Behavioral Implications of Causal Misperceptions Part III

Behavioral Implications of Causal Misperceptions Part III

Ran Spiegler (TAU & UCL)

ES Winter School, Delhi

December 2019

• So far we have taken agents’ causal model as given.

• Let us now endogenize it.

• Various forces that generate the agent’s causal model:

– Extrapolating from observations

– Selecting between models offered (opportunistically) by others, according to their anticipatory value

Endogenous Causal Models

• Political beliefs are shaped by narratives.

• Battles over public opinion involve “competing narratives”.

• A policy is popular when backed by an attractive narrative.

• People are attracted to hopeful narratives (stories with a “happy ending”…).

Political Narratives

"Barack Obama was a writer before he became a politician, and he saw his Presidency as a struggle over narrative.”

The New Yorker, 2018

“There can be little doubt then that people think narratives are important and that crafting, manipulating, or influencing

them likely shapes public policy …by telling a story that includes assertions about what causes what, who the victims are,

who is causing the harm, and what should be done.”

LSE Impact blog, 2018

• Eliaz-Spiegler 2018: Conceptualizing narratives as causal models that map policies to public consequences.

• An equilibrium model of how narrative-policy pairs compete for public popularity.

Narratives as Causal Models

• 𝑋𝑋 = {0,1}^𝑚𝑚, 𝑚𝑚 > 2

• 𝑥𝑥₁ (also denoted 𝑎𝑎) represents an action.

• 𝑥𝑥_𝑚𝑚 (also denoted 𝑦𝑦) represents a consequence.

• 𝑝𝑝 ∈ ∆(𝑋𝑋) is a joint distribution over 𝑥𝑥₁, … ,𝑥𝑥_𝑚𝑚.

The Model

• 𝑝𝑝 𝑎𝑎 = 1 = 𝛼𝛼

– Historical action frequency, endogenized later.

• 𝑝𝑝 𝑦𝑦 = 1 = 𝜇𝜇 ∈ (0,1), exogenously

– Exogenous and independent of 𝑎𝑎.

• (𝑝𝑝 𝑥𝑥₂, … ,𝑥𝑥_𝑚𝑚−1 𝑎𝑎, 𝑦𝑦 ) is exogenous with full support for any 𝑎𝑎, 𝑦𝑦.

• A narrative is a DAG (𝑁𝑁, 𝑅𝑅), where:

– 𝑁𝑁 ⊆ {1, … , 𝑚𝑚} is the set of nodes, 𝑁𝑁 ≤ 𝑛𝑛 ≤ 𝑚𝑚.

The Model

Narratives as Causal Models

• The set of feasible narratives is a set of DAGs ℛ that include the nodes 1 and 𝑚𝑚, such that 1 is an ancestral node.

• 𝑛𝑛 is an exogenous limit on DAG size.

• When 2 < 𝑛𝑛 < 𝑚𝑚, the narrative selects variables (in addition to the action and the consequence).

• I identify the DAG with 𝑅𝑅 unless specified otherwise.

• 𝑑𝑑 ∈ [𝜀𝜀, 1 − 𝜀𝜀] is a policy.

– Proposed frequency of playing 𝑎𝑎 = 1

• A representative agent’s utility is 𝑦𝑦 − 𝐶𝐶(𝑑𝑑 − 𝑑𝑑^∗).

• 𝑑𝑑^∗ ∈ [0.5,1 − 𝜀𝜀) is the agent’s intrinsically ideal policy.

• 𝐶𝐶 is a symmetric, convex cost function, with 𝐶𝐶 0 = 𝐶𝐶^′ 0 = 0.

Policies

• The narrative-policy pair (𝑅𝑅, 𝑑𝑑) induces a gross anticipatory expected payoff (perceived probability of a good outcome):

𝑉𝑉 𝑅𝑅, 𝑑𝑑 𝛼𝛼 = 𝑑𝑑 � 𝑝𝑝_𝑅𝑅 𝑦𝑦 = 1 𝑎𝑎 = 1 + (1 − 𝑑𝑑) � 𝑝𝑝_𝑅𝑅 𝑦𝑦 = 1 𝑎𝑎 = 0

• The notation highlights that 𝑝𝑝_𝑅𝑅 𝑦𝑦|𝑎𝑎 - and therefore 𝑉𝑉 - may depend on the steady-state frequency 𝛼𝛼.

• How does the representative agent choose between conflicting narrative-policy pairs?

Anticipatory Payoff

Definition: An action frequency 𝛼𝛼 and a distribution 𝜎𝜎 over narrative-policy pairs 𝑅𝑅, 𝑑𝑑 form an equilibrium if:

1. 𝑆𝑆𝑆𝑆𝑝𝑝𝑝𝑝 𝜎𝜎 ⊆ 𝑎𝑎𝑎𝑎𝑎𝑎𝑚𝑚𝑎𝑎𝑥𝑥 𝑅𝑅,𝑑𝑑 ∈ℛ×[0,1] 𝑉𝑉 𝑅𝑅, 𝑑𝑑 𝛼𝛼 − 𝐶𝐶 𝑑𝑑 − 𝑑𝑑^∗ 2. 𝛼𝛼 = ∑_{(𝑅𝑅,𝑑𝑑)} 𝜎𝜎(𝑅𝑅, 𝑑𝑑) � 𝑑𝑑

Proposition: An equilibrium exists. (Proof method: Construct an auxiliary game that has a Nash equilibrium.)

Equilibrium

• Suppose ℛ only consists of the DAG 𝑅𝑅: 𝑎𝑎 → 𝑦𝑦.

• Then, 𝑝𝑝_𝑅𝑅 𝑦𝑦 = 1 𝑎𝑎 = 𝜇𝜇 for all 𝑎𝑎 (rational expectations).

– 𝑉𝑉 𝑅𝑅, 𝑑𝑑 𝛼𝛼 = 𝜇𝜇 for every feasible 𝑅𝑅, 𝑑𝑑 .

– In equilibrium, 𝜎𝜎 assigns probability one to 𝑑𝑑 = 𝑑𝑑^∗.

Rational-Expectations Benchmark

• 𝑚𝑚 = 3

• The three variables are 𝑎𝑎, 𝑦𝑦, 𝑠𝑠:

– 𝑎𝑎 = 1 Hawkish policy toward rival country – 𝑦𝑦 = 1 Weak regime in rival country

– 𝑠𝑠 = 1 Strong nationalistic sentiment in rival country

An Example: Foreign-Policy Narratives

• Long-run distribution 𝑝𝑝 (with full support) over the variables – 𝑝𝑝 𝑎𝑎 = 1 = 𝛼𝛼

– 𝑝𝑝 𝑦𝑦 = 1 = ¹₂ , independently of 𝑎𝑎

Foreign Policy

Foreign Policy

𝑝𝑝 𝑠𝑠 = 1 𝑎𝑎, 𝑦𝑦 = ^{𝑎𝑎+1−𝑦𝑦}₂

• Hawkish policy tends to strengthen nationalism in the rival country.

• Nationalism and regime weakness are negatively correlated.

– This correlation is not causal; it is due to latent confounding variables.

Examples of Narratives

• 𝑎𝑎 → 𝑠𝑠 → 𝑦𝑦 is a “lever narrative” that incorporates 𝑠𝑠 into the story as a mediator.

• 𝑎𝑎 → 𝑦𝑦 ← 𝑠𝑠 is a “threat/opportunity narrative” that incorporates 𝑠𝑠 as an exogenous force.

• 𝐶𝐶 ≡ 0

• Therefore, the agent’s anticipatory utility is 𝑉𝑉(𝑅𝑅, 𝑎𝑎;𝛼𝛼) = 𝑝𝑝_𝑅𝑅 𝑦𝑦 = 1 𝑎𝑎

Anticipatory Payoff

Unique Equilibrium

• 𝛼𝛼 ≈ 0.57

• 𝑆𝑆𝑆𝑆𝑝𝑝𝑝𝑝(𝜎𝜎) consists of two narrative-action pairs:

1. Lever narrative 𝑅𝑅^𝐿𝐿: 𝑎𝑎 → 𝑠𝑠 → 𝑦𝑦 coupled with the dovish action 𝑎𝑎 = 0.

2. Opportunity narrative 𝑅𝑅^𝑂𝑂: 𝑎𝑎 → 𝑦𝑦 ← 𝑠𝑠 coupled with the hawkish action 𝑎𝑎 = 1.

Why does 𝑅𝑅 ^𝐿𝐿 Support a Dovish Policy?

𝑝𝑝 𝑠𝑠 = 1 𝑎𝑎, 𝑦𝑦 = 𝑎𝑎 + 1 − 𝑦𝑦 2

• (𝑎𝑎, 𝑠𝑠) are positively correlated

• (𝑠𝑠, 𝑦𝑦) are negatively correlated

• 𝑎𝑎 → 𝑠𝑠 → 𝑦𝑦 induces a (perceived, indirect) negative causal effect of 𝑎𝑎 on 𝑦𝑦.

Why does 𝑅𝑅 ^𝑂𝑂 Support a Hawkish Policy?

𝑝𝑝 𝑠𝑠 = 1 𝑎𝑎, 𝑦𝑦 = ^{𝑎𝑎+1−𝑦𝑦}₂

• 𝑝𝑝 𝑠𝑠 𝑎𝑎, 𝑦𝑦 is a function of 𝑎𝑎 − 𝑦𝑦.

⇒ For any given 𝑠𝑠, 𝑝𝑝 𝑦𝑦 = 1 𝑎𝑎, 𝑠𝑠 increases with 𝑎𝑎.

• But 𝑝𝑝_𝑅𝑅^𝑂𝑂 regards 𝑠𝑠 as independent of 𝑎𝑎.

• Therefore, 𝑝𝑝_𝑅𝑅^𝑂𝑂(𝑦𝑦 = 1|𝑎𝑎) increases with 𝑎𝑎.

Why Two Narratives?

• “Diminishing returns” to belief manipulation

– 𝑉𝑉(𝐺𝐺,𝑎𝑎;𝛼𝛼) goes down as 𝛼𝛼 moves toward 𝑎𝑎.

– As the action that one narrative promotes gets implemented more frequently, the competing narrative becomes more appealing.

• Other DAGs induce rational expectations (can’t sell illusions).

1 𝛼𝛼

0 1

2 𝛼𝛼

𝑎𝑎 → 𝑦𝑦 ← 𝑠𝑠 𝑎𝑎 → 𝑠𝑠 → 𝑦𝑦

Hawkish Bias

• Objective distribution treats 𝑎𝑎 = 1 and 𝑎𝑎 = 0 symmetrically.

• And yet, 𝛼𝛼 > ¹₂ : Average equilibrium policy is hawkish.

• Why?

– At 𝛼𝛼 = ¹₂ , 𝑅𝑅^𝑂𝑂 generates higher false correlation between 𝑎𝑎 and 𝑦𝑦 than 𝑅𝑅^𝐿𝐿. Then, 𝛼𝛼 needs to go up to restore

indifference between the two narratives.

• Tension between “easy fix” and rational-expectations narratives

– True narrative: The consequence is determined by a “deep cause”

that is impervious to policy

– The “Easy fix” narrative falsely attributes the consequence to a symptom, which is affected by policy.

– When these are the feasible narratives, both prevail in equilibrium;

the false narrative feeds off the true one.

Another Example: “Easy-Fix Narratives”

Proposition: Let ℛ be a set of perfect DAGs that contains a DAG 𝑅𝑅 such that 𝑝𝑝_𝑅𝑅(𝑦𝑦|𝑎𝑎) is non-constant in 𝑎𝑎.

Then, in any equilibrium (𝛼𝛼, 𝜎𝜎), 𝜎𝜎 assigns positive probability to exactly two policies: 𝑑𝑑₁ ≥ 𝑑𝑑^∗ and 𝑑𝑑₀ ≤ 𝑑𝑑^∗.

• The proof makes heavy use of the “no distortion of marginals”

property of perfect DAGs - 𝑉𝑉 𝑅𝑅, 𝛼𝛼 𝛼𝛼 = 𝜇𝜇

General Result: Equilibrium Polarization

• Suppose 𝜎𝜎 assigns probability one to 𝛼𝛼.

• 𝑉𝑉 𝑅𝑅, 𝛼𝛼 𝛼𝛼 = 𝜇𝜇 for any feasible narrative 𝑅𝑅.

– 𝛼𝛼 = 𝑑𝑑^∗: Deviate to lever narrative with 𝑑𝑑 ≈ 𝑑𝑑^∗.

– 𝛼𝛼 ≠ 𝑑𝑑^∗: Deviate to 𝑎𝑎 → 𝑦𝑦 with 𝑑𝑑^∗.

• Therefore, there must be mixing over policy.

• No-marginal-distortion implies 𝑚𝑚𝑎𝑎𝑥𝑥_𝑅𝑅𝑉𝑉 𝑅𝑅, 𝑑𝑑 𝛼𝛼 only depends on whether 𝑑𝑑 > 𝛼𝛼 or 𝑑𝑑 < 𝛼𝛼; this gives two-policy equilibrium.

Proof Strategy

Competing Narratives: Comments

• Open question: When does the two-narrative result hold when we allow for imperfect DAGs?

• Extension: State-dependent evaluation of narratives (example:

denialist vs. exaggerationist narratives)

• Eliaz-Spiegler-Weiss 2019: What is the maximal predicted correlation that a misspecified DAG can generate?

Other Relevant Works

• Schumacher-Thysen 2017: Contracting with an agent who has a

wrong model of the mapping from effort to output/cost

• Spiegler 2018, Eliaz-Spiegler-Thysen 2018: A dual approach

(Bayesian-network factorization as a representation of extrapolating a belief from partial statistical data)

Conclusion

• A framework for equilibrium models with non-rational

expectations, highlighting the role of subjective causal perceptions

• DAGs represent causal models and capture systematic errors of causal attribution

– Bayesian-network tools (equivalent DAGs, perfection, d- separation) for analyzing behavioral implications

Conclusion

• Purely graphical (and hence non-parametric) characterizations of behavioral features:

– Equilibrium effects in individual choice – Systematically biased estimates

– Multiple prevailing narratives when the criterion for selecting narratives is based on anticipatory utility

Research Directions

• How do decision makers form causal models?

• Connections to other approaches to decision making under misspecified models

• Applications (macro, IO, political economics, contract theory…)

• Complexity considerations in probabilistic inference

• Boundedly rational causal inference