Bayesian Networks and Boundedly Rational Expectations ∗ Ran Spiegler
advertisement
Bayesian Networks and Boundedly Rational
Expectations∗
Ran Spiegler†
June 23, 2015
Abstract
I present a framework for analyzing decision making under imperfect understanding of correlation structures and causal relations. A
decision maker (DM) faces an objective long-run probability distribution over several variables (including the action taken by previous
DMs). He is characterized by a directed acyclic graph over variables’
labels. I interpret the graph primarily as a subjective causal model.
The DM attempts to fit this model to , resulting in a subjective belief that factorizes according to via the standard Bayesian-network
formula. As a result of this belief distortion, the DM’s evaluation of actions can vary with their long-run distribution. Accordingly, I define a
"personal equilibrium" notion of individual behavior. The framework
provides simple graphical representations of errors of causal attribution (such as coarseness or reverse causation), as well as tools for
checking rationality properties of the DM’s behavior. I demonstrate
the framework’s scope of applications with examples covering diverse
areas, from demand for education to fiscal policy.
∗
This paper has benefitted from ESRC grant no. ES/L003031/1. I am grateful to
Yair Antler, In-Koo Cho, Philip Dawid, Kfir Eliaz, Erik Eyster, Philippe Jehiel, Ehud
Lehrer, Heidi Thysen and Michael Woodford, an editor and referees, as well as seminar
and conference audiences, for many helpful conversations and comments.
†
Tel Aviv University,
University College London and CFM. URL:
http://www.tau.ac.il/~rani. E-mail: rani@post.tau.ac.il.
1
1
Introduction
The standard rational-expectations postulate means that agents in an economic model perfectly understand its equilibrium statistical regularities - in
particular, the structure of correlations among variables. In recent years,
economists have become increasingly interested in equilibrium models that
relax this extreme assumption. This paper proposes an approach to modeling
decision makers with an imperfect understanding of correlation structures,
based on the borrowed concept of Bayesian networks.
Consider a decision maker (DM) whose vNM utility function is defined
over a collection of variables = (1 ), where 1 represents the DM’s
action. Imagine that before choosing how to act, the DM gets access to a
"historical database" consisting of (infinitely) many joint observations of all
variables - including the actions taken by previous DMs who faced the same
decision problem. The long-run joint distribution over , given by the
historical database, can be written in the form of the textbook chain rule:
() ≡ (1 )(2 | 1 )(3 | 1 2 ) · · · ( | 1 −1 )
(1)
where the enumeration of the variables is arbitrary.
To represent a DM with an imperfect understanding of the correlation
structure of , I propose the following extension of the chain rule. The DM
is characterized by a directed acyclic graph (DAG) defined over the set of
nodes {1 }, namely the set of variable labels. The set of directed links
in the DAG is . The DM’s subjective belief distorts every objective long-run
distribution by "factorizing it according to ", via the formula
() =
Y
=1
( | () )
(2)
where () denotes the set of direct parents of the node in the DAG, and
() is the projection of on (). For instance, if : 1 → 2 → 3 ← 4, then
() = (1 )(4 )(2 | 1 )(3 | 2 4 ). Thus, represents the DM’s
subjective, possibly distorted view of the long-run distribution .
2
A DAG and the set of distributions representable by (2) define what is
known as a Bayesian network. This concept was introduced by statisticians
as a representation of conditional-independence assumptions, and has become
ubiquitous in AI as a platform for studying algorithmic aspects of reasoning
about uncertainty. This paper puts Bayesian networks to novel use, namely
as a representation of "boundedly rational expectations" in a model of decision making. When the DM’s DAG is fully connected, it reduces formula
(2) to a standard chain rule, thus representing a DM with rational expectations - and specifically, a perfect understanding of the correlation structure
of . At the other extreme, when is empty, we have a DM who cannot
perceive any correlation that might exhibit: () = (1 ) · · · ( ).1
Pearl (2009) advocated the view of DAGs as causal structures that underlie observed statistical regularities, such that the link → means that
is considered to be a direct cause of . This causal interpretation intuitively
justifies the asymmetry and acyclicity properties of . I follow Pearl and interpret primarily as the DM’s prior subjective causal model, such that
is the outcome of the DM’s attempt to fit his (possibly misspecified) causal
model to "historical data". The DM thus extracts the correlation between
and () (for each = 1 ) from the "historical database" represented
by . He focuses on these particular correlations because they are the relevant ones according to his subjective causal model. Data does not "speak
for itself"; it takes effort to extract correlations from the data, and the DM
will exert it only if he thinks it is relevant. The following example illustrates
this idea and its potential behavioral implications.
An example: The Dieter’s Dilemma
The DM considers starting a diet that involves abstaining from a food he
likes, in order to improve his health. In reality, the DM’s choice and his
health are statistically independent, yet they share a common consequence:
the level of some chemical in the DM’s blood is normal when he is healthy
1
For textbooks on Bayesian networks, see Cowell et al. (1999) and Koski and Noble
(2009). Graphical probabilistic models were introduced into economics in other contexts:
to facilitate computation of Nash equilibria (Kearns et al. (2001), Koller and Milch (2003)),
or to discuss causality in econometric models (White and Chalak (2009)).
3
or when he diets (otherwise, it is abnormally high). Let 1 2 3 denote the
DM’s dieting decision, state of health and chemical level. Any objective
can thus be written as () = (1 )(2 )(3 | 1 2 ). In other words, is
consistent with a "true causal model" given by the DAG ∗ : 1 → 3 ← 2.
Suppose the chemical level as such is payoff-irrelevant. Therefore, a DM with
rational expectations would choose not to diet.
Now assume that our DM’s DAG is 1 → 3 → 2 - i.e., he inverts the causal
link between the chemical level and his health. The DM’s attempt to fit the
long-run distribution to his causal model will generate the distorted view:
(1 2 3 ) = (1 )(3 | 1 )(2 | 3 )
Guided by this view, the DM will choose 1 to maximize
X
2
(2 | 1 )(1 2 ) =
XX
2
3
(3 | 1 )(2 | 3 )(1 2 )
(3)
The DM’s prior belief in the causal chain → → is
manifest in (3).
The DM’s subjective causal model postulates that 2 is independent of
1 conditional on 3 . However, this property does not hold under the true
process: if we know that the chemical level is normal, learning the DM’s
action should affect our prediction of his health. Therefore, the term (2 |
3 ) in (3) can be sensitive to the marginal of over 1 . Our assumptions
regarding the true process imply that as the long-run frequency of dieting
goes down, the estimated correlation between 2 and 3 will strengthen, and
the DM will consequently feel an increased pressure to diet. It follows that
the DM’s evaluation of actions is affected by the historical behavior of DMs in
his situation. We are therefore led to think of the DM’s steady-state behavior
as an equilibrium concept: the long-run distribution is in equilibrium if it
only assigns positive probability to actions 1 that maximize the expectation
of (1 2 ) w.r.t (2 | 1 ).
In Section 3.1, I show that when the disutility from dieting is not too
large, the DM’s equilibrium probability of dieting is positive. The reason
4
is that the DM correctly perceives the steady-state correlation between the
chemical level and his health, yet he mistakes this correlation for a causal
effect of the former on the latter; and since he correctly grasps the effect of
dieting on the chemical level, he may erroneously perceive a large benefit from
dieting. For a range of parameter values, the unique equilibrium is mixed - a
genuine equilibrium effect that is impossible under standard expected-utility
maximization.
In Section 2, I present the equilibrium model of individual choice. For
expositional simplicity, I assume that the DM cannot condition his action
on any signal (Appendix C relaxes this assumption). An objective steadystate distribution is a "personal equilibrium" if whenever an action 1 is
played with positive probability, it maximizes the expectation of w.r.t the
conditional distribution (2 | 1 ). A conventional "trembling hand"
criterion handles zero-probability events.2
Section 3 presents applications of personal equilibrium in various economic domains: fiscal policy, demand for education and managerial decisions.
As in the Dieter’s Dilemma, in each example there is a true causal model
given by a DAG ∗ . The DM’s subjective DAG is obtained by performing
a basic operation on ∗ - removing, adding, inverting or reorienting links.
Thus, the Bayesian-network formalism provides a language for representing
errors of causal attribution. Different basic operations capture different errors: inverting a link captures "reverse causation" (as shown by the Dieter’s
Dilemma), removing a link captures "coarseness", etc.
The framework is more than a language - it also provides technical tools
for characterizing rationality properties of personal equilibrium. Section 4.1
characterizes the possibility of equilibrium effects in terms of the structural
relation between and ∗ . This result can be easily operationalized, thanks
to a basic Bayesian-networks tool called -separation (explained in Appendix
2
The idea that the DM’s long-run behavior may affect his evaluation of actions when he
misperceives correlation structures has precedents in the literature (see Sargent (2001) and
Esponda (2008)). To my knowledge, this is the first general articulation of this idea. The
term "personal equilibrium" was introduced by Kőszegi (2010) in the context of decision
making with reference-dependent preferences, when the reference point is a function of the
DM’s expectation of his own choice.
5
B). Section 4.2 presents a necessary and sufficient condition on for the
DM’s behavior to be universally consistent with rational expectations, for
any specification of the payoff-relevant variables. Section 4.3 shows that two
subjective DAGs never dominate one another in terms of objective expected
payoff, unless one of them is a linear ordering.
This paper offers a fresh look at the literature on equilibrium models with
non-rational expectations. Osborne and Rubinstein (1998) studied games
with players who misperceive the consequences of their actions as a result of
naive extrapolation from small samples. Eyster and Rabin (2005) assumed
that players in Bayesian games underestimate the correlation between opponents’ actions and signals. Madarasz (2012) modeled players who suffer the
"curse of knowledge". In Esponda (2008) and Esponda and Pouzo (2014a),
agents neglect the counterfactual effect of their actions on the distribution
of payoff consequences. Piccione and Rubinstein (2003), Jehiel (2005), Jehiel and Koessler (2008), Mullainathan et al. (2008), Eyster and Piccione
(2013) and Schwartzstein (2014) studied models in which agents’ beliefs are
measurable w.r.t a coarse representation of the set of contingencies (by omitting variables from their subjective model or by clumping contingencies into
"analogy classes").3 Section 5 shows that parts of this literature can be described as special cases of the present framework, defined by specifications of
∗ and (or as refinements and variations of such special cases). Bayesian
networks thus seem to offer a partially unifying framework, highlighting the
thread of flawed causal reasoning that runs through equilibrium models with
non-rational expectations.
2
The Modeling Framework
Let = 1 × · · · × be a finite set of states, where ≥ 2. I refer to each
as a variable, and = {1 } is the set of variable labels. For every
⊆ , denote = ( )∈ . Let 1 = be the set of feasible actions
available to a decision maker (DM). I use 1 or interchangeably to denote
3
Similar elements of "coarse reasoning" appeared in macroeconomics under the title
"restricted perceptions equilibrium" (Evans and Honkapohja (2001), Woodford (2013)).
6
an action. I often write the state as a pair ( ), where = (2 ).
Thus, the DM’s action is part of the description of a state. Assume that the
DM is uninformed of the realization of any of the other variables in when
he takes his action (Appendix C relaxes this assumption). Let ∈ ∆() be
an objective probability distribution over states.
2.1
Beliefs
To capture limited understanding of the correlation structure of , I introduce
a new primitive. Let be an irreflexive, asymmetric and acyclic binary
relation over , and for every ∈ define () = { ∈ | }. The pair
( ) is a directed acyclic graph (DAG), where is the set of nodes and
is the set of directed links. I write and → interchangeably. Abusing
notation, I identify the DAG with , and use ̃ to denote the skeleton of
(namely its undirected version - ̃ if and only if or ). A subset
⊆ is a clique in if ̃ for every ∈ . A clique is ancestral
if () ⊂ for every ∈ . A node is a descendant of another node if
there is a directed path in from to .
The DM is characterized by a "subjective DAG" . For any objective
distribution , the DM’s subjective belief over is , given by the factorization formula (2). Thus, is a short-hand for a mapping that assigns a
subjective belief to every objective distribution. It is instructive to compare
this to the traditional notion of subjective priors. Under the latter approach,
the DM has a fixed belief which is independent of the objective distribution
. In contrast, according to the DAG representation (2), the DM’s subjective
belief changes systematically with .4
We will say that is consistent with a DAG if () = () for all
∈ . If is consistent with , it is necessarily consistent with every DAG
that contains . For any three disjoint subsets ⊂ , the notation
⊥ | means that for every that is consistent with , and
are independent conditional on . Appendix B presents a basic tool
4
The possibility that (() ) = 0 does not pose difficulties, because when we sum over
in expected-utility calculations, we exclude zero-probability realizations.
7
from the Bayesian-networks literature, called -separation, which characterizes the conditional-independence properties satisfied by all distributions that
are consistent with a given DAG. This means that a DAG represents the set
of distributions that satisfy a certain list of conditional-independence properties. For example, the DAG 1 → 3 ← 2 represents the property 2 ⊥ 1 ,
while the DAG 1 → 3 → 2 represents the property 2 ⊥ 1 | 3 . (However, not every collection of conditional-independence properties has a DAG
representation.)
For any , the DM’s subjective distribution over conditional on is
defined as usual,
( | ) =
as long as () 0.
( )
( )
=P
0
()
0 ( )
The interpretation of
Following Pearl (2009), I interpret the DM’s DAG as a prior subjective
causal model (alternative interpretations are discussed in Section 6). According to this interpretation, () is the set of variables that the DM perceives
to be direct causes of the variable . The subjective belief is the outcome
of the DM’s attempt to fit his causal model to data generated by . The DM
does not have any preconception regarding the sign or magnitude of causal
relations - he infers those from the data; his causal model merely postulates
causal links and their direction. The causal interpretation justifies the asymmetry and acyclicity of . It also justifies the inclusion of the DM’s action
in the description of a state - the DM’s subjective model establishes causal
relations between his action and the other variables.
The following image makes the causal interpretation more concrete. The
DM has access to a rich "historical database" consisting of many observations
of joint realizations of and , independently drawn from . He poses a
sequence of questions to the database, where question is: "What is the
distribution over conditional on () ?" The DM then pastes the answers
by taking the product of the conditional distributions ( | () ). The DM
8
poses these particular questions because he looks for the correlations that
are implied by his subjective causal model; he does not look for correlations
it rules out, due to implicit data-processing costs.
Equivalent DAGs
A given can be consistent with multiple DAGs, even when they do not
contain one another. For instance, when = 2, the DAGs 1 → 2 and
2 → 1 are both consistent with rational expectations, due to the basic identity
(1 2 ) ≡ (1 )(2 | 1 ) ≡ (2 )(1 | 2 ). This suggests a natural
equivalence relation: two DAGs are equivalent if they represent the same
mapping from objective distributions to subjective beliefs.
Definition 1 Two DAGs and are equivalent if () ≡ () for
every ∈ ∆().
Thus, two different causal models can be indistinguishable in terms of
the statistical regularities they induce. In particular, a DAG that involves
intuitive causal relations can be equivalent to a DAG that makes little sense
as a causal model (e.g., it postulates that the DM’s action is caused by his
final payoffs).
The following characterization of equivalent DAGs will be useful in the
sequel. Define the -structure of a DAG to be the set of all ordered triples
of nodes ( ) such that , ,
/ and
/ (that is, contains links
from and into , yet and are not linked to each other).
Proposition 1 (Verma and Pearl (1991)) Two DAGs and are equivalent if and only if they have the same skeleton and the same -structure.
To illustrate this result, all linear orderings have the same skeleton (every
pair of nodes is linked) and a vacuous -structure, hence they are all equivalent (indeed, they all induce rational expectations because they reduce
(2) to the textbook chain rule). In contrast, the DAGs 1 → 2 → 3 and
1 → 2 ← 3 are not equivalent because they have identical skeletons but different -structures (vacuous in the former case, and consisting of the triple
(1 3 2) in the latter).
9
True and subjective DAGs
I will often restrict the domain of possible objective distributions to be
those that are consistent with some DAG ∗ . In this case, I will simply say
that the "true DAG" is ∗ . Such domain restrictions arise naturally when
reality has an underlying causal structure, such that ∗ captures the true
causal relations in the DM’s environment. A linear ordering corresponds to
an unrestricted domain of objective distributions.
In applications, it is often the case that the DM’s subjective DAG can
be obtained from ∗ via one of the following simple operations: inverting, removing, reorienting or adding links. These operations intuitively correspond
to basic errors of causal attribution. We saw that inverting a link captures
reverse causation.5 Let us briefly discuss the others:
() Removing a link captures coarseness. For example, if ∗ : 1 → 3 ← 2
and : 1 → 3
2, then captures a coarse perception of the causes of 3 :
in reality, 3 is a function of both 1 and 2 , yet acknowledges only 1 .
Thus, while in reality () = (1 )(2 )(3 | 1 2 ), the DM’s subjective
belief is () = (1 )(2 )(3 | 1 ).
() Changing the origin of a link captures misattribution. For instance, if
3 ← 2 and : 1 → 3
2, then errs by attributing 3 to the
∗ : 1
wrong cause (1 instead of 2 ). Compare this example to the one used to
illustrate coarseness: the same can capture different errors, depending on
its exact relation to ∗ .
() Adding a link captures spurious causation - i.e., the DM postulates a
direct causal relation that does not exist in reality. However, recall that if
is consistent with ∗ then it is also consistent with any DAG that adds links
to ∗ . It follows that incorrect subjective beliefs that arise from spurious
causation are outside the framework’s scope.6
5
Although inversion and reorientation can be decomposed into addition and removal
of links, it makes sense to view them as "primitive" operations because they correspond
to intuitive errors of causal attribution.
6
The operations need not preserve equivalence between DAGs. Let = {1 2 3} and
omit the link 2 → 3 from two linear orderings that contain it - one in which 1 → 2
and another in which 3 → 1. The obtained DAGs are not equivalent: 2 ← 1 → 3 and
2 → 1 ← 3.
10
Throughout the paper, I will assume that ∗ (1) = (1) = ∅ - i.e. the
node that represents the DM’s action is ancestral in both true and subjective
DAGs. This restriction is not necessary: a DAG in which 1 is ancestral can
be equivalent to a DAG in which it is not. Nevertheless, I will adhere to it
for several reasons. First, it fits the causal interpretation: the DM’s action
cannot be caused by a variable if he does not get a signal about its realization.
Second, the restriction often simplifies definitions and expressions - e.g. it
can be shown that
(2 | 1 ) =
Y
=2
( | () )
Finally, the restriction suggests a natural definition of endogenous variables:
is endogenous according to if is a descendant of 1.
A "Mixed" DAG representation
The DAG representation can be extended, by allowing the DM to be characterized by a probability distribution over DAGs, such that his subjective
belief is
X
() ()
(4)
() =
This representation captures magnitudes of belief errors. For example, when
mixes between the true DAG ∗ and some other DAG , the magnitude of
the DM’s error increases with (). An example of a mixed representation is
"partial cursedness" (Eyster and Rabin (2005)) - see Section 5. The analysis
in this paper will focus on the "pure" DAG representation (2).
2.2
Decisions
Let us turn to decision making under the DAG representation of subjective
beliefs. Our DM is an expected utility maximizer, with a vNM utility function : → R. Recall that represents a long-run joint distribution over
all variables. We will require the DM’s long-run behavior (given by the marginal of over ) to be optimal w.r.t his subjective belief . As we shall
see, the belief distortion inherent in means that the marginal long-run
11
distribution over actions can influence the DM’s evaluation of each course of
action, because ( | ) is generally not invariant to (()) . Therefore, we
are led to define individual choice as an equilibrium notion.
To motivate the definition of equilibrium, suppose the above-mentioned
"historical database" is created by a long sequence of short-lived agents facing
the decision problem. The distribution represents the historical joint distribution over the agents’ actions and all other variables. Each agent forms a
subjective (possibly distorted) view of historical data given by , and takes
an action that maximizes his expected payoff according to this subjective
view. Equilibrium will correspond to a steady state of this dynamic.
As usual, equilibrium will require the DM to optimize w.r.t his subjective
belief. It would be conventional to treat (()) as the object of the definition
and take (( | )) as given. Instead, I will treat the entire joint distribution as the object of the formal definition. This is a contrivance that
merely simplifies notation (in applications, I will fix (( | )) and find
(()) ). Also, the need for an equilibrium definition requires us to consider
off-equilibrium actions. I address this concern with a conventional "trembling
hand" criterion. We say that 0 is a perturbation of if 0 ( | ) ≡ ( | )
and the marginal of 0 on has full support. A perturbation fixes every
aspect of except the DM’s behavior, such that every action is played with
positive probability.
Definition 2 (Personal equilibrium) A distribution ∈ ∆() with full
support on is an -perturbed personal equilibrium if
∈ arg max
0
X
( | 0 )(0 )
whenever () . A distribution ∗ ∈ ∆() is a personal equilibrium if
there exists a sequence → ∗ of perturbations of ∗ , as well as a sequence
→ 0, such that is an -perturbed personal equilibrium for every .
The concept of -perturbed personal equilibrium allows the DM to experiment with sub-optimal actions with probability at most. Personal equilibrium takes the → 0 limit.
12
Proposition 2 For every (( | )) , there exists (()) such that =
((() (( | )) ) is a personal equilibrium.
As we shall see in Section 3, "pure" personal equilibria (where the marginal of over is degenerate) need not exist. Note that if we restrict the
domain of possible objective distributions to those that are consistent with
a true DAG ∗ , then all personal equilibria are consistent with this DAG.
3
Illustrations
This section analyzes personal equilibria for various specifications of true and
subjective DAGs and ∗ . In each example, is obtained from ∗ by one
of the basic operations discussed in Section 2. Each sub-section presents
the material in a different concrete economic context, thus illustrating the
framework’s scope of applications. The analyses follow a two-step "recipe".
First, I provide a basic characterization of the DM’s personal-equilibrium
behavior, involving a formula for ( | ) that is based entirely on the
structure of . Certain properties of this characterization can be gleaned
from this formula (aided by knowledge of the true DAG ∗ and the payoffrelevant variables). In the second step, I impose parametric assumptions
on and ( | ) that fit the economic scenario, and use them to obtain
closed-form expressions for the conditional-probability terms in the formula
for ( | ). This enables me to complete the characterization of personal
equilibria. In this section alone, I abuse notation and identify variables with
their labels, for expositional ease.7
3.1
Reversing Causation: Fiscal Policy
The DM is a government facing a fiscal-policy dilemma. There are three variables, denoted , representing the levels of government spending, GDP
7
Throughout this section, the trembling-hand criterion in the definition of personal
equilibrium merely ensures that equilibria are well-defined, but the equilibria do not rely
on the selection of the sequence of perturbations. Trembles play a more interesting role
in the example analyzed in Appendix C.
13
growth and public debt. When the government chooses , it is uninformed
of or . The variables and are statistically independent, and is a
consequence of and . Thus, the true DAG is ∗ : → ← , such
that every possible objective distribution can be written as ( ) =
()()( | ). If the government had rational expectations, it would
choose to maximize
XX
()( | )( )
Suppose that the government’s subjective DAG is : → → , such
that ( ) = ()( | )( | ). Thus, relative to the true DAG ∗ ,
inverts the direction of the causal link between growth and debt. If is a
personal equilibrium, then for every 0 for which (0 ) 0,
0 ∈ arg max
XX
( | )( ) =
XX
( | )( | )( )
According to the true DAG ∗ , is not independent of conditional on
. That is, the conditional probability ( | ) implicitly involves summing
over the government’s spending levels, where the weights are affected by the
marginal of over ; if (()) were to change, so could ( | ), and so
could the government’s optimal action. Thus, the equilibrium aspect of the
government’s choice is not redundant.
For the rest of this sub-section, I impose additional structure. All variables take two values, low (denoted ) and high (denoted ). I refer to
the actions = and = as "austerity" and "expansion". The objective distribution satisfies ( = ) = 12 (independently of , in line
with the true DAG ∗ ), and is a deterministic function of : =
if and only if = and = . Thus, high debt can only result from
a combination of fiscal expansion and low growth. Finally, assume that
( ) = 1( = ) + · 1( = ), where 0 is a constant. That
is, the government cares about social programs (which it provides via fiscal
expansion) and growth; it does not care about debt per se. Under rational
expectations, the government would choose = with certainty.
14
Let us now characterize personal equilibria under the government’s subjective DAG. Denote ( = ) = . We have specified ( | ), hence it
remains to find . Before stating the result, let us calculate a few relevant
conditional probabilities: ( = | = ) = 1, ( = | = ) = 12 ,
( = | = ) = 0 and
( = | = ) =
1
2
·1
1
·1
2
1
+ 2 · (1
− )
=
1
2−
The fact that appears in the last expression demonstrates our earlier observation that it is not true that ⊥∗ | . The intuition is simple. If we
learn that the government has chosen expansion, we can predict a perfect
negative correlation between debt and growth. In contrast, if we learn that
the government has chosen austerity, growth is equally likely to be high or
low, independently of the level of debt. The higher the long-run frequency
of expansion , the stronger the estimated (negative) correlation between
debt and growth. Thus, the model exhibits strategic substitutability: when
the steady-state frequency of expansion is low (high), the perceived effect
of on is large (small), and this strengthens (weakens) the government’s
tendency to choose expansion.
Proposition 3 There is a unique personal equilibrium:
⎧
⎪
⎨
1
( = ) =
2−
⎪
⎩
0
2
≤2
∈ (2 4)
4
Proof. (Proofs of analogous results in later sub-sections are relegated to
Appendix A). The government’s evaluation of = given is
1 + · [( = | = )( = | = )
+( = | = )( = | = )] = 1 + ·
15
1
1
·
2 2−
The government’s evaluation of = given is
· [( = | = ) · ( = | = )
+( = | = ) · ( = | = )] = · 1 ·
1
2−
Note that when the government evaluates an action, it takes as given, as
required by the definition of personal equilibrium. In equilibrium, 0
( 1) only if = () attains the highest evaluation. This immediately
gives the result.
Thus, although expansion is the unique optimal action under rational
expectations, it is inconsistent with personal equilibrium given the government’s subjective DAG, as long as it cares enough about growth. The possibility of a "mixed" unique equilibrium provides a further demonstration that
individual choice in this model is fundamentally an equilibrium notion. This
effect would be impossible under conventional subjective expected-utility
maximization.
What is the intuition for this result? Suppose the government plays =
in a putative equilibrium. Then, it learns a perfectly negative correlation between debt and growth. It correctly grasps the effect of fiscal policy on debt.
And since it misperceives debt-growth correlation as a causal effect of debt
on growth, it erroneously believes that the low debt resulting from austerity
will lead to high growth. This is not the usual logic of self-confirming expectations (Fudenberg and Levine (1993)): the DM’s reasoning does not rest
on off-equilibrium beliefs, but on incorrect causal inference from statistical
regularities on the equilibrium path.
Comment: The Dieter’s Dilemma revisited. The example from the Introduction can be cast in the same terms as the fiscal policy example. Suppose that
= indicates that the DM diets; = indicates good health; and =
indicates an abnormal chemical level. Personal equilibrium is then given by
Proposition 3.
16
3.2
Coarseness I: Demand for Education
In this sub-section I present an example in which the DM’s subjective DAG
commits a "coarseness" error of ignoring an exogenous confounding variable.
I describe it in the concrete context of demand for education. The DM is
a parent who chooses how much to invest in his child’s education (I use a
female pronoun for the child). There are four variables, denoted ,
representing the parent’s investment, the child’s innate ability, her school
performance and her labor-market outcome. The parent receives no information regarding the other variables at the time of choice. The true DAG
∗ is
→ ←
(5)
& ↓
Note that is an exogenous "confounder": it causes variation in both and
. The parent’s utility is purely a function of and . Therefore, if the
parent had rational expectations, he would choose to maximize
X
()
X
( | )
X
( | )( )
Now suppose the parent’s subjective DAG removes the links
Because is payoff-irrelevant, we can equivalently assume that
altogether - i.e., : → → . One interpretation is that
neglects because it is unobservable. In personal equilibrium,
will choose to maximize
X
( | )
X
of in ∗ .
omits it
the parent
the parent
( | )( )
This expression can be elaborated into
Ã
X X
!
()( | )
Ã
X X
0
0
0
!
( | )( | ) ( )
Thus, the parent’s objective function involves two implicit summations over
17
the latent variable. The term (0 | ) is sensitive to (()) - e.g., if the
child performs well at school despite low parental investment, then she probably has high ability. It follows that individual choice in this example is
fundamentally an equilibrium notion.
In the rest of this sub-section I impose additional structure. Assume can
take any value in [0 1].8 All other variables have two possible realizations:
high (denoted 1) or low (denoted 0). Let ( ) = − (), where is a
twice-differentiable, increasing and weakly convex cost function, with 0 (0) =
0, and 0 (1) ≥ 1. Finally, assume ( = 1 | ) = and ( = 1 | ) =
, where 1 0 . Thus, high ability is necessary for both school and
labor-market success; conditional on high ability, success at school increases
the probability of labor-market success. Denote ( = 1) = 0. If the
parent had rational expectations, he would choose to maximize 1 +
(1 − ) 0 − (). The optimal action ∗ would be given by the first-order
condition:
0 (∗ ) = [ 1 − 0 ]
The following result characterizes personal equilibria under the parent’s subjective DAG.
Proposition 4 In every personal equilibrium, the parent chooses an action
∗∗ which solves the equation
∙
( ) = 1 − 0 ·
0
∗∗
(1 − ∗∗ )
(1 − ∗∗ ) + 1 −
¸
(6)
If 0 is either a weakly convex function or a weakly concave function, personal
equilibrium is unique.
According to Equation (6), the parent over-invests relative to the rationalexpectations benchmark. The reason is that he interprets the positive correlation between and as a pure causal effect of on , whereas in reality,
8
Although the formal analysis so far has assumed finite action sets, the extension to a
continuum of actions is straightforward.
18
the correlation is partly due to the confounder . This biases the perceived
marginal benefit from education upward.
Equation (6) also exemplifies the equilibrium aspect of individual behavior in this model, because the parent’s perceived marginal benefit from
education (given by the R.H.S) is a function of his equilibrium choice ∗∗ .
Although the parent’s subjective DAG postulates that ⊥ | , the true
DAG violates this property, and as a result the perceived causal effect of
on is sensitive to the parent’s equilibrium behavior. In particular, higher
long-run investment raises the perceived marginal benefit of education. Such
strategic complementarity can lead to multiple personal equilibria, depending
on how the curvature of 0 behaves.
The specification of ∗ and in this sub-section fits other contexts.
Esponda (2008) presents a bilateral-trade example (where the buyer makes
a take-it-or-leave-it offer), which can be translated as follows: , , and
represent the buyer’s offer, the seller’s valuation, the final allocation and
the buyer’s gross payoff from it. The buyer fails to realize that both the
probability of trade and the transaction value are affected by the seller’s valuation. The buyer’s behavior in Esponda’s example is equivalent to personal
equilibrium under the current specification of ∗ and , although Esponda
analyzes it using a different solution concept (see Section 5).9
3.3
Coarseness II: Managerial Decisions
The previous sub-section examined a DM who ignores the causal effects of an
exogenous confounding variable. Now I turn to a DM who misunderstands
the role of an endogenous variable in the causal chain from his action to some
target variable. Once again, consider the dilemma of whether to abstain
from a certain type of food. Because the food contains many ingredients,
the effect of abstention on a target health indicator can run through multiple
physiological channels, and the DM need not be aware of all of them. Other
examples lie in the area of public policy. When a government evaluates a tariff
9
Esponda and Pouzo (2014) study an electoral model, in which invdividual voters’
behavior can be translated into an informed-DM variation on the current specification.
19
reform, it may take certain consumption quantities as given, whereas in fact
they are endogenous variables that will respond to the reform. Likewise, when
a higher-education regulator considers changing the minimal accreditation
requirement of some degree, he may neglect the possible composition effect
on the pool of applicants, and therefore on the graduates’ ultimate quality.10
I analyze an example, in which a manager considers a costly action having
a beneficial effect on a measurable indicator of workers’ productivity (e.g.
output per hour on a certain task). However, the decision also impacts
another variable that affects total output - e.g. worker morale, the effort
they exert on another task, or their total number of working hours. This
variable is harder to measure, or presumed by the manager to be exogenous
or irrelevant.
Formally, there are four variables, 1 2 , where represents the manager’s action; 1 and 2 are two effects of his action on workers’ productivity;
and represents their final output. Assume that is purely a function of
and . The true DAG ∗ is
→ 1 →
&
%
2
(7)
If the manager had rational expectations, he would choose to maximize
XX
1
2
(1 | )(2 | )
X
3
( | 1 2 )( )
Now suppose that the DM’s subjective DAG differs from ∗ by removing at least one of the links of 2 (or by eliminating this node and its links
altogether). In personal equilibrium, if () 0, then maximizes
X
1
(1 | )
X
( | 1 )( )
10
(8)
Sargent (2001) models a central bank that believes in a classical Phillips curve, whereas
the true process is given by an expectational Phillips curve. The steady state in Sargent’s
model can be reformulated as personal equilibrium under this sub-section’s specification
of ∗ and .
20
which can be elaborated into
⎞
⎛
X XX
X
⎝
(1 | )
(0 | 1 )(02 | 0 )( | 1 02 )⎠ ( )
1
0
02
Thus, the manager’s failure to account for the causal channel that passes
through 2 means that when he calculates ( | 1 ), he effectively sums over
and 2 , weighted according to his equilibrium behavior.
Let us impose additional structure. The variables , 1 and 2 take values
in {0 1}; ( = 1 | ) = , where 1 2 0; and is a deterministic
function of 1 and 2 , given by = 1 + 2 , where 6= 0. Finally, ( ) =
− , where 0 is a constant. When 0, 2 is a reinforcing effect.
E.g., = 1 can represent an investment that improves working conditions,
such that 1 and 2 capture the direct effect on workers’ productivity and the
indirect effect on their morale. The manager neglects the effect of working
conditions on morale or the effect of morale on output, or he simply fails to
consider morale altogether. When 0, 2 is a countervailing effect. E.g.,
= 1 can represent the introduction of a device for monitoring a certain
task. In this case, 2 represents the worker’s output at another, unmonitored
task. The worker responds to the new regime by shifting effort from the
unmonitored task to the monitored task, an endogeneity the manager fails
to take into account.
If the manager had rational expectations, he would choose = 1 whenever
1 + 2 . The next result characterizes personal equilibria under his
subjective DAG.
Proposition 5 () When 1 ≥ , there is a personal equilibrium in which
( = 1) = 1. () When 1 (1 + 2 ) ≤ , there is a personal equilibrium in
which ( = 1) = 0. () When is between 1 and 1 (1 + 2 ), there is a
personal equilibrium with ( = 1) ∈ (0 1), given by the equations
= 1 (1 + 2 )
( = 0)
=
( = 0) + ( = 1) · (1 − 1 )
21
Thus, as expected, the manager’s neglect of a reinforcing (countervailing)
effect biases his behavior toward = 0 ( = 1). However, there is another
qualitative difference between the two cases, which is perhaps less obvious:
when 0 there is a unique personal equilibrium, whereas multiple equilibria
are possible under 0. In both cases, there is a mixed equilibrium for a
range of cost values.
4
General Analysis
In this section I characterize rationality properties of personal equilibrium,
using elementary tools from the Bayesian-networks literature. The characterizations are stated in terms of structural features of true and subjective
DAGs, and do not involve parametric restrictions on and ( | ). For
expositional ease, I present the simplest versions of the results. In Sections
4.1 and 4.2, I use 1 (rather than ) to denote the DM’s action.
4.1
Strategic Rationality
In the examples of Section 3, individual optimization under a misspecified
subjective DAG led to genuine equilibrium effects (mixed unique equilibrium,
multiple equilibria). This sub-section offers a systematic treatment of this
idea, based on the following notion of "strategic rationality".
Definition 3 A DAG is strategically rational w.r.t a true DAG ∗
if the following holds for every pair of objective distributions that are
consistent with ∗ : if (2 | 1 ) = (2 | 1 ) for every , then
(2 | 1 ) = (2 | 1 ) for every .
Strategic rationality requires that if we modify an objective distribution
that is consistent with ∗ only by changing its marginal over 1 - without changing the stochastic mapping from 1 to 2 - then the DM’s
perception of this mapping should remain unchanged as well. When is
strategically rational w.r.t ∗ , we can rewrite the definition of personal equilibrium as a maximization problem, because (2 | 1 ) is independent
22
of the DM’s long-run behavior given by ((1 ))1 . When strategic rationality is violated, individual behavior will exhibit equilibrium effects for some
specifications of . Clearly, any DAG is strategically rational w.r.t itself.
From now on, I will take it for granted that 6= ∗ .
Proposition 6 The subjective DAG is strategically rational w.r.t ∗ if
and only if for every 1, 1 ∈
() implies ⊥∗ 1 | () .
Thus, a necessary and sufficient condition for strategic rationality is the
following: whenever omits the DM’s action as a direct cause of some
other variable , it must be the case that for every distribution that is
consistent with ∗ , is independent of 1 conditional on () . Note that
when ∗ is a linear ordering - i.e., when the domain of objective distributions
is unrestricted - the condition for strategic rationality becomes 1 ∈ () for
every 1.
To illustrate this result, recall the "reversing causation" example of Section 3.1, where the true and subjective DAGs are ∗ : 1 → 3 ← 2 and
: 1 → 3 → 2. Since 1 ∈ (3) and 1 ∈
(2) = {3}, Proposition 6 implies
that we only need to check whether 2 ⊥∗ 1 | 3 . As observed in Section
3.1, this property does not hold, hence strategic rationality is violated.
Alternatively, for the same ∗ : 1 → 3 ← 2, suppose the DM’s subjective
DAG is : 1 → 3
2. This DM is "fully coarse/cursed" in the sense of
Eyster and Rabin (2005) and Jehiel and Koessler (2008): he fails to perceive
the effect of the exogenous state 2 on the final consequence 3 . The DM
will choose 1 to maximize
X
2 3
(2 3 | 1 )(1 2 3 ) =
XX
2
3
(2 )(3 | 1 )(1 2 3 )
If the DM had rational expectations, he would maximize the same expression,
except that (3 | 1 ) would be replaced with (3 | 1 2 ). To see why
strategic rationality holds in this case, note that (2) = ∅ and (3) = {1};
by Proposition 6 we only need to check that 2 ⊥∗ 1 , which clearly holds.
23
When and ∗ are large, checking the conditional-independence conditions of Proposition 6 can be a daunting task. However, it is greatly facilitated by a Bayesian-networks tool called -separation (see Koski and Noble
(2009, pp. Ch.2) and Pearl (2009, pp. 16-18)). Appendix B presents the tool
and illustrates its applicability in the present context.
4.2
Behavioral Rationality
Strategic rationality is a weak rationality requirement, which allows the DM
to choose an objectively sub-optimal action. In this sub-section I look for
a structural property of the DM’s subjective DAG that will ensure fully
rational behavior in terms of objective payoffs. I impose no restriction on
the set of possible objective distributions - i.e., the true DAG ∗ is some
linear ordering. Instead, I assume that is purely a function of , where
1 ∈ ⊂ (all the examples in Section 3 had this feature).
Definition 4 A DAG is behaviorally rational if in every personal equiP
librium , (1 ) 0 implies 1 ∈ arg max01 −1 (−1 | 01 )(01 −1 ).
As a first step toward characterizing behavioral rationality, I examine the
following question: when is the DM’s subjective marginal distribution over
some collection of variables guaranteed to be unbiased, despite his misspecified subjective DAG? By definition, if is not a linear ordering, then there
exists an objective such that 6= . However, may agree with on
some projections. The next result characterizes which ones.
Proposition 7 (Spiegler (2015)) Let be a DAG and let ⊂ . Then,
( ) ≡ ( ) for every if and only if is an ancestral clique in some
DAG in the equivalence class of .
For instance, let = 3 and : 1 → 2 ← 3. Then, (2 ) is biased
for some , because the node 2 is not ancestral in (and no other DAG
is equivalent to , by Proposition 1). In contrast, when : 1 → 2 → 3,
24
(3 ) coincides with (3 ) because 3 is ancestral in the equivalent DAG
0 : 3 → 2 → 1. In both examples, (1 3 ) does not coincide with
(1 3 ) for every , because the nodes 1 and 3 are not linked and therefore
cannot form a clique (let alone an ancestral one) in any equivalent DAG, by
Proposition 1.
Proposition 8 The DM is behaviorally rational if and only if is an ancestral clique in some DAG in the equivalence class of .
Thus, when all payoff-relevant variables are causally linked and have no
other cause (according to some DAG in the equivalence class of the DM’s subjective DAG), the DM is behaviorally rational. Otherwise, there are specifications of for which his behavior is inconsistent with rational expectations.
Application: When can coarseness lead to sub-optimal behavior?
The modeling framework enables us to formulate the following question:
When does a specific error of causal attribution (captured by a basic operation on the true DAG) violate behavioral rationality? The following is
an example of such an exercise. Let = {1 }. Recall that the true DAG
∗ is a linear ordering, and assume that is a terminal node in ∗ (i.e.,
∈
∗ () for all ). The interpretation is that the variable is an
ultimate consequence, such that the DM’s payoff depends only on his action
and the ultimate consequence. Now suppose that the DM’s subjective DAG
differs from ∗ only by omitting a single link → .
Proposition 9 The DM is behaviorally rational if and only if = and
6= 1.
Thus, even if the DM performs the "coarseness" error of neglecting the
direct effect of some intermediate variable ( 6= 1 ) on the ultimate
consequence , he will be behaviorally rational. In any other case, there are
specifications of (2 | 1 ) and for which the DM’s error will have
payoff implications.
25
4.3
Payoff Ranking of DAGs
A more complete subjective DAG represents a more thorough understanding of correlation structures, hence it intuitively captures "more rational"
expectations. Does this mean that it will always lead to better objective
performance? The following example shows the answer to be negative. Let
= 4, and suppose is a linear ordering satisfying 1234. Suppose further that is purely a function of 1 and 4 . Obtain the DAG 0 by removing
the link 2 → 3 from . By Proposition 9, 0 is not behaviorally rational i.e., it is weakly dominated by in terms of the expected performance it
induces. Now obtain 00 by removing the link 2 → 4 from 0 . It is easy to
00
verify that is equivalent to a DAG in which {1 4} is an ancestral clique.
By Proposition 8, a DM whose subjective DAG is 00 performs exactly like
a DM whose subjective DAG is . Thus, removing a link from the DM’s
subjective DAG can result in better performance for some .11
I now examine the question with some generality. I return to the notation
1 = , −1 = . For expositional simplicity, suppose that is independent
of under every possible (i.e., is consistent with a true DAG ∗ in which
1 is an isolated node). This means that ( | ) ≡ (), for all relevant
subjective DAGs , which are therefore strategically rational w.r.t ∗ .
Definition 5 Let 0 be two DAGs in which the node that corresponds to
the DM’s action is isolated. We say that is more rational than 0 if for
every 0 , the pair of inequalities
X
()( )
X
()(0 )
0 ()(0 )
implies
X
X
0 ()( )
X
()( )
X
()(0 )
11
Eyster and Piccione (2013) made an observation in the same spirit, in the context of
their model of competitive asset markets in which traders hold diversely coarse theories.
26
That is, if ranks above 0 and 0 ranks 0 over , then the rationalexpectations payoff ranking of the two actions necessarily sides with . When
is a linear ordering and 0 is not, the property holds trivially. The following result shows that this is the only case in which two DAGs can be
unambiguously ranked in terms of their expected performance.
Proposition 10 Let 0 be two DAGs that are not linear orderings (and
in which the node that corresponds to the DM’s action is isolated). Then,
neither DAG is more rational than the other.
Definition 5 is extendible to mixed DAG representations (as defined in
Section 2). Let ∗ be a distribution that assigns probability one to linear
orderings. Consider two distributions 0 that do not assign probability
one to linear orderings, and satisfy = ∗ + (1 − )0 , where ∈ (0 1). It
is easy to see that is more rational than 0 . Thus, mixed representations
enable us to rank some types according to their performance.
5
Relation to Existing Concepts
This paper follows a growing literature on equilibrium models with nonrational expectations. Most of this literature proceeded by postulating gametheoretic solution concepts that capture different aspects of limited understanding of correlations. In this section I discuss a few of the concepts that
were defined for static games, and offer an admittedly perfunctory discussion
of their relation to the Bayesian-networks formalism. For expositional simplicity, I consider two-player games and examine the behavior of one player,
to whom I refer as the DM. Throughout the discussion, represents the opponent’s action and is a state of Nature (known by the opponent). The
comparison can be extended to the case of partially informed DMs.
Analogy-based expectations
This concept was introduced by Jehiel (2005) in the context of extensive-form
games with complete information, and was later adapted to static Bayesian
games by Jehiel and Koessler (2008). The latter can be translated into the
27
Bayesian-networks language, as the following example illustrates. The true
← → , where represents the "analogy class" to which
DAG ∗ is
belongs. For a DM with rational expectations, is irrelevant and can be
omitted from his subjective model altogether. The DM’s subjective DAG
is
← ← - i.e., differs from ∗ only by changing the origin of the
link that goes into , from to . Thus, the DM interprets the opponent’s
equilibrium behavior as if it were a measurable function of the analogy class,
rather than the actual state of Nature. The definition of Analogy-Based
Expectations Equilibrium (ABEE) by Jehiel and Koessler (2008) requires
the DM’s action to be subjectively optimal w.r.t . Thus, we can reduce
individual best-replying under ABEE in static Bayesian games to subjective
optimization under the misspecified DAG .12
Naive behavioral equilibrium
Although the leading example in Esponda (2008) is translatable into the
Bayesian-networks language (see Section 3.2), it originally illustrated a solution concept called "naive behavioral equilibrium", whose relation to the
current framework is subtler. The true DAG ∗ is
→
& ↓ .
(9)
where is a "learning feedback" variable.13 The DM’s subjective DAG
differs from ∗ by removing the link → . Like personal equilibrium in the
present framework, naive behavioral equilibrium requires the DM’s action
to be subjectively optimal w.r.t . However, it goes further by requiring
() = () - i.e., the subjective marginal distribution over the feedback
variable should be unbiased. Because is not an ancestral node in any DAG
in the equivalence class of , Proposition 7 implies that Esponda’s additional
12
Esyter and Piccione’s (2013) formalism can be similarly translated. Suppose that
is purely a function of , ⊂ . The DM’s subjective DAG omits some of the nodes
outside . This DAG corresponds to Eyster and Piccione’s notion of an incomplete
theory. The DM in their model best-replies to the subjective belief .
13
Esponda (2008) employs three distinct feedback variables. My simplification does not
violate the spirit of Esponda’s approach.
28
requirement is not vacuous. It follows that naive behavioral equilibrium in
a single-player game can be viewed as a refinement of personal equilibrium
under certain specifications of ∗ and .
Naive behavioral equilibrium is a concept in the tradition of self-confirming
equilibrium (Fudenberg and Levine (1993)). In the present example, selfconfirming equilibrium would require the DM to play a best-reply to a subjective belief that satisfies () = ( ) - that is, the only restriction on the
DM’s subjective belief is that it is consistent with the designated feedback
variable. Relative to these two solution concepts, the present framework
takes a different approach to modeling feedback: rather than adding explicit
feedback variables, it implicitly assumes that the DM’s feedback consists of
all the correlations that he perceives to be relevant given his subjective causal
model (Section 6 elaborates on the limited-feedback interpretation).
Partial cursedness
← . Let differ from ∗ only by
Suppose the true DAG is ∗ :
removing the link from into . The DM is characterized by a mixed DAG
representation that assigns weight to and weight 1 − to ∗ . This DM
is "partially cursed" in the sense of Eyster and Rabin (2005); the parameter
captures the extent to which the DM ascribes his own (null) information
to the opponent. The discussion at the end of Section 4.3 implies that DMs
with different values of can be unambiguously ordered in terms of their
objective payoff performance: a DM with a lower is "more rational".
S (k ) equilibrium
Osborne and Rubinstein (1998) presented a solution concept in which each
player postulates a direct mapping from his action to the game’s outcome,
without forming an explicit belief regarding opponents’ strategies. The player
estimates this mapping by sampling each action times against the opponents’ distribution, and selecting the action that performs best in his sample.
In the context of a two-person game, the player’s misperception can be described in terms of the Bayesian-networks formalism. The true DAG ∗ is
as given by (9), except that the node and its links are omitted. The DM’s
utility is purely a function of , and his subjective DAG is : → . It
29
is easy to show that in our model, the DM’s behavior in this case would be
consistent with rational expectations. However, this is because he perfectly
learns ( | ), whereas in Osborne and Rubinstein (1998), he uses a finite
sample to estimate it and naively neglects the sampling error.
6
Concluding Remarks
The Bayesian-networks framework developed in this paper enables us to analyze behavioral implications of systematic errors in statistical reasoning and
causal attribution - mistaking correlation for causation, ignoring confounding variables, etc. Statistics teachers regularly try to uproot these fallacies
from students. Indeed, one motivation behind the Bayesian-networks literature has been to systematize correct causal reasoning. Instead, this paper
employed the tool descriptively, to capture the very errors statisticians and
econometricians warn us against.
Alternative interpretations of the subjective DAG
The "Q&A" story used to illustrate the causal interpretation of suggests
a different one: the questions need not reflect an explicit prior causal model,
but rather a boundedly rational attempt to find statistical patterns in the
data. The DM is unable to grasp the multivariate distribution in its totality,
and poses a sequence of partial queries that examine slices of . DAG
captures the questions that are natural to pose in the relevant context. The
DM behaves as if he has an explicit subjective causal model, but the essence
of this bounded rationality is that he fails to ask the right questions about
. For instance, recall Esponda’s (2008) bilateral-trade example discussed
in Section 3.2. The buyer learns the observable variables’ realizations in
chronological order: first he submits his bid, then he learns whether trade
takes place, and finally he learns the traded object’s value. It is therefore
natural for him to pose questions that track this order: What is the likelihood
of trade as a function of my bid? What is the expected value of the object
if trade takes place? These same partial queries are implied by a subjective
30
DAG that omits the seller’s valuation.14
Certain DAGs can be interpreted as representations of objective feedback
limitations that are faced by the DM as he tries to learn , given the method
he employs for extrapolating from limited feedback. Imagine that the DM only
manages to learn the marginals of over some collections of variables, and
that he wishes to extend these marginals to a fully specified distribution over
. Hajek et al. (1992) showed that when is perfect (i.e., direct parents
of any node are linked), there exists S ⊂ 2 such that is the maximalentropy extension of the known marginals of over , ∈ S. Moreover,
S is the set of maximal cliques in . In this sense, the DAG representation
(2) can be justified as the outcome of systematic extrapolation from limited
learning feedback. In Spiegler (2015), I further develop the limited-feedback
foundation for the DAG representation, by considering a more "behaviorally
motivated" method of extrapolation.
Finally, the DM’s attempt to fit to data generated by can be described in terms of Bayesian learning with misspecified priors. Esponda and
Pouzo (2014b - EP henceforth) proposed a game-theoretic framework, where
each player is characterized by a "subjective model" - a set of stochastic
mappings from his action to a primitive set of payoff-relevant consequences
(for simplicity, assume players are uninformed). EP’s definition of equilibrium requires the player’s subjective distribution over to be the closest
(in the set implied by his subjective model) to the true equilibrium distribution; distance is measured by a weighted Kullback-Leibler divergence. EP
justify this concept as the steady state of a process of Bayesian learning by
forward-looking agents, extending classical results on Bayesian learning with
misspecified priors (Berk (1966)). Personal equilibrium in the present paper
can be viewed as an EP equilibrium in single-player games, where the player’s
subjective model is the set of all conditional probabilities ( ( | )) induced by the DAG for some objective .15
14
Esponda and Vespa (2014) interpret a pivotal-voting experiment in this spirit.
Relatedly, there is a strand in the literature that views boundedly rational agents as
"time-series econometricians" who work with a misspecified model (e.g., Bray (1982), Cho,
Sargent and Williams (2002), Rabin and Vayanos (2010)).
15
31
Multi-agent models
As we saw in Section 5, existing notions of equilibrium with non-rational
expectations were typically presented as distinct solution concepts in some
class of games. The Bayesian-networks formalism reduces the element of
non-rational expectations in such concepts to types of individual agents (their
subjective DAGs) within a single modeling framework. This has several advantages as we move to multi-agent models. First, the formalism enables
us to analyze interactions among agents who exhibit different kinds of belief
errors, and perform comparative statics along this axis. Second, a model
in which agents are characterized by distinct DAGs would exhibit "structured belief heterogeneity", because their beliefs are different deterministic
transformations of the same objective distribution . In particular, when is
consistent with the empty DAG, all agents’ beliefs will coincide with . Third,
since the DAG representation is not tied to a particular economic model, it
can be incorporated into diverse families of models (games, competitive markets). Finally, it provides a language for studying "high-order" reasoning
about boundedly rational expectations. By incorporating one agent’s DAG
as a variable in another agent’s subjective model, we can express statements
such as "player does not understand the correlation between player ’s understanding of correlations and his information". This element is beyond the
reach of current approaches, and I plan to explore it in future research.
References
[1] Berk, R. (1966), “Limiting Behavior of Posterior Distributions when the
Model is Incorrect,” Annals of Mathematical Statistics 37, 51-58.
[2] Bray, M. (1982), “Learning, Estimation, and the Stability of Rational
Expectations,” Journal of Economic Theory, 26, 318-339.
[3] Cho, I., N. Williams and T. Sargent (2002), “Escaping Nash Inflation,”
Review of Economic Studies, 69, 1-40.
32
[4] Cowell, R., P. Dawid, S. Lauritzen and D. Spiegelhalter (1999), Probabilistic Networks and Expert Systems, Springer, London.
[5] Esponda, I. (2008), “Behavioral Equilibrium in Economies with Adverse
Selection,” The American Economic Review, 98, 1269-1291.
[6] Esponda, I. and D. Pouzo (2014a), “Conditional Retrospective Voting
in Large Elections,” mimeo.
[7] Esponda, I. and D. Pouzo (2014b), “An Equilibrium Framework for
Modeling Bounded Rationality,” mimeo.
[8] Esponda, I. and E. Vespa (2014), “Hypothetical Thinking and Information Extraction in the Laboratory,” American Economic Journal: Microeconomics, forthcoming.
[9] Evans, G. and S. Honkapohja (2001), Learning and Expectations in
Macroeconomics, Princeton University Press.
[10] Eyster, E. and M. Piccione (2013), “An Approach to Asset Pricing Under
Incomplete and Diverse Perceptions,” Econometrica, 81, 1483-1506.
[11] Eyster, E. and M. Rabin (2005), “Cursed Equilibrium,” Econometrica,
73, 1623-1672.
[12] Fudenberg, D. and D. Levine (1993), “Self-confirming equilibrium,”
Econometrica, 61, 523-545.
[13] Geiger, D., T. Verma and J. Pearl (1990), “Identifying independence in
Bayesian networks,” Networks 20.5, 507-534.
[14] Jehiel, P. (2005), “Analogy-Based Expectation Equilibrium,” Journal of
Economic Theory, 123, 81-104.
[15] Jehiel, P. and F. Koessler (2008), “Revisiting Games of Incomplete Information with Analogy-Based Expectations,” Games and Economic Behavior, 62, 533-557.
33
[16] Kearns, M., M. Littman and S. Singh (2001), “Graphical models for
game theory,” Proceedings of the Seventeenth conference on Uncertainty
in artificial intelligence. Morgan Kaufmann Publishers Inc.
[17] Koller, D. and B. Milch (2003), “Multi-Agent Influence Diagrams for
Representing and Solving Games,” Games and Economic Behavior 45,
181-221.
[18] Koski, T. and J. Noble (2009), Bayesian Networks: An Introduction,
John Wiley and Sons.
[19] Köszegi, B. (2010), “Utility from Anticipation and Personal Equilibrium,” Economic Theory, 44, 415-444.
[20] Langer, E. (1975), “The illusion of control,” Journal of personality and
social psychology 32, 311-328.
[21] Madarasz, K. (2012), “Information projection: Model and Applications,” Review of Economic Studies 79, 961-985.
[22] Mullainathan, S. J. Schwartzstein and A. Shleifer (2008), “Coarse
Thinking and Persuasion,” Quarterly Journal of Economics 123, 577619.
[23] Osborne, M. and A. Rubinstein (1998), “Games with Procedurally Rational Players,” American Economic Review, 88, 834-849.
[24] Pearl, J. (2009), Causality: Models, Reasoning and Inference, Cambridge University Press, Cambridge.
[25] Piccione, M. and A. Rubinstein (2003), “Modeling the Economic Interaction of Agents with Diverse Abilities to Recognize Equilibrium Patterns,” Journal of the European Economic Association, 1, 212-223.
[26] Rabin, M. and D. Vayanos (2010), “The gambler’s and Hot-Hand Fallacies: Theory and Applications,” Review of Economic Studies, 77, 730778.
34
[27] Sargent, T. (2001), The conquest of American inflation, Princeton University Press.
[28] Schwartzstein, J. (2014), “Selective Attention and Learning,” Journal
of European Economic Association, 12, 1423-1452.
[29] Spiegler, R. (2015), “Bayesian Networks and Missing-Data Imputation,”
mimeo.
[30] Verma, T. and J. Pearl (1990), “Causal Networks: Semantics and Expressiveness,” Uncertainty in Artificial Intelligence, 4, 69-76.
[31] Verma, T. and J. Pearl (1991), “Equivalence and Synthesis of Causal
Models,” Uncertainty in Artificial Intelligence, 6, 255-268.
[32] White, H. and K. Chalak (2009), “Settable Systems: an Extension of
Pearl’s Causal Model with Optimization, Equilibrium, and Learning,”
Journal of Machine Learning Research 10, 1759-1799.
[33] Woodford, M. (2013), “Macroeconomic Analysis without the Rational
Expectations Hypothesis,” Annual Review of Economics, forthcoming.
Appendix A: Proofs
Proposition 2
Fix (( | )) . For a fixed ∈ (0 1), let be the set of distributions
∈ ∆() such that () ≥ for every . Denote = ( (( | )) ), and
define accordingly. Define
() = arg max
∈
X
()
X
( | )( )
If is an -perturbed personal equilibrium, then ∈ (). Because ( |
) is continuous in , is continuous as well. Also, the target function in
the definition of is linear in , hence () is a convex set. Since the
set is compact and convex, has a fixed point, by Kakutani’s theorem.
35
Therefore, an -perturbed personal equilibrium exists for any 0. By
standard arguments, there is a convergent sequence of -perturbed personal
equilibria.
Proposition 4
P
The parent’s objective function is =01 ( | )( = 1 | ) − (). By our
assumptions on , = 0 implies that = = 0 with certainty, whereas = 1
implies that = 1 with probability . Let denote the parent’s probability
measure over actions . This implies the following conditional probabilities:
( = 1 | ) = , ( = 1 | = 1) = 1 , and
R
0 0 (0 )(1 − 0 )
R
( = 1 | = 0) =
= 0
1 − + 0 (0 )(1 − 0 )
The derivative of the parent’s objective function is ( 1 − 0 ) − 0 (). The
first term, which represents the parent’s perceived marginal benefit from
education, lies in (0 1). The parent takes it as given when choosing . The
second term is continuous and strictly increasing, with 0 (0) = 0 and 0 (1)
1. Therefore, there is a unique best-reply ∗∗ , which means that assigns
probability one to ∗∗ . In equilibrium, ∗∗ solves the first-order condition
0 (∗∗ ) = ( 1 − 0 ), where
=
(1 − ∗∗ )
1 − + (1 − ∗∗ )
(10)
This gives the equation (6). Observe that 0 (∗∗ ) is continuous in ∗∗ , with
0 (0) = 0, 0 (1) ≥ 1, whereas ( 1 − 0 ) is a strictly convex and increasing
function of ∗∗ , which attains values strictly between 0 and 1. Therefore, the
two functions must cross at least once, such that (6) has a solution. When
0 is either weakly convex or weakly concave, the two functions cross exactly
once, such that the solution is unique.
Proposition 5
Denote ( = 1) = . To calculate personal equilibria, we need to compute
the conditional probabilities that appear in (8). All of them were defined
36
up-front, except ( | 1 ), which is given by ( = 1 | 1 = 1) = 1 and
( = 1 | 1 = 0) =
(1 − 1 )
(1 − 1 ) + 1 −
Because ( = 1 | 1 = 1) = 1, the manager’s evaluation of = 1 conditional
on 1 = 1 is
X
2
(2 | = 1)(1 + 2 ) = 1 + 2
independently of .
Let us try to sustain = 1 in equilibrium. Because (1 = 1 | = 0) = 0,
the manager’s evaluation of = 0, given by (8), is simplified into
XX
0
2
(0 | 1 = 0)(2 | 0 )2 =
X
2
(2 | 0 = 1)2 = 2
where the substitution follows from plugging = 1 into the expression for
(8). It follows that the manager’s evaluation of = 1 conditional on 1 = 0
is also 2 . Because ( = 1 | 1 = 1) = 1, the manager’s evaluation of = 1
is simplified into
(1 − 1 ) · 2 + 1 · (1 + 2 ) − = 1 + 2 −
Thus, the manager will weakly prefer = 1 to = 0 if 1 + 2 − ≥ 2 ,
i.e. if 1 ≥ .
Let us try to sustain = 0 in equilibrium. Since (1 = 1 | = 0) = 0,
the manager’s evaluation of = 0, given by (8), is simplified into
XX
0
02
(0 | 1 = 0)(02 | 0 )02 =
X
02
(02 | 0 = 0)02 = 0
where the substitution follows from plugging = 0 into the expression for
(8). It follows that the manager’s evaluation of = 1 conditional on 1 = 0
is also 0. Because ( = 1 | 1 = 1) = 1, the manager’s evaluation of = 1
is simplified into
(1 − 1 ) · 0 + 1 · (1 + 2 ) −
37
Thus, the manager will weakly prefer = 0 to = 1 if 1 (1 + 2 ) − ≥ 0.
Finally, let us try to sustain ∈ (0 1) in equilibrium. The manager’s
evaluation of = 0 is
XX
0
02
(0
|
1 = 0)(02 | 0 )02
(1 − 1 )
1−
· 2 +
·0
(1 − 1 ) + 1 −
(1 − 1 ) + 1 −
= (1 − ) 2
=
His evaluation of = 1 is
(1 − 1 ) · (1 − ) 2 + 1 · (1 + 2 ) −
The manager must be indifferent between the two actions, i.e.
(1 − ) 2 = (1 − 1 ) · (1 − ) 2 + 1 · (1 + 2 ) −
yielding the characterization of the mixed equilibrium. It is well-defined (i.e.,
∈ (0 1)) if and only if is between 1 and 1 (1 + 2 ).
Proposition 6
The conditional probability (2 | 1 ) can be written as
(1 ) ·
X
02 0
(1 ) ·
Y
=2
Y
=2
( | () )
(0 | ()∩{1} 0()−{1} )
and the term (1 ) cancels out. Recall that we are considering a modification
of that changes the marginal of on 1 , while leaving (2 | 1 )
intact for all . We need to check whether the term (0 | ()∩{1} 0()−{1} )
is affected, for = 2 . If 1 ∈ (), the term is clearly unchanged. In
38
contrast, when 1 ∈
(), the term can be written as
(0 | 0() ) =
X
00
1
(001 )(0 | 001 0() )
The term (0 | 001 0() ) is unaffected by the modification of . If it is not
constant in 001 , we can find a modification of (01 ) for some values of 01
such that the expression for (2 | 1 ) will change. In contrast, if
the probability is constant in 001 (i.e., (0 | 001 0() ) = (0 | 0() ), the
expression for (2 | 1 ) is necessarily unchanged.
Proposition 8
By assumption, 1 is an ancestral node in . By Proposition 7, (1 ) ≡ (1 )
for every . We can write (−{1} | 1 ) = ( ) (1 ). Therefore,
(−{1} | 1 ) ≡ (−{1} | 1 ) if and only if ( ) ≡ ( ). By
Proposition 7, the latter holds if and only if is an ancestral clique in a
DAG in the equivalence class of . If ( ) ≡ ( ), the definition of
behavioral rationality is trivially satisfied. If ( ) 6= ( ) for some
and , then we can easily construct such that the DM will strictly prefer
an action that is objectively sub-optimal.
Proposition 9
() If = 1 and = , then 1,
/ hence {1 } cannot be a clique (let alone
an ancestral one) in any DAG in the equivalence class of .
() If 6= , then , and yet and are not linked. By Proposition
1, these properties must hold in any DAG in the equivalence class of , which
means that {1 } cannot be an ancestral clique in any such DAG.
() If = and 6= 1, is a perfect DAG - i.e., if and , then ̃.
It is well-known that every clique in a perfect DAG is ancestral in some DAG
in its equivalence class (see Spiegler (2015) for details). Note that () ∪{}
is a clique because the only link that was removed from the original linear
ordering was → . Therefore, {1 } is an ancestral clique in some DAG in
the equivalence class of .
The result follows from () − (), by Proposition 8.
39
Proposition 10
If and 0 are equivalent, the claim holds trivially. Now assume there
exist non-equivalent 0 that are not linear orderings, such that is more
rational than 0 . Fix and denote = ( ()) , = (0 ()) . Both and
are probability vectors of length − 1. I use to denote the -th
component of the (−1) vectors . Define the (−1)-vector as follows:
for each , = ( ) − (0 ). Consider the ( − 1) × 3 matrix
=
1
..
.
− 1
..
.
−1
..
.
−1 − −1 −−1
Let = (− − −) be a vector in R3 , where 0 is arbitrarily small. The
assumption that is more rational than 0 thus implies that there exists
no that satisfies the inequality . By Farkas’s Lemma, this means
that there is a vector 0 in R3 , such that there = 0 (and since
2
3
0, 0). Thus, = 1 + 1 for every = 1 − 1. Since
P−1
P−1
P−1
1
2
3
=1 =
=1 =
=1 = 1 by assumption, = + , such that
the claim holds with = 3 (2 + 3 ).
We have thus established that for any , we can find ∈ (0 1) such that
= + (1 − )0 . In particular, for any that is consistent with ,
= and so the equation reduces to = 0 . Likewise, for any that
is consistent with 0 , 0 = and again we obtain = 0 . It follows
that the sets of distributions that are consistent with and 0 are identical,
contradicting the assumption that and 0 are not equivalent.
Appendix B: d-Separation
In this appendix I present the concept of -separation, which is useful for
applying the characterization of strategic rationality in Section 4.1. A path
in a DAG is a sequence of directly connected nodes in , ignoring the
links’ directions.
Definition 6 (Blocking a path) A subset ⊂ blocks a path in if
either of the following two conditions holds: (1) the path contains a segment
40
of the form → → or ← → such that ∈ ; (2) the path
contains a segment of the form → ← such that neither nor any of
its descendants are in .
To illustrate this definition, consider the DAG : 1 → 2 ← 3 → 4. The
path between nodes 1 and 4 is blocked by {3} - either because it contains
the segment 2 ← 3 → 4 (thus satisfying condition (1)) or because it contains
the segment 1 → 2 ← 3 (thus satisfying condition (2), as the node 2 has no
descendants and 2 ∈
{3}). However, the path between 1 and 4 is not blocked
by {2}, because it does not contain a segment of the form → 2 → or
← 2 → , and the only segment of the form → ← that it contains
satisfies = 2.
Definition 7 (-separation) Let be disjoint subsets of . We say
that and are -separated by (in a DAG ) if blocks every path
between any node in and any node in .
Proposition 11 (Verma and Pearl (1990)) Let be disjoint subsets of . Then, ⊥ | if and only if and are -separated by
in .
Thus, -separation provides a convenient rule for checking whether a conditional independence property is satisfied by all the distributions that are
consistent with a DAG. Moreover, the rule is computationally simple: Geiger
et al. (1990) presented a linear-time algorithm for checking -separation.
Armed with this result, I illustrate Proposition 6, using two specifications
from Section 3.
Ignoring a confounder (Section 3.2). The true DAG ∗ is given by (5), and
: → → . Because ∈
() = {}, Proposition 6 requires us to
check whether ⊥∗ | . To see why this condition fails, observe that
→ ← → is a path in ∗ that connects and . This path is
not blocked by (as we saw in the example that illustrated blocking), and
therefore and are not -separated by .
41
Ignoring an effect (Section 3.3). The true DAG ∗ is given by (7) and
() = {1 }, we only need to
: → 1 → . Since ∈ (1 ) and ∈
check that ⊥∗ | 1 - i.e., that and are -separated by 1 in ∗ . This
condition is violated, because ∗ contains the path → 2 → , which is not
blocked by 1 . It follows that strategic rationality is violated.
Appendix C: The Case of an Informed DM
In this appendix I extend the decision model to the case in which the DM receives a signal 0 ∈ 0 prior to making his decision. Thus, = (0 1 ).
I use the notations 0 and interchangeably, and often write = ( ).
All DAGs are now defined over = {0 1 2 }, such that
() =
Y
=0
( | () )
Assume that in all DAGs, whether true or subjective, the DM’s signal is the
sole direct cause of his action - i.e., (1) = ∗ (1) = {0}.
The extension of the definition of personal equilibrium is straightforward.
Let have full support on . We say that 0 is a perturbation of if 0 () ≡
(), 0 ( | ) ≡ ( | ), and 0 has full support on × .
Definition 8 A distribution ∈ ∆() with full support on × is an
-perturbed personal equilibrium if
∈ arg max
0
X
( | 0 )( 0 )
for every for which ( | ) . A distribution ∗ ∈ ∆() with full
support on is a personal equilibrium if there exists a sequence → ∗
of perturbations of ∗ , as well as a sequence → 0, such for every , is
an -perturbed personal equilibrium.
The existence result given by Proposition 2 extends to this case.
42
An example: Illusion of control
As in Section 3.2, consider a parent who makes a decision regarding his
child’s education. There are three variables, denoted , representing the
parent’s investment decision, the child’s school performance, and the parent’s
valuation of success at school. The parent is informed of before making
his choice. The true DAG ∗ : ← → . Thus, is independent of
conditional on . One story behind the causal link → is that the parent’s
values are imbued in the child and affect her attitude to learning. If the
P
parent had rational expectations, he would choose to maximize ( |
)( ).
Now suppose that the parent’s subjective DAG is : → → . Thus,
departs from ∗ by changing the origin of the link that goes into . This
link reorientation captures a misattribution error often referred to as "illusion
of control" (Langer (1975)): in reality, is caused by the exogenous variable
, yet the parent attributes to his own action. As in Section 3.2, the
parent’s error is that he mishandles a confounding variable. The difference
is that while in the previous example the parent neglected the confounder
altogether, here he is aware of it (indeed, he conditions his action on it),
yet he fails to perceive its role as a confounder; he interprets any correlation
between and as a causal effect of on , whereas in reality the correlation
is due to the confounder. In personal equilibrium, whenever (0 | ) 0,
0 ∈ arg max
X
( | )( ) =
X
( | )
X
( | )( )
(11)
As the term ( | ) indicates, the equilibrium aspect of individual behavior
is fundamental in this example.
Let us add structure to the example. All variables take values in {0 1}.
Assume ( = 1) = and ( = 1 | ) = , where ∈ (0 1). Finally, let
( ) = − , where ∈ (0 1) is a constant. Thus, the child succeeds
at school if and only if her family thinks it is important. The action = 1 is
a useless investment in the child’s education, and is its cost. If the parent
had rational expectations, he would choose = 0 for every , because the
costly action does not affect the child’s school performance. The following
43
result characterizes personal equilibria under the parent’s subjective DAG.
Proposition 12 There are multiple personal equilibria. In one equilibrium,
( = 1 | ) = 0 for all . In another, ( = 1 | ) = . Finally, if 1 − ,
there is a third equilibrium, in which
( = 1 | ) = ·
+−1
Proof. When the parent’s information is = 0, playing = 0 is optimal
regardless of his beliefs. Therefore, in any personal equilibrium, ( = 1 | =
0) = 0. Let us try to sustain ( = 1 | = 1) = 0 in equilibrium. Because
the action = 1 is never taken in this putative equilibrium, we need to check
whether there is a sequence of -perturbed personal equilibria that converges
to . For every 0, define ( = 1 | ) = for all , () ≡ () and
( | ) ≡ ( | ). Then, ( = 1 | = 0) ≡ ( = 1 | = 1), hence
playing = 0 is subjective optimal, which is consistent with the definition
of -perturbed personal equilibrium.
Now, let us try to sustain personal equilibria in which ( = 1 | = 1) =
0. Then, ( = 1 | = 1) = 1 and
( = 1 | = 0) =
−
(1 − )
=
1 − + (1 − )
1 −
It follows that when the parent’s information is = 1, his evaluation of = 1
is 1 − , whereas his evaluation of = 0 is ( − )(1 − ). Therefore,
0 if and only if 1 − ≥ ( − )(1 − ). This inequality is binding
if ∈ (0 1). Since 1, the inequality holds for = 1. If 1 − , the
inequality can hold bindingly with = ( + − 1).
This result demonstrates several effects. At the substantive level, it is
possible to sustain a sub-optimal equilibrium, in which parents make a useless investment in their children’s education if and only if they care about
school performance. Thanks to the positive correlation between and , this
behavioral pattern is consistent: the parent mistakenly attributes success at
school to the investment, and since only high-valuation parents make the
44
investment, and will be positively correlated. The parents’ erroneous
causal interpretation of this correlation aligns it with their incentives. At
the methodological level, Proposition 12 highlights the role of "trembles"
in sustaining personal equilibria: the rational-expectations outcome is sustainable because the parent’s off-equilibrium experimentation is uncorrelated
with his information, in which case he is not led to attribute variations in
school performance to variations in investment.
Strategic rationality
Let us extend the notion of strategic rationality to the case of an informed
DM.
Definition 9 A DAG is strategically rational w.r.t a "true DAG" ∗
if for every pair of distributions that are consistent with ∗ , if () = ()
and ( | ) = ( | ) for every , then ( | ) = ( | ) for
every .
Proposition 13 The subjective DAG is strategically rational w.r.t the
true DAG ∗ if and only if:
1. For every 1, if 1 ∈
(), then ⊥∗ 1 | ()∪{0} .
2. For every 1, if 0 ∈
(), then ⊥∗ 0 | ()∪{1} .
3. If (0) 6= ∅, then 1 ⊥∗ (0) | 0 .
Proof. Recall that = 0 , = 1 , = (2 ). By assumption, (1) =
(0) ∗ (0). We
∗ (1) = {0}. Hence, by the asymmetry of and ∗ , 1 ∈
can thus write ( | ) = (2 | 0 1 ) as
Y
=0
X
02 0
=
(0 | 0(0) ) · (1 | 0 ) ·
Y
6=1
X
02 0
( | () )
(0 | 0(0) ) ·
Y
=2
(0 | ()∩{01} 0()−{01} )
( | () )
Y
=2
(12)
(0 | ()∩{01} 0()−{01} )
45
Recall that we are considering modifications of that change (1 | 0 )
for some 0 1 , while leaving the marginal of on 0 and the conditional
distributions (2 | 0 1 ) intact. Both and its modification are
required to be consistent with ∗ . Let us now examine each of the terms in
(12). If all terms are invariant to any eligible modification of , so will be
(12) itself; otherwise, there is an eligible modification of that changes (12).
() For any (0) 0 , the term (0 | (0) ) can be written as follows:
P
(0 ) 01 (01 | 0 )((0) | 01 0 )
P
(0 | (0) ) = P
0
0
0
0
0
0 (0 )
0 (1 | 0 )((0) | 1 0 )
0
1
The term (0 ) is by definition invariant to the modification of . As to
((0) | 01 0 ), we saw that (0) ⊆ {2 }. By definition, (2 |
1 0 ) is invariant to the modification of . Since
((0) | 1 0 ) =
X
{2}−(0)
(2 | 1 0 )
It follows that ((0) | 01 0 ) is invariant to the modification of . Suppose
that ((0) | 01 0 ) is not constant in 1 . Then, we can find a distribution
that is consistent with ∗ and an eligible modification of that will change
(0), be independently
(0 | (0) ). In particular, let all variables , ∈
distributed under . The only term in (12) that will change as a result of
the modification of is (0 | (0) ), and therefore (12) will change, too. In
contrast, if ((0) | 01 0 ) is constant in 1 , then (0 | (0) ) is invariant
to any eligible modification of .
() and 0 ∈ (), then the term
() For any 0 and any () , if 1 ∈
( | () ) can be written as follows
(
|
() ) =
=
X
1
Ã
X
1
(1 | () )( | 1 () )
( )(1 | 0 )( | 1 0 )
P 0
0
0
0 (0 )(1 | 0 )( | 1 0 )
1
46
!
( | 1 0 )
where = () − {0}. By definition, the terms (0 ), ( | 1 0 ) and
( | 1 0 ) are invariant to the modification of . Suppose that ( |
1 0 ) is not constant in 1 . Then, we can find a distribution that is
consistent with ∗ and an eligible modification of that will change ( |
() ). In particular, let all variables , 0, 6= , be independently
distributed under . The only term in (12) that will change as a result of
the modification of is ( | () ), and therefore (12) will change, too. In
contrast, if ( | 1 0 ) is constant in 1 , then ( | () ) is invariant
to any eligible modification of .
() and 1 ∈ (), then the term
() For any 0 and any () , if 0 ∈
( | () ) can be written as follows
( | () ) =
X
0
Ã
( )( | )( | )
P 0 0 1 0 0 1 0 0
00 (0 )(1 | 0 )( | 1 0 )
!
( | 1 0 )
where = () − {1}. By definition, the terms (0 ), ( | 1 0 ) and
( | 1 0 ) are invariant to the modification of . Using essentially the
same argument as in (), we can show that if ( | 1 0 ) is constant
in 0 , then ( | () ) is invariant to any eligible modification of ; and if
( | 1 0 ) is not constant in 0 , we can find a distribution that is
consistent with ∗ and an eligible modification of that will change ( |
() ).
(), then the term ( | () )
() For any 0 and any () , if 1 0 ∈
can be written as follows
( | () ) = (0 | () )(1 | 0 () )( | 1 0 () )
where
(0
P
(0 )(01 | 0 )(() | 01 0 )
| () ) = P P
0
0
0
0
0
0
0 (0 )(1 | 0 )(() | 1 0 )
01
0
(1
1
(0 )(1 | 0 )(() | 1 0 )
| 0 () ) = P
0
0
01 (0 )(1 | 0 )(() | 1 0 )
47
By definition, the terms (0 ), (() | 1 0 ) and ( | 1 0 () ) are
invariant to the modification of . Using essentially the same argument as in
(), we can show that if ( | 1 0 () ) is constant in 1 0 , then ( |
() ) is invariant to any eligible modification of ; and if ( | 1 0 () )
is not constant in 1 0 , we can find a distribution that is consistent with
∗ and an eligible modification of that will change ( | () ).
The first condition in Proposition 13 is a minor variation on the condition
for strategic rationality in the uninformed-DM case. The second is an analogous condition for the new variable 0 . The third condition is that under
the true DAG, the DM’s action is independent of the signal’s direct (subjective) causes conditional on the signal. Let us illustrate this result using two
previous specifications.
Cursedness. The true DAG ∗ is ← ← → → , where represents a
state of Nature, and represents the signals obtained by the DM and his
opponent, and and represent their actions. The subjective DAG differs
from ∗ by changing the origin of the link that goes into , from to . This
specification captures a "fully cursed", partially informed DM, as in Eyster
and Rabin (2005). Condition (1) in Proposition 13 holds because the node
blocks any path in ∗ between the node and any other node. Condition (2)
holds vacuously because is the sole cause of under . Finally, Condition
(3) holds because ⊥∗ | . Therefore, strategic rationality holds.
Illusion of control. The true DAG is ∗ : ← → , while the subjective
DAG is : → → . Since ∈
() = {}, the second condition for
strategic rationality requires that ⊥∗ | , which is clearly false.
48
