Selling Experiments. Dirk Bergemann 1 Alessandro Bonatti 2 Alex Smolin 1. BFI Chicago July 10th, Yale University.

Transcription

1 Selling Experiments Dirk Bergemann 1 Alessandro Bonatti 2 Alex Smolin 1 1 Yale University 2 MIT Sloan BFI Chicago July 10th, 2015

2 Introduction A decision maker under uncertainty: has partial (private) information; seeks to acquire additional information ( data buyer ). Data seller can augment data buyer s information. How much information to provide at what price? How to provide different information to different data buyers? Interpretation: selling access to a database.

3 Leading Example: Selling Information about Consumers firms can tailor online advertising levels to individual users targeting requires information about characteristis of individual users. data seller, data broker has information on individual users attributes. attributes differ in their sign and explanatory power. data broker may offer to reveal certain attributes. firms may have first-party information on users heterogeneous valuation for a given attribute.

4 Setting data seller offers information product = experiment (Blackwell) reveals information about the decision maker s payoff-relevant state value of experiment depends on decision maker s private information. decision maker s beliefs are his type data seller only has (common) prior over types

5 Analysis optimal versioning of an information product only information product itself is contractible by contrast, action of decision maker or realized state are not contractible data seller has database already installed seller has zero marginal cost of data provision.

6 Related Literature Selling Information Admati and Pfleiderer (1986, 1990), Eső and Szentes (2007), Selling Cookies AEJ Micro (2015). Bundling Information and Products Johnson and Myatt (2006), Bergemann and Pesendorfer (2007), Targeting in Advertising Markets RAND (2011). Persuasion Rayo and Segal (2010), Kamenica and Gentzkow (2011), Lin (2013), Kolotilin et al. (2014) Screening Mussa and Rosen (1978), Myerson (1981), Jullien (2000).

7 Results A menu of experiments is offered. The menu only contains simple experiments. Menu itself is much coarser than possible types. Systematic distortions in information provided. Screening facilitated by directional information. Linearity (in probabilities) limits the use of versioning.

8 Model Single decision-maker (buyer of information). Finite states Finite actions ω Ω. a A. For most of the talk Ω = A = 2. Richer specification of buyer s private information.

9 Private Information Common prior probability over states µ Ω. Decision maker privately observes an informative signal. Decision maker forms an initial belief θ Ω given the signal. Initial beliefs are private information. Alternatives Distribution of initial beliefs F (θ) (from seller s point of view).

10 Database and Experiments An experiment (information structure) I consists of I = {S, π} π : Ω S. Seller offers a menu of experiments M = {I, t}, with I = {I} t : I R +. Database query or machine interpretation: contracting before the state ω is realized; costless provision of information (data is already stored); initial and incremental signals are independent conditionally on the state

11 Initial and Incremental Information interim probability θ i = Pr (ω = ω i ) likelihood function under experiment I: perfect information: extremal information: π ij = Pr (s j ω i ) π ij = { 1, if i = j 0, if i j π ij = 1 for some i = j

12 Buyer s Payoff ex-post utility of decision maker u (a, ω) without loss of generality: matching action and state: normalization to {0, 1} : u (a, ω) = 1 [a=ω].

13 Value of Experiments Buyer s payoff under partial information u (θ) max a A E θ [u (a, ω)]. Value of experiment (net value of augmented information) V (I, θ) E I,θ [max a A E s,θ [u (a, ω)]] u (θ).

14 Value of Experiments with symmetry and matching the state value of experiment I for buyer θ : V (I, θ) = j max i {θ i π ij } max {θ i }. i

15 Geometry of Value of Experiments three states ω 1, ω 2, ω 3 interim beliefs θ = (θ 1, θ 2 ) perfect information experiment

16 Geometry of Value of Experiments II three states ω 1, ω 2, ω 3 interim beliefs θ = (θ 1, θ 2 ) imperfect but extremal experiment π s 1 s 2 s 3 ω 1 2/3 1/6 1/6 ω ω

17 Seller s Problem Seller can offer a menu of experiments. Items intended to match initial and incremental information. Direct mechanism by the Revelation Principle M = {I (θ), t (θ)}. Seller s objective function, IC and IR constraints: t (θ) df (θ), max {I(θ),t(θ)} s.t. V (I (θ), θ) t (θ) V ( I ( θ ), θ ) t ( θ ) θ, θ, V (I (θ), θ) t (θ) 0 θ.

18 Optimality of Coarse Signals Continuum of information structures I(θ). Each one a potentially complicated map: states signals. Optimally coarse signals (and, later, optimally coarse menu). Merge signals in I(θ) leading to the same action for type θ. V (I(θ), θ) stays constant but V (I(θ), θ ) decreases θ θ. Lemma (Coarse Signals) In an optimal menu, every experiment consists of A signals.

19 Binary Model binary action, binary state a = a L a = a H ω = ω L 1 0 ω = ω H 0 1 wlog restrict attention to experiments I = s L s H ω L β 1 β ω H 1 α α with the convention α + β 1.

20 Value of Experiment with Binary Model Let θ = Pr (ω = ω H ). Value of experiment (α, β) V (α, β, θ) = [αθ + β (1 θ) max{θ, 1 θ}] +.

21 Informative Signals Perfectly informative experiment: α = β = 1. I = s L s H ω L 1 0 ω H 0 1 Directionally informative experiment, e.g., α = 1, β < 1. I = s L s H ω L β 1 β ω H 0 1 Directional information valuable for some types, but not for others. Useful to decision maker who believes that ω H is very likely, but not useful to agent who believes that ω L is very likely, because he would still choose a L under both signals.

22 Value of a Perfectly Informative Experiment Value of experiment (α, β) = (1, 1) for type θ. (α, β) = (1, 1) Highest-value buyer is θ = 1/2.

23 Value of a Directionally Informative Experiment Value of experiment (α, β) = (1, 2/2) for type θ. (α, β) = (1, 1/2) Distance θ 1/2 is not suffi cient to describe the value of experiment

24 Preferences over Experiments Value of experiment (α, β) for type θ V (α, β, θ) = (α β) θ + β max{θ, 1 θ}. β = baseline informativeness (from payoff normalization). α β = relative informativeness. Two goods that cannot be produced independently.

25 Feasible Set of Experiments

26 Indifference Curves for Given Type θ value of experiment is V (α, β, θ) = (α β) θ + β max {θ, 1 θ} higher types θ have stronger preference for relative

27 Value of Baseline Information incremental change in the baseline information β while keeping the relative informativeness α β constant V (α + δ, β + δ, θ) V (α, β, θ) = δ, uniform increase in value of experiment for all types θ

28 we can reduce the choice of experiment to a one-dimensional problem: Set of Optimal Experiments maximal baseline informativeness for any given relative informativeness

29 Value of an Experiment Value of experiment (α, β) for type θ V (α, β, θ) = [(α β) θ + β max{θ, 1 θ}] +. Holding α β constant, can increase value uniformly θ. Lemma (Partially Revealing Signals) In an optimal menu, every experiment has either α = 1 or β = 1. At least one signal perfectly reveals one state. One-dimensional allocation (relative informativeness) q(θ) α(θ) β(θ) [ 1, 1].

30 Buyer s Utility Value of experiment q [ 1, 1] for type θ [0, 1]: V (q, θ) = [θq max{q, 0} + min{θ, 1 θ}] + Experiments q { 1, 1} entirely uninformative. Single crossing in (θ, q): as in classic nonlinear pricing. Experiment q = 0 is the most valuable for all types θ. Type θ = 1/2 has the highest value for any experiment q. Types disagree on ranking of imperfect experiments.

31 Optimal Experiments directionally informative if q > 0 : useful for agent who thinks ω H is very likely I = s L s H ω L 1 q q ω H 0 1 if q < 0 : useful for agent who thinks ω L is very likely I = s L s H ω L 1 0 ω H q 1 + q

32 TOptimal Experiments with Two Types Two types: θ {0.2, 0.7} with equal probability. Sell full information at reservation value to type θ = 0.7. Net Value of Experiment q (θ)

33 Two-Type Example IC does not rule out serving type θ = 0.2. Net Value of Experiment q (θ)

34 Two-Type Example Optimal allocation: q (0.2) = 1/5, and q (0.7) = 0. Posterior beliefs p (0.2) {1/21, 1}, and p (0.7) {0, 1}. Net Value of Experiment q (θ)

35 Two-Types: Summary The optimal menu contains either one or two experiments. One experiment: one or both types may purchase. Two experiments: requires asymmetry in types on opposite sides of 1/2; no distortion at the top (θ closer to 1/2); no rent at the bottom; corner solution no rent at the top.

36 Geometry of the Problem Highest type is interior (θ = 1/2). Optimal menu on each side of 1/2: single experiment q = 0. Easy cases: truncated support or symmetric distribution. More generally, worry about IC on both sides. Price discrimination within and across two groups.

37 Continuum of Types Recall the value of experiment q [ 1, 1] for type θ [0, 1]: V (q, θ) = [θq max{q, 0} + min{θ, 1 θ}] +. Single-crossing suggests q increasing in θ. Types θ = 0 and θ = 1 receive zero rents. Consider type θ = 1/2, derive additional condition.

38 Incentive Compatibility Rent of type θ = 1/2 U(1/2) = U(0) + 1/2 0 1 V θ (q, θ)dθ = U(1) V θ (q, θ)dθ. 1/2

39 Incentive Compatibility Rent of type θ = 1/2 U(1/2) = 1/2 0 1 (q (θ) + 1)dθ = (q (θ) 1)dθ. 1/2 Lemma (Implementable Allocations) The allocation q (θ) is implementable if and only if q (θ) [ 1, 1] is non-decreasing, and 1 0 q (θ) dθ = 0. Note: a different kind of constraint.

40 Seller s Problem max q(θ) 1 0 [( θ 1 F (θ) ) ] q (θ) max {q (θ), 0} f (θ) dθ, f (θ) s.t. q (θ) [ 1, 1] non-decreasing, 1 0 q (θ) dθ = 0. Piecewise linear (concave) problem with integral constraint. Absent the integral constraint, corner solutions: q { 1, 0, 1}, i.e., all-or-nothing information, flat price. E.g., truncated support or symmetric distribution.

41 Optimal Menu Jump to Properties Proposition (Optimal Menu) An optimal menu consists of at most two experiments. The first experiment is fully informative. The second experiment (contains a signal that) perfectly reveals one state.

42 Seller s Problem max q(θ) 1 0 [( θ 1 F (θ) ) ] q (θ) max {q (θ), 0} f (θ) dθ, f (θ) s.t. q (θ) [ 1, 1] non-decreasing, 1 0 q (θ) dθ = 0.

43 Seller s Problem max q(θ) 1 0 [(θf (θ) + F (θ)) q (θ) max {q (θ), 0} f (θ)] dθ, s.t. q (θ) [ 1, 1] non-decreasing, 1 0 q (θ) dθ = 0. Consider virtual values for each experiment q separately: { θf (θ) + F (θ) for q < 0, φ(θ, q) (θ 1)f (θ) + F (θ) for q > 0. φ = marginal value of going from q(θ) = 1 to 0 to 1. Problem is not separable: virtual value φ depends on q.

44 General Case Let λ denote the multiplier on the integral constraint (shadow cost of providing higher quantity ). Let φ (θ, q) denote the ironed virtual value for experiment q. Proposition (Optimal Allocation Rule) Allocation q (θ) is optimal if and only if there exists λ 0 s.t. [ q q (θ) arg max ( φ (θ, x) λ ) ] dx for all θ, q [ 1,1] 0 has the pooling property, and satisfies the integral constraint. Myerson (1981), Toikka (2011), Luenberger (1969).

45 Example 1: Uniform Distribution Virtual Values: φ (θ, 1) in blue; φ (θ, 1) in red.

46 Example 1: Uniform Distribution Virtual Values: φ (θ, 1) in blue; φ (θ, 1) in red.

47 Example 1: Uniform Distribution Optimal Menu: Single Experiment

48 Optimality of Flat Pricing Proposition (Flat Pricing) The optimal menu contains only the fully informative experiment when any of the following conditions hold: 1 the density f(θ) = 0 for all θ > 1/2 or θ < 1/2; 2 the density f(θ) is symmetric around θ = 1/2. 3 F (θ) + θf(θ) and F (θ) + (θ 1)f(θ) are strictly increasing. A second experiment is offered only if ironing is required.

49 Non-monotone Density Probability Density Function: informed types are frequent

50 Example 2: Combination of Beta Distributions Probability Density Function

51 Example 2: Beta Distributions Virtual Values: φ (θ, 1) in blue; φ (θ, 1) in red.

52 Example 2: Beta Distributions Ironed Virtual Values

53 Example 2: Beta Distributions Ironed Virtual Values

54 Example 2: Beta Distributions One More Example Optimal Menu: Two Experiments (q {.21, 0})

55 Properties of the Optimal Menu Optimal mechanism involves 2 bunching intervals. Ideally, would sell q = 0 at two different prices (for θ 1/2). Symmetric distribution or truncated support flat pricing. Second-best menu may contain q = 0 only or distort the least profitable side. No further versioning is optimal. Least informed types most valuable to the seller. Type θ = 1/2 need not get effi cient q = 0.

56 Conclusions: Selling Information Selling incremental information to privately informed buyers. Costless acquisition & transmission, free degrading. Uninterested seller packaging problem. Bayesian problem for buyers: Linear in probabilities: limited use of versioning. Screening across groups through directional information. More general approach to nonlinear pricing within and across.

57 Implications for Observables How to damage an information good Should not observe arbitrarily damaged goods. Directional information: only type-i or II errors. Should not observe multiple distortions of the same kind. Directional distortions disclosure of specific attributes (correlated with high- or low- value consumers).

58 Enriching the Model More than two states and actions. Cost of providing information. Heterogeneous tastes for actions. Other sources of buyer heterogeneity. Investment problem with continuous states. Alternatives Investment Comparison with richer contracts spaces. Role of orthogonal vs. correlated information.

59 Many States Many states: ω 1,..., ω N Ω Many actions: a 1,..., a N A Experiments π ij with Σ j π ij = 1: s 1 s 2 s 3 ω 1 π 11 π 12 π 13 ω 2 π 21 π 22 π 23. ω N π N1 π N2 π N3 Perfectly informative experiment: π ii = 1 for every i. Extreme points: π ii = 1 for some i.

60 Many States Structure of incentive problem remains piecewise linear. Proposition (Optimal Experiments) 1 Every optimal experiment contains at most N signals. 2 Every optimal experiment is an extreme point. 3 Any optimal menu contains the perfectly informative experiment.

61 Three States, Three Actions, Continuous Types 1 uniform distribution towards the vertices

62 Three States, Three Actions, Continuous Types 1 Flat pricing under the uniform distribution

63 Three States, Three Actions, Continuous Types 1 beta distribution on the ω 1 ray richer menu if distribution differs across rays

64 Partial and Complete Information Uninformed types receive complete information: s 1 s 2 s 3 ω I ω ω Partially informed types t 1 receive partial information s 1 s 2 s 3 ω I ω ω Remain partially confused after signal s 1.

65 Three States, Three Actions, Continuous Types 1 Prior and Posteriors

66 Three States, Three Actions, Continuous Types 2 Types are uniformly distributed on the orthogonal lines. Richness of the allocation (vs. types). Partial information is provided.

67 Three States, Three Actions, Continuous Types 2

68 Partial and Complete Information Uninformed types receive complete information: I 0 = s 1 s 2 s 3 ω ω ω Partially informed types t i, here t 1 receive partial information I 1 = s 1 s 2 s 3 ω 1 1/2 1/4 1/4 ω ω Remain partially confused after signals s 2, s 3.

69 Sources of Buyer Heterogeneity Back Alternative sources of heterogeneity: Heterogeneous tastes. Heterogeneous stakes of the game. Heterogeneous relevance. Differential signal-processing abilities. Buyer s private information = beliefs. Belief heterogeneity: role of mean vs. variance?

70 The (not so) Natural Model Back Buyer faces prediction problem with linear-quadratic payoffs. Normally distributed state and signals. Buyers differ in interim beliefs (mean and variance). Willingness to pay proportional to incremental precision, independent of interim mean belief. No role for directional information, or for versioning.

71 Investment / Product Quality Model Back Continuous state ω R; binary action a {0, 1}. Normalize u(a, ω) = a ω. Buyer s interim pdf is g (ω θ), with E [ω θ] = θ. Assume g (ω θ) ordered by rotation around ω = 0. Restrict attention to binary partitions ω {[, q], [q, ]}. Resulting V (q, θ) has single crossing in (q, θ).

72 Investment / Product Quality Model A menu is a collection of experiments and prices {q(θ), t(θ)}. Highest-value experiment for any θ is q = 0. Robust features of menu with binary state: Different ranking of imperfectly informative experiments. Integral constraint. Coarse (2-item) optimal menu. Compared to contractible-action model: information distortions (cf. Eső and Szentes (2007), Li and Shi (2015)).

73 Example 3: Power Distribution Probability Density Function

74 Example 3: Power Distribution Ironed Virtual Values

75 Example 3: Power Distribution Back Optimal Menu: Two Experiments (q {0, 1/5})

76