Behavioral Implications of Causal Misperceptions Part I
Ran Spiegler (TAU & UCL)
ES Winter School, Delhi
December 2019
Equilibrium without Rational Expectations
β’ Standard equilibrium analysis in economics:
1. Steady state approach (even in dynamic models) 2. Agents best-reply to their beliefs.
3. Agentsβ beliefs reflect perfect understanding of the steady- state empirical regularities.
β’ Research program: Keep 1+2, relax 3
Examples from the Literature
β’ Sampling: Players evaluate action-consequence mapping via finite samples (Osborne-Rubinstein 1998)
β’ Coarseness: Beliefs are measurable w.r.t a partition of all contingencies (Piccione-Rubinstein 2003, Jehiel 2005)
β’ βCursednessβ: Players cannot perceive dependence of endogenous variables on factors other than their information (Eyster-Rabin 2005)
β’ NaΓ―ve extrapolation from selective samples (Esponda 2008)
Common Feature
β’ Agents interpret statistical regularities through the prism of a (wrong) subjective model.
β The model involves errors of causal attribution.
β The concepts differ in the kind of causal misattribution they assume and the data agents use to βestimateβ their model.
β’ In this lecture series: A formalism of equilibrium with non-rational expectations that takes this description as a starting point
In this Lecture Seriesβ¦
β’ Decision makers are endowed with subjective causal models, formalized as directed acyclic graphs (DAGs).
β Relying heavily on a rich Statistics/AI literature on probabilistic graphical models (βBayesian networksβ)
β’ Capturing agents who βmistake correlation for causationβ
β’ Partial unification of earlier approaches
In this Lecture Seriesβ¦
β’ Bayesian networks offer tools for representing causal
misperceptions and analyzing their behavioral implications.
β’ Applications: Health/lifestyle/occupational decisions, demand for education, monetary policy, narratives and political beliefs,
contracting with agents who misperceive their production function
β’ Opening the door for the study of causal reasoning by people other than Joshua Angrist...
Sources
β’ Forthcoming article in Annual Rev. of Econ.
β’ Three specific papers:
1. βBayesian Networks and Boundedly Rational Expectationsβ (QJE 2016)
2. βCan Agents with Causal Misperceptions be Systematically Fooled?β (JEEAforthcoming) 3. βA Model of Competing Narrativesβ (joint with Kfir Eliaz)
β’ Bayesian networks in Statistics and AI (Lauritzen 1996, Cowell et al.
1999, Pearl 2009, Koller-Friedman 2009, Pearl-Mackenzie 2018)
β’ Psychology of causal reasoning (Sloman 2005, Lagnado-Sloman 2015)
Lecture Plan
β’ Lecture 1: Individual Behavior
β Using DAGs to represent causal misperceptions
β Individual decision making as βpersonal equilibriumβ
β’ Lecture 2: Interaction
β Leader-follower model
β A βmonetary policyβ application
β’ Lecture 3: Endogenous Causal Models
β A model of competing political narratives
β’ An agent chooses whether to consume a dietary supplement.
β’ Three variables take values in {0,1}:
β ππ represents the agentβs action (1 means consuming) β β represents state of health (1 means good health) β ππ represents blood chemical level (1 means abnormal)
β’ The agentβs payoff is β β ππππ, where ππ > 0 is constant.
Example: The Dieterβs Dilemma
β’ ππ is a long-run (steady-state) distribution over ππ, β, ππ.
β’ ππ β = 1 = 0.5, independently of ππ.
β The agentΚΉs rational choice would be ππ = ππ.
β’ ππ ππ = 1 ππ, β) = (1 β ππ)(1 β β)
β Chemical level is normal if the agent is healthy or if he takes the supplement.
The Dieterβs Dilemma
β’ The agent has a subjective causal model, represented by a directed acyclic graph (DAG) π π over the three variables:
ππ β ππ β β
β A causal chain from action to health via chemical level
β’ The agent fits his causal model to the long-run distribution:
πππ π ππ, β, ππ = ππ ππ ππ ππ ππ ππ(β|ππ)
The Dieterβs Dilemma
β’ The agent relies on πππ π ππ, β, ππ = ππ ππ ππ ππ ππ ππ(β|ππ) to compute
πππ π β ππ = οΏ½
ππππ ππ ππ ππ(β|ππ)
β’ Why doesnβt the agent directly estimate ππ β ππ ? β The benefit of using models
β Differential availability of data about various correlations
The Dieterβs Dilemma
β’ π π : ππ β ππ β β assumes β β₯ ππ | ππ. This assumption is false:
β Given normal chemical level, if we learn that the agent didnβt take the supplement, we infer that he must be healthy.
β’ ππ ππ, β, ππ = ππ ππ ππ β ππ(ππ|ππ, β)
β This is consistent with a βtrueβ causal model ππ β ππ β β. β π π exhibits reverse causality w.r.t the true model.
The Dieterβs Dilemma
β’ The agentβs subjective expected utility from ππ:
οΏ½βπππ π (β|ππ)(β β ππππ) = πππ π β = 1 ππ β ππππ
β’ Does πππ π β ππ correctly measure the causal effect of ππ on β given the agentβs subjective model?
β’ Mistaking correlation for causation
The Dieterβs Dilemma
πππ π β ππ = οΏ½
ππππ ππ ππ ππ(β|ππ)
β’ The agent computes the effect of ππ on β as if β β₯ ππ | ππ.
β’ But we saw that ππ does not satisfy this property.
β It follows that ππ(β|ππ) is not invariant to ππ(ππ).
β Calls for an equilibrium notion of subjective maximization!
The Dieterβs Dilemma
β’ βPersonal equilibriumβ : If ππ ππ > 0, then ππ maximizes subjective expected utility w.r.t πππ π β ππ .
β’ Need to introduce trembles for definition to be precise.
The Dieterβs Dilemma
β’ Look for a personal equilibrium with full support:
ππππππ = πππ π β = 1 ππ = 1 β πππ π β = 1 ππ = 0 = ππ
πππ π β = 1|ππ = οΏ½
ππππ ππ ππ ππ β = 1 |ππ
β’ Just calculate the relevant conditional probabilities!
β’ πΌπΌ = ππ ππ = 1 denotes the probability of taking the supplement.
The Dieterβs Dilemma
β’ Recall: β = 1 is good health, ππ = 0 is normal chemical level.
β’ ππ ππ = 0 ππ = 1 = 1 ππ ππ = 0 ππ = 0 = 0.5
β’ ππ β = 1 ππ = 1 = 0 ππ β = 1 |ππ = 0 = 0.5+0.5πΌπΌ0.5
β’ Condition for interior equilibrium
ππππππ = ππ ππ = 0 ππ = 1 β ππ ππ = 1 |ππ = 0 οΏ½ ππ β = 1 ππ = 0
= 0.5 οΏ½ 0.5+0.5πΌπΌ0.5 = ππ
The Dieterβs Dilemma
β’ πΌπΌ = (1 β 2ππ)/2ππ
β’ An interior (unique) equilibrium exists for ππ β 0. 25, 0.5 .
β’ Summary:
β The agent misreads ππ β β correlation as a causal effect of ππ on ππ. β This results in equilibrium βmixingβ (with sub-optimal
consumption with positive long-run frequency).
The Dieterβs Dilemma
Bayesian Networks
β’ π₯π₯0, π₯π₯1, β¦ , π₯π₯ππ is a collection of variables, π₯π₯ππ β ππππ.
β’ ππ = ππ0 Γ β―Γ ππππ
β’ ππ β β(ππ) is an objective long-run probability distribution.
β’ Standard chain rule (arbitrary enumeration of variables):
ππ π₯π₯ = ππ π₯π₯0 ππ(π₯π₯1|π₯π₯0) β― ππ(π₯π₯ππ|π₯π₯0, β¦ ,π₯π₯ππβ1)
Bayesian Networks
β’ A causal model is a directed acyclic graph (DAG) (ππ, π π ): β ππ is a set of nodes that represent variables.
β π π is a set of directed links (use πππ π ππ or ππ β ππ interchangeably) that represent perceived causal relations.
β π π ππ = {ππ β ππ|πππ π ππ} is the set of βimmediate causes of ππβ.
β Iβll often suppress ππ and identify causal models with π π .
Bayesian Networks
β’ Factorize ππ according to π π :
ππ
πΉπΉππ = οΏ½
ππβπ΅π΅
ππ(ππ
ππ|ππ
πΉπΉ(ππ))
β’ 1 β 0 β 3 β 2 β πππ π π₯π₯ = ππ π₯π₯1 ππ π₯π₯2 ππ π₯π₯0 π₯π₯1 ππ(π₯π₯3|π₯π₯0, π₯π₯2)
β’ Fully connected DAG β Standard chain rule
β’ Empty DAG β πππ π π₯π₯ = ππ(π₯π₯0) β― ππ(π₯π₯ππ)
Bayesian Networks
β’ The set of distributions that are consistent with πππ π constitute a Bayesian network.
β Representing conditional-independence assumptions
β Platform for algorithmic probabilistic inference (Pearl 1988, Cowell et al. 1999, Koller-Friedman 2009)
β Platform for algorithmic causal inference (Pearl 2009)
What does π π Mean in Present Context?
β’ π π encodes a systematic distortion of objective distributions into subjective beliefs.
β’ Unlike the subjective-priors approach, here the primitive is not a belief but a belief distortion function.
β’ In larger models, π π will be part of an agentβs βtypeβ.
β’ Suppose ππ = 2 and ππ π₯π₯ β‘ ππ π₯π₯0 ππ π₯π₯1 ππ(π₯π₯2|π₯π₯0, π₯π₯1). β Consistent with a βtrue DAGβ: 0 β 2 β 1
β’ Subjective DAGs that exhibit specific errors:
Coarse reasoning 0 β 2 1 (omitting a link) Reverse causality 0 β 2 β 1 (inverting a link)
Capturing Belief Errors
β’ Suppose ππ = 2 and ππ π₯π₯ β‘ ππ π₯π₯0 ππ π₯π₯1|π₯π₯0 ππ(π₯π₯2|π₯π₯0).
β Consistent with a βtrue DAGβ: 2 β 0 β 1
β Interpret 0 as a state of Nature and 2 as an opponentβs move
β’ A subjective DAG that exhibits an attribution error (reorienting a link): 0 β 1 β 2
β βIllusion of controlβ: 1 represents a decision makerβs action β Analogical reasoning: 1 is an βanalogy classβ
Capturing Belief Errors
β’ Suppose ππ = 3 and ππ π₯π₯ β‘ ππ π₯π₯0 ππ π₯π₯1 ππ(π₯π₯2|π₯π₯0, π₯π₯1)ππ(π₯π₯3|π₯π₯1).
1
β Consistent with a βtrue DAGβ: 0 β 2 β 3
β’ Subjective DAG that exhibits confounder neglect: 0 β 2 β 3
β Omitting a node and its links
β Causal interpretation of confounding-based correlation
Capturing Belief Errors
β’ Suppose ππ = 2 and ππ π₯π₯ β‘ ππ π₯π₯0 ππ π₯π₯1|π₯π₯0 ππ(π₯π₯2|π₯π₯0)ππ(π₯π₯3|π₯π₯1, π₯π₯2).
1
β Consistent with a βtrue DAGβ: 0 β 2 β 3
β’ Subjective DAG that neglects a causal channel: 0 β 2 β 3
β Unawareness of some causal channels
β Neglecting indirect/equilibrium effects of economic policies
Capturing Belief Errors
Observationally Equivalent DAGs
Definition: π π and ππ are equivalent if πππ π β‘ ππππ for every ππ.
β’ 1 β 2 and 2 β 1 are equivalent since
ππ π₯π₯1 ππ π₯π₯2 π₯π₯1 = ππ π₯π₯2 ππ(π₯π₯1|π₯π₯2)
β’ Observational equivalence β Causal equivalence
β’ Proposition (Frydenberg 1990, Verma-Pearl 1991): Two DAGs are equivalent if and only if they have the same undirected version and the same set of βimmoralitiesβ (ππ β ππ β ππ
without a direct link between ππ and ππ).
β 0 β 2 β 1 and 0 β 2 β 1 are equivalent.
β 0 β 2 β 1 and 0 β 2 β 1 are not equivalent.
Observationally Equivalent DAGs
Decision Model
β’ For simplicity, let us focus on an uninformed DM.
β π₯π₯0 is the DMΚΉs action.
β 0 is an ancestral node in true and subjective DAGs.
β’ ππ π₯π₯β0 π₯π₯0 π₯π₯ is exogenous and fixed.
β’ ππ(π₯π₯0) π₯π₯0 is the DMβs (endogenous) strategy.
β’ π’π’: ππ βΆ β is the DMβs vNM function.
β’ Subjective EU maximization: If ππ(π₯π₯0) > 0, π₯π₯0 should maximize
οΏ½
π₯π₯β0
πππ π π₯π₯β0 π₯π₯0 π’π’(π₯π₯0, π₯π₯β0)
β’ The DM treats πππ π π₯π₯β0 π₯π₯0 as the causal effect of π₯π₯0 on π₯π₯β0.
β’ βMistaking correlation for causationβ?
β’ The assumption that 0 is ancestral in π π ensures the DM does not err in this regard (Pearlβs βdo-calculusβ (2009))
Decision Model
β’ Subjective EU maximization: If ππ(π₯π₯0) > 0, π₯π₯0 should maximize
οΏ½
π₯π₯β0
πππ π π₯π₯β0 π₯π₯0 π’π’(π₯π₯0, π₯π₯β0)
β’ Do-calculus is left outside the scope of this lecture series.
β’ The DM has a wrong causal model, but he draws correct causal inferences from observational data given the model.
Decision Model
β’ Subjective EU maximization: If ππ(π₯π₯0) > 0, π₯π₯0 should maximize
οΏ½
π₯π₯β0
πππ π π₯π₯β0 π₯π₯0 π’π’(π₯π₯0, π₯π₯β0)
β’ As we saw in the βDieterβs Dilemmaβ, πππ π π₯π₯β0 π₯π₯0 may be ill- defined without knowing ππ(π₯π₯0) (the DMβs strategy).
β’ Need for an equilibrium model of individual choice
Decision Model
Definition: A strategy ππ(π₯π₯0) π₯π₯0 with full support is a personal ππ-equilibrium if whenever ππ(π₯π₯0) > ππ,
π₯π₯0 β πππππππππππ₯π₯ π₯π₯0β² οΏ½
π₯π₯β0
πππ π π₯π₯β0 π₯π₯0β² π’π’(π₯π₯0β², π₯π₯β0)
β’ A personal equilibrium is a limit of personal ππ-equilibria.
β Because 0 is an ancestral node, the exact perturbation doesnβt matter (it would in an informed-DM model).
Personal Equilibrium
β’ Consider a three-variable environment:
β ππ is the DMβs action.
β ππ is an exogenous state of nature.
β π§π§ is a consequence.
β’ ππ ππ, ππ, π§π§ β‘ ππ ππ ππ ππ ππ(π§π§|ππ, ππ)
β’ ππ is consistent with a βtrue DAGβ ππ β π§π§ β ππ.
When is Personal Equilibrium Needed?
β’ True DAG: ππ β π§π§ β ππ β ππ ππ, π§π§ ππ = ππ ππ ππ π§π§ ππ, ππ
β’ Subjective DAG: ππ β π§π§ ππ
β πππ π ππ, π§π§ ππ = ππ ππ ππ π§π§ ππ = ππ ππ βππβ² ππ ππβ² ππ π§π§ ππ, ππβ²
β Invariant to (ππ ππ )ππ ; personal equilibrium is reducible to maximization w.r.t a wrong belief.
Coarse Reasoning
β’ True DAG: ππ β π§π§ β ππ β ππ ππ, π§π§ ππ = ππ ππ ππ π§π§ ππ, ππ
β’ Subjective DAG: ππ β π§π§ β ππ
β’ πππ π ππ, π§π§|ππ = ππ π§π§ ππ ππ ππ π§π§ = ππ π§π§ ππ βππβ² ππ ππβ²|ππ ππ(ππ|ππβ², π§π§)
β’ πππ π ππ, π§π§|ππ is sensitive to (ππ ππ )ππ ; personal equilibrium is not reducible to maximization (as in the βDieterβs Dilemmaβ).
Reverse Causality
β’ When is personal equilibrium reducible to maximization?
β πππ π οΏ½ π₯π₯0 shouldnβt change with the marginal of ππ over π₯π₯0.
Definition: π π is c-rational w.r.t a βtrue DAGβ π π β if for every ππ, ππ that are consistent with π π β, if ππ οΏ½ π₯π₯0 = ππ(οΏ½ |π₯π₯0) for every π₯π₯0, then πππ π οΏ½ π₯π₯0 = πππ π (οΏ½ |π₯π₯0) for every π₯π₯0.
C-Rationality
β’ π₯π₯ππ β₯π π π₯π₯ππ | π₯π₯π΄π΄ denotes a conditional-independence property that holds for all distributions that are consistent with π π .
β’ This property has a computationally simple graphical characterization known as d-separation.
β Patterns of βpath blockingβ
β Basic material in any introduction to Bayesian networks
C-Rationality
Proposition: Let π π β 0 = π π 0 = β . Then, π π is c-rational w.r.t π π β iff 0 β π π (ππ) implies π₯π₯ππ β₯π π β π₯π₯0 | π₯π₯π π (ππ).
β’ Illustrating the result for π π β : 0 β 2 β 1 β Coarse reasoning 0 β 2 1 β π π 1 = β ; and indeed, π₯π₯1 β₯π π β π₯π₯0 .
C-Rationality
Proposition: Let π π β 0 = π π 0 = β . Then, π π is c-rational w.r.t π π β iff 0 β π π (ππ) implies π₯π₯ππ β₯π π β π₯π₯0 | π₯π₯π π (ππ).
β’ Illustrating the result for π π β : 0 β 2 β 1 β Reverse causality 0 β 2 β 1 β π π 1 = {2} ; but not π₯π₯1 β₯π π β π₯π₯0| π₯π₯2 .
C-Rationality
β’ True DAG π π β: ππ β π π β ππ β π€π€
β’ The DM is a parent.
β ππ β [0,1] is the parentβs investment in his childβs education.
β ππ β {0,1} is the childβs βinnate abilityβ.
β π π β {0,1} is the childβs school performance.
β π€π€ β {0,1} is the childβs labor-market outcome.
Example: Spurious Demand for Education
β’ True DAG π π β: ππ β π π β ππ β π€π€
β’ π’π’ is purely a function of ππ and π€π€.
β’ The parentβs subjective DAG is π π : ππ β π π β π€π€.
β’ Interpretation: ππ is an unobservable confounder; the parentβs error is that he ignores it.
Demand for Education
β’ True DAG π π β: ππ β π π β ππ β π€π€
β’ The parentβs DAG π π : ππ β π π β π€π€ violates c-rationality:
β π π π€π€ = {π π } , but not π€π€ β₯π π β ππ|π π
β Reason: ππ π€π€ π π = βππ ππ ππ π π ππ(π€π€|ππ) , and ππ ππ π π is sensitive to parental investment.
β Easy to check graphically using d-separation
Demand for Education
β’ Put more structure on exogenous components of ππ.
β’ ππ π π = 1 ππ, ππ = ππππ ; ππ π€π€ = 1 ππ = ππππ
β π π = 1 represents success at school.
β π€π€ = 1 represents success in the labor market.
β High ability (ππ = 1) is necessary for both.
β School success as such is irrelevant for labor-market success.
Demand for Education
β’ ππ ππ = 1 = πΏπΏ ; ππ π π = 1 ππ, ππ = ππππ ; ππ π€π€ = 1 ππ = ππππ
β’ π’π’ ππ, π€π€ = π€π€ β 0.5ππ2
β’ Under rational expectations, the parent chooses ππβ = 0.
β Parental investment doesnβt affect the childβs ability, which is the sole determinant of his labor-market outcome.
Demand for Education
Result: Under π π : ππ β π π β π€π€, there is a unique personal equilibrium. The parent plays ππββ > 0 given by
ππββ = πππΏπΏ(1 β πΏπΏ)
1 β πΏπΏ + πΏπΏ(1 β ππββ)
β Marginal cost = Perceived marginal benefit
β Long-run behavior affects perceived marginal benefit.
β ππββ increases with ππ and as πΏπΏ moves away from 0 or 1.
Demand for Education
β’ ππ π π = 1 ππ = πΏπΏππ. Perceived benefit from investment ππ is:
ππ π π = 1 ππ οΏ½ ππ π€π€ = 1 π π = 1 + ππ π π = 0 ππ οΏ½ ππ π€π€ = 1 π π = 0 = ππ π€π€ = 1 π π = 0 + ππ π π = 1 ππ οΏ½ ππ π€π€ = 1 π π = 1 β ππ π€π€ = 1 π π = 0
β’ ππ π€π€ = 1 π π = 1 = ππ because high ability is necessary for success at school.
β’ In contrast, π π = 0 can result from ππ = 0 or low ππ.
Sketch of Proof
ππ π€π€ = 1 π π = 0 = ππ(ππ = 1) οΏ½ βππβ² ππ(ππβ²) οΏ½ (1 β ππβ²) οΏ½ ππ
ππ ππ = 0 + ππ(ππ = 1) οΏ½ βππβ² ππ(ππβ²) οΏ½ (1 β ππβ²) = πΎπΎππ
β The parentβs chosen value of ππ does not feature in this term!
β’ Perceived benefit of investment ππ is πΎπΎππ + πΏπΏππ οΏ½ ππ β πΎπΎππ .
β’ Marginal perceived benefit from ππ is πΏπΏ ππ β πΎπΎππ .
β Overestimation gets worse when ππ(ππβ²) shifts to the right. This complementarity could lead to multiple equilibria for other ππ.
Sketch of Proof
β’ DAGs represent subjective causal models.
β’ The factorization formula πππ π represents systematic belief
distortion due to fitting a wrong causal model to long-run data Personal equilibrium: Subjective EU maximization w.r.t πππ π
β’ Bayesian-network tool (d-separation) helps understanding when to expect equilibrium effects
