A Survey of Psychological Games: Theoretical
Findings and Experimental Evidence?
Giuseppe Attanasiyand Rosemarie Nagelz x
1. Introduction: the role of psychological games
Most economic models assume that agents maximize their expected material payo¤.
However, subjects in the lab exhibit persistent and signi…cant deviations from this self-
interested maximizing behavior. A reasonable explanation for this behavior is that
players can be motivated not only by material (monetary) payo¤s but also by what are
sometimes referred to as ‘psychological’ utilities. These are related to preferences that
are in some degree ‘other regarding’: they take others into account. Traditional game
theory does not provide enough tools to adequately describe many of these preferences:
the traditional approach assumes that utilities only depend on the actions that are chosen
by the players. By contrast, when players are emotional or motivated by reciprocity or
social respect, their utilities may also directly depend on the beliefs (about choices,
?Published in “Games, Rationality and Behaviour. Essays on Behavioural Game Theory and Ex-
periments”, A. Innocenti and P. Sbriglia (eds.), Palgrave McMillan, Houndmills, February 2008 (Ch.
9, pp. 204-232).
yToulouse School of Economics (Georges Meyer Chair in Mathematical Economics, LERNA) and
Bocconi University (Department of Economics).
zUniversitat Pompeu Fabra (Department of Economics).
xWe are grateful to Pierpaolo Battigalli for the useful suggestions encouraging this survey and to
Martin Dufwenberg for the helpful starting points we found in his doctoral dissertation. We are also
grateful to Nikolaos Georgantzis and the participants in several seminars for helpful discussions and to
Michel Sera…nelli for useful research assistance. O¤ course the authors are responsible for any error in the
chapter. Giuseppe Attanasi acknowledges …nancial support from Bocconi University and thanks Univer-
sitat Jaume I of Castelló for its hospitality during part of this project. Rosemarie Nagel acknowledges
…nancial support from the Spanish Ministry of Education and Science under grant SEC2002-03403, and
thanks the Barcelona Economics Program of CREA for support. Both authors thank for the hospitality
of HSS in Caltech where part of the chapter was written.
beliefs, or information) they hold. This is not to say that traditional game theory is
not able to analyze the in‡uence of feelings, emotions and social norms on the players’
behavior. Distribution-dependent preferences à la Fehr and Schmidt (1999), for example,
can be addressed by the traditional game theory. But when we deal with intention-
based feelings, emotions and social norms, i.e. belief-dependent motivations, we need to
turn to psychological game theory. This new framework focuses on strategic settings
where at least one player has belief-dependent motivations or believes, with a certain
probability, that one of his opponents has belief-dependent motivations. Nonetheless, it
allows for every other kind of social preferences. In that sense, it can be interpreted as
a generalization of the traditional game theory.
In games with belief-dependent motivations there are clearly two channels through
which beliefs and information a¤ect behavior: the direct (psychological) impact of beliefs
on preferences over terminal histories, and the (traditional) impact of (updated) beliefs
about the opponents on the preferences over own strategies. Geanakoplos, Pearce and
Stacchetti (1989) is the seminal paper that shows the inadequacy of traditional methods
in representing the involved preferences, and develops extensions of the traditional game
theory in order to deal with the matter. Battigalli and Dufwenberg (2005) generalize
and extend Geanakoplos, Pearce and Stacchetti (1989), thus providing the framework
we use to analyze games with belief-dependent motivations both from a theoretical and
from an experimental point of view.
Experimental evidence gives support to the theories of belief-dependent motivations.
Even when not explicitly designed to test such motivations, several experimental works
provide results that are in line with psychological game theoretical predictions. Some of
these experiments can be seen as an indirect proof of the relevance of psychological games
in explaining certain forms of strategic interaction. Some others have been explicitly
designed to test the relevance of psychological game theory.
The plan of the chapter is as follows. In section 2, we provide the main theoretical
insights of the general psychological game framework, through the example of a simple
trust game in which belief-dependent motivations are involved. In section 3, we describe
and discuss the main experimental papers that directly or indirectly refer to psycholog-
ical game-theoretic explanations as being able to rationalize their experimental results.
Throughout the chapter, we try to clarify ideas and to dispel misconceptions emerged
among economists about this new …eld, in order to stress the role and the importance
of psychological games both for the strategic interaction analysis and for the related
2. Main Theoretical Findings
2.1 Theoretical Literature
Geanakoplos, Pearce and Stacchetti (1989; henceforth GPS) introduce belief-dependent
motivation into strategic decision making. They develop a new analytical framework cen-
tred on the concept of psychological game (or game with belief-dependent motivations),
i.e. a strategic interactive situation in which players’ utilities do not only depend on ter-
minal nodes but also on the beliefs (about choices, beliefs, or information) they hold.1
GPS also present several examples that illustrate the inadequacy of traditional methods
in representing preferences that re‡ect various forms of belief-dependent motivation.
GPS framework may be seen as a generalization of a traditional game able to model
only some of the speci…c belief-dependent motivations in strategic settings: as stated
by Battigalli and Dufwenberg (2005; henceforth BD), ‘GPS’s toolbox of psychological
games incorporates several restrictions that rule out many plausible forms of belief-
dependent motivation’ (p. 41). In particular, GPS only allow initial beliefs to enter
the domain of a player’s utility; however, many seemingly important forms of belief-
dependent motivations require the introduction of updated beliefs.
BD generalize and extend GPS, by allowing updated higher-order beliefs, beliefs of
others, planned strategies, and incomplete information to in‡uence motivation. Among
other advances, BD address the issue of how beliefs about others’ beliefs are revised as
the play unfolds: they are able to model dynamic psychological e¤ects that are ruled
out when epistemic types are identi…ed with hierarchies of initial beliefs. They also
de…ne a notion of Psychological Sequential Equilibrium, which generalizes the sequential
equilibrium notion for traditional games, for which they prove existence under mild
assumptions. In the next paragraph we will use their notion of psychological sequential
equilibrium to solve a simple two-stage psychological game.
The most well-known example of a psychological-game based application is Rabin’s
(1993) model of intention-based reciprocity, according to which players wish to act kindly
(unkindly) in response to kind (unkind) actions. The key notion of kindness depends on
beliefs in such a way that reciprocal motivation can only be described using psychological
However, the range of topics that have been explored in models of belief dependent
1As Dufwenberg (2006) correctly points out, ‘the term game with belief-dependent motivation would
be more descriptive than the term psychological game, but we stick with the latter which has become
established’ (p. 2).
motivation is still limited. BD suggest that there is a variety of interesting forms of
belief-dependent motivations waiting to be analytically explored. In his survey paper
on ‘Emotions and Economic Theory’, Elster (1998) argues that a key characteristic of
emotions is that ‘they are triggered by beliefs’ (p. 49). He discusses, inter alia, anger,
hatred, guilt, shame, pride, admiration, regret, rejoicing, disappointment, elation, fear,
hope, joy, grief, envy, malice, indignation, jealousy, surprise, boredom, sexual desire,
enjoyment, worry, and frustration.
2.2. An example: trust games as psychological games
Let us show and analyze some of the more important features of BD’s psychological
game-theoretic framework by means of a simple trust game where some belief-dependent
motivations can emerge.
Figure 1 Trust Game with material payo¤s
We build on a simple two-stage game representing the following economic situation of
strategic interaction. Player A (the truster, ‘he’) and B (the trustee, ‘she’) are partners
on a project that has thus far yielded total pro…ts of 2e. Player A has to decide whether
to withdraw from the project or not. If player A dissolves the partnership, the contract
dictates that the players split the pro…ts …fty-…fty. If player A leaves his resources in the
project, total pro…ts would be higher (4e); however, according to the contract, in that
case player B has the right to share or not the pro…ts after the project is completed.
So, player A must decide whether to Dissolve or to Continue the partnership, without
knowing if there will be pro…t sharing in case he continues. After knowing player A’s
choice and only in case player A has chosen to Continue the partnership, player B has
to decide whether to Take or Share the higher pro…ts. The game tree with the material
payo¤s is represented in Figure 1. Payo¤s are in euros and do not necessarily represent
preferences. For this reason we call them ‘material payo¤s’.
We know from traditional game theory that the unique subgame perfect equilibrium
of the two-stage trust game (with material payo¤s) in Figure 1 is player A choosing
Dissolve and player B choosing Take if A would Continue.
Now, suppose that we represent this game in a laboratory and let participants play
it in pairs. According to the experimental literature on this subject,2one should expect
quite a high percentage of pairs with outcomes (Continue;Share).
A reason for this could be the bounded rationality of some A or B players (or both).
However, it seems very di¢cult to support that in a so simple and clear game a quite
high number of players (or pairs) are not able to understand the rules of the game or
to calculate their pro…t-maximizing action given their expectations (…rst-order beliefs)
about their opponent’s choice.3
Another reason could be that player A or player B are motivated not only by self-
interest, but also, at least some of them, by ‘social preferences’.
Let us …rst concentrate on distribution-dependent preferences à la Fehr and Schmidt
(1999). As stated above, this kind of preferences can be addressed by traditional game
theory. More speci…cally, let us suppose that B is inequity averse (i.e. motivated by
fairness) and that this is common knowledge among players. Formally, B’s preferences
are represented by the utility function
uB(sA;sB) = ?A(sA;sB)
?k maxf0;?B(sA;sB) ? ?A(sA;sB)g ? hmaxf0;?A(sA;sB) ? ?B(sA;sB)g
where siis the strategy of player i = A;B, ?i(sA;sB) is the material payo¤ of player
i = A;B and k;h are positive parameters such that k 2 [0;1] and h 2 [0;k].
is a self-interested player and knows uB(sA;sB).
Given the payo¤ structure of our
simple trust game, the utility function of B reduces to uB(sA;sB) = ?A(sA;sB) ?
2Among others, Charness and Dufwenberg (2006). See section 3 for a complete review of the exper-
imental literature on this family of trust games.
3A reasonable explanation for outcomes that partially di¤er from the subgame perfect equilibrium
one could be the ‘uncorrectness of beliefs’ of some players. For example, suppose that A and B are
both self-interest and rational, i.e. they maximize their (expected) material payo¤. Suppose also that
A believes that B is not rational and so he chooses Continue, being quite sure that B will choose Share
after Continue. Being rational and self-interested, B instead chooses Take after Continue and so the
outcome (Continue;Take) takes place.
hmaxf0;?A(sA;sB) ? ?B(sA;sB)g and so (Continue;Share) is the (unique) equilib-
rium outcome of the trust game in case B is highly inequity averse, i.e. k 2
and h 2
tally independently of B’s expectation of A’s expectation on B choosing Share after
2;k?. In that case, (Continue;Share) outcomes should emerge experimen-
However, as we shall report below, experimental evidence on trust games shows
that there is a certain correlation between B’s expectation of A’s trust on her and
trust ful…lment; that can be explained by a particular kind of social preferences: those
expressed as belief-dependent motivations. As said above, traditional game theory is
ill-equipped to address such preferences. For that reason, we need to introduce some
psychological game tools.
Consider the game with material payo¤s in Figure 1.
Let us …rst suppose that A is a self-interested player, while B is (also) a guilt-averse
Guilt aversion can be de…ned as follows: people su¤er from guilt if they in‡ict
harm on others; although guilt could have a variety of sources, one preeminent way
to in‡ict harm is to let others down (see Tangney, 1995). Battigalli and Dufwenberg
(2007) develop a general theory of guilt aversion and show how to solve for sequential
equilibria. Their approach can be easily applied to our simple trust game, whenever
we assume that B is a¤ected by guilt. Moreover, Charness and Dufwenberg (2006)
suggest that sensitivity to guilt aversion imposes a speci…c behavior to the trustee (B):
suppose that in an experiment it is possible to truthfullly elicit B’s expectation of A’s
expectation that B would Share after Continue (B’s second-order beliefs of Share); given
that the (Continue;Share) outcome indicates trust ful…lment, as we shall report below,
experimental evidence on trust games shows positive correlation between B’s second-
order beliefs of Share (after Continue) and trust ful…lment and thus higher expected
mean guess of Bs who choose Share after Continue than the mean guess of Bs who
choose Take after Continue.
We …rst consider the psychological utility of player A. Since A is only motivated by
self-interest, his feeling sensitivity to each possible belief-dependent motivation is equal
to zero. Therefore, A’s total utility function reduces to his material payo¤.
Next, let us concentrate on the psychological utility of player B. Since B is moti-
vated by guilt aversion, she takes into account the disappointment of player A when the
material payo¤ he expects to receive after Continue does not match the one he actu-
ally receives. The material payo¤ A expects to receive after Continue depends on his
…rst-order beliefs on B’s strategy.
Let us analyze the situation from a formal point of view: de…ne A’s initial …rst-
order belief that B will Share if A chooses Continue as ?A= PrA[Share if Continue].
Hence, ?Ameasures A’s trust on B before the game starts. De…ne also B’s conditional
2nd-order belief that she would Share if A would Continue as ?B= EB[?AjContinue].
Hence, ?Bmeasures B’s expectation of A’s trust on her given that B knows that A has
Given the notation we introduced, we can express A’s expected material payo¤ after
Continue as ECont;?A[?A] = 2 ? ?A+ 0 ? (1 ? ?A) = 2?A. How much would A feel ‘let
down’ after (Continue;Take)? According to BD, the amount of his disappointment is
exactly ?2?A, i.e. the di¤erence between the payo¤ he receives after (Continue;Take),
which is zero, and the payo¤ he would have received after (Continue;Share).
B’s guilt is given by her expectation of A’s disappointment, given that A has chosen
Continue, i.e. B’s expectation of (?2?A), given Continue. This amount, ?2?B, mul-
tiplied by B’s sensitivity to guilt aversion, ?g
B? 0, is exactly B’s psychological utility
when A chooses Continue and she chooses Take. B’s total utility after (Continue;Take)
is given by 4 ? ?g
The psychological game representing the trust game with a self-interested truster and
B2?B, i.e. the sum of her material payo¤ and her psychological utility.
a guilt-averse trustee is depicted in Figure 2. What appears at the terminal histories
should be thought of as utilities, not as material payo¤s, although the two notions
coincide for all but one of the terminal histories.
4 ? ?g
Figure 2 Trust Game with guilt aversion.
4In general, we use the Greek letter ? to refer to …rst-order beliefs, and the letter ? to refer to
Looking at Figure 2 we can conclude that in the simple trust game we are analyzing, B
exhibits guilt aversion if her expected utility from playing Take after Continue depends
negatively on her expectation of ?A, conditional on A choosing Continue.
Let us now suppose that A is again a self-interested player, while B is (also) a
Reciprocity has two sides: positive reciprocity, where a player is kind in return to
another one’s kind choice, and negative reciprocity, where a player is unkind in return
to another one’s unkind choice. Rabin’s (1993) theory of reciprocity, in which players
reciprocate belief dependent (un)kindness with (un)kindness, is probably the most well-
known application of GPS’s psychological game theory. Rabin works with the normal
form version of GPS’s theory. His goal is to highlight certain key qualitative features
of reciprocity, and he does not address issues of dynamic decision making, although he
points out that this is important for applied work (p. 1296). Dufwenberg and Kirch-
steiger (2004), while building on the same framework as Rabin in de…ning reciprocity,
depart from it, developing a theory of reciprocity for extensive games. As BD, in dealing
with sequential reciprocity they argue that it is necessary to deviate from GPS’s exten-
sive form framework: GPS only allow initial beliefs to enter the domain of a player’s
utility, while the modeling of reciprocal responses at various nodes of a game tree requires
kindness to be re-evaluated using updated belief.
We build on the same intuition as Rabin (1993) according to which modeling reci-
procity may require belief-dependent utilities, since kindness and perceived kindness de-
pend on beliefs. And we also take into account the suggestion of Dufwenberg and Kirch-
steiger (2004) that reciprocity in a dynamic setting is a ‘conditional’ belief-dependent
motivation, in the sense that updated belief matters when evaluating the kindness of a
player. Nonetheless, following Battigalli (2007), we model reciprocity in an easier and
more direct way compared to the formal de…nition of Rabin (1993), although we build
on the same form of belief-dependency.
Let us start by de…ning A’s kindness (and B’s perceived kindness) through a simple
question: if A chooses Continue can we safely postulate that he has been kind to B?
The answer is ‘possibly not’, because, in the case A is quite certain that B would choose
Share after Continue (say, ?A>
2), he is just maximizing his material payo¤. Thus,
when A chooses Continue, he is (and is perceived to be) more kind, the less he thinks
that B would choose Share after Continue. Obviously, in case A chooses Dissolve, he is
not kind to B and we can intuitively state, without loss of generality, that A’s kindness
to B is negative (see the exact calculation below). Hence, Continue may be deemed
‘kind’, if ?Ais low. Continue is perceived as kind by B if ?Bis low. If ?Bis low and
B is highly motivated by reciprocity considerations, she reciprocates and chooses Share
Formally, we de…ne the ‘kindness’ of player i as a function of his strategy and beliefs:5
i is kind (unkind) to j if he intends to make j get more (less) money than a context-
dependent ‘equitable payo¤’ ?ei
j, with i;j = A;B, i 6= j. For example, the equitable
payo¤ ascribed by B to A, given B’s belief, can be expressed with the formula
sBEB[?A;?B;sB] + min
where ?B= PrB[Continue], i.e. it measures B’s initial …rst-order belief that A would
choose Continue. The equitable payo¤ is de…ned in this case as the average between the
highest and the lowest expected payo¤ that B can allow to A, given her initial belief
on the action chosen by A. Therefore, we can de…ne ‘Kindness’ of B towards A as the
di¤erence between the expected payo¤ B would allow to A and the equitable payo¤, i.e.
KB(?B;sB) = EB[?A;?B;sB] ? ?eB
Notice that B’s kindness depends on B’s intentions.
In the same way, we can de…ne EB[KA;?B], i.e. player B’s perceived kindness of
A. Since A’s kindness depends on the …rst-order belief of A (his belief about sB), then
player B’s perceived kindness depends on the second-order belief of B (belief of B about
belief of A):
Given B’s sensitivity to reciprocity, ?r
B? 0, we express the psychological part of B’s
(total) utility function when she is sensitive to reciprocity as in Battigalli (2007), i.e. as
the product of B’s sensitivity to reciprocity, her perceived A’s kindness and A’s material
payo¤.6Then, the total expected utility function of B when she is (also) concerned with
5We point out a subtle issue. Here we follow BD and assume that the kindness of a player depends
on his beliefs and his actual strategy. This means that observing a player’s behavior may force a change
in the perception of his kindness. Battigalli (2007) note that kindness should be more appropriately
modeled as a function of a player’s beliefs about the other’s behavior and about his own behavior.
When coupled with the trembling hand logic of sequential equilibrium (according to which deviations
are unintentional and players never change their beliefs about the beliefs of others), this would imply
that the perception of the co-player’s kindness is always the same, on and o¤ the equilibrium path, thus
trivializing the equilibrium analysis of sequential reciprocity.
6Battigalli (2007) himself acknowledges that several functional forms could be able to adequately
represent the psychological preferences of a player concerned with intention-based reciprocity. Among
other reasons, we chose to use his formulation because of its simplicity and tractability.
uB((?B;sB);?B) = ?B(?B;sB) + ?r
B? EB[KA;?B] ? ?A(?B;sB)
Given that we assumed A being a self-interested player, we have ?r
A= 0, hence again
A’s total utility function reduces to his material payo¤.
Next, let us concentrate on the psychological utility of player B. According to Bat-
tigalli’s (2007) formulation, A’s Kindness when he chooses Continue is
KA(Continue;?A) = 2?A+ 4(1 ? ?A) ?1
2(2?A+ 4(1 ? ?A) + 1) =3
and so B’s perceived Kindness when A chooses Continue is
? B’s total utility after (Continue;Share) is given by
uB((Continue;Share);?B) = 2 + ?r
? 2 = 2 + ?r
B(3 ? 2?B)
? B’s total utility after (Continue;Take) is given by
uB((Continue;Take);?B) = 4 + ?r
? 0 = 4
Using the same procedure, we can calculate B’s total utility after Dissolve, i.e.
uB((Dissolve);?B) = 1 + ?r
which is no greater than 1, given that ?r
The psychological game representing the trust game with a self-interested truster
and a reciprocity concerned trustee is depicted in Figure 3. Again, what appears at the
terminal histories should be thought of as utilities, not as material payo¤s, although the
7From now on, we denote with the ‘bold’ labels Take and Share player B’s strategies ‘Take if
Continue’ and ‘Share if Continue’ respectively.
two notions coincide for one of the terminal histories.
2 + ?r
B(3 ? 2?B)
1 + ?r
Figure 3 Trust Game with reciprocity.
2.3. Psychological games: some misconception dispelled
In this subsection we want to analyze more deeply some of the features of our psycho-
logical game framework in order to make the main concepts clearer and to dispel some
misconceptions that could emerge while comparing psychological game theoretical tools
to the well-known features of traditional game theory. We try to answer the questions
that could be asked by a reader with a good background in game theory that approaches
for the …rst time the games depicted in Figure 2 and in Figure 3. We bear in mind the
main assumptions of the general framework of psychological games as described in BD.
Question 1. In Figure 2, why do we suppose that player B cares about A’s disap-
pointment even though A is just an expected material payo¤ maximizer?
Answer. Our analysis is descriptive, not prescriptive. Utility functions are meant
to help analyzing players’ behavior. They describe hypothetical preferences (i.e. condi-
tioned to some hypothesis). In that sense, utilities in Figure 2 are only ‘instruments’.
We do not use them to represent players’ happiness at the end of the game. It is pos-
sible that A feels disappointment but his behavior is not a¤ected by the anticipation of
such disappointment. Nonetheless, even though A’s feelings do not in‡uence his own
behavior, it is possible that B cares about them.
Moreover, for the trust game it can be shown:
? that the qualitative analysis does not change if we put disappointment in A’s utility
? that the set of pure strategy equilibria of the psychological game is the same both
in case we add A’s disappointment in his utility function and in case we do not
Hence, we choose not to consider it in order to simplify both the equilibrium calcu-
lations and the analysis of the utility functions of the game.
Question 2. Given the high number of unknown parameters in a psychological game,
under which conditions can we say that we are in a situation of complete information?
Answer. If we assume that the material game form in Figure 1 is common knowl-
edge, and that players are expected material payo¤ maximizers and this is also common
knowledge, then the trust game in Figure 1 has complete information. If we assume that
players have psychological (belief-dependent) preferences, then it is more reasonable to
allow for incomplete information, which comes from two sources:
(i) player i does not know which feeling j is sensitive to or, even if i knows what is the
feeling, j does not know i knows that and so on. In our trust game, this happens
for example if A does not know whether B is only self-interested, or (also) sensitive
to guilt, or to reciprocity or to both. This means that A does not know if he is
playing the trust game in Figure 1, that one in Figure 2, that one in Figure 3 or
a trust game in which B’s total utility after (Dissolve) is 1 + ?r
(Continue;Take) is 4 ? ?g
i.e. a psychological game which includes both guilt and reciprocity of player B.8
B2?Band after (Continue;Share) is 2 + ?r
B(3 ? 2?B),
(ii) even if i knows which feeling j is sensitive to, i does not know j’s sensitivity to
that speci…c feeling. This happens, for example, in case A and B know they are
playing the psychological game in Figure 2, but A does not know the value of ?g
In strategic settings in which the two players do not know each other very well, con-
dition (i) applies. In all these cases, we deal with a Psychological game with Incomplete
Now suppose that we are in a setting in which condition (i) does not apply, i.e. the set
of belief-dependent motivations to which each player is sensitive to is common knowledge
among players. This is the case, for example, in which the game structure in Figure 2
8Another case would be that A knows with certainty that B is sensitive to, say, guilt aversion, but
B does not know A knows that.
9Another case would be that in which A and B know they are playing the psychological game in
Figure 2, A knows the value of ?g
B, but B does not know that A knows it.
is common knowledge. Then, we say that the trust game in Figure 2 is a Psychological
Game with Complete Information when ?g
Bis commonly known among players. In case
Bis not commonly known among players, we deal again with a Psychological game
with Incomplete Information.10
In both cases, ?Bis unknown to A and is endogenous. Therefore, despite information
is complete, in a psychological game there is still a source of uncertainty, that is not solved
even after the end of the game. This is because A will not know the realized value of
B’s conditional second-order belief in any of the three end nodes of the game of Figure
Question 3. According to BD, an easier way to represent and analyze the psycholog-
ical game in Figure 2 is to write B’s utility after (Continue;Take) as 4 ? ?g
given that B does not know A’s …rst-order beliefs about her strategy, suppose that she
maximizes the expected value of the previous expression. In other words, B maximizes
the expected value of a state-dependent utility function. It may be argued that it does
not make any sense to insert a state variable in a utility function if that state is not
revealed ex post or if that utility is not ‘experienced’ ex-post. This is because even after
the end of the game, B would not know the exact value of ?Aand so she would not
know her exact total utility after (Continue;Take). Therefore, a reasonable question
could be: does it make sense to express B’s belief-dependent motivations by inserting
?Ainto B’s utility function?
Answer. As emphasized above, the utility functions are analytical tools used to
explain players’ behavior, not to measure their happiness: uB does not represent the
utility ‘experienced’ by B. In the simple trust game we analyze, we just want to describe
the behavior of player B, who maximizes the expected value of her total utility (material
payo¤ and psychological utility). Therefore, it is not necessary for the state variable or
the induced utility to be indeed ‘experienced’ ex-post. Their only role is to help us
analyzing players’ motivations. Doing that by using (initial) …rst-order beliefs is much
simpler than using (conditional) second-order ones.
Question 4. What is the di¤erence between psychological games (with complete or
incomplete information) and traditional games with incomplete information?
Answer. The payo¤s of a (traditional) game with incomplete information depend on
players’ actions and on an exogenous parameter representing the state of nature, about
which players are asymmetrically informed. In psychological games, players’ beliefs in
10The same de…nitions can be extended to the trust game in Figure 3, given that ?r
knowledge or not, respectively.
the utility functions are not (exogenous) parameters, but rather endogenous variables.
Consider the psychological game in Figure 2: ?Bis endogenous because ?Ais an en-
dogenous variable and any belief on an endogenous variable is endogenous. ?Bis a
conditional second-order belief (conditional on A choosing Continue) and, at the same
time, a terminal belief. Moreover, what B chooses to do is endogenous, but, by con-
struction, ?Bis not in‡uenced by B’s choice: B’s beliefs on her opponent’s beliefs and
actions do not depend on what she decides to do after A has chosen Continue.
2.4. Solving a Psychological Game with Complete Information
In this subsection, we show how to solve a Dynamic Psychological Game with Complete
Information. More speci…cally, we will solve the two psychological games in Figure 2 and
in Figure 3, supposing respectively that ?g
Bare commonly known among players.
Although it could seem unrealistic to suppose that each others’ feelings sensitivity is
common knowledge, this assumption is needed as a …rst step in order to predict players’
behavior when they hold belief-dependent motivations.
In general, the equilibrium concepts we use in order to solve psychological games
are not ‘di¤erent’ from those used in traditional game theory. Starting from GPS, BD
simply apply the same logic as Nash Equilibrium to solve static psychological games and
the standard logic of backward induction and of the (traditional) sequential equilibrium
to solve dynamic psychological games. They are somehow ‘forced’ to generalize the
previous concepts in order to consider the fact that in psychological games beliefs enter
players’ utility function. That creates two additional problems that need to be taken
into account when dealing with equilibrium behavior:
(a) the condition of correctness of beliefs requires also correctness of stated utilities.
Given that di¤erent orders of beliefs are involved in the analysis, we have to ex-
plicitly impose that they are correct in equilibrium (for example, in each of the
game in Figure 1, 2 and 3, it must be ?A= ?Bin equilibrium). Nonetheless, when
checking that in equilibrium players maximize their total utilities at all decision
nodes given their correct beliefs about one another’s actions, we must take into
account that some of these beliefs (independently of the order) are part of (some
of) the utilities of (some) players. Hence, in equilibrium, players maximize their
total utilities measured according to the correct beliefs they hold on actions and
beliefs of their opponents and, at the same time, (correct) beliefs of di¤erent orders
match players’ best replies calculated according to the ‘correctly’ stated total util-
ities. This issue is addressed by GPS and generalized by BD, who allow for both
own beliefs and beliefs of others in the (total) utility function; they also provide
a general framework able to account for conditional beliefs of whatever order in
players’ (total) utility function.
(b) as the play unfolds, beliefs about the beliefs of others are revised; but players’
beliefs updating leads also to players’ utilities updating.
Building on the theory of hierarchies of conditional beliefs due to Battigalli and
Siniscalchi (1999), BD model a universal belief space that accounts for updating
beliefs about others’ beliefs. The idea is that, in a psychological game, in order to
decide on the best course of action, player i may need to form (conditional) beliefs
about the in…nite hierarchies of (conditional) beliefs of other players, because they
enter her psychological utility function. For example, in the trust games in Figure
2 and Figure 3, since ?Benters B’s psychological utility function, in order to decide
if she prefers to Take or to Share after Continue, B may need to form conditional
beliefs about the …rst-order beliefs of A. But when B updates her second-order
beliefs after A has chosen Continue, her utility from choosing Take after Continue
is also updated.
GPS propose a notion of psychological equilibrium that takes into account (a), but
does not solve (b), since they allow only initial pre-play beliefs to enter players’ util-
ity functions. Hence, GPS’s notion of equilibrium cannot encompass belief-dependent
motivations as those considered here.
Building on the previous considerations, BD propose a notion of Psychological Se-
quential Equilibrium (PSE henceforth), which generalizes the sequential equilibrium
concept of Kreps and Wilson (1982). As stated in BD, ‘Kreps and Wilson (1982) argue
that an appropriate de…nition of equilibrium in extensive form games must refer to as-
sessments, that is, pro…les of (behavior) strategies and conditional (…rst-order) beliefs.
They formulate a de…nition of sequential equilibrium in two steps: …rst, they put forward
a consistency condition for assessments, and then they stipulate that an assessment is
a sequential equilibrium if it is consistent and satis…es sequential rationality’ (p. 18).
BD follow a similar two-step approach, adding to it a third requirement concerning the
higher-order beliefs that need to be speci…ed in psychological games. This third condi-
tion is analogous to a condition used by GPS. Essentially, it requires that players hold
at each node common, correct beliefs about each others’ beliefs. This implies that in
equilibrium players do not change their beliefs about the beliefs of the opponents.11
BD’s PSE is the equilibrium concept that we use to solve (static and) dynamic
psychological games; it reduces to GPS’s equilibrium when dealing with their restricted
class of psychological preferences, i.e. when the utility of a terminal node for a player
depends only on the hierarchy of initial beliefs of that player. BD also show that
the equivalence can be extended to some dynamic psychological games not covered by
GPS, i.e. those where the initial (but not the updated) beliefs of others enter the
utility function. Nonetheless, this extended equivalence result only concerns sequential
equilibria, and BD argue that the non-equilibrium analysis of psychological games is
important. Moreover, their de…nition of sequential equilibrium makes essential use of
conditional higher-order beliefs whereas GPS have to take an indirect route whereby
they look at the subgame perfect equilibria of a …ctitious standard game associated to
initial belief hierarchies that satisfy a …xed point condition.12
In the next subsections, we will use the PSE as solution concept for our simple
psychological game of trust, pointing out, whenever possible, the di¤erences with the
Psychological Subgame Perfect Equilibrium à la GPS.
2.4.1. PSE of the Trust Game with Guilt Aversion
Let us …rst …nd the set of PSE for the psychological game in Figure 2, according to all
possible values of the positive parameter ?g
B. We will focus on the equilibrium values
of the key beliefs ?A, ?B(already de…ned above) and ?B = PrB[Continue]. Since in
equilibrium beliefs are correct, the …rst-order belief ?Amay be interpreted as the mixed
strategy of B, and the …rst-order belief ?Bmay be interpreted as the mixed strategy of
A. Furthermore, the correctness condition also imply that in equilibrium ?A= ?B.
B2 (0;1), it is
uB((Continue;Take);?B) > uB((Continue;Share);?B)
11BD argue that this feature is implicit in the sequential equilibrium concept as de…ned for standard
games, although they …nd it objectionable.
12The GPS equilibrium concept for dynamic psychological games requires correctness of initial pre-
play beliefs and that players optimize their beliefs and choices at all given decision nodes. In both games
in Figure 1 and Figure 2, B’s payo¤ depends directly on ?B, and it is necessary to impose explicitly that
the beliefs ?Band ?Bare correct in equilibrium, i.e. they match the (behavior) strategy of player B.
To this end, note that given ?B, both psychological games have real numbers characterizing payo¤s at
each end-node. In this sense, they reduce to a ‘standard game’. Therefore players must play a subgame
perfect equilibrium in this (correct) belief-reparameterized game.
In games with imperfect information about past moves GPS look at the sequential equilibria of the
…ctitious standard game.
Thus, B always chooses Take and, going backward, A chooses Dissolve; hence, the
only PSE pro…le of beliefs is ?B= 0;?A= ?B= 0.
B2 [1;+1), it could be uB((Continue;Take);?B) 7 uB((Continue;Share);?B)
depending on the value of ?B.
Let us …rst look at ‘pure strategy’13equilibria.
For ?B= 0, it is uB((Continue;Take);?B) > uB((Continue;Share);?B), hence
B chooses Take and, since A holds correct beliefs in equilibrium (?A= 0), he chooses
Dissolve. Hence, ?B= 0, ?A= ?B= 0 is a PSE in beliefs.
For ?B= 1, it is uB((Continue;Take);?B) < uB((Continue;Share);?B), hence B
chooses Share and, since A holds correct beliefs in equilibrium (?A = 1), he chooses
Continue. Hence, ?B= 1, ?A= ?B= 1 is a PSE in beliefs.
Let us now look for mixed equilibria.
B is indi¤erent and can optimally play a mixed strategy if and only if 4??g
mixed strategy equilibrium it must be ?A= ?B=
B. By the requirement of correctness of …rst and second-order beliefs, in every
B.14What about A? Given that
> 1, i.e. ?g
B2 (1;2), he chooses
B= 2. By the requirement
Dissolve if ?g
of correctness of …rst-order beliefs in equilibrium, it is ?B= 1 if ?g
Keeping in mind that in every equilibrium we require (by de…nition) ?A= ?B, we
summarize in the following table all the possible PSE of the psychological game in Figure
2, for all values of the positive parameter ?g
B, A chooses Continue if 2 ?
B2 (2;+1) and he randomizes if and only if ?g
B+ 0 ?
B2 (1;2), ?B= 0 if
B2 (2;+1) and ?B2 [0;1] if ?g
B2 (0; 1)
B2 (1; 2)
?B= 1; ?A=
?B2 [0;1]; ?A=1
?B= 0; ?A=
Table 1 Psychological Sequential Equilibria, given a guilt averse B
13We use the word strategy in an improper way, since we are looking for equilibria in beliefs. The
word pure, in fact, refers not only to strategies, but also to beliefs, in the sense that ?A;?B;?B2 f0;1g.
14In order for the equilibrium to be in mixed strategies it must be ?g
the pure strategy Share. Thus, we look for mixed strategy equilibria only for ?g
B6= 1, otherwise B would play
Note that (Dissolve;Take) is always an equilibrium, but if B is su¢ciently sensitive
to guilt (?g
B? 1) there are multiple equilibria. Is there a reasonable selection criterion?
Like in Dufwenberg (1995, 2002), consider the following intuitive ‘psychological forward
induction’ argument: it is rational for A to trust B only if he assigns at least 50%
probability to strategy Share. In other words, A’s optimal decision depends on B’s
choice: he has to compare the expected payo¤ of choosing Continue, that is 2?A, with
the (certain) payo¤ of choosing Dissolve, that is 1. Therefore, a rational A chooses
to trust B and so to Continue the partnership only if it is ?A?
B observes the action Continue, and if he believes A is rational, then it must be the
2. Then, whenever
case that ?B?
on A choosing Continue. The argument cannot be recast using GPS’s theory since it
2. The logic of this argument depends on belief ?Bbeing conditional
allows only initial beliefs, but it can be captured in BD’s framework, since it allows all
hierarchies of conditional beliefs15and considers a slightly di¤erent concept of sequential
The forward induction story involves somehow a change of beliefs, and, above all, the
beliefs entering the utility functions are those held by B when his node is reached (thus
not only the initial ones, as it was in GPS). Such a criterion, in which players strategically
a¤ect their opponents’ beliefs, has an even more pervasive strength in psychological
games than it has in traditional games, because of the direct e¤ect of beliefs on utilities.
We conclude this subsection with an important remark: even if A has the opportunity
to signal his own beliefs and to force B to hold certain second-order beliefs, it would be
incorrect to infer that A chooses his own beliefs along with his own strategy. A a¤ects
?B(a conditional second-order belief of B) and in equilibrium it must be the case that
?A= ?B, but A takes ?Aas given.16
2.4.2. PSE of the Trust Game with Reciprocity
Next we compute the set of PSE for the psychological game in Figure 3, for all possible
values of the positive parameter ?r
15Battigalli and Siniscalchi (2002) characterize the forward induction logic in term of strong belief in
rationality: the intuitive idea is that when players update their beliefs, they do this in a way that is
consistent with the assumption of rationality of their opponents. They try to ‘rationalize’ the opponents’
actions, that is to infer, from their choices, the beliefs of their opponents. These beliefs are such that the
observed moves are best responses to them. BD show how this forward induction logic can be extended
to psychological games.
16Similarly, in the perfect equilibrium of a leader-follower game, the leader a¤ects the action of the
follower, which is a best response, but the equilibrium choice of the leader takes as given the best-
response strategy (contingent choice) of the follower.
uB((Continue;Take);?B) > uB((Continue;Share);?B)
?, it is
Thus, B always chooses Take and, going backward, A chooses Dissolve; hence, the
only PSE in beliefs is ?B= 0;?A= ?B= 0.
3, it is
uB((Continue;Take);?B) = uB((Continue;Share);?B)
uB((Continue;Take);?B) > uB((Continue;Share);?B)
for ?B2 (0;1]
for ?B= 0
Therefore, ?B= 0;?A= ?B= 0 is again the only PSE in beliefs.
3;2?; no ‘pure’ PSE exists. The intuitive reason is that the more B thinks
that A expected him to Share, the less B perceives A as kind, the higher B’s incentive
to Take. Recall that in any PSE we have ?A = ?B. The indi¤erence condition for
B, given that A chooses Continue, yields ?B=
of A’s kindness, (3
2? ?B), is high and so she prefers to Share. Since ?A has to be
correct (?A= ?B), this would imply ?A= 1, which contradicts ?A= ?B<
to a similar contradiction. The mixed equilibrium is easily obtained as a function of
Bfrom the indi¤erence condition ?B=
taking the best response of A to ?A.
B2 (2;+1), it is
perception of A’s kindness, Therefore, the only PSE in beliefs is ?B= 1, ?A= ?B= 1.
In order for ?B= 0;?A= ?B= 0 to be a PSE in beliefs it must be ?A= ?B>
which is impossible, given that ?A;?B2 [0;1].17
Therefore, if B is su¢ciently sensitive to reciprocity (?r
B: If ?B<
B, B’s perception
B, then B’s perception of A’s kindness is low and she prefers to Take, leading
and the equilibrium condition ?A= ?B,
> 1. Then, B prefers to Share, independently of her
B2 [2;+1)), (Continue;Share)
is the unique equilibrium. This is because, according to Dufwenberg and Kirchsteiger
(2004) formulation of perceived kindness, player A is perceived as kind by B when he
chooses an action ‘increasing’ B’s payo¤, even in case A’s own payo¤ increases too. If
B is highly sensitive to reciprocity (?r
Bhigh), she does not mind if the action chosen
by A leads to a (relatively) higher material payo¤ also to him. She looks only at her
own payo¤ and if it is higher than the (equitable) payo¤ that she believes it would be
possible to ‘receive’ from A, then she reciprocates by playing the ‘kind’ action Share,
independently of her perception of A’s kindness.
17For the limit case ?r
B= 2, the unique PSE in beliefs is again ?B= 1, ?A= ?B= 1.
Keeping in mind that in every equilibrium we require (by de…nition) ?A= ?B, we
summarize in the following table all the possible PSE of the psychological game in Figure
3, for all values of the positive parameter ?r
Table 2 Psychological Sequential Equilibria, given a reciprocity concerned B
Yes No No
?B= 0; ?A=
?B2 [0;1]; ?A=1
?B= 1; ?A=
Finally, we remark that these results depend on the adopted de…nition of perceived
kindness, which is inspired by Dufwenberg and Kirchsteiger (2004). Rabin (1993) pro-
poses a di¤erent de…nition, which yields di¤erent psychological utilities and partially
di¤erent equilibria. In particular, for no value of the (positive) parameter ?r
B’s sensitivity to reciprocity the equilibrium (Continue;Share) exists. This is because
according to Rabin’s formulation A is perceived as kind by B only when he ‘increases’
B’s payo¤ by decreasing his own payo¤. When the action chosen by A increases both
his payo¤ and the payo¤ of B, he is perceived as ‘neutral’ by B, who does not play the
‘kind’ action Share, even in case she is highly sensitive to reciprocity.
2.5. The ‘incomplete information’ case: some hints
It is probably not realistic to assume that players know one another’s psychological
propensities. Unless one models interaction within a family (or amongst friends) and
there is not an ex-ante stage in which all players’ feelings sensitivities are (elicited and)
transmitted to their co-players, we are dealing with a psychological game with (some
form of) incomplete information.
In order to extend the analysis of psychological games to include incomplete infor-
mation, let ? = (?A;?B) denotes a vector of parameters that summarize all the payo¤-
relevant aspects of the game that are not common knowledge; ?i(i = A;B) is a compo-
nent known to player i only (such as his sensitivity to certain psychological motivations).
It is common knowledge that ? belongs to a parameter space ? = ?A??B. Elements of
? are called states of Nature. As underlined by BD, in general, in dynamic psychological
games, ‘one can assume for simplicity that players do not get more re…ned information
about the states of Nature as the play unfolds, they only observe the actions chosen in
previous stages of the game’ (p. 37). In the simple two-stage game we analyze we can
maintain this assumption without loss of generality. It is relatively easy to generalize
our construction of the belief space in order to include beliefs about the state of Nature.
However, since states of Nature are exogenous, also players’ hierarchies of initial beliefs
about the state of Nature are exogenous, and the model may specify assumptions about
such exogenous beliefs.
With this extended framework in place, one can regard the trust game with guilt
aversion and reciprocity of B analyzed in the previous subsection as psychological games
with incomplete information.
Attanasi, Battigalli and Nagel (2007) provide a speci…c framework in order to analyze
and solve (one-stage) psychological games with incomplete information. In the simple
trust game under consideration (assuming that ?A= 0 for each possible belief-dependent
motivation and that this is common knowledge), none of the parameters ?g
known to A: he supposes that (?g
B) is drawn from a joint distribution. Thus, A does
not know the true values of ?g
B, but has a prior on each of the two. With this
extended framework in place, they can regard trust games in which there is not an ex-
ante communication of feeling sensitivities between truster and trustee (standard case)
as psychological games with incomplete information.
Attanasi and Nagel (2006) and Attanasi, Battigalli and Nagel (2007) remark that
another source of incomplete information has to be taken into account. It is possible
that psychological utilities of players sensitive to guilt and/or to reciprocity are not
linear, di¤erently from what we assume throughout this chapter. Finally, Attanasi and
Nagel (2006) shows that repeated psychological games with incomplete information have
a non-standard signaling game structure.
3. Review of the Experimental Literature on Psycho-
In this section, we review and interpret some experimental papers giving support to
theories of belief-dependent motivations. We mainly concentrate on social dilemma
games belonging to the same family of the one presented in section 2. We divide those
contributions into two subgroups: in section 3.1 we discuss the two main behavioral
theories aiming to explain the causal relations between beliefs and actions; in section
3.2 we present in deeper detail the main experimental studies analyzing the relations
between emotions (expressed as belief-dependent motivations) and actions.
3.1. Relationship between Actions and Beliefs
Strategic settings in which players’ private interest con‡icts with their collective interest
(like trust games) have been studied both by experimental economists and psychologists.
One regularity among all these studies involves the relationship between individual’s
beliefs and actions: in particular, expectations of others’ cooperation and one’s actual
cooperation are robustly and positively correlated. However, two competing classes of
(causal) theories have been proposed to explain this relationship.
The …rst class of theories, coming from economics, suggests that beliefs cause ac-
tions. These theories are de…ned by Croson and Miller (2004) reaction theories, since
individuals choose actions to react to beliefs.
The second class of theories, proposed by psychologist, suggest that actions cause
beliefs. There is a number of di¤erent speci…c theories in this class: Croson and Miller
(2004) provide a critical analysis of many of them. They conclude that, whichever
of these theories one adopts, they all have the similar property that expectations are
projected from own (anticipated) behavior. They refer to these theories as projection
There are several studies (e.g. Messick et al. (1983), Schroeder et al. (1983) and
Weimann (1994)) showing that manipulated expectations can a¤ect choices, thus pro-
viding evidence to reaction theories. However, other studies support projection theories.
Yamagishi and Sato (1986), for example, argue that choices are justi…ed ex-post by ad-
justing expectations. Dawes et al. (1977) state that the di¤erence between the variance
of the players’ estimates of others’ actions and the observers’ estimates of others’ actions
demonstrates ‘false’ consensus.
Almost all the experimental works on the relation between actions and beliefs in
social dilemmas have assumed one theory or another in the experimental design (for
example, psychologists tend to elicit beliefs after individuals make decisions in strategic
settings, economists before). However, none of these previous experiments were designed
to explicitly distinguish between these competing classes of theories. Starting from the
fact that both classes of theories are consistent with the existing evidence of a positive
relationship between beliefs and actions in social dilemmas, Croson and Miller (2004)
design a particular social dilemma experiment in order to explicitly provide a clean
separation of these two classes of explanations: their results are consistent with the
reaction hypothesis as opposed to the projection hypothesis. This is not to conclude
that people don’t project in dilemma settings, but that in settings where reaction and
projection predict di¤erent outcomes, reaction has the stronger e¤ect.
That would lead us to conclude that, introducing belief elicitation in an experimental
setting, the in‡uence of belief elicitation on subsequently chosen actions should be greater
in size with respect to the one that actions could have on subsequent beliefs. However,
the experimental literature surveyed in the next subsection, among other things, shows
that in trust games (a particular case of social dilemma games) the in‡uence of belief
elicitation (and transmission) on subsequent chosen actions is small: di¤erent kinds of
‘belief manipulation’ do not a¤ect players’ behavior.
Nonetheless, our prediction is that reaction theories and projection theories should
apply even more strongly to psychological games, because of a structural ‘casual relation
system’: actions cause beliefs (projection); since beliefs are part of players’ psychological
utilities, when they update as the play unfolds, players’ (total) utilities update; therefore,
the optimal action is chosen according to the ‘new’ updated utilities. Hence, beliefs cause
actions (reaction) also through the psychological utilities.
3.2. Relationship between Actions and Beliefs-dependent Moti-
In this subsection, we mainly concentrate on the relations between actions and feelings
in trust games.18In the review below, we also analyze the techniques of belief elicita-
tion and transmission used in the previous experimental settings to investigate players’
sensitivity to some feelings. Belief elicitation is essential to measure the in‡uence of
belief-dependent motivations on players’ behavior in psychological games with incom-
plete information. Nonetheless, feeling elicitation and transmission among players is
essential to represent a psychological game with complete information in a laboratory.
We …rst analyze the main belief elicitation techniques in the experimental literature.
Then, we discuss some feeling elicitation and transmission techniques.
Dufwenberg and Gneezy (2000) analyze the relevance of trust responsiveness19in a
18In order to ensure uniformity while summarizing the di¤erent experiments, in each trust game
described in this section, we call truster the player who has to choose to place trust (or not) and trustee
the player who has to choose to ful…ll trust (or not).
19Trust responsiveness (or the self-ful…lling property of trust) is ‘a tendency to ful…l trust because
you believe that it has been placed on you’ (Bacharach, Guerra and Zizzo (2001), p. 1). Actually,
Dufwenberg and Gneezy (2000) talk about ‘let-down aversion’, which could be thought of as synonymous
simple trust game, in which the truster may take x 2 [0;20] Dutch guilders, or leave them
and let the trustee split 20 guilders between them. After the truster and the trustee
have simultaneously made their choices (strategy method), there is beliefs elicitation
both from truster (…rst-order beliefs) and from trustee (second-order beliefs), but there
is no belief transmission. They conclude that trust responsiveness, which predicts that
trustees who have higher guesses will be more likely to ful…l trust, should hold even if
belief transmission did not take place.20This result is not in line with Croson (2000),
which shows how belief elicitation distorts behavior in the context of social public goods
and prisoner dilemma experiments.
Morrison and Rutstrom (2002) explore the possibility that psychological games are
associated with the relationship between prior beliefs about others’ behavior and the
actual revealed actions of those individuals. In other words, they investigate whether
a surprise or a con…rmation of prior beliefs have ‘payo¤’ consequences. The authors
develop and implement an experimental design that allows for belief elicitation in an
investment game, where material self-interest would predict the absence of either trust or
reciprocity. They elicit truster’s …rst-order beliefs before he chooses and trustee’s second-
order beliefs before she learns about the decision made by her partner. Their results,
supporting existing evidence that rejects the Nash equilibrium in monetary payo¤s, are
consistent with Rabin (1993), given that most trustees engage in reciprocity as a way
of repaying kindness by the trusters. They also support somehow Fehr and Schmidt’s
(1999) inequality aversion model. However the strongly signi…cant correlation between
beliefs and amounts sent back (by the trustees) cannot be explained by the inequality
Bacharach, Guerra and Zizzo (2001) test trust responsiveness in basic 2 ? 2 trust
games with di¤erent payo¤ structures. In their experimental setting, beliefs are elicited
in an incentive-compatible way on the part both of trusters and of trustees, using the
same technique of Dufwenberg and Gneezy (2000), but (a descriptive statistics about)
to the subsequently adopted term of ‘guilt aversion’. Comparing Bacharach, Guerra and Zizzo (2001)
and Guerra and Zizzo (2004) with Dufwenberg and Gneezy (2000) and Charness and Dufwenberg
(2005), it seems clear that trust responsiveness and guilt aversion are two sides of the same coin: they
describe the same feeling, the former from a positive point of view and the latter from the negative
one. Moreover, they both predict the same behavior for the trustee: positive correlation between her
second-order beliefs (as statement of truster’s con…dence that she would ful…l trust) and trust ful…lment.
However, the term ‘trust responsiveness’ seems too much ‘context-dependent’, whereas the term ‘guilt
aversion’ looks more ‘general’, being a statement about a motivation that can be extended also to
games outside the family of trust ones.
20Dufwenberg and Gneezy (2000) do not only look at trust games but also at dictator games. They
report evidence in favor of let-down aversion for dictator games too.
trusters’ …rst-order beliefs are transmitted to each trustee, before eliciting her second-
This is common knowledge among all participants. We think that
because of beliefs (elicitation and) transmission from trusters to trustees, Bacharach,
Guerra and Zizzo’s (2001) results may have not been very general. Belief transmission
is motivated in the paper by the desire to enable trustees to form de…nite enough beliefs
about the trusters’ con…dence on them. While this is reasonable, this may have biased
trust responsiveness upwards for at least two reasons: …rst of all, it may draw the atten-
tion of subjects to the truster’s con…dence in them, possibly making any psychological
factor leading to trust responsiveness more salient than otherwise; secondly, it is not
obvious that in real world people typically receive this kind of information. Guerra and
Zizzo (2004) use two simple trust games to measure directly or indirectly the robustness
of trust responsiveness in three experimental treatments: beliefs are not elicited; beliefs
are elicited but not transmitted; beliefs are elicited and transmitted from trusters to
trustees, as in Bacharach, Guerra and Zizzo (2001). Since trusting and ful…lling rates
are quite similar in all three treatments, they conclude that trust responsiveness is ro-
bust with respect to any kind of ‘belief manipulation’, hence strengthening the case for
the real-world signi…cance of trust responsiveness. Di¤erently from Bacharach, Guerra
and Zizzo (2001) and Guerra and Zizzo (2004), we do not agree with the ‘neutrality’
of their beliefs (elicitation and) transmission procedure on players’ behavior, while we
completely agree with the ‘neutrality’ of the beliefs elicitation alone.
In this review of belief-elicitation (and transmission) experimental works, we mainly
concentrated on 2 ? 2 trust games. However, there are some other experimental papers
on belief-elicitation in other kinds of (social dilemma) games, which states interesting
rules aiming to provide the right incentive for subjects in the lab to reveal their true
…rst-order beliefs: Nyarko and Schotter (2002), who test behavior in two-player 2 ?
2 constant-sum games with a unique mixed equilibrium; Costa-Gomes and Weizsäcker
(2006), who test behavior in two-player 3x3 games with a unique equilibrium in pure
strategies; Rey Biel (2006), who tests behavior in two-player 3x3 constant-sum games
with a unique equilibrium in pure strategies and with di¤erent number of rounds of iter-
ated deletion of (strictly) dominated strategies required to reach the Nash equilibrium.
Let us now examine the role of pre-play communication as a technique of feeling
(elicitation and) transmission, able to reproduce psychological games with complete
information in the lab.
A number of experiments (e.g., Dawes et al., 1977) provide evidence that face-to-
21Each trustee receives a report of the mean value of …rst-order beliefs of her non-coplayers.
face communication can greatly improve cooperation, even when the dominant strategy
(with sel…sh preferences) is ‘defection’. However, as Roth (1995) points out, there may
be many confusing and uncontrolled e¤ects in a face-to-face contact.
Clark et al. (2001) show that non-binding pre-play communication moves outcome in
the direction of the Pareto-dominant equilibrium in games which converge to a Pareto-
dominated equilibrium in the absence of communication. They …rst reproduce the ex-
perimental …ndings of Cooper et al. (1990) suggesting an important role for pre-play
communication for the coordination on the Pareto-dominant equilibrium in a simple
two-player coordination game. Then they go on and study other games in which co-
ordination failures take place in the absence of communication, in order to evaluate
the argument in Aumann (1990) according to which the e¤ectiveness of communication
is sensitive to the payo¤ structure. The authors …nd that when players have a strict
preference over the opponent’s choices, communication still in‡uences outcomes, but its
impact is considerably lessened. In line with Aumann’s argument, they also …nd that
agreements to play Nash equilibrium are fragile. Finally, they …nd that when communi-
cation is e¤ective, it does not always work in the direction one might expect, i.e. it does
not necessarily lead to the Pareto-dominant equilibrium.
Charness and Dufwenberg (2006) run one-shot interactions between participants in
a simple experimental trust game (not so di¤erent from the one presented in section
2)22in order to test for guilt aversion. The authors examine the e¤ect of non-binding
pre-play communication (cheap talk from trustee to truster) on players’ trust and co-
operation. Their experimental results suggest that there are di¤erences in the level of
e¢ciency among the various forms of communication. Beliefs are higher when promises
are made. It seems that statements of intent are instrumental in changing perceptions
and facts about what people do. As the authors admit, measuring beliefs is crucial to the
guilt-aversion story and also to the reciprocity model. After collecting the strategic de-
cisions made by participants (they use the strategy method), the authors elicit truster’s
…rst-order and trustee’s second-order beliefs. As the game is one-shot and beliefs are
transmitted only after strategies are chosen, the belief elicitation should not have any
impact on participants’ prior choices. The authors do not allow face-to-face interaction,
but instead allow the trustee to send a free-form message to the truster. This permits
a clean, controlled test of the function of communication: they only permit a single
message from one party to the other. Moreover, as they are interested in whether some
22The simple one-shot trust game they represent in the lab di¤ers from the one in section 2, because,
after the action pro…le (Continue;Share) they allow for a random draw, introduced in order to capture
the essence of ‘hidden action’.
particular type of message has an impact or not, they give the sender a blank piece of
paper to write any (anonymous) message, instead of restricting the message space.
Masclet et al. (2003) test informal sanctions contribution in public good games.
Each agent can express disapproval on others’ contribution decision thus showing how
this non-monetary system is able (as much as monetary sanctions) to increase contribu-
tion levels. Moreover, they stress the fact that punishment may be a particular form of
communication and, in a repeated game, it serves as a form of pre-play communication
for future periods.
Starting from Charness and Dufwenberg’s (2006) intuition on the importance of pre-
play communication in psychological games and from Masclet et al. (2003) result of
a non-monetary punishment scheme able to create ‘psychological pressure’, Attanasi,
Battigalli and Nagel (2007) build a simple mechanism to elicit and transmit agents’
sensitivity to some particular feelings in a psychological game of trust similar to the one
analyzed in section 2. Di¤erently from Charness and Dufwenberg (2006), who introduce
a form of ‘free’ pre-play communication in which the truster can send any kind of signal
(maybe having no relations with social preferences) to the trustee, Attanasi, Battigalli
and Nagel (2007) introduce a well-structured feelings’ elicitation scheme which enables
to elicit trustee’s sensitivity to both distribution-dependent preferences à la Fehr and
Schmidt (1999) and to belief-dependent motivations à la GPS. In particular, they design
a reliable mechanism of elicitation and transmission (from trustee to truster) of a good
approximation of trustee’s feelings sensitivities (to guilt, reciprocity or both). Hence,
when this mechanism is in place it is like the two parameters ?g
Bin Figures 2 and
3 are both public information between the two players. In that way, they are able to
compare players’ behavior in the ‘almost complete information treatment’ with the one
in the ‘incomplete information treatment’, i.e. when they play the (one-stage) game of
trust without any kind of ex-ante belief-dependent motivations transmission. Their ex-
perimental results show that eliciting B’s psychological preferences without transmitting
them to A does not modify players behavior. This could be interpreted as an indirect
proof of the ‘existence’ of the psychological preferences: the authors do not induce them,
they only elicit something that was already in place. Moreover, their experimental results
show that eliciting and transmitting B’s psychological preferences and thus letting the
two players playing the (almost) complete information one-stage psychological game of
trust leads them to behave in a visibly di¤erent manner (with respect to their behavior in
the corresponding incomplete information psychological game setting). More precisely,
the public information of the trustee’s psychological preferences in the one-stage psy-
chological game results in truster’s perception of trustee’s belief-dependent motivations
(like guilt aversion and/or reciprocity) which would be otherwise underestimated. That
in turn ends in a more cooperative behavior for both players. The authors provide also
theoretical …ndings on psychological games with complete and incomplete information
that are well matched by their experimental results.
Using the same feeling elicitation and transmission procedure, Attanasi and Nagel
(2006) test players’ behavior in a …nitely repeated game, in which they elicit beliefs at
the beginning of each period. This is the …rst experimental work allowing for reputation
or repeat-game e¤ects in psychological games too. The authors are able to disentangle
between the (standard) e¤ect of reputation and the (non-standard) e¤ect of public in-
formation of trustee’s feeling sensitivity on players’ behavior. They also provide speci…c
theoretical …ndings for repeated psychological games. These games present several fea-
tures that one cannot …nd in framing and solving other kinds of dynamic psychological
games. They deeply investigate such features. Moreover, by concentrating on repeated
psychological games, they both de…ne speci…c intertemporal psychological utility func-
tions and introduce two (behaviorally derived) monotonicity conditions on the dynam-
ics of players’ beliefs. These conditions allow them to select among the multiplicity
of sequential psychological equilibria coming out by applying Battigalli and Dufwen-
berg (2005) and Dufwenberg and Kirchsteiger (2004) solution concepts for psychological
games with guilt aversion and with reciprocity, respectively. The subset of equilibria
satisfying their monotonicity conditions matches very well the behavior of participants
in their experimental sessions.
In this chapter, we provided the main theoretical insights of the general psychological
game framework, through an example of a simple trust game in which belief-dependent
motivations are involved.
From a theoretical point of view, our conclusion is that there is little reason to assume
that equilibrium coordination is easier in psychological games than in standard games.
In fact, since psychological games often seem more complicated, and since problems of
equilibrium multiplicity are likely to be enhanced in psychological games (as shown in
section 2), assuming equilibrium may be assuming too much, especially in psychological
games. Another reason to feel skeptical about a fully ‡edged equilibrium analysis in
psychological games is the following: it is often argued that players learn to play Nash
equilibrium because, through recurrent strategic interaction, they come to hold correct
beliefs about the actions of the opponents. This is not enough for a psychological equi-
librium: since utilities depend on hierarchical beliefs, players would have to be able to
learn the beliefs of others, but, unlike actions, beliefs are typically not observable ex post.
GPS, the …rst paper on psychological games, restrict attention to equilibrium analysis,
but in many strategic situations there is little compelling reason to expect players to
coordinate on an equilibrium, hence one may wish to explore alternative assumptions.
However, giving up the equilibrium assumption does not necessarily mean giving up on
predictive power. Following this direction, BD develop a framework for analyzing in-
teractive epistemology in psychological games, without postulating equilibrium play. In
particular, building on an epistemic theme due to Battigalli and Siniscalchi (2002), they
extend Pearce’s (1984) classical notion of (extensive form) rationalizability to psycho-
logical games. The concept captures psychological forward induction in simple games
like those in Figure 1 and Figure 2 (section 2), and in more complicated games for which
long chains of beliefs about beliefs may be needed to get clear-cut predictions.
In the second part of the chapter, we discuss the main experimental works that
directly or indirectly refer to psychological game-theoretic explanations as able to ratio-
nalize their experimental results. In the last two decades, many experimental economists
have studied the relation between actions and feelings in social dilemma games, allow-
ing for ex-post explanations claiming for belief-dependent motivations as source of the
‘irregularities’ in the experimental results. Attanasi, Battigalli and Nagel (2007) are the
…rst to provide a direct test for psychological games in the lab, clearly distinguishing
between psychological games with complete and incomplete information. Using a simi-
lar feelings elicitation and transmission method, Attanasi and Nagel (2006) extend the
analysis of the relationships among actions, beliefs and feelings to dynamic (repeated)
settings: they state the evolution of beliefs, their correlation with the played action pro-
…le and the way through which they incorporate players’ belief-dependent motivations.
These works are only a starting point, although they are indicative of the viability of
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