Eliciting information from a large population Kohei Kawamura, - - PowerPoint PPT Presentation

”Eliciting information from a large population”

Kohei Kawamura, Journal of Public Economics, July 2013 Kayoung Choi, Josh Lanier, Fikri Pitsuwan December 5, 2013

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Survey: Funding for Economics research

You are one of n=500 UK residents whose opinion is sampled. Policymaker would implement the average opinion.

1 Decrease it staggeringly 2 Decrease it somewhat 3 Keep it at the current amount 4 Increase it somewhat 5 Increase it staggeringly Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Motivation

Information transmission in social surveys: potential problem

• f people responding to survey strategically

effective ways to ask survey questions? effect of size of survey? — larger sample → better estimation?

How cheap talk communication changes according to:

1

sample size

2

quality of prior belief about preference distribution

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Model: Timing

1 Individuals and DM endowed with a common prior of pref

distribution

2 Individuals privately learn their types 3 DM randomly samples n individuals - each reports

costless/non-verifiable message

4 DM estimates population distribution and chooses y 5 payoffs realized Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Model: Set up

Continuum of individuals a ∈ [0, 1]

each with type (pref) θa ∈ Θ ⊂ R |Θ| = H ≥ 3 — number of types −(y − θi)2 — payoff for type i when policy y ∈ R chosen

Decision Maker (DM)

qi ≥ 0 — proportion of individuals with type θi, H

i=1 qi = 1

unable to implement ideal policy for everyone — optimal policy to max utilitarian social welfare: max

y∈R

−

H

• i=1

qi(y − θi)2

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Model: Set up

DM does not observe freq vector q ≡ (q1, ..., qH) estimate q by randomly sample n (finite) individuals — each independently send cheap talk messages to DM q ∼ Dir(α) — Dirichlet dist. with parameters α ≡ (α0p1, α0p2, ..., α0pH), H

i=1 α0pi = α0

E[qi] = α0pi

α0 = pi

p ≡ (p1, ..., pH) — expected prior population dist of pref prior expected mean of individuals’ types µ ≡ H

i=1 piθi

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Dirichlet Distribution

Figure: (α0p1 = α0p2 = α0p3), effect of changing α0 on shape of

• distribution. [from wikipedia]

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Model: Set up

Respondents reveal truthfully — let x ≡ (x1, ..., xH) be the count vector for each type For DM: E[qi|xi] = α0pi+xi

α0+n

For each individual a ∈ [0, 1] : E[qi|θa = θi] = α0pi+1

α0+1

For i = j, E[qj|θa = θi] = α0pj

α0+1

α0: ”strength” of the prior belief or evel of lex ante aggregate uncertainty

as α0 → ∞, E[qi|xi] → E[qi] = pi E[qi|θa = θi] ↓ as α0 ↑ as α0 → 0, E[qi|θa = θi] → 1

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Equilibrium

Dynamics not considered PBE has respondents and policy maker behave optimally. May induce exaggeration An aside on PBE with infinite players

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Equilibrium

Focus on equilibria where respondents choose pure symmetric strategies Let z ≡ (z1, ..., zK) be count vector of responses, K ≤ H

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Equilibrium

Decision makers best response to info set z is: ¯ y(z) = H

i=1 E[qi|z]θi

In EQ policy maker gets E[qi|z] correct A respondent may be tempted to exaggerate if ¯ y(z) not very sensitive to one agent changes in z ¯ y(z) sensitivity to z:

1

Decrease in n

2

Decrease in α0

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Proposition 1

Proposition 1 If α0 is sufficiently large, a binary equilibrium exists for any n and is the only informative equilibrium in partitional strategy.

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Proposition 2

Proposition 2 For sample size n sufficiently large, either i no informative equilibrium in partitional strategy exists; or ii the most informative equilibrium in partitional strategy is binary.

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Figure 1

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Table

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Conclusion

Conclusion Even if communication and information processing are costless, trade-off between the quality and quantity in communication: due to respondents strategic incentive to misreport

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Criticisms and Extensions

Criticisms Kawamura’s motivating examples DM and respondents all have correct prior beliefs (in expectation) Common knowledge of sample size n Extensions Experts’ advice as n → ∞ individual might as well tell the truth? What if sample size n is not common knowledge?

Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”