Is It the Message or the Messenger? Examining Movement in Immigration Beliefs

Is It the Message or the Messenger? Examining Movement in Immigration Beliefs was written by Hassan Afrouzi, Carolina Arteaga, and Emily Weisburst. It was published in the Journal of Political Economy Microeconomics volume 2 (2024).

The authors design an experiment to isolate the effects of political messages and the effects of who gave the political message. Pro- and anti-immigration speeches, as well as non-ideological (turkey pardoning) speeches, are selected from Obama and Trump. The same speeches are reproduced with a voice actor. The non-ideological speeches enable control for effects of political messages. The reproduction enables control for the effects of who gave the political message.

Probability Theory

A sample space Ω represents an individual's opinion about immigration (binary; favorable or unfavorable). Subjective opinions--as in given S sources of political messages and M expected messages from each source--are given in Ω‾ = Ω * M * S.

The unconditional probability of belief ω is given by P(ω) = Σ P(ω,s',m') for all s' ∈ S and m' ∈ M.

For a 'treatment' s' ∈ S and m' ∈ M, the conditional probability of belief ω is given by P(ω|s',m').

It is more realistic to measure relative probabilities of beliefs in an experiment, however. Therefore the treatment belief relative to the unconditional belief is given by:

Decompose with Bayes rule and log probabilities into separate anonymous message effects and source persuasion effects:

Again by Bayes rule, decompose the conditional probabilities:

P(ω|m) = P(ω)P(m|ω) / P(m)

...or using an odds ratio (Θ) formulation:

P(ω|m) = P(ω)Θ(m|ω)

This is an immediately useful formulation: the anonymous message effect β_m increases with the odds ratio Θ(m|ω) and is zero iff the odds ratio is one.

Similarly now:

P(ω|m,s) = P(ω|m)P(s|ω,m) / P(s|m) = P(ω|m)Θ(s|ω,m)

Note furthermore that:

Θ(s|ω,m) = P(s|ω,m) / P(s|m) = P(ω|m,s) / P(ω|m)

Lastly, understand that for all s ∈ S there is a complementary ¬s such that:

P(w|m) = P(w,s|m) + P(w,¬s|m) = P(s|m)P(w|s,m) + P(¬s|m)P(w|¬s,m)

This all comes together to calculate the source persuasion effect β_sm which increases with the odds ratio Θ(s|m,ω) and is zero iff the odds ratio is one. As an intermediary step:

There is a source persuasion effect iff:

Θ(s|m,ω) > 1

...which can be more easily worked with as:

Θ(s|m,ω) - 1 > 0

Substitute the left-hand side's odds ratio Θ(s|m,ω) and combine terms:

Θ(s|m,ω) - 1 = [P(ω|m,s) / P(ω|m)] - 1 = [P(ω|m,s) / P(ω|m)] - [P(ω|m) / P(ω|m)] = [P(ω|m,s) - P(ω|m)] / P(ω|m)

Substitute the numerator's term P(ω|m) with the expanded expression above and factor out P(w|s,m):

[P(ω|m,s) - P(ω|m)] / P(ω|m) = [P(ω|m,s) - [P(s|m)P(w|s,m) + P(¬s|m)P(w|¬s,m)]] / P(ω|m) = [P(ω|m,s) - P(s|m)P(w|s,m) - P(¬s|m)P(w|¬s,m)] / P(ω|m) = [P(w|s,m)(1 - P(s|m)) - P(¬s|m)P(w|¬s,m)] / P(ω|m)

Decompose with the complementary rule and factor out that term:

[P(w|s,m)(1 - P(s|m)) - P(¬s|m)P(w|¬s,m)] / P(ω|m) = [P(w|s,m)(1 - P(s|m)) - (1 - P(s|m))P(w|¬s,m)]] / P(ω|m) = (1 - P(s|m))(P(w|s,m) - P(w|¬s,m)) / P(ω|m)

This is interpreted as:

• P(ω|m) is a prior belief given messages

• (1 - P(s|m)) is a surprise term that captures how unexpected a message is from a source

• (P(w|s,m) - P(w|¬s,m)) is a reliability of source term that captures the reliability of the source relative to all others

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