Consumer theory for cheap information Gary Baker University of - PowerPoint PPT Presentation

Consumer theory for cheap information Gary Baker University of Wisconsin–Madison 16 October 2020 UW–Madison, Theory Seminar

Question Consider a constrained decision-maker who has to make a decision under uncertainty. Before acting, she has access to multiple, costly sources of information about the state of the world. She must decide not only ▶ from which sources to buy information, but also ▶ how much information to buy from each.

Question For example: some vaccine (say, for COVID-19): ▶ Voter trying to decide on a party: ▶ State: true optimal policy ▶ Action: for which party to vote ▶ Info sources: difgerent newspapers ▶ Amount of info: how many articles to read ▶ Constraint: Limited time to read the news ▶ A researcher trying to determine the efgectiveness of ▶ State: true efgectiveness ▶ Action: whether to introduce the vaccine or not ▶ Info sources: available tests for the condition ▶ Amount of info: how many trial participants ▶ Constraint: Grant budget

Goal We’d like to have a consumer theory for information. ▶ Tradeofgs between difgerent sources ▶ marginal rate of substitution ▶ Demand for information in constrained settings ▶ Elasticities

Potential Applications resources (e.g. time) between difgerent news/info sources ▶ Media and rational inattention: how people allocate their ▶ Research design and optimal treatment allocation

Problems with information as a good Going back to Blackwell [1951]: compared ordered by garbling. ▶ Information from difgerent sources can’t easily be ▶ In the broadest sense, information sources can only be

Problems with information as a good Another example: In a quasilinear setting, marginal values of information is typically upward sloping at small samples. ▶ First-order condition analysis doesn’t easily work

Problems with information as a good In general, information value doesn’t have a nice, closed-form expression.

What I do To answer these questions I develop an (approximate) ordinal theory of tradeofgs between information source. That is, I will values, valid at large samples (when info is cheap) information sources constrained setting ▶ Find an approximate ordinal expression for information ▶ Characterize the marginal rate of substitution between ▶ Explore implications for information demand in a budget

What I do characteristics (prior, utility function). bundle at large samples. ▶ This approximation will not depend on decision-maker ▶ Everyone facing the same costs will agree on the optimal

Method: large deviations Information is valuable insofar as it prevents you from taking a suboptimal action. misleads the decision maker. afuer seeing the info. With a lot of information (large samples), the probability of being misled is very small (a tail event). ▶ We can characterize info values by the probability that it ▶ That is, when the decision maker takes the wrong action Approximating this is the realm of large deviations theory.

Method: large deviations So my approximations will be valid when the DM purchases a lot of info. So a scenario with ▶ Cheap information, ▶ Large budgets, or ▶ Some combination

Agenda Preview of results Literature Model Large deviations approximations The two-state case The many-state case Consumer theory “Marginal” rate of substitution Implications for information demand Is the approximation useful? Future work

Literature Statistics: perform as well as 𝑜 from another Contribution: ▶ Chernofg [1952] asymptotic relative efgiciency ▶ How many samples from one statistical test are necessary to ▶ Comparison of extremes: all one or all the other ▶ extend to local (interior) comparisons (MRS), and ▶ to arbitrary finite-action/finite-state decision problems.

Literature Economics: approximation of value, and thus demand for information in the single source case multiple sources and explore implications for tradeofgs between them. gives tighter bounds on convergence rate and implies a full, asymptotic expansion. ▶ Moscarini and Smith [2002] ▶ Apply methods similar to Chernofg to write an asymptotic ▶ Economic contribution: extend this to environment with ▶ Technical contribution: Proof approach for the approximation

Other related literature Value of and comparisons between information sources: Börgers et al. [2013], Athey and Levin [2018] Rational inattention: Sims [2003], and many, many others Optimal experiment design / treatment assignment: Elfving [1952], Chernofg [1953], Dette et al. [2007] (another huge literature)

Agenda Preview of results Literature Model Large deviations approximations The two-state case The many-state case Consumer theory “Marginal” rate of substitution Implications for information demand Is the approximation useful? Future work

Model – Environment action. the state ▶ Finitely many possible underlying states, θ ∈ Θ ▶ DM has prior p ∈ ΔΘ (no degenerate beliefs) ▶ Finitely many possible actions 𝑏 ∈ A . ▶ For the presentation, assume each state has a unique optimal ▶ DM has state-dependent utility function 𝑣(𝑏, θ) ▶ Chooses action, 𝑏 , to maximize ∑ θ 𝑞 θ 𝑣(𝑏, θ) ▶ Prior to acting, the DM can purchase information about

Model – Information sources (AKA: tests, signals, or experiments) strict subset of states ▶ Two information sources, ℰ 1 , ℰ 2 ▶ ℰ 𝑗 ≡ ⟨F 𝑗 (𝑠 | θ)⟩ ( 𝑠 ∈ R realizations) ▶ Assume: No signal realization perfectly rules in or rules out any ▶ Assume for exposition: each has conditional density 𝑔 𝑗 (𝑠 | θ)

Model – Information sources sample. information. observes the vector of realizations, and updates via Bayes Rule. ▶ DM can purchase an arbitrary number of conditionally independent samples, 𝑜 𝑗 , from each source at cost 𝑑 𝑗 per ▶ DM has a large, but finite, budget to spend on ▶ Afuer choosing a bundle of information (𝑜 1 , 𝑜 2 ) , DM

Model – Value with information {∑ Goal: Maximize subject to budget constraint Payofg to acting afuer updating ⎦ ⎥ ⎥ ⎤ 𝑔 𝑜 1 ,𝑜 2 (𝑠 | θ) ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ 𝑣(𝑏, θ)ℙ(θ | 𝑠)} θ 𝑏 Expected value of acting afuer observing signal realizations max 𝑠∈R ∫ ⎣ ⎢ ⎢ ⎡ 𝑞 θ θ 𝑤(𝑜 1 , 𝑜 2 ) = ∑ 𝑑 1 𝑜 1 + 𝑑 2 𝑜 2 ≤ Y

Agenda Preview of results Literature Model Large deviations approximations The two-state case The many-state case Consumer theory “Marginal” rate of substitution Implications for information demand Is the approximation useful? Future work

Two states – setup ▶ States: ▶ Null hypothesis – H 0 ▶ Alternative hypothesis – H 1 ▶ Prior that the alternative is true – 𝑞 ▶ Actions: ▶ Accept the null – 𝒝 ▶ Reject the null – ℛ

Two states – setup 𝑤(𝑜 1 , 𝑜 2 ) = (1 − 𝑞)(α I (𝑜 1 , 𝑜 2 )𝑣(ℛ, H 0 ) + (1 − α I (𝑜 1 , 𝑜 2 ))𝑣(𝒝, H 0 )) + 𝑞 (α II (𝑜 1 , 𝑜 2 )𝑣(𝒝, H 1 ) + (1 − α II (𝑜 1 , 𝑜 2 ))𝑣(ℛ, H 1 )) ▶ α I – Type I error probability ▶ α II – Type II error probability

Full-information gap We get a bit of simplification by considering the full info-gap instead of value: FIG(𝑜 1 , 𝑜 2 ) ≡ payofg from perfect info ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ (1 − 𝑞)𝑣(𝒝, H 0 ) + 𝑞𝑣(ℛ, H 1 ) −𝑤(𝑜 1 , 𝑜 2 ) = (1 − 𝑞)α I (𝑜 1 , 𝑜 2 ) loss from Type-I ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ (𝑣(𝒝, H 0 ) − 𝑣(ℛ, H 0 )) loss from Type-II ⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞ (𝑣(ℛ, H 1 ) − 𝑣(𝒝, H 1 )) (Minimizing the FIG is equivalent to maximizing the value) + 𝑞 α II (𝑜 1 , 𝑜 2 )

Roadmap Goal: Find a nice ordinally-equivalent expression for value Method: 1. Approximate error probability 2. Simplify with a monotone transformation of value

Error probabilities Consider the one info source case from MS02 > ̄ 𝑞 ∏ 𝑜 𝑗=1 𝑔 (𝑠 𝑗 | H 1 ) α I (𝑜) = ℙ ( 𝑞 | H 0 ) 𝑞 ∏ 𝑜 𝑗=1 𝑔 (𝑠 𝑗 | H 1 ) + (1 − 𝑞) ∏ 𝑜 𝑗=1 𝑔 (𝑠 𝑗 | H 0 )

Error probabilities 𝑞 This is a large deviation . when the sample average LLR is far from its mean. 𝔽(𝑡 𝑗 |H 0 ) < 0 , so at large sample size, a mistake only occurs 𝑗=1 ∑ 𝑜 ̄ Change to log-likelihood ratios: 1 − ̄ 𝑗=1 ∑ 𝑜 𝑞 More info log (𝑔 (𝑠 𝑗 | H 1 ) α I (𝑜) = ℙ ( log ( 1 − 𝑞) + 𝑔 (𝑠 𝑗 | H 0 )) > log ( 𝑞) | H 0 ) 𝑡 𝑗 > ̄ ≡ ℙ ( 𝑚 − 𝑚 | H 0 )

Large deviations – Chernofg Index 𝑢 moment generating function: ρ ≡ min 𝑢 M(𝑢) = min 𝑢 Properties: Large deviations probabilities ofuen depend on the minimized 𝑔 (𝑠 | H0) ) 𝑔 (𝑠 | H 0 )𝑒𝑠 = min 𝑢 log ( 𝑔 (𝑠 | H1) ∫ 𝑓 ∫ 𝑔 (𝑠 | H 1 ) 𝑢 𝑔 (𝑠 | H 0 ) 1−𝑢 d 𝑠 Call ρ the Chernofg index of the info source ▶ ρ ∈ (0, 1) ▶ Blackwell more informative ⇒ lower Chernofg index ▶ a source composed of 𝑜 i.i.d. samples has index ρ 𝑜