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 of 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: sample size 1 quality of prior belief about preference distribution 2 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) θ a ∈ Θ ⊂ R | Θ | = H ≥ 3 — number of types − ( y − θ i ) 2 — payoff for type i when policy y ∈ R chosen Decision Maker (DM) q i ≥ 0 — proportion of individuals with type θ i , � H i =1 q i = 1 unable to implement ideal policy for everyone — optimal policy to max utilitarian social welfare: H � q i ( y − θ i ) 2 max − y ∈ R i =1 Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Model: Set up DM does not observe freq vector q ≡ ( q 1 , ..., q H ) estimate q by randomly sample n (finite) individuals — each independently send cheap talk messages to DM q ∼ Dir ( α ) — Dirichlet dist. with parameters α ≡ ( α 0 p 1 , α 0 p 2 , ..., α 0 p H ) , � H i =1 α 0 p i = α 0 E [ q i ] = α 0 p i α 0 = p i p ≡ ( p 1 , ..., p H ) — expected prior population dist of pref prior expected mean of individuals’ types µ ≡ � H i =1 p i θ i Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Dirichlet Distribution Figure: ( α 0 p 1 = α 0 p 2 = α 0 p 3 ), effect of changing α 0 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 ≡ ( x 1 , ..., x H ) be the count vector for each type For DM: E [ q i | x i ] = α 0 p i + x i α 0 + n For each individual a ∈ [0 , 1] : E [ q i | θ a = θ i ] = α 0 p i +1 α 0 +1 E [ q j | θ a = θ i ] = α 0 p j For i � = j , α 0 +1 α 0 : ”strength” of the prior belief or evel of lex ante aggregate uncertainty as α 0 → ∞ , E [ q i | x i ] → E [ q i ] = p i E [ q i | θ a = θ i ] ↓ as α 0 ↑ as α 0 → 0 , E [ q i | θ a = θ i ] → 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 ≡ ( z 1 , ..., z K ) 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 [ q i | z ] θ i ¯ In EQ policy maker gets E [ q i | 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 : Decrease in n 1 Decrease in α 0 2 Kayoung Choi, Josh Lanier, Fikri Pitsuwan ”Eliciting information from a large population”

Proposition 1 Proposition 1 If α 0 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”