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de travail
Bureau d’Économie
Théorique et Appliquée
BETA
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@beta_economics
Contact :
jaoulgrammare@beta-cnrs.unistra.fr
«Distance in Beliefs and
Individually-Consistent
Sequential Equilibrium»
Auteurs
Gisèle Umbhauer, Arnaud Wolf
Document de Travail n° 2022 – 37
Décembre 2022
Distance in Beliefs and Individually-Consistent
Sequential Equilibrium
Gisèle Umbhauer∗Arnaud Wolff†
December 2022
Abstract. The concept of Individually-Consistent Sequential-Equilibrium
broadens the concept of Sequential Equilibrium by allowing players to have
different beliefs on potential deviations. This heterogeneity spontaneously
gives rise to a notion of distance between beliefs. Yet, studying distance
between beliefs in a strategic context reveals to be intricate. Announced
beliefs may be different from revealed beliefs and the meaning of distance
depends on the role assigned to beliefs. If out-of-equilibrium beliefs help
getting a larger payoff at equilibrium, then we might need to reconsider the
traditional definition of sequential rationality: more than just requiring that
players behave optimally at every information set given their beliefs and the
strategies played by others players, we might additionally require that there
does not exist another perturbation scheme that is individually-consistent
and which provides higher payoffs to the players.
Keywords: AGM-Consistency, Distance in Beliefs, Heterogeneous Beliefs, Individually-
Consistent Sequential Equilibrium, Revealed Beliefs
JEL: C72
∗Bureau d’Economie Théorique et Appliquée (BETA) - Université de Strasbourg - 61 Avenue de
la Forêt Noire, 67000 Strasbourg - France. e-mail: umbhauer@unistra.fr.
†Bureau d’Economie Théorique et Appliquée (BETA) - Université de Strasbourg - 61 Avenue de
la Forêt Noire, 67000 Strasbourg - France. e-mail: arnaudwolff@unistra.fr.
1
1. Introduction
In this paper, we extend the concept of Individually-Consistent Sequential Equi-
librium (ICSE, Umbhauer and Wolff 2019), which builds on the Sequential Equilibrium
(SE, Kreps and Wilson 1982), a solution concept commonly used to solve extensive-
form games. The SE requires consistency of beliefs at all information sets, even at those
that find themselves out of the equilibrium strategy path. This implies that players are
required to share the same beliefs at out-of-equilibrium information sets, even about the
numerical values of mathematical artifacts used to generate perturbations of strategy
profiles, which are arbitrary by nature. Since there is no a priori basis for requiring
players to agree on the probabilities of other players’ possible mistakes (or deviations),
the ICSE accepts different perturbation systems for different players.
This paper focuses on games with n≥3players since these are the games in which
the ICSE solution concept can differ from the SE. In particular, we focus on games
in which some players might belong to a same social group or a same community.
Therefore, although out-of-equilibrium beliefs are never directly confronted to reality
(so that players can in some sense agree to disagree), players that belong to a same
social group may feel ill at ease when adopting different beliefs. In fact, research in
political science has shown that individuals are often motivated to shift their beliefs
towards the ones associated with the social groups they belong to (Barber and Pope,
2019,Gould and Klor,2019,Slothuus and Bisgaard,2021). Therefore, rather than being
completely arbitrary, beliefs at out-of-equilibrium information sets might be correlated
among players. This leads us to develop a notion of distance between the beliefs of
different players as well as the idea of maximally allowed heterogeneity between the
players’ beliefs.
The notion of distance between beliefs introduced in this paper can not be properly
studied without further delving into the function of beliefs and their intrinsic link to
actions. For instance, a player may publicly declare to hold some beliefs but his actions
contradict the proclaimed beliefs. That is to say, the revealed beliefs of the player are
different from the announced beliefs. The question then arises as to whether we should
measure the distance between the player’s announced beliefs or between their revealed
beliefs. Furthermore, in a game as in real life, the purpose of out-of-equilibrium beliefs
may be to help a player maximize his own payoffs. In this sense, beliefs become strategic
and they may in some way belong to the strategy set of the players. For instance, it
might be in Player 1 and Player 2’s interests to have similar (or different) beliefs so as
2
to incentivize Player 3 to adopt a strategy that maximizes their own payoffs. In this
context, players build their beliefs with a strategic purpose and the distance between
their beliefs becomes irrelevant. These considerations lead us to revisit the notion of
sequential rationality in dynamic games of incomplete information. More than just
requiring that players behave optimally at every information set given their beliefs and
the strategies played by other players, we might additionally require that there does not
exist another perturbation scheme that is individually-consistent and which provides
higher payoffs to the players.
The paper is organized as follows. In Section 2, we describe the concept of ICSE and
discuss the way it introduces heterogeneity in beliefs at out-of-equilibrium information
sets. In Section 3, we compare our concept with other often-used solution concepts such
as PBE or AGM-consistency. In Section 4, we turn to the notion of distance between
beliefs. We present two ways of measuring distance. The first is an order relation
on beliefs, while the second is an Euclidean notion of distance, in that we measure
the minimal payoff perturbations necessary to ensure convergence in beliefs. Yet, the
main difficulties remain elsewhere. In Section 5, we introduce the distinction between
revealed and announced beliefs and discuss the strategic function of beliefs. Section 6
discusses the findings of this paper by revisiting the definition of sequential rationality.
The last section concludes.
2. Individually-Consistent Sequential Equilibrium
In this paper, we consider finite extensive-form games and focus on games with
n≥3players. Let Nrepresent the finite set of players (with typical element n∈N),
Xthe set of non-terminal decision nodes (with typical element x∈X) and Hthe set
of all possible information sets (with h∈Ha specific information set). Let Hi⊆H
denote the set of all possible information sets at which Player imight be called upon
to play. We call i(h)the player playing at hand for every h∈Hi, we note Ahthe set
of actions available to player iat information set h.
Abehavioral strategy for player i, noted πi, is a probability distribution over her
possible actions at each of her information sets. That is, a behavioral strategy for
player iis a member of "h∈Hi∆(Ah). The set of behavioral-strategy profiles is therefore
"i∈N"h∈Hi∆(Ah), with typical element π= (πi)i∈N. A system of beliefs is a function
µ:X→[0,1] such that ∀h∈H, Σx∈hµ(x)=1.
3
The Sequential Equilibrium (SE) requires consistency of beliefs at all information
sets, even at those that find themselves out of the equilibrium strategy path. To generate
beliefs that are consistent at every information set, Kreps and Wilson (1982) require that
the belief vector µbe the limit of a sequence of belief vectors derived from Bayes’ rule
applied to a sequence of fully mixed strategy profiles (strategy profiles that put positive
probability to every action in every information set). Let us denote by "h∈H∆0(Ah)
the set of all fully mixed behavioral strategies. Formally, a pair (µ, π) is consistent if
and only if there exists some sequence (ˆµk,ˆπk)∞
k=1 such that:
(i) ˆπk∈"h∈H∆0(Ah),∀k∈ {1,2,3, ...},
(ii) ˆµk
h(x) = P(x|ˆπk)
Σ
y∈h
P(y|ˆπk),∀h∈H, ∀x∈h, ∀k∈ {1,2,3, ...},1
(iii) πi(h)(ah) = lim
k→∞ˆπk(ah),∀i∈N, ∀h∈H, ∀ah∈Ah,
(iv) µh(x) = lim
k→∞ˆµk
h(x),∀h∈H, ∀x∈h.
A SE is defined to be any pair (µ, π) that is consistent and sequentially rational
(Kreps and Wilson 1982, p.872).
What is crucial in the concept of SE is that the players are required to implicitly
agree on the value of the ϵused to generate perturbations of the strategy profiles. That
is, while the ϵare arbitrary in nature (they only represent mathematical artifacts),
players still need to share the same beliefs about their numerical values. In some way,
it is as if an external player shakes the strategies for everybody. In Umbhauer and Wolff
(2019), we argue that this requirement is too strong. Indeed, we argue that there is
no a priori basis for requiring that players agree on the probabilities of other players’
possible mistakes (or deviations).
Formally, what distinguishes our Individually-Consistent Sequential Equilibrium
(ICSE) concept from the SE is that we do not require the existence of only one sequence
of perturbed strategy profiles on which all players need to agree but allow for different
perturbation systems for different players. In other words, each player jintroduces
his own perturbations on the actions at each information set h∈H. So, ˆπk
j,i(h)(ah)
is the value player jassigns to the probability with which player i(h)plays ahat his
information set h, while ˆπk
jis player j’s profile of perturbed strategies in the whole game.
Of course, for consistency, we require that πi(h)(ah) = lim
k→∞ˆπk
j,i(h)(ah)for all j∈N, so
that each player j’s perturbed strategy profile has to fit with the played actions in the
game.
1With P(x|·)being computed using Bayes’ rule.
4
Therefore, a pair (µ, π) is individual ly-consistent if and only if there exist some
sequences (ˆµk
j,ˆπk
j)∞
k=1, for all j∈N, such that:
(i) ˆπk
j,i(h)∈"h∈H∆0(Ah),∀k∈ {1,2,3, ...},∀j∈N,
(ii) ˆµk
i(h)(x) = P(x|ˆπk
i(h))
Σ
y∈h
P(y|ˆπk
i(h)),∀h∈H, ∀x∈h, ∀k∈ {1,2,3, ...},
(iii) πi(h)(ah) = lim
k→∞ˆπk
j,i(h)(ah),∀h∈H, ∀ah∈Ah,∀j∈N,
(iv) µi(h)(x) = lim
k→∞ˆµk
i(h)(x),∀h∈H, ∀x∈h.
An Individually-Consistent Sequential Equilibrium (ICSE) is any pair (µ, π) that
is both individually-consistent and sequentially rational.
Let us illustrate the consequences of such a concept. In the game in Figure 1, there
does not exist any SE leading Player 1 to play C1(see Appendix 1), so the players can
not reach the Pareto optimal payoffs (5.99, 10, 10). As a matter of fact, to sustain
B2, Player 2 has to believe that Player 1 trembles toward B1at least 4 times more
often than toward A1(µ(x2)≤1
5), whereas to be willing to play B3, Player 3 has to
believe that Player 1 trembles toward B1at most 3 times more often than toward A2
(µ(y2)≥1
4, hence µ(x2)≥1
4). This is not possible in a SE, in that all the players shake
the strategies in the same way. Yet this becomes possible with the ICSE.
What is new, in comparison with the SE, is the fact that players can have different
beliefs at the same out-of-equilibrium information set. So, in the above example, we can
set: µ2(x2) = 0.1,µ2(x3) = 0.9for Player 2, and µ3(y1) = 0,µ3(y2) = 0.3,µ3(y3) = 0
and µ3(y4)=0.7for Player 3. This implicitly means that Player 3 assigns the belief
0.3 to x2and the belief 0.7 to x3, given that Bayes’ Rule requires that Player 3’s beliefs
are consistent: µ(y2) = µ(x2)and µ(y4) = µ(x3). This is due to the fact that each
player shakes the strategies in the way he wants. So, for example, Player 2 may have
in mind the perturbed strategy profile {(1 −ϵk−9ϵk)C1+ 9ϵkB1+ϵkA1,(1 −ϵk)B2+
ϵkA2,(1 −ϵk)B3+ϵkA3}, while Player 3 may have in mind {(1 −3ϵk−7ϵk)C1+ 7ϵkB1+
3ϵkA1,(1−ϵk)B2+ϵkA2,(1−ϵk)B3+ϵkA3}.2Heterogeneous beliefs at out-of-equilibrium
information sets can therefore sustain the Pareto optimal payoffs at equilibrium in this
strategic context.
2Player 1’s perturbations have no impact on the game, so we can suppose that he has the same
profile of perturbed strategies as Player 2 for example.
5
Player 1
x1
5.99
10
10
C1
x3
y4
0
6
0
B3
6
3
1
A3
B2
y3
0
5
3
B3
1
4
4
A3
A2
B1
x2
y2
1
0
4
B3
0
1
1
A3
B2
y1
6
4
2
B3
6
2
3
A3
A2
A1
Player 2 (h)
Player 3 (h’)
Figure 1: An example of the distinction between ICSE and SE.
3. Connections between ICSE and other solution concepts
3.1. Links between ICSE, SE, SPNE, PBE and SCE
Few solution concepts support the idea that people may share different beliefs at
out-of-equilibrium information sets.3In fact, researchers tend to require that "players
can not agree to disagree", the logical result stemming from Aumann (1976)’s paper.
Yet, in our context, there is no true state to discover since the beliefs are about de-
viations that will never occur at equilibrium. So Player 2 (respectively Player 3) can
durably think that, if they should face a deviation, then surely Player 1 played A1with
a probability lower than 1/5 (respectively with a probability larger than 1/4). Nothing
will contradict their beliefs given that Player 1 never deviates.
In the following Proposition, we enumerate the existing links between our ICSE so-
lution concept and other well-known and often-used solution concepts, such as Subgame-
Perfect Nash Equilibrium (SPNE), Sequential Equilibrium (SE), Perfect Bayesian Equi-
librium (PBE) and Self-Confirming Equilibrium (SCE).
3We thank Giacomo Bonanno for informing us that Greenberg et al. (2009) developed a similar
idea in their MACA concept.
6
Proposition 3.1. (i) The set of ICSE is included in the set of Subgame-Perfect Nash
Equilibria (SPNE).
(ii) By construction, the set of SE is included in the set of ICSE, so the existence of
an ICSE in a finite extensive-form game follows from the existence of a SE in a
finite extensive-form game.
(iii) The set of ICSE is equal to the set of SE in a two-player game.
(iv) There is no inclusion relation between the set of Perfect Bayesian Equilibria (Fu-
denberg and Tirole,1991) and the set of ICSE.
(v) The set of ICSE is included in the set of Self-Confirming Equilibria (Fudenberg
and Levine,1993).
Proof. In Appendix 2. ■
3.2. Links between ICSE and AGM-consistency
The ICSE also shares some links with the concept of AGM-consistency (Bonanno,
2013,2016). AGM-consistency introduces a plausibility order on stories of actions and
belief revision is based on this plausibility. This plausibility concept grants a large
degree of freedom to the way beliefs are computed after deviations; this liberty differs
from heterogeneous perturbation systems but it shares a partial link with the ICSE.
The following Proposition describes these links.
Proposition 3.2. i) The set of ICSE beliefs is almost included in the set of AGM-
consistent beliefs.
ii) There is no inclusion relationship between the set of ICSE beliefs and Bonanno’s
Perfect Bayesian Equilibrium beliefs.
Proof. In Appendix 2. ■
We come back to Bonanno’s concept when studying the notion of distance, so we
illustrate this concept on the game in Figure 1. Consider the ICSE with µ2(x2) = 0.1,
µ2(x3)=0.9,µ3(y1) = 0,µ3(y2) = 0.3,µ3(y3)=0and µ3(y4) = 0.7. These probabilities
are compatible with AGM-consistency since they give positive weight to the actions
(stories) A1and B1and to the stories A1B2and B1B2. Therefore, they respect the
plausibility-preserving action B2. AGM-consistency is a qualitative notion, so the values
of the beliefs are not important. What matters is that if the support of the beliefs are
7
the stories A1and B1, then the support of the stories reaching h′are the stories A1B2
and B1B2. So, every ICSE, which by definition lead to µ2(x2)≤1
5,µ2(x3) = 1−µ2(x2),
and µ3(y1)=0,µ3(y2)≥1
4,µ3(y3)=0,µ3(y4)=1−µ3(y2)respect AGM-consistency,
except for the assessment that puts a 0 on µ(x2)or a 1 on µ(y2).
As a matter of fact, let us consider the "extreme" ICSE with µ2(x2)=0,µ2(x3) = 1,
µ3(y1)=0,µ3(y2)=1,µ3(y3)=0and µ3(y4)=0. According to AGM-consistency,
plausible histories can not sustain these beliefs, given that if µ(B1)(the probability
assigned to story B1) is equal to 1 (because µ(x2)=0and µ(x3)=1), then µ(B1B2)
(the probability assigned to story B1B2) is also 1, since B2is played with probability
1 (it is the plausibility-preserving action); so the story B1B2is as plausible as the
story B1. Given that A1is a less plausible story (in fact µ(A1)=0) and given that
µ(A1) = µ(A1B2), we get µ(A1B2) = µ(y2)< µ(B1B2) = µ(y4), so µ(y2)can not be
equal to 1. With AGM-consistency, all happens as if an external observer deals with
the possible beliefs of every player, upholding the planned equilibrium actions (as in
the SE and in the ICSE) but possibly changing his view on an earlier out-of-equilibrium
way of playing each time he faces a new deviation. So, if by observing that Player 1
does not play C1, he becomes convinced that he plays B1(µ(x3)=1), then, given that
Player 2 plays B2at equilibrium, he necessarily assigns belief 1 to y4.
We now consider Bonnano’s PBE concept. The above ICSE, with µ(x2) = 0.1,
µ(x3)=0.9,µ(y1)=0,µ(y2)=0.3,µ(y3)=0, and µ(y4)=0.7is not a PBE
(Bonnano’s version) in that, via Bayes’ Rule, µ(x2) = 0.1and µ(x3) = 0.9lead to
µ(y1)=0,µ(y2)=0.1,µ(y3)=0and µ(y4) = 0.9.
We finally show that, conversely, many AGM-consistent stories, and even Bo-
nanno’s PBE consistent stories, are not compatible with the concept of ICSE. So
consider the game in Figure 2. Assume that the planned actions are the bold lines
(in red) and that the beliefs (in blue) are given by µ(x2)≥0.7,µ(x3) = 1 −µ(x2),
µ(y1) = µ(x2),µ(y2) = 1 −µ(x2),µ(y3) = µ(y4)=0.5.
These beliefs are AGM-consistent and they check Bonanno’s PBE consistency.
This is due to the fact that Bayes’ Rule applies when switching from hto h′but it
does not apply when switching from hto h′′, since B2is not a plausibility-preserving
action (by contrast to A2). An external observer, when observing the unexpected
action B2, may completely reconsider the stories of the game. At h, he believes that
Player 1 more often deviates to A1than to B1, but after observing the new deviation
B2, he changes his mind and thinks that Player 1 deviates to B1as often as to A1.
This is not possible in an ICSE because the same player, Player 3, plays at h′and h′′ .
Regardless of Player 3’s perturbations on Player 1 and Player 2’s actions, we necessarily
8
1
x1
C1
x3
1−µ(x2)
y4
0.5
b3a3
B2
y21−µ(x2)
B3A3
A2
B1
x2µ(x2)≥0.7
y3
0.5
b3a3
B2
y1µ(x2)
B3A3
A2
A1
2h
3h” 3h’
Figure 2: An example illustrating the difference between ICSE and AGM-consistency.
have µ3(y1) = µ3(y3)and µ3(y2) = µ3(y4), because Player 2 plays A2with the same
probability at x2and x3and B2with the same probability at x2and x3. Therefore,
AGM-consistency allows an external player to have different evolving beliefs at a same
information set (before Player 2’s deviation, the external player sets µ(x2)≥0.7, but
after Player 2’s deviation he sets µ(x2)=0.5), whereas the ICSE does not allow evolving
beliefs at a same information set. Rather, it only allows different beliefs among the
players (so µ3(y1) = µ3(y3) = µ(x2)because these three probabilities express the way
Player 3 evaluates the deviation from Player 1 towards A1and B1(before and after
Player 2’s choice of action), but Player 3’s way of evaluating Player 1’s deviations may
be different from Player 2’s way of evaluating these deviations, that is to say µ2(x2)
can be different from Player 3’s beliefs on x2).
4. A physical distance between beliefs
While we defend the point of view that there is no logical reason that constrains
people to have the same beliefs with respect to out-of-equilibrium actions, social groups
often (implicitly, if not explicitly) require their members to have rather similar beliefs
(Barber and Pope,2019,Gould and Klor,2019,Slothuus and Bisgaard,2021), so the
pressure to modify one’s beliefs is increasing in the difference between the player’s
9
beliefs and the beliefs of the group they belong to. Therefore, if all the players in
a game belong to a same community, it makes sense for them to seek to reduce the
distance between beliefs.
To approach the distance between beliefs, we start with a first observation. In a
game, very often, the equilibrium payoffs are sustained by sets of beliefs. For example,
the ICSE equilibrium payoffs (5.99, 10, 10) in the game in Figure 1are sustained by
Player 2’s beliefs µ2(x2)≤1
5and Player 3’s beliefs µ3(y2)≥1
4. So Player 2 has to assign
a probability lower than 1
5to Player 1’s deviating action A1whereas Player 3 has to
assign a probability larger than 1
4to this deviation, a fact we reproduce in Figure 3, in
which we highlight Player 2 and Player 3’s sustaining beliefs (SB) on the action A1.
01
5
1
4
1
21
P2’s SB P3’s SB
Figure 3: Player 2 and Player 3’s sustaining beliefs.
We are concerned with the closest possible beliefs sustaining an ICSE, here 1
5and
1
4. Figure 3illustrates three facts:
(i) First, if Player 2’s and Player 3’s sustaining beliefs have a non-empty intersection,
then common beliefs can sustain the ICSE payoffs and there exists a sequential
equilibrium with these payoffs. So the minimum distance would reduce to 0.
(ii) Second, Player 2’s and Player 3’s sustaining beliefs have an empty intersection,
but both sets are close ( 1
4-1
5is small in comparison to 1), so that few changes in
the game may lead both sets to have a non-empty intersection.
(iii) Third, we observe that both players can assign a probability lower than 1
2to A1
to sustain the ICSE outcome (red probabilities for Player 2 and Player 3).
4.1. Ordered ICSE
Our first way to consider distance starts with the third observation. Imagine that
Player 2 and Player 3 meet and discuss together: they can easily agree on the fact that
both think that Player 1 deviates more often to B1than to A1(red probabilities lower
than 1
2). Player 2 is sure that Player 1 deviates at least 4 times more often to B1than
to A1. Player 3 can agree that Player 1 deviates more often to B1than to A1but at
most 3 times more often. So there is a possible consensus between Player 2 and Player
10
3 despite the fact that they can not have the same beliefs. This consensus is on the
way they order the deviations: both players believe that Player 1 more often deviates
to B1than to A1, yet only the "intensity" of this deviation is not shared among them.
It derives from this observation that a soft notion of proximity between beliefs con-
sists in requiring that all the players order the perturbed strategies at each information
set in the same way.
Definition 1. An ordered ICSE is an ICSE that checks the additional condition:
v) ∀h∈H, ∀a, a′∈Ah,∀j, j′∈N, ˆπk
j,i(h)(a)≥ˆπk
j,i(h)(a′)⇒ˆπk
j′,i(h)(a)≥ˆπk
j′,i(h)(a′).
In the game in Figure 1, it is easy to find an ordered ICSE that sustains the
equilibrium actions (C1, B2, B3) and that checks Definition 1. As a matter of example,
with Player 2’s distribution (ϵ, 4ϵ, 1−5ϵ) on the actions A1,B1and C1and with
Players 3’s distribution (ϵ, 3ϵ, 1−4ϵ) on the actions A1,B1and C1, beliefs converge to
sustaining beliefs and ϵ= ˆπk
2,1(x1)(A1)<4ϵ= ˆπk
2,1(x1)(B1)<1−5ϵ= ˆπk
2,1(x1)(C1)and
ϵ= ˆπk
3,1(x1)(A1)<3ϵ= ˆπk
3,1(x1)(B1)<1−4ϵ= ˆπk
3,1(x1)(C1), for ϵclose to 0+.
Yet, not every ICSE equilibrium actions can be sustained by probabilities that
check Definition 1. For example, if Player 3’s payoff 4 after A1B2B3is replaced by the
payoff 1.9, then (C1, B2, B3) will still be an ICSE, but no ICSE checks the condition in
Definition 1. This is due to the fact that we necessarily have ˆπk
2,1(x1)(A1)<ˆπk
2,1(x1)(B1)
since we need µ2(x2)≤1
5< µ2(x3), and ˆπk
3,1(x1)(A1)>ˆπk
3,1(x1)(B1)since we need
µ3(y2)≥1
1.9>1
2> µ3(y4).
4.2. How to make ICSE beliefs SE-consistent?
Our second way to consider distance consists in exploiting the small size of the
interval between Player 2’s sustaining beliefs and Player 3’s sustaining beliefs (second
observation). Clearly, with respect to the game in Figure 1, the ICSE payoffs could
become SE payoffs (and so the minimum distance between beliefs could collapse) by
changing the game in a very smooth way. By replacing the payoff 1 after B1B2A3by
0.87 and the payoff 6 after B1B2B3by 6.17, it is possible to build an ICSE that is also
a SE. We get µ2(x2) = 4.5
20 ,µ2(x3) = 15.5
20 , and µ3(y1) = 0,µ3(y2) = 4.5
20 ,µ3(y3) = 0,
µ3(y4) = 15.5
20 , that is to say Player 2 and Player 3 have the same beliefs on Player 1’s
deviations.
It follows from this observation that another way to study the proximity between
beliefs consists in looking at how much we need to shake the payoffs in order to get an
ICSE that is also a SE; that is, how much we should shake payoffs to get beliefs that
are consistent in a SE way. In other terms, after observing that it is not possible to
11
get a SE with the equilibrium actions of a given ICSE, we can look if small changes in
payoffs can allow us to get a SE with the ICSE payoffs.
Definition 2. The ISCE beliefs are close if they can become SE-compatible with very
small changes in payoffs. In that sense, the distance in beliefs becomes the distance in
payoffs required to change the ICSE equilibrium payoffs into SE equilibrium payoffs.
The steps are the following ones. We start with ICSE equilibrium behavioral
strategies that can not be part of a SE. Then we introduce variables that express
changes in payoffs and we minimize the changes in payoffs under the constraint that
the ICSE actions and the associated beliefs become a SE.
Let us illustrate the procedure for the game in Figure 1, which is represented again
in Figure 4. A first observation is that it is always possible to change the payoffs in
order to get a SE with the ICSE played actions. The optimization program makes sense
only if it leads to small payoff changes. In that case, we can say that the ICSE beliefs
are not much distant one from another. If so, players will not feel under pressure to
change them, namely because in real life there is always some incomplete information
on the exact payoffs, so the smoothly changed payoffs (needed to share the same beliefs)
belong to the set of possible payoffs.
Player 1
x1
5.99
10
10
C1
x3
y4
0
6 + x4
0 + y4
B3
6
3
1 + y3
A3
B2
y3
0
5 + x3
3
B3
1
4
4
A3
A2
B1
x2
y2
1
0 + x2
4 + y2
B3
0
1
1 + y1
A3
B2
y1
6
4 + x1
2
B3
6
2
3
A3
A2
A1
Player 2
Player 3
Figure 4: Absolute changes in payoffs that can make ICSE beliefs SE-compatible.
12
A second observation is that there are only a limited number of changes to introduce
in that many payoffs have no role to play in the studied equilibrium. In our game, the
payoffs to work on are the ones underlined. What matters is that, for (C1, B2, B3) to
become a SE, the SE consistency requires that µ(x2) = µ(y2) = µand µ(x3) = µ(y4) =
1−µ. So the program we solve is Program 1:
min
x1,x2,x3,x4,y1,y2,y3,y4,µ x12+x22+x32+x42+y12+y22+y32+y42
s.t. (0 + x2)µ+ (6 + x4)(1 −µ)≥(4 + x1)µ+ (5 + x3)(1 −µ)(1)
(4 + y2)µ+ (0 + y4)(1µ)≥(1 + y1)µ+ (1 + y3)(1 −µ)(2)
µ≥0(3)
1−µ≥0(4)
This objective function is one among the many possible ways to measure the
changes in payoffs, surely the easiest one. We will propose later a more proportional
one. Equations (1) and (2) are the necessary equations ensuring sequential rationality
and SE consistency. Equation (1) ensures sequential rationality for Player 2, Equation
(2) ensures sequential rationality for Player 3 and sequential rationality for Player 1 is
ensured in that nothing has changed for himself with respect to Figure 1. What matters
is that conditions (1) and (2) also ensure the SE consistency, which requires that Player
2 and Player 3 put the same belief µon x2(and therefore on y2), the same belief 1−µ
on x3(and therefore on y4) and a null belief on y1and y3.
Given that the objective function goes to +∞when ∥x∥and/or ∥y∥goes to +∞,
and given that the admissible set is closed and that µis limited by 0 and 1, it is easy
to adapt Weierstrass’ corollary to ensure that Program 1 has a global minimum. The
only solution (see Appendix 3) is:
x2=−x1≃0.0158
x4=−x3≃0.0564
y2=−y1≃0.0206
y4=−y3≃0.0736
µ≃0.219
x12+x22+x32+x42+y12+y22+y32+y42= 0.0185
The necessary errors are quite small, since the largest one does not exceed 0.074,
which is quite small given that we work with integers ranging from 0 to 6. In other
13
terms, we can say that our ICSE payoffs are easily SE-compatible (because the needed
payoffs changes are very small). Observe that the SE belief µ(x2)becomes 0.219, which
is between 1
5and 1
4.
If we switch to a more proportional way to see payoff adjustments, we can choose
to switch to Figure 5and to the maximization Program 2:
min
x1,x2,x3,y1,y2,y3,µ (x1
4)2+ (x2
5)2+ (x3
6)2+y12+ (y2
4)2+y32
s.t. (6 + x3)(1 −µ)≥(4 + x1)µ+ (5 + x2)(1 −µ)(1)
(4 + y2)µ≥(1 + y1)µ+ (1 + y3)(1 −µ)(2)
µ≥0(3)
1−µ≥0(4)
Player 1
x1
5.99
10
10
C1
x3
y4
0
6 + x3
0
B3
6
3
1 + y3
A3
B2
y3
0
5 + x2
3
B3
1
4
4
A3
A2
B1
x2
y2
1
0
4 + y2
B3
0
1
1 + y1
A3
B2
y1
6
4 + x1
2
B3
6
2
3
A3
A2
A1
Player 2
Player 3
Figure 5: Proportional changes in payoffs that can make ICSE beliefs SE-compatible.
The only solution (see Appendix 3) is: x1=−0.0258,x2=−0.1232,x3= 0.1774,
y1=−0.0021,y2= 0.0338,y3=−0.0065 and µ= 0.2466.
We can observe that (x1
4)2+ (x2
5)2+ (x3
6)2+y12+ (y2
4)2+y32= 0.0016, which is
again quite small, and that no term (perturbation/payoff) is larger than 0.00645. The
SE belief µ(x1)now becomes 0.247, which is again between 1
5and 1
4. Several remarks
have to be made.
14
This concept of distance is rather simple to employ given that the program is easy
to write (sequential rationality and the SE consistency give the set of constraints and
the objective is a function increasing in the introduced payoff perturbations). But the
interpretation of the result is necessarily a little subjective. For example, in Program
2, what should we require in order to say that beliefs are close? What is the threshold
|dx
x|we should accept (where dx is the variation of payoff and xthe payoff)? We can of
course impose the constraint |dx
x| ≤ 0.1in order to prevent too strong payoff changes,
but this gives us no way to appreciate the optimal value of the objective function. Also,
should we introduce a fixed threshold on the objective function and/or the ratios |dx
x|?
Or should the thresholds depend on the payoffs that divergent beliefs allow to get at
equilibrium? We think that, when opting for a proportional approach (Program 2),
rather than asking for |dx
x| ≤ 0.1or another small value, we should ask for a threshold
whose value rises with the benefit linked to the ICSE. This way of doing is motivated
by the following fact: when a player earns a large payoff, he is less induced to change
things (e.g., his beliefs) and he is less induced to ask that other persons change their
beliefs.
4.3. Distance in ICSE beliefs and AGM-consistent beliefs
Another remark, which brings us back to Bonanno (2013,2016)’s concept of AGM-
consistency, is that this notion of distance does not take into account the number of
deviations required to reach an information set. Let us consider the game in Figure
6. Suppose that it is possible to build an ICSE (C1, A2, B3) with beliefs checking
µ2(x2)≥0.6and µ3(y1)≤0.3. Of course, SE consistency requires that µ2(x2) = µ3(y1),
so that we potentially have to strongly shake the payoffs in order to make the ICSE
payoffs SE-compatible. In other terms, if we measure the distance as in Program 1 or
Program 2, we will surely conclude that Player 2 and Player 3’s beliefs are distant from
one another. But this conclusion does not take into account a strong difference between
hand h′:hneeds one deviation to be reached (Player 1’s deviation) whereas h′needs
two deviations to be reached (Player 1’s deviation and Player 2’s deviation).
This fact explains that Player 2 and Player 3’s beliefs are AGM-consistent. As
a matter of fact, the story A1is as plausible as the story A1A2(because A2is the
plausibility-preserving action), the story B1is as plausible as the story B1A2for the
same reason, and so an external observer can judge that the stories A1and A1A2are
more plausible than the stories B1and B1A2. Yet, he may also judge that the story
B1B2is more plausible than the story A1A2because B2is a new deviation that totally
15
1
x1
C1
x3
1−µ(x2)
y2
1−µ(y1)
B3A3
B2A2
B1
x2µ(x2)≥0.6
y1
µ(y1)≤0.3
B3A3
B2A2
A1
2h
3h’
Figure 6: Revisiting the links between ICSE and AGM-consistency.
shakes his understanding of the game: at h, after Player 1’s deviation, he thinks that
Player 1 probably deviated toward A1, so that Player 2 should play A2, but after Player
2’s unexpected deviation toward B2(at h′), he changes his mind and finally thinks that
Player 1 probably deviated to B1.
By contrast to Bonanno, we do not work with an external observer but only with
the players themselves. This amounts, in Figure 6, in translating Bonanno’s switch
in beliefs into the following question: can we reasonably require that somebody who
observes more deviations needs to share the same beliefs than somebody who observes
less deviations? Player 2 observes only one deviation (Player 1’s deviation) whereas
Player 3 observes two deviations (Player 1’s and Player 2’s deviations). Facing more
deviations may introduce more doubts and therefore allow for different beliefs. To say
it differently, the distance in beliefs might also depend on the distance (in the number
of deviations to reach it) between an information set and the equilibrium path. This
amounts to saying that players facing more deviations can be expected to have more
distant beliefs. In some way, if we note Player 2’s payoff changes by dx, and Player 3’s
payoff changes by dy, it could make sense, in the game in Figure 6, to weight |dx
x|more
strongly than |dy
y|in the distance function to minimize (for example 2(dx
x)2and (dy
y)2).
It would perhaps even make more sense to change the power assigned to |dx
x|and |dy
y|
given that the probability of several deviations exponentially decreases with the number
of deviations (we could work with (dx
x)2and (dy
y)4). In this way, given that |dx
x|and
16
|dy
y|are lower than one, Player 3 can afford more payoff changes without increasing too
much the value of the distance function: this amounts to saying that we do not judge
his beliefs very distant from the other players’, even if in fact they are very different.
5. Revealed beliefs and strategic beliefs
Taking into account the number of deviations to reach an out-of-equilibrium in-
formation set puts into light a new problem when studying the notion of distance in
beliefs. What exactly is the link between beliefs and deviations? May there be a differ-
ence between announced beliefs and the beliefs revealed through the players’ behavior?
Let us again consider the game in Figure 6. Are Player 3’s beliefs really distant
from Player 2’s revealed beliefs? A2is optimal when µ(x2)≥0.6. Yet, when Player
3 is called on to play, Player 2 played B2and not A2. Given that Player 2 plays B2
when his beliefs check µ(x2)<0.6(because the complementary beliefs lead to the
play of A2), we can say that if Player 3 is called on to play (if Player 2 played B2),
then Player 2’s revealed beliefs contradict the beliefs announced at the information
set h. And the revealed beliefs (µ(x2)<0.6), are compatible with Player 3’s beliefs,
µ(y1) = µ(x2)≤0.3. So Player 3’s beliefs are in reality compatible with Player 2’s
revealed beliefs. By the way, this might provide a logical reason for the reversal of
beliefs of Bonanno’s external observer at h′in the game in Figure 6. Given that Player
2 does not play A2(the action compatible with the observer’s beliefs µ(x2)≥0.6),
Player 2 reveals to the observer that his beliefs are the reversed ones, which induces
the observer to change his beliefs.4
This way of coping with beliefs may seem attractive but it clearly leads to difficul-
ties. First, it is often incompatible with the ICSE beliefs. In the game in Figure 2for
example, Player 3, with the ICSE concept, necessarily has the same beliefs at h′and
h′′ (µ(y1) = µ(y3)), so his beliefs are not reversed at h′′ despite the fact that he knows,
at h′′, that Player 2 did not behave in conformity with his beliefs at h(given that he
did not play A2). In fact Player 3, at h′′, sees Player 2’s action B2as a trembling hand
action that has no informative content and his beliefs only follow from his own per-
turbations on Player 1’s actions A1and B1. Secondly, AGM-consistent beliefs in this
game are compatible with the notion of revealed beliefs, given that Player 3’s beliefs
at h′′ take into account that Player 2, by playing B2, revealed that his beliefs are such
4But Bonanno (2013,2016) does not require this reversal. AGM-consistent assessments also allow
beliefs such as µ(x2) = µ(y1)and µ(x3) = µ(y2).
17
that µ(x2)≤0.7. But AGM-consistent beliefs could also assign probability 0.7 to y3
in that the external observer is in no way compelled to change his view on Player 1’s
played actions after an unexpected action from Player 2.
Thirdly, taking into account revealed beliefs puts into question the measure of the
distance we proposed previously. If a player tries to be close to a previous player’s
beliefs, then the notion of revealed beliefs requires that his beliefs must be different
depending on whether the previous player played the planned action or not. In Figure
2for example, a SE requires µ(x2) = µ(y1) = µ(y3), so if an equilibrium starts with
µ(y1)=µ(y3), the distance is necessarily strictly positive despite the fact that revealed
beliefs by definition require µ(y1)=µ(y3). This suggests that taking into account the
number of deviations required to reach an information set must change our measure of
distance.
Finally, we should consider the notion of revealed beliefs with suspicion. Let us
consider the game in Figure 7.
1
x1
3
6
3
C1
x3
y2
6
4
6
B3
2
1
2
A3
B2
2
2
2
A2
B1
x2µ(x2)≤1
3
y1
≥1
2
2
1
2
B3
6
4
6
A3
B2
2
2
2
A2
A1
2h
3h’
Figure 7: Revealed and announced beliefs.
In every sequential equilibrium, we have µ(x2) = µ(y1), and it follows that Player
2 always plays B2(because either µ(x2) = µ(y1)>1
2, Player 3 plays A3and therefore
Player 2 plays B2or µ(x2) = µ(y1)<1
2, Player 3 plays B3and therefore Player 2
plays B2, or µ(x2) = µ(y1) = 1
2, Player 3 plays A3with probability qand therefore
18
Player 2 plays B2). This implies that Player 1 plays A1if µ(x2) = µ(y1)>1
2,B1if
µ(x2) = µ(y1)<1
2, and A1or B1depending on the value of qif µ(x2) = µ(y1) = 1
2.
Hence, both Players 1 and 3 get 6 at any sequential equilibrium and Player 2 gets at
best 4.5
Now consider the ICSE given in Figure 7, with the SE incompatible beliefs µ(x2)≤
1
3and µ(y1)≥1
2. This new profile of actions and beliefs checks AGM-consistency (and
Bonanno’s PBE consistency) and is conform to revealed beliefs. As a matter of fact,
given his beliefs and Player 3’s action A3, Player 2 should play A2, which he does at
equilibrium and which induces Player 1 to play C1. So, if Player 3 is called on to play,
this means that Player 2 played B2which, according to revealed beliefs, reveals that he
does not believe that Player 1 played A1with a probability lower than 1
3. Therefore,
Player 3, if he wishes to share similar beliefs to Player 2, can believe that Player 1
played A1with a probability larger than 1
3and possibly larger than 1
2. These beliefs
induce him to play A3. So revealed beliefs sustain a profile of strategies where Players
1 and 3 only get 3 and Player 2 gets 6, by leading Player 1 to play C1.
But let us look more closely at Player 2’s revealed beliefs. By playing B2instead
of the planned action A2, Player 2 sends the following message to Player 3: "Normally
I play A2because it is my best response to your action A3, since I believe that Player
1 much more deviates to B1than to A1; so, if you see me playing B2, this means that I
changed my opinion about Player 1’s deviation, so that you are right in believing that he
more often deviated to A1, and so you are right in playing A3." The problem is that this
message is at best "cheap talk" because Player 2 has no information to reveal to Player
3. Player 2 has no idea about the potential deviations of Player 1, given that Player
1 does not deviate when he expects Player 2 to play A2. One more time, by contrast
to Aumann (1976), Geanakoplos and Polemarchakis (1982), and Hart and Tauman
(2004), the out-of-equilibrium actions played by the players do not provide information
on a true state to discover but on an action that will never be observed at equilibrium.
Player 1 does not deviate and so there is nothing to learn about his deviation. If Player
3 "naively" believes that Player 2 can reveal something about Player 1’s deviations with
his behavior, then he gives Player 2 the power of manipulating Player 3’s beliefs to his
advantage. As a matter of fact, the beliefs in this ICSE clearly are advantaging Player
2 because they induce Player 1 to play C1, so they lead to the payoff profile (3,6,3)
which is exclusively in advantage of Player 2.
5There also exists a SE where Player 1 plays A1and B1with probability 1
2and Player 3 plays A3
and B3with probability 1
2, so they both get 4 given that Player 2 plays B2.
19
6. Discussion
Our analysis suggests that out-of-equilibrium beliefs might be here to justify the
players’ behaviour at out-of-equilibrium information sets. Choosing them in a given
way may help to get a higher payoff, which means that they belong to the strategy
set. Let us recall that in a SE, everybody builds the beliefs on a same profile of
perturbations, so a player does not really choose his beliefs given that he applies Bayes’
Rule to the same perturbations. So in a SE, out-of-equilibrium beliefs do not belong to
the player’s strategy set. By contrast, in an ICSE, each player chooses his own profile
of perturbations, which means that he chooses his own beliefs at out-of-equilibrium
sets. This degree of liberty can be exploited to build beliefs that lead to interesting
payoffs. In the game in Figure 7, it is good for Player 2 to build beliefs (about Player
1’s deviation) that are strikingly different from those of Player 3 in order to justify the
action A2that prevents Player 1 from deviating from C1, the most interesting action
for Player 2. By contrast, for Player 3, it is better to have similar beliefs than Player 2,
in order to lead Player 2 to play B2, which ultimately leads to the payoffs (6,4,6). So
we are tempted to say that, given that each player chooses his set of perturbations, out-
of-equilibrium beliefs belong to the strategy set. This fact induces two consequences:
we have to reconsider the notion of sequential rationality and we have to reconsider the
notion of distance between beliefs.
We start with the notion of distance by coming back to the game studied in Figure
1. We already made the observation, in Section 3, when opting for a proportional
approach (Program 2), that rather than asking for |dx
x| ≤ 0.1or another threshold,
we should ask for a threshold whose value rises with the benefit linked to the ICSE
payoffs. As a matter of fact, if everybody benefits from the ICSE payoffs then nobody
cares about the distance between the beliefs necessary to sustain the equilibrium. By
contrast, in the game in Figure 7, Player 3 may require that Player 2 has beliefs that
are close to his own beliefs (especially if Player 2 is a newcomer in the community)
because similar beliefs among Player 2 and Player 3 are necessary for Player 3 to get
the nice payoff 6. Conversely, if Player 3 is the newcomer in the community, then Player
2 might not pressure Player 3 to adopt beliefs close to his own. The (social) pressure to
modify beliefs so as to be close to another player’s beliefs must therefore be contrasted
with the benefits (in terms of actions played) of holding different beliefs.
Concerning sequential rationality, given that each player chooses his perturbation
scheme, these perturbation schemes belong to his strategy set, that is, each player
20
will build (in a consistent way) beliefs at his out-of-equilibrium information sets to
get a better payoff. This changes nothing with respect to the definition of individual
consistency, but this should lead us to reconsider Kreps and Wilson (1982)’s notion
of sequential rationality. In some way, we should add that, for each player, given the
strategies played by the other players, there does not exist a perturbation scheme that
is sequentially rational (as defined by Kreps and Wilson 1982), individually-consistent
and that leads to a larger payoff for the player.
However, sequential rationality rests on unilateral deviations and this additional
condition might thus not always help. For example, the ICSE in Figure 7would resist
such an additional condition despite the fact that Player 3 would like to adapt his
beliefs to Player 2’s beliefs to compel him to play B2. The problem is that, as long as
Player 2 plays A2and Player 1 plays C1, Player 3’s beliefs and actions have no impact
on his equilibrium payoff. The idea is that each player selects the perturbation scheme
associated to the ICSE that leads him to his largest equilibrium payoff. If so, in the
game in Figure 7, Player 3 should opt for a perturbation scheme (on Player 1’s actions)
similar to Player 2’s, to push Player 1 and Player 2 to play A1and B2for example (he
can choose the SE (A1, B2, A3) with the beliefs µ(x2) = µ(y1)=1). Yet, Player 2 would
of course choose another perturbation scheme, namely the one leading to the ICSE in
Figure 7. So this additional condition, in the game in Figure 7, leads to the non-
existence of a system of perturbation schemes both selected by Player 2 and Player 3.
In the game in Figure 1, Player 2 and Player 3 can select the same perturbation scheme,
namely the one of an ICSE leading to the payoffs (5.99, 10,10). Player 1 can not counter
Player 2 and Player 3’s selection, in that if they play B2and B3he is constrained to
play C1. Yet, even in this game, Player 1 may opt for another equilibrium in which he
plays A1, therefore constraining Player 2 and Player 3 to play A2and A3. Therefore,
there is no obvious way, even if we switch to coalitions of players, to clearly formalize
and express the wish to select payoff-optimizing perturbation schemes. The only trivial
configuration is a game such that one ICSE ensures the best payoff to all the players,
so that the grand coalition of all the players will be incentivized to select it.
7. Conclusion
The paper started with an obvious observation: there is no reason that leads
every player to build the perturbed strategies similarly. Each player has to respect
21
the probabilities assigned to actions that are in the support of the equilibrium, but,
given that there does not exist an external observer who can decide for the profiles of
ϵ-perturbations, each player is free to build the perturbations assigned to the actions
out of the support of the equilibrium. It follows that in an ICSE, players can have
different beliefs at out-of-equilibrium information sets.
This led us first to evaluate the distance between different beliefs, because players
in the same community are often expected to share similar beliefs. We did this in
Section 3 in two ways: (i) an ordering of perturbations and (ii) the minimization of
changes in payoffs necessary to make the ICSE beliefs SE-compatible.
We then focused on the function held by beliefs at out-of-equilibrium sets. Since
players can build their beliefs at out-of-equilibrium sets, they might build them strate-
gically in order to improve their payoffs. This observation led us to reconsider the
traditional concept of sequential rationality, by further requiring that there does not
exist a perturbation profile that is individually-consistent and that provides greater
payoffs to the player, even though such an additional constraint is not always easy to
cope with.
8. Appendix 1
We show that there are no sequential equilibria supporting the action C1for player
1 (and therefore the socially optimal situation) in the game shown in Figure 1.
Case 1: Player 2 plays A2. Therefore player 1 plays A1.
Case 2: Player 2 plays B2.
(i): If player 3 plays A3, player 1 plays B1.
(ii): If player 3 plays B3, then player 1 might want to play C1, but we have already
shown that the beliefs that would support this equilibrium are not mutually consistent.
(iii): If player 3 plays A3and B3, then γ=1
4, and so necessarily α=1
4. But
then, player 2 would prefer to play A2. To show this, first let rbe the probability
that player 3 plays A3. By playing A2, player 2’s expected payoff is 1
4(2r+ 4(1 −r) +
3
4(4r+ 5(1 −r)) = 1
4(14r+ 19(1 −r)). By playing B2, player 2’s expected payoff is
1
4r+3
4(3r+ 6(1 −r)) = 1
4(10r+ 18(1 −r)), which is strictly inferior to the expected
gain of playing A2.
Case 3: Player 2 plays A2and B2.
(i): Player 3 plays A3. In this case, it would not be profitable for player 2 to
22
randomize, given that playing only A2would allow her to always gain strictly more.
(ii): Player 3 plays B3. Given player 2’s indifference between A2and B2, it is
necessary that α=1
5. To show that with these beliefs, player 3 would want to deviate,
first note qthe probability that player 2 would play A2. Then by playing A3, player 3’s
expected gain would be 1
5(1 + 2q) + 4
5(1 + 3q) = 1
5(5 + 14q). By playing B3, player 3’s
expected gain would be 1
5(4 −2q) + 4
5(3q) = 1
5(4 + 10q), which is strictly inferior to the
expected gain player 3 would receive by playing A3.
(iii): Player 3 plays A3and B3. Let rbe the probability that player 3 plays
A3, and qthe probability that player 2 plays A2. Let ϵ0and ϵ1be the perturbations
associated to A1and B1respectively. The expected gain of playing A2for player 2 is
ϵ0(2r+ 4(1 −r)) + ϵ1(4r+ 5(1 −r)) = ϵ0(4 −2r) + ϵ1(5 −r). The expected gain of
playing B2for player 2 is ϵ0r+ϵ1(3r+ 6(1 −r)) = ϵ0r+ϵ1(6 −3r). Equalizing these
expected gains yields ϵ0(4 −2r) + ϵ1(5 −r) = ϵ0r+ϵ1(6 −3r), or ϵ0(4 −3r) = ϵ1(1 −2r).
The expected gain of playing A3for player 3 is ϵ0(3q+ (1 −q)) + ϵ1(4q+ (1 −q)) =
ϵ0(1+ 2q) +ϵ1(1 + 3q). The expected gain of playing B3for player 3 is ϵ0(2q+4(1 −q))+
ϵ13q=ϵ0(4−2q) +ϵ13q. Equalizing these expected gains yields ϵ0(1+ 2q)+ ϵ1(1+ 3q) =
ϵ0(4 −2q) + ϵ13q, or ϵ0(3 −4q) = ϵ1. It follows that ϵ1=ϵ0(3 −4q) = ϵ04−3r
1−2r. Therefore,
3−4q=4−3r
1−2r, so 4q= 3 −4−3r
1−2r=3−6r−4+3r
1−r=−1−3r
1−r, which is strictly inferior to 0; an
impossible event.
9. Appendix 2
Proof of Proposition 3.1. (i) This follows from the fact that in an ICSE, all strate-
gies are sequentially rational and beliefs are obtained via Bayes’ Rule applied
to strategies close to the true ones (even if the perturbations are not the same
among players). In an ICSE, players agree on the planned actions, even those at
unreached subgames, so the ICSE induces a Nash equilibrium in each subgame.
(ii) This follows directly from the definition of both concepts.
(iii) This follows from the fact that the perturbations required by Player 2 (to build
his beliefs) are about actions played by Player 1, and the perturbations required
by Player 1 are about actions played by Player 2. Both players do not work with
different perturbations about actions played by another (third) player. So we
can work with one set of perturbations for the game, which is the same for both
23
players (by taking Player 1’s perturbations (about Player 2’s actions) and Player
2’s perturbations (about Player 1’s actions)).
(iv) To show why an ICSE is not necessarily a PBE, we choose a game closer to the
games studied by Fudenberg and Tirole (1991), by changing our main example in
the following way. θ1and θ′
1are Player 1’s two possible types, unknown to Player
2 and to Player 3 (prior probabilities ρand 1−ρ). So we get the game in Figure
8.
Figure 8: Distinction between ICSE and PBE (1/3).
According to Fudenberg and Tirole (1991)’s PBE equilibrium concept, Player
2 and Player 3 share the same beliefs everywhere (Condition B(iv) p.332), and
these beliefs are build using the history of play whenever possible. So, if µ(x4) =
µ2(θ′
1/h) = µ(θ′
1/h) = 1
5, we get:
µ(y3) = µ(θ′
1/h)π2(B2)
µ(θ1/h)π2(B2) + µ(θ1/h)π2(A2) + µ(θ′
1/h)π2(B2) + µ(θ′
1/h)π2(A2)
=µ(θ′
1/h)
=1
5,
(due to the Condition B(ii) p.332). Therefore, we can not get µ(y3) = 1
4. Player
24
3’s beliefs are built like Player 2’s. Given that B2is an expected action, and given
that the beliefs at hare not 0, the beliefs at y3are necessarily the same than the
ones at x4.
The ICSE concept takes into account that B2is an expected action, but it al-
lows Player 3 not to share Player 2’s beliefs at h. Therefore, we keep Condition
B(ii) but not Condition B(iv), in that Player 2 (respectively Player 3) assigns
probability 1
5(respectively 1
4) to θ′if his reached.
Yet, all PBE are not necessarily ICSE. For example, consider Fudenberg and
Tirole (1991)’s example reproduced in Figures 9 and 10 (Figure 8.9 p.346 in their
book). The beliefs are in red. The beliefs assigned to the states θhave been
obtained by Bayesian inference from previous play. Player 1 is the player who
plays the actions a∗
1,a′
1and a′′
1, while ek,e′
kare perturbations going to 0.
Figure 9: Distinction between ICSE and PBE (2/3).
Figure 10: Distinction between ICSE and PBE (3/3).
Fudenberg and Tirole (1991) say that the beliefs (in red) at hand h′belong to
a PBE because the PBE places no restrictions on the beliefs at hand h′, since
both beliefs that lead to hand h′are 0. The only condition is that these beliefs
have to be common to all players.
25
Yet the beliefs at hand h′can not belong to an ICSE when the same player,
Player 2 in Figure 9, plays at hand h′. As a matter of fact, by calling ϵ′kand ϵ′′k
the probabilities to reach y1and y2, respectively, which go to 0 given the beliefs,
we get µ2(x3) = lim
ϵ→0(ϵ′ke′
k
ϵ′ke′
k+ϵ′′k(1−ek−ek′)), which can go to 1 only if ϵ′′ k(1−ek−ek′)
ϵ′ke′
k
goes
to 0, which requires that ϵ′′ k
ϵ′k→0. But then µ2(x2) = lim
ϵ→0(ϵ′′kek
ϵ′′kek+ϵ′k(1−ek−ek′))→0.
So the PBE is not a ICSE.
By contrast, when there are two different players at hand h′′, like in Figure 10
(Player 2 and Player 3 respectively), the PBE is an ICSE, since we can take
different perturbations leading to y1and y2for Player 2 (ϵ′k
2and ϵ′′k
2) and Player
3 (ϵ′k
3and ϵ′′k
3), respectively. Therefore, we get: µ3(x3) = lim
ϵ→0(ϵ′k
3e′
k
ϵ′k
3e′
k+ϵ′′k
3(1−ek−ek′)),
which can go to 1 if ϵ′′ k
3
ϵ′k
3e′
k→0. And we get µ2(x2) = lim
ϵ→0(ϵ′′k
2ek
ϵ′′k
2ek+ϵ′k
2(1−ek−ek′)), which
can also go to 1 if ϵ′k
2
ϵ′′k
2ek→0. Given that the set of perturbations is different for
the two players, both conditions can be fulfilled and the PBE becomes an ICSE.
(v) To formally describe the concept of SCE, we need to introduce some more no-
tation. Let a mixed strategy profile be represented by σ= (σi)i∈N, while a
behavioral strategy profile will be denoted, as before, by π= (πi)i∈N. We refer
to information sets that are reached with positive probability under the mixed
strategy profile σas ¯
H(σ), and write hifor a specific information set controlled by
player i. We write the behavioral representation of a mixed strategy σias ˆπi(·|σi),
such that ˆπi(hi|σi)represents the probability distribution over actions induced by
the mixed strategy σiat information set hi. Furthermore, let µirepresent player
i’s beliefs about their opponents’ play, such that µiis a probability distribution
over Π−i, the set of other players’ behavioral strategies, with typical element π−i.
A (mixed) strategy profile σis a Self-Confirming Equilibrium (Fudenberg and
Levine,1993) if, ∀i∈Nand ∀si∈support(σi), there exists beliefs µisuch that:
(i) simaximizes ui(·, µi), and
(ii) µi[{π−i|πj(hj) = ˆπj(hj|σj)}]=1,∀j=i, ∀hj∈¯
H(si, σ−i).
In words, every players’ subjective probability distribution needs to put proba-
bility 1 to strategy profiles that are compatible with observed play (reached with
positive probability). That is, players’ expectations need to be right on the equi-
librium path, but need not be right at information sets that are never reached.
26
Importantly, since each player has to best respond only to the observed actions
of other players, the SCE only requires that players play a best response to their
beliefs about other player’s actions out of the equilibrium strategy path. This
implies that a SCE is not necessarily an ISCE, since the ICSE requires that
players play best responses to other player’s actions, even at information sets
that are out of the equilibrium strategy path. On the other hand, an ICSE is
always a SCE, since it is always possible to create a SCE in which players play
best responses, and have accurate beliefs, even at out-of-equilibrium information
sets.
■
Proof of Proposition 3.2. (i) AGM-consistency works with plausibility orders on sto-
ries. It is a qualitative notion that focuses on plausibility-preserving actions, which
are actions played with a positive probability in the game. Given that the ICSE
also respects the actions played with a positive probability in the game, an ICSE
is usually AGM-consistent, except if it leads to 0-1 assessments.
(ii) Bonanno’s PBE concept transforms the qualitative notion into a quantitative one,
by requiring that the beliefs respect Bayes’ Rule when it applies, i.e., in presence of
plausibility-preserving actions. Given that Bonanno’s concept only introduces one
probability distribution at each out-of-equilibrium information set, it immediately
follows that it cannot intersect with the ICSE concept, which works with the
Bayesian rule applied to several perturbation distributions, one for each player.
■
10. Appendix 3
Program 1:
min
x1,x2,x3,x4,y1,y2,y3,y4,µ x12+x22+x32+x42+y12+y22+y32+y42
s.t. (0 + x2)µ+ (6 + x4)(1 −µ)≥(4 + x1)µ+ (5 + x3)(1 −µ)(1) λ1
(4 + y2)µ+ (0 + y4)(1 −µ)≥(1 + y1)µ+ (1 + y3)(1 −µ)(2) λ2
µ≥0(3) λ3
1−µ≥0(4) λ4
27
Equations (1) and (2) can be rewritten:
1−x3+x4−µ(5 + x1−x2−x3+x4)≥0(1)
−1−y3+y4−µ(−4 + y1−y2−y3+y4)≥0(2)
The KT function becomes:
x12+x22+x32+x42+y12+y22+y32+y42−λ1(1 −x3+x4−µ(5 + x1−x2−x3+x4))
−λ2(−1−y3+y4−µ(−4 + y1−y2−y3+y4)) −λ3µ−λ4(1 −µ).
The KT equations are:
2x1+λ1µ= 0 (a)
2x2−λ1µ= 0 (b)
2x3+λ1(1 −µ) = 0 (c)
2x4−λ1(1 −µ) = 0 (d)
2y1+λ2µ= 0 (e)
2y2−λ2µ= 0 (f)
2y3+λ2(1 −µ) = 0 (g)
2y4−λ2(1 −µ) = 0 (h)
λ1(5 + x1−x2−x3+x4) + λ2(−4 + y1−y2−y3+y4)−λ3+λ4= 0 (i)
It follows that x2=−x1≥0,x4=−x3≥0,y2=−y1≥0, and y4=−y3≥0, due
to the positivity of the KT multipliers.
Both Conditions (1) and (2) are necessarily checked with equality, because λ1= 0
leads to λ2(−4−2y2+ 2y4) = 0, hence y4≥2, which can clearly not lead to a global
minimum given our numerical introduction, and λ2= 0 leads to λ1(5 −2x2+ 2x4) = 0,
hence x4≥2.5, which can not lead to a global minimum for the same reason.
We seek a solution that checks 0< µ < 1, and so λ3=λ4= 0. It follows that
λ1=2x2
µ=2x4
1−µ, hence µ=x2
x2+x4, and λ2=2y2
µ=2y4
1−µ, hence µ=y2
y2+y4. As a result:
x2y4=y2x4(j)
2(x2
2+x2
4) = 4x2−x4(k)
2(y2
2+y2
4) = −3y2+y4(l)
x2(1 + 2x4) + y2(−1+2y4) = 0 (m)
28
The only solution is:
x2=−x1=−53√29 −290
290 ≃0.0158
x4=−x3=33√29 −145
580 ≃0.0564
y2=−y1=83√29 −435
580 ≃0.0206
y4=−y3=−19√29 −145
580 ≃0.0736
µ=−106√29 + 580
435 −73√29 ≃0.219
λ1=435 −73√29
290 ≃0.1444
λ2=64√29 −290
290 ≃0.1885.
Program 2:
min
x1,x2,x3,y1,y2,y3,µ (x1
4)2+ (x2
5)2+ (x3
6)2+y12+ (y2
4)2+y32
s.t. (6 + x3)(1 −µ)≥(4 + x1)µ+ (5 + x2)(1 −µ)(1) λ1
(4 + y2)µ≥(1 + y1)µ+ (1 + y3)(1 −µ)(2) λ2
µ≥0(3) λ3
1−µ≥0(4) λ4
Equations (1) and (2) can be rewritten:
1 + x3−x2+µ(−5−x1+x2−x3)≥0(1)
−1−y3+µ(4 −y1+y2+y3)≥0(2)
The KT function becomes:
(x1
4)2+ (x2
5)2+ (x3
6)2+y12+ (y2
4)2+y32−λ1(1 + x3−x2+µ(−5−x1+x2−x3))
−λ2(−1−y3+µ(4 −y1+y2+y3)) −λ3µ−λ4(1 −µ).
29
The KT equations are:
2x1
16 +λ1µ= 0 (a)
2x2
25 +λ1(1 −µ)=0 (b)
2x3
36 −λ1(1 −µ)=0 (c)
2y1+λ2µ= 0 (e)
2y2
16 −λ2µ= 0 (f)
2y3+λ2(1 −µ)=0 (g)
λ1(5 + x1−x2+x3) + λ2(−4 + y1−y2−y3)−λ3+λ4= 0 (i)
We look for a solution such that Conditions (1) and (2) are checked with equality,
and such that 0< µ < 1(so that λ3=λ4= 0. The only solution gives: x1=−0.0258,
x2=−0.1232,x3= 0.1774,y1=−0.0021,y2= 0.0338,y3=−0.00645,λ1= 0.01308,
λ2= 0.01712 and µ= 0.24657.
References
Aumann, R. J. (1976): “Agreeing to disagree,” The annals of statistics, 4, 1236–1239.
Barber, M. and J. C. Pope (2019): “Does party trump ideology? Disentangling party and
ideology in America,” American Political Science Review, 113, 38–54.
Bonanno, G. (2013): “AGM-consistency and perfect Bayesian equilibrium. Part I: definition
and properties,” International Journal of Game Theory, 42, 567–592.
——— (2016): “AGM-consistency and perfect Bayesian equilibrium. Part II: from PBE to
sequential equilibrium,” International Journal of Game Theory, 45, 1071–1094.
Fudenberg, D. and D. K. Levine (1993): “Self-confirming equilibrium,” Econometrica:
Journal of the Econometric Society, 523–545.
Fudenberg, D. and J. Tirole (1991): Game theory, MIT press.
Geanakoplos, J. D. and H. M. Polemarchakis (1982): “We can’t disagree forever,”
Journal of Economic theory, 28, 192–200.
Gould, E. D. and E. F. Klor (2019): “Party hacks and true believers: The effect of party
affiliation on political preferences,” Journal of Comparative Economics, 47, 504–524.
Greenberg, J., S. Gupta, and X. Luo (2009): “Mutually acceptable courses of action,”
Economic Theory, 40, 91–112.
30
Hart, S. and Y. Tauman (2004): “Market crashes without external shocks,” The Journal
of Business, 77, 1–8.
Kreps, D. M. and R. Wilson (1982): “Sequential equilibria,” Econometrica: Journal of
the Econometric Society, 863–894.
Slothuus, R. and M. Bisgaard (2021): “How political parties shape public opinion in the
real world,” American Journal of Political Science, 65, 896–911.
Umbhauer, G. and A. Wolff (2019): “Individually-Consistent Sequential Equilibrium,”
BETA Working Paper.
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