higher the types of A the higher the subjective probability that B ’s guilt
type is high, and that, for each type of A, the preference component and the belief component of
B’s type are stochastically independent.
Belief map B ( ) is given by a parametrized family of strictly increasing and continuous CDFs
fFeB : [0; 1] ! [0; 1]geB 2[0;1] (FeB (0) = 0, FeB (1) = 1) according to the following equation: for all
GB , eB and x
x] = FeB (x);
(10)
B (GB ; eB )[tA
hence, the beliefs of B depend only on his epistemic type eB . We assume that the continuous65
map eB 7 ! FeB ( ) is strictly increasing with respect to …rst-order stochastic dominance, i.e.
63
(T i ) is endowed with the topology of weak convergence of measures and the Borel sigma algebra generated
by this topology.
64
We put in square brackets events about a player i, that is, measurable sets of types ti that satisfy the bracketed
conditions.
65
Map eB 7 ! FeB (x) has to be continuous for each x because FeB (x) is continuous in x for each eB and and
the belief map B ( ) is continuous.
41
FeB (x) is strictly decreasing in eB for each x 2 (0; 1). This means that higher epistemic types of
B have higher prior beliefs about the epistemic type of A. All of the above is common knowledge.
For every given pair of measurable behavioral rules (strategies of the Bayesian game) A :
TA ! fD; Cg, B : TB ! fT; Sg, we obtain …rst and second-order beliefs about actions.66
Speci…cally, letting
Pti [aj ] = i (ti )[ j (tj ) = aj ]
(11)
denote the subjective probability assigned by type ti to the opponent’s action aj given
have:
(tA ) = PtA [S],
Z 1
0
(tA ) B (tB )[dtA ]
(tB ) = EtB [ ] =
(tB ) = EtB [ jC] = (PtB [C])
1
Z
ftA :
j,
we
(12)
(13)
0
(tA )
B (tB )[dtA ],
A (tA )=Cg
if PtB [C] > 0
(14)
(to keep notation simple, we do not emphasize the dependence of these functions on A ; B ).
Since (GB ; eB ) depends only on the epistemic component eB of B ’s type, from now on we replace
tB with eB in the expressions above.
A Bayesian equilibrium is a pro…le of functions ( A ; B ; ; 0 ; ) such that eq.s (11)-(14)
hold and, for all tA , GB and eB ,67
A (tA )
B (GB ; eB )
= D i¤
A (tA )
1
,
2
= T i¤ PeB [C] = 0 or GB (eB )
(15)
1 .
(16)
Eq. (15) says that A maximizes her expected monetary payo¤, eq. (16) says that B maximizes
his expected psychological utility.
The Bayesian game thus obtained has an equilibrium in which every type of A chooses Dissolve
and every type of B chooses Take. We look instead for equilibria where a positive fraction of types
of each player chooses the cooperative action. It turns out that there is only one such equilibrium,
which –obviously –Pareto dominates the previous no-trust equilibrium.
Proposition 6 There is a unique Bayesian equilibrium ( A ; B ; ; 0 ; ) such that a positive
fraction of A’s types Continue (i.e., ftA : A (tA ) = Cg has positive Lebesgue measure), and it
has the following properties:
(a) for every type tA , A (tA ) = tA and A (tA ) = D i¤ tA 21 ;
(b) the prior second-order belief 0 ( ) is strictly increasing, and, for every epistemic type eB ,
Z
1
1
(eB ) = (1 FeB (1=2))
tA dFeB (tA ) > ,
1
2
( ;1]
2
B (G
L; e
B)
= T and
B (G
H; e
B)
= S.
66
D is the strategy Dissolve, C is the strategy Continue, and so on.
We are assuming that an indi¤erent type takes the uncooperative action. This is without essential loss of
generality within the given model.
67
42
Intuitively, every type (GL ; eB ) of B with low guilt aversion chooses Take. As for the types
B ) with high guilt aversion, under the stated assumption, we can use Bayes rule to derive
the conditional second-order belief (eB ) given Continue. Since, in equilibrium, only types tA
that believe that B chooses Share with more than 50% probability choose Continue, (eB ) =
EeB [ jC] 1=2; therefore (GH ; eB ) wants to share because GH > 2. Thus, B chooses Share if
and only if he is (highly) guilt averse. This simple decision function of B determines A and ,
which in turn determine 0 and . A more formal proof follows.
(GH ; e
Proof of Proposition 6 We start from the conjecture that a strictly positive fraction of types
of A choose C and derive a unique equilibrium that veri…es this property. By eq. (15), the set of
types of A choosing C is ftA : tA > 1=2g. By assumption, this set has positive Lebesgue measure.
Since every probability measure B (eB ) has full support, PeB [C] = B (eB )[ (tA ) > 1=2] > 0.
Therefore, (eB ) = EeB [ jC] is well de…ned and (eB ) > 1=2. With this, eq. (16) implies that
L
H
L (e ) < 1 < GH (e ) for each
B (G ; eB ) = T and B (G ; eB ) = S for every eB , because G
B
B
eB . Then, for every tA , (tA ) = A (tA )[GB = GH ] = tA and A (tA ) = C if and only if tA > 1=2.
This implies that (i) the prior-…rst order belief
Z 1
0
(eB ) = EeB [ ] =
tA dFeB (tA )
0
is strictly increasing, because FeB (x) is strictly decreasing in eB for each x 2 (0; 1); and that (ii)
the conditional second-order belief of epistemic type eB is
Z 1
1
1
(eB ) = (1 FeB (1=2))
tA dFeB (tA ) > .
1
2
2
The Bayesian model we just described does not assume that agents know the true distribution
of types. Subjective beliefs are su¢ cient to derive the equilibrium; but, to obtain predictions
about the distribution of behavior and endogenous beliefs, we need to add assumptions about
the actual joint distribution of A-types and B -types.
First, since we want to model behavior in an experiment where A-subjects are randomly
matched to B -subjects, we assume that A-types and B -types are statistically independent. Second, we assume that the epistemic type of B, eB , is statistically independent of the guilt type of B,
G. Finally, we assume that every (non-empty) open set of types has strictly positive probability.
Speci…cally, we assume that the actual joint distribution of types has the form
Z x
Z y
H
H
P tA x; GB = G ; eB y =
fB (eB )dtA
p
fB (tB )dtB ,
0
0
pH
where 0 <
< 1, and fA and fB are bounded away from zero on [0; 1]. From this, one can
derive qualitative predictions about behavior and endogenous beliefs.
4.3.2
Incomplete information about guilt and reciprocity
It is common knowledge that A is sel…sh and risk-neutral, and that the belief-dependent utility
function of B has the form (3):
G
uB (mB ; mA ; ) = ln(1 + mB )
[DA ( ; mA )]2 + R KA ( ) mA , G; R 2 [0; L],
4
43
where L > ln (5=3) is a large upper bound on parameters. Recall that the willingness to share of
the utility-type (G; R) of B in the Trust Minigame, when he is certain that the …rst-order belief
of A is , is
W S( ; G; R) = uB (2; 2; )
uB (4; 0; ) = G
2
2R + 3R
`,
where we write ` := ln (5=3) to ease notation. Player B chooses Share if EB [W S(e ; G; R)jC] > 0,
and Take if EB [W S(e ; G; R)jC] < 0.
As in the linear-guilt model, we consider a type structure and a Bayesian equilibrium where
a positive fraction of A-types choose Continue, and the epistemic type of B, eB , is independent
of the utility-type of B, (G; R). Since an A-type tA chooses Continue only if (tA ) 1=2, this
means that each B -type is certain that
1=2 conditional on Continue. In particular, every B type (G; R; eB ) such that W S( ; G; R) > 0 (respectively, W S( ; G; R) < 0) for every 2 [1=2; 1]
de…nitely chooses Share (respectively, Take). With this, we determine the dominance regions for
Share and Take in the (G; R) space, given
1=2:
(
)
S :=
T :=
(G; R) 2 [0; L]2 : min W S( ; G; R) > 0 ,
2[ 21 ;1]
(
)
(G; R) 2 [0; L]2 : max W S( ; G; R) < 0 .
2[ 21 ;1]
Since W S is strictly convex in , it attains a maximum on [1=2; 1] either at = 1=2, or at =
1. Therefore, W S( ; G; R) < 0 for every 2 [1=2; 1] i¤ maxfW S(1=2; G; R); W S(1; G; R)g < 0,
where
1
1
W S(1=2; G; R) < 0 () G + R < `,
8
2
W S(1; G; R) < 0 () G + R < `.
Thus,
1
1
(G; R) 2 R2+ : G + R < `; G + R < ` .
8
2
Dominance region S is the union of three sub-regions: Function W S attains its minimum at
= R=G, which may be to the left of 1=2, to the right of 1 , or in the interval [1=2; 1]; each case
gives raise to a sub-region.
T=
If R=G < 1=2, W S is strictly increasing on [1=2; 1], hence it attains its minimum at
Letting W S(1=2; G; R) > 0 in this case, we obtain
S1 :=
= 1=2.
1 1
1
(G; R) 2 [0; L]2 : R < G; G + R > ` .
2 8
2
If 1=2 R=G 1, W S attains its minimum in the interval [1=2; 1]. Letting W S(1=2; G; R) >
0, in this case, we obtain
S2
:
=
1
R
(G; R) 2 [0; L]2 : G R G; W S
; G; R > 0
2
G
1
(G; R) 2 [0; L]2 : G R G; R2 3GR + `G < 0 .
2
=
44
If R=G > 1, W S is strictly decreasing on [1=2; 1], hence it attains its minimum at
Letting W S(1; G; R) > 0 in this case, we obtain
= 1.
S3 := (G; R) 2 [0; L]2 : R > G; G + R > ` .
Thus, S=S1 [ S2 [ S3 .
We assume that the Harsanyi-type structure (Ti ;
1. TA =
i=dA
i
i=1 [0; tA ]
2. TB = [0; L]2
di¤erent).
3.
i )i2fA;Bg
is as follows:
RdA .
EB , where EB =
j=dB
j
j=1 [0; eA ]
RdB (the dimensions dA ; dB 2 N may be
: TA ! (TB ) is continuous and such that, for every (G; R) 2 [0; L]2 and open set
;=
6 OB EB
[0; R] OB ) = FtA (G; R) (OB ) > 0,
A (tA ) ([0; G]
A
where
2 (EB ) is absolutely continuous with respect to the Lebesgue measure, with
density function bounded away from zero. Furthermore,
t A 2 TA :
4.
A (tA ) (S
EB ) >
1
2
6= ;.
: TA ! (TB ) is continuous and such that B (G; R; eB ) depends only on eB (hence
we write B (eB ) to ease notation), furthermore, for every eB 2 EB , B (eB ) is absolutely
continuous with respect to the Lebesgue measure, with density function bounded away from
zero.
B
The last assumption implies that, for every open set ; =
6 OA
TA and every eB 2 EB ,
(e
)(O
)
>
0.
Since
the
belief
map
is
continuous
and
each
(e
)
B B
A
A
B B is absolutely continuous,
the set of types
1
tA 2 TA : A (tA ) (S EB ) >
2
is open.68 By assumption, this set is also non-empty, therefore
B (eB )
tA 2 TA :
A (tA ) (S
EB ) >
1
2
>0
for each eB 2 EB . (A similar argument shows that tA 2 TA : A (tA ) (T EB ) > 21 is open.)
We conjecture that there exists an equilibrium with measurable decision functions ( A ; B )
where A (tA ) = C for every tA in some non-empty, open set of A-types, and we describe its
68
The boundary of S EB has zero probability for each type tA . Hence, the portmanteu theorem and continuity
of A with respect to the weak convergence of measures imply that, for each converging sequence tn
A ! tA ,
limn!1 A (tn
A ) (S EB ) = A (tA ) (S EB ). This in turn implies that, for each p 2 [0; 1], the set of types tA such
that A (tA ) (S EB ) > p is open.
45
qualitative features.69 Since each B -type assigns positive probability to every non-empty, open
set of A-types, in such an equilibrium, the conditional belief
B (eB )(
jCont:) =
B (eB ) (
j ftA 2 TA :
A (tA )
= Cg)
is well de…ned for each eB . Recall that
(tA ) =
A (tA ) (f(G; R; eB )
2 TB :
B (G; R; eB )
= Sg) ;
hence
ftA 2 TA :
A (tA )
= Cg
tA 2 TA :
and
B (eB )
B (eB )
t A 2 TA :
A (tA )
1
C
2
eA
1
2
1
2
A (tA )
=
j ftA 2 TA :
A (tA )
= Cg
= 1.
Therefore, for every B -type (G; R; eB ),
(G; R) 2 S )
(G; R) 2 T )
B (G; R; eB )
= S,
B (G; R; eB )
= T.
This implies that
t A 2 TA :
A (tA ) (T
t A 2 TA :
A (tA ) (S
1
2
1
EB ) >
2
EB ) >
ftA 2 TA :
A (tA )
= Dg ,
ftA 2 TA :
A (tA )
= Cg .
As explained above, the sets on the left-hand side are open in TA . Hence, in particular, the set
of A-types that assign more than 50% probability to dominance region S is an open set of types
who choose Continue, and our assumptions on B imply that each eB assigns positive probability
to this set.
Since each type space is connected and each belief map is continuous, it follows that every
value between and minimum and maximum of (tA ) is attained by some tA , and every value
between the minimum and maximum of 0 (eB ) = EeB [ ] (respectively (eB ) = EeB [ jC]) is
attained by some eB :
(TA ) =
0
(EB ) =
min
(tA ); max
min
0
tA 2TA
eB 2EB
tA 2TA
(eB ); max
eB 2EB
69
(tA ) ,
0
(eB ) .
For the incomplete-information model with linear guilt of the previous sub-section, we could prove the existence
and uniqueness of such equilibrium. We can not do the same here without adding more structure.
46
The belief of every type tA –including the minimizers of A (tA ) (S EB ) and A (tA ) (T EB ) –is
determined by a probability density function bounded away from zero; hence,
min
(tA )
max
(tA )
tA 2TA
tA 2TA
min
tA 2TA
1
A (tA ) (S
min
EB ) > 0,
A (tA ) (T
tA 2TA
EB ) < 1.
This in turn implies
min
0
max
0
eB 2EB
eB 2EB
(eB )
min
(tA ) > 0,
max
(tA ) < 1.
tA 2TA
(eB )
tA 2TA
The following proposition summarizes these observations.
Proposition 7 Every Bayesian equilibrium where a non-empty, open set of A-types choose Continue has the following features:
(a) For every tA 2 TA ,
A (tA ) (S
EB ) >
A (tA ) (T
EB ) >
1
)
2
1
)
2
A (tA )
= C,
A (tA )
= D,
and
A (tA ) (S
EB )
(tA )
1
A (tA ) (T
EB ) .
(b) For every (G; R; eB ) 2 TB ,
(G; R) 2 S )
(G; R) 2 T )
B (G; R; eB )
= S,
B (G; R; eB )
= T,
and
B (eB )
hence (eB ) 1=2.
(c) There are values
> 0 and
1
C
2
e
= 1,
< 1 such that,
(tA );
0
(eB )
,
for every tA 2 TB and eB 2 EB .
(d) All values between mintA 2TA (tA ) and maxtA 2TA (tA ) are attained by some tA , and all values
between mineB 2EB 0 (eB ) and maxeB 2EB 0 (eB ) are attained by some eB .
As explained above, we need to add assumptions about the actual distribution of types in
order to derive implications about the distribution of behavior and endogenous beliefs from the
47
equilibrium analysis. We assume that the distribution of types is such that the type of A and the
type of B are statistically independent, the epistemic type of B is independent of the utility-type
of B, and the marginal distributions are determined by density functions bounded away from
^ R)
^ 2 [0; L]2 , e^B 2 EB ,
zero: For all t^A 2 TA , (G;
=
Z
[0;t^A ]
fA (tA )dtA
P ([0; tA ]
!
Z
[0; G]
^ [0;R]
^
[0;G]
[0; R]
[0; eA ]) =
!
Z
1
fB (G; R)dGdR
[0;^
eB ]
fB2 (eB )deB
!
,
where the densities fA , fB1 and fB2 are bounded away from zero, respectively, on TA , [0; L]2
and EB .70 From this, one can derive the qualitative predictions about behavior and endogenous
beliefs of Proposition 5. In particular, the positive correlation between the fraction of B -types
with G 2R choosing Share and the conditional second-order belief holds if the utility-type and
epistemic type of B are statistically independent. This however, is just a su¢ cient condition to
obtain such positive correlation.
70
Recall that, in the present model, tA and eB are vectors in RdA and RdB , respectively. Hence [0; t^A ] and [0; e^B ]
2
are Cartesian products of intervals, and fA , fB
are multivariate density functions.
48