When to Ask for an Update: Timing in Strategic Communication. National University of Singapore June 5, 2018

Transcription

1 When to Ask for an Update: Timing in Strategic Communication Ying Chen Johns Hopkins University Atara Oliver Rice University National University of Singapore June 5, 2018

2 Main idea In many communication situations, the sender learns about the state over time. Examples: A manager who reports to an executive learns about an investment opportunity by visiting various markets to investigate local conditions. An advisor to a politician consults different experts on the implications of a proposed policy on different dimensions. Natural instrument for eliciting information: timing of report frequent updates or a summary report?

3 Overview A project s value depends on two aspects. An agent (sender) potentially learns about the value of one aspect in each period. When asked, the sender chooses whether or not to disclose his signal. A principal (receiver) decides whether to take on the project. Each party wants the project accepted if its value exceeds his/her own threshold; the sender has a lower threshold.

4 Overview We compare the following reporting protocols: Infrequent updating: receiver asks for one report at the end. Frequent updating: receiver asks for a report in each period.

5 Main findings Crucial: whether nondisclosure leads to acceptance or rejection. If nondisclosure leads to rejection, frequency of reporting does not matter.

6 Main findings Crucial: whether nondisclosure leads to acceptance or rejection. If nondisclosure leads to rejection, frequency of reporting does not matter. If nondisclosure leads to acceptance, frequency matters. Frequent reporting facilitates information transmission if (i) the interests are sufficiently aligned; (ii) it is unlikely that the sender observes an informative signal in the second period. Infrequent reporting is better otherwise.

7 Static disclosure game An agent (sender) privately observes a signal s about the value of a project v, distributed according to continuous cdf F on [0, 1]. s = with probability p; s = v with probability 1 p. The sender can either disclose (m = s) or not (m = ). (Sender cannot prove that he has observed s =.) A principal (receiver) then decides whether to accept the project (Y or N). If project is rejected, both receive payoff 0; if accepted, receiver s payoff is v c R and sender s payoff is v c S. 0 c S < c R : the sender is biased in favor of acceptance. Special case: c S = 0. The sender always prefers acceptance.

8 Equilibrium in static disclosure game The receiver chooses Y if m c R and N if m < c R in equilibrium. If m = :?

9 Equilibrium in static disclosure game The receiver chooses Y if m c R and N if m < c R in equilibrium. If m = :? To address problem of multiple equilibria, Consider a perturbation s.t. the sender gains ε > 0 by revealing s. Look at limit equilibrium as ε 0. Analogous to truth-leaning" equilibrium in Hart, Kremer and Perry (2016). The sender reveals s if s c S or if s c R. If s (c S, c R ):?

10 Case I: c R above expectation of v full revelation Let v denote expectation of v. Suppose c R > v (rejection under prior). m = induces N. The sender reveals all s.

11 Case II: c R below expectation of v Suppose c R v. v 0 : posterior expectation of v when receiving if the sender conceals s (c S, c R ). v 0 p v + (1 p) c R c = S vdf p + (1 p)[f (c R ) F (c S )] < v.

12 Case II (a): Nondisclosure leads to acceptance Suppose c R v 0. m = induces Y. The sender conceals s (c S, c R ) and induces Y.

13 Case II (b): Nondisclosure leads to rejection. Suppose v 0 < c R < v. Is concealing s (c S, c R ) a (limit) equilibrium? No, since would induce N in this case profitable deviation for s (c S, c R ) (in the perturbed game).

14 Case II (b): Nondisclosure leads to rejection. Suppose v 0 < c R < v. Is concealing s (c S, c R ) a (limit) equilibrium? No, since would induce N in this case profitable deviation for s (c S, c R ) (in the perturbed game). Is revealing s (c S, c R ) an equilibrium? No, since would induce Y in this case profitable deviation for s (c S, c R ).

15 Case II (b): Nondisclosure leads to rejection. For v 0 < c R < v, There exists ŝ (c S, c R ) such that the sender reveals s (c S, ŝ) and conceals s [ŝ, c R ). The receiver is indifferent between Y and N when receiving. In the limit equilibrium, m = induces N with prob. 1.

16 Summary of equilibria in static game High c R : nondisclosure leads to rejection; full disclosure of s. Intermediate c R : nondisclosure leads to rejection; partial disclosure of s (c S, c R ). Low c R : nondisclosure leads to acceptance; no disclosure of s (c S, c R ).

17 Dynamic model A project has two aspects: v 1, v 2. Total value of the project is v = g(v 1, v 2 ). g : [0, 1] 2 R + is continuous and increasing in v 1 and v 2. Normalize g(0, 0) = 0. v i is distributed according to continuous cdf F i on [0, 1]; v 1 and v 2 are independent. Preferences of players are the same as in the static game. Assume that c R < g(1, 1): it is optimal for R to accept the project for some realizations of v. Special case: c S = 0, player S always wants the projected accepted.

18 Dynamic model In each period t = 1, 2, the sender observes a signal s t about v t. s t = with probability p t (0, 1); s t = v t with probability 1 p t. We compare two games that differ in the timing of updates.

19 Game Γ I : only one report In Γ I, receiver solicits a summary report at the end of period 2. After observing both s 1 and s 2, the sender sends m {(s 1, s 2 ), (s 1, ), (, s 2 ), (, )}. After receiving m, the receiver chooses a {Y, N}.

20 Game Γ F : frequent report In Γ F, receiver solicits a report from the sender in every period. In each period t, the sender makes a report m t {s t, } after observing s t. After receiving m 1 and m 2, the receiver chooses a {Y, N}.

21 Strategies in Γ I The sender s (pure) reporting strategy is µ I (s 1, s 2 ) {(s 1, s 2 ), (s 1, ), (, s 2 ), (, )}. The receiver s action strategy is α I (m) [0, 1]: probability of choosing Y when receiving m.

22 Strategies in Γ F The sender s (pure) reporting strategy consists of µ F 1 (s 1) {s 1, }; µ F 2 (s 1, s 2, m 1 ) {s 2, }; The receiver s action strategy is α F (m 1, m 2 ) [0, 1]: probability of choosing Y when receiving (m 1, m 2 ).

23 Equilibrium Solution concept: PBE with refinement. Again, consider the perturbation such that the sender gains ε > 0 by revealing s t. Look at limit equilibria as ε 0.

24 Equilibrium outcome Equilibrium outcome: mapping between S signals and the actions that R is induced to take. In Γ I, equilibrium outcome function y I (s 1, s 2 )= α I (µ I (s 1, s 2 )). In Γ F, y F (s 1, s 2 )= α F (µ F 1 (s 1), µ F 2 (s 1, s 2, µ F 1 (s 1))). We say that the equilibrium outcomes in Γ I and Γ F are equivalent if y I (s 1, s 2 ) = y F (s 1, s 2 ) with probability 1.

25 Example: v = min{v 1, v 2 } Perfect complements: each aspect has to be above c i for player i to prefer acceptance.

26 Some cutoffs for equilibrium characterization. Consider message (v 1, ). Suppose R believes that either s 2 = or s 2 = v 2 (c S, c R ). Let v 1 be defined by E(v m = (v 1, )) = c R if a solution in [0, 1] exists. (If no solution, let v 1 > 1.) That is, v 1 is such that R s posterior expectation of v when receiving (v 1, ) is equal to c R. For perfect complements, v 1 > c R. v 2 is similarly defined.

27 Cutoffs v 1 and v 2

28 High threshold for acceptance Suppose under prior, R chooses N.

29 High threshold for acceptance Suppose under prior, R chooses N. Consider Γ I first.

30 High threshold for acceptance Suppose under prior, R chooses N. Consider Γ I first. Receiver s equilibrium strategy is 1 if min{m 1, m 2 } c R, α I 1 if m 1 v 1 (m 1, m 2 ) = and m 2 =, 1 if m 1 = and m 2 v 2, 0 otherwise.

31 High c R : Equilibrium in Γ I If min{s 1, s 2 } c R : reveal both signals and induce Y. If min{s 1, s 2 } c S : reveal both signals and induce N. If min{s 1, s 2 } (c S, c R ): induce Y if s 1 v 1 or if s 2 v 2. If only one signal received, induce Y if s t v t.

32 High c R : Equilibrium in Γ I If min{s 1, s 2 } c R : reveal both signals and induce Y. If min{s 1, s 2 } c S : reveal both signals and induce N. If min{s 1, s 2 } (c S, c R ): induce Y if s 1 v 1 or if s 2 v 2. If only one signal received, induce Y if s t v t. Equilibrium outcome:

33 High c R : receiver s equilibrium strategy in Γ F If both aspects are revealed, α F is same as α I ; If only the first aspect is revealed, α F is same as α I ; To be determined for α F : only the second aspect is revealed. But note that α F (, m 2 ) = 0 if m 2 < c R.

34 High c R : equilibrium in Γ F S s strategy: reveal s 1 if s 1 c S or if s 1 c R ; conceal s 1 (c S, c R ).

35 High c R : equilibrium in Γ F S s strategy: reveal s 1 if s 1 c S or if s 1 c R ; conceal s 1 (c S, c R ). If s 1 c S, sender prefers N and therefore optimal to reveal. If s 1 v 1 : by revealing s 1, sender can induce his preferred action regardless of s 2. If s 1 [c R, v 1 ]: by revealing s 1, sender can induce Y if s 2 c R ; there is no gain from concealing s 1. If s 1 (c S, c R ): revealing s 1 can only induce N, concealing s 1 may induce Y if s 2 is sufficiently high. The sender therefore conceals.

36 High c R : equilibrium in Γ F If m 1 =, player R believes that either s 1 = or s 1 (c S, c R ). Hence, if only second aspect is revealed, R chooses Y iff m 2 v 2 : same as in Γ I.

37 High c R : equilibrium outcome in Γ F Receiver s strategy is the same as in Γ I. Y if both aspects are revealed to be higher than c R ; Y if only aspect t is revealed and it is higher than v t. Sender s strategy in period 1: conceal s 1 (c S, c R ). Equilibrium outcome:

38 High c R : equilibrium outcome independent of timing of updates Proposition Suppose v = min{v 1, v 2 }. If player R prefers to reject the project under her prior, then the equilibrium outcomes are equivalent in Γ I and Γ F.

39 High c R : equilibrium outcome independent of timing of updates Proposition Suppose v = min{v 1, v 2 }. If player R prefers to reject the project under her prior, then the equilibrium outcomes are equivalent in Γ I and Γ F. Implications: When R s threshold of acceptance is high, if there is a cost of soliciting a report, frequent reporting is inefficient. But if there is gain from early resolution, then frequent reporting is preferable. In the paper, we generalize the equivalence result by providing conditions under which it holds.

40 Low c R : (, ) induces Y in equilibrium For the equivalence result, we looked at the case in which nondisclosure leads to rejection in equilibrium. Now we consider low c R such that nondisclosure leads to acceptance in equilibrium. Consider Γ I first.

41 Low c R : equilibrium in Γ I Receiver s equilibrium strategy is the same as before, except that α I (, ) = 1. Note that (i) whenever S prefers N, he can just reveal his signals to induce N; (ii) any type can induce Y given that nondisclosure induces Y. Hence, S always induces his preferred action in Γ I.

42 Low c R : equilibrium outcome in Γ I Equilibrium outcome: Definition of v S t : When (s 1, s 2 ) = (v S 1, ) or (s 1, s 2 ) = (, v S 2 ), player S is indifferent between Y and N.

43 Low c R : equilibrium in Γ F s 1 c S : reveal since S prefers N for any s 2. s 1 v 1 : reveal since S can again induce his preferred action for any s 2. Consider s 1 (v S 1, v 1 ). If S conceals s 1, he can induce his preferred action for any s 2. If S reveals s 1, N will be induced if s 2 =, but he prefers Y in that case. Optimal for S to conceal.

44 Low c R : Sender s equilibrium strategy in Γ F Consider s 1 (c S, v1 S ): countervailing incentives. Since s 1 < v1 S, player S prefers N without further information on the second aspect. If s 2 =, player S is better off by revealing s 1. But since s 1 > c S, player S prefers Y if s 2 > c S. If s 2 > c S, player S is better off by concealing s 1. When nondisclosure leads to acceptances, concealing bad news early on has a cost.

45 Low c R : equilibrium in Γ F At s 1 = v 1 (c S, v1 S ), S is indifferent. Below v 1, the loss from concealing is greater. Above v 1, the loss from revealing is greater. Hence, reveal below v 1 and conceal above it.

46 Low c R : equilibrium in Γ F At s 1 = v 1 (c S, v1 S ), S is indifferent. Below v 1, the loss from concealing is greater. Above v 1, the loss from revealing is greater. Hence, reveal below v 1 and conceal above it. Equilibrium outcome:

47 When are outcomes different in Γ I and Γ F? (i) If s 1 v 1 and min{s 1, s 2 } (c S, c R ), then N is induced in Γ F whereas Y is induced in Γ I. Γ F Γ I

48 When are outcomes different in Γ I and Γ F? (ii) If s 1 (v 1, v S 1 ) and s 2 =, then Y is induced in Γ F whereas N is induced in Γ I. Γ F Γ I

49 Optimal frequency of update In both cases, R prefers N. In case (i), R is better off in Γ F. In case (ii), R is better off in Γ I. Intuitively, R prefers frequent updating when v 1 is high (more revelation of bad news early on) and prefers infrequent updating when v 1 is low. Higher c S and higher p 2 (probability that s 2 = ) imply stronger incentive to reveal higher v 1.

50 Optimal frequency of update for low c R Suppose v = min{v 1, v 2 } and v i is uniformly distributed on [0, 1]. Proposition Suppose (, ) induces Y in equilibrium in both games. (i) If the players interests are sufficiently close or p 2 is sufficiently high, then player R is better off in Γ F ; (ii) if c S > 0 is sufficiently far from c R and p 2 is sufficiently low, then player R is better off in Γ I.

51 Optimal frequency of update for low c R Suppose v = min{v 1, v 2 } and v i is uniformly distributed on [0, 1]. Proposition Suppose (, ) induces Y in equilibrium in both games. (i) If the players interests are sufficiently close or p 2 is sufficiently high, then player R is better off in Γ F ; (ii) if c S > 0 is sufficiently far from c R and p 2 is sufficiently low, then player R is better off in Γ I. If c S = 0, then S has no incentive to reveal any bad news in period 1 and we have equivalence again.

52 Optimal frequency of update: remarks Frequent updating exploits S s willingness to reveal bad news early on for fear that concealment will lead to acceptance of a bad project. For this incentive to work, it is important that the only opportunity for S to reveal s 1 is in period 1. Ex post inefficient. Plausible if the cost for the sender to admit hiding information is high. It can also be regarded as the receiver s commitment to not hearing or acting upon late disclosure. Example: Federal Rule of Civil Procedure 37 says if a party fails to disclose timely materials in discovery, it is sanctioned by the exclusion of the undisclosed information in a later motion/hearing/trial.

53 Alternative interpretation Multiple senders observe different aspects. Senders have common interest different from the receiver s. Γ I : senders pool their information together and make a joint report. Γ F : individual senders are consulted sequentially and publicly.

54 Summary High c R (nondisclosure leads to rejection): equilibrium outcome is same regardless of frequency of update. Low c R (nondisclosure leads to acceptance): frequency of updating matters. If divergence of interest is sufficiently low or it is unlikely for the sender to observe an informative signal in a later period, frequent updating is beneficial. Otherwise, the receiver should ask for only a summary report.

55 Related Literature Static disclosure with uncertainty about information endowment: Dye (1985); Jung and Kwon (1988), Shin (1994, 2003). Dynamic disclosure: Eihorn and Ziv (2008); Dye (2010); Acharya, Demarzo and Kremer (2011); Guttman, Kremer and Skrypacz (2014), Gratton, Holden and Kolotilin (2016).

56 Extensions and variations Multiple senders who observe signals of different aspects: in addition to joint report and sequential report, what about simultaneous report? What happens if the receiver can commit to a dynamic mechanism?