When to Ask for an Update: Timing in Strategic Communication

Transcription

1 When to Ask for an Update: Timing in Strategic Communication Work in Progress Ying Chen Johns Hopkins University Atara Oliver Rice University March 19, 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. 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. When asked, the sender chooses whether or not to disclose his signal.

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. 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, s 2 ))). 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 E(v m = (v 1, )) = c R v 1 > 1.) if a solution exists. (If no solution, let 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.

40 Generalizing the equivalence result Suppose m = (v 1, ) is sent if s 2 = or if s 2 = v 2 with v = g(v 1, v 2 ) (c S, c R ). Let e 1 (v 1 ) be R s expectation of v when receiving m = (v 1, ). Define e 2 (v 2 ) analogously. Define v i by e i (v i ) = c R if e i (0) c R e i (1). If e i (1) < c R, let v i > 1; if e i (0) > c R, let v i < 0.

41 SCP and characteristics of equilibrium strategies Suppose e i (v i ) c R satisfies the single crossing property (SCP). SCP implies that v 1 and v 2 are uniquely determined. it is optimal for R to play a cutoff strategy since e i (v i ) > c R above v i and e i (v i ) < c R below v i.

42 SCP and characteristics of equilibrium strategies Suppose e i (v i ) c R satisfies the single crossing property (SCP). SCP implies that v 1 and v 2 are uniquely determined. it is optimal for R to play a cutoff strategy since e i (v i ) > c R above v i and e i (v i ) < c R below v i. In the example where g(v 1, v 2 ) = min{v 1, v 2 }, e i (v i ) is strictly increasing and therefore SCP is satisfied. Other examples: g(v 1, v 2 ) = v 1 + θv 2, and v i is uniform on [0, 1].

43 Equivalence in equilibrium outcome Proposition Suppose that R prefers to reject the project under her prior. If e i (v i ) c R satisfies the SCP and g(v 1, v 2 ) c R, then the equilibrium outcomes are equivalent in Γ I and Γ F.

44 Equivalence in equilibrium outcome Proposition Suppose that R prefers to reject the project under her prior. If e i (v i ) c R satisfies the SCP and g(v 1, v 2 ) c R, then the equilibrium outcomes are equivalent in Γ I and Γ F. Equilibrium outcome: Y is induced in the following scenarios: 1 g(s 1, s 2 ) c R ; 2 g(s 1, s 2 ) (c S, c R ), s 1 v 1 or s 2 v 2 ; 3 s 1 v 1, m 2 = ; 4 v 1 =, v 2 v 2.

45 The condition g(v 1, v 2 ) > c R

46 When equivalence breaks down In the proposition, we have the condition g(v 1, v 2 ) c R. It is satisfied when g(v 1, v 2 ) = min{v 1, v 2 }. It is also satisfied if g is additively separable: v = g 1 (v 1 ) + g 2 (v 2 ); or multiplicatively separable: v = g 1 (v 1 ) g 2 (v 2 ). An example in which this condition fails and equivalence breaks down: g(v 1, v 2 ) = max{v 1, v 2 }. jump to low c R

47 Example: v = max{v 1, v 2 }. For simplicity, assume that c S = 0 and p 1 = p 2 = p. Recall that e 1 (v 1 ) is R s expectation of v when receiving m = (v 1, ). v = max{v 1, v 2 } v 1 < c R. Similarly, v 2 < c R. Hence, g(v 1, v 2 ) < c R.

48 Cutoffs v10 and v20.

49 Cutoffs v1 and v2.

50 v = max{v 1, v 2 }: equilibrium in Γ I Receiver s equilibrium strategy is 1 if max{m 1, m 2 } c R, α I 1 if m 1 v1 (m 1, m 2 ) = and m 2 =, 1 if m 1 = and m 2 v2, 0 otherwise.

51 v = max{v 1, v 2 }: equilibrium in Γ I If max{s 1, s 2 } c R : reveal both signals and induce Y. If max{s 1, s 2 } (0, c R ): induce Y if s 1 v1 or if s 2 v2. If s 1 = : induce Y if s 2 v2. If s 2 = : induce Y if s 1 v1.

52 v = max{v 1, v 2 }: equilibrium in Γ F The receiver s equilibrium strategy α F is not the same as α I. In particular, α F (m 1, ) = 1 iff m 1 v 1. Sender strategy in period 1: reveal s 1 if s 1 v 1 otherwise. and conceal Given this, α F (, m 2 ) = 1 iff m 2 v 2.

53 v = max{v1, v2 }: equilibrium in ΓF

54 v = max{v 1, v 2 }: equilibrium outcomes are different in Γ I and Γ F Γ I Γ F

55 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. Back to v = min{v 1, v 2 }. Consider Γ I first.

56 Low c R : equilibrium in Γ I Receiver s equilibrium strategy is the same as before, except that α I (, ) = 1.

57 Low c R : equilibrium in Γ I For c S > 0, let v1 S be such that when (s 1, s 2 ) = (v1 S, ), S is indifferent between Y and N. Note that c S < v1 S < v 1. If c S = 0, let v1 S = 0. Define v S 2 analogously.

58 Low c R : equilibrium outcome in Γ I (i) Whenever S prefers N, he can just reveal his signals to induce N + (ii) (, ) induces Y S always induces his preferred action. Equilibrium outcome:

59 Low c R : equilibrium in Γ F s 1 c S : reveal since S prefers N. s 1 v 1 : reveal since S can 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.

60 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.

61 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.

62 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:

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

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

65 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.

66 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 c S is sufficiently close to c R 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.

67 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 c S is sufficiently close to c R 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. The result also holds if g(v 1, v 2 ) = v 1 + θv 2, v i uniformly distributed. We conjecture that it holds more generally. If c S = 0, then S has no incentive to reveal any bad news in period 1 and we have equivalence again.

68 Optimal frequency of update Remarks: When the players interests are close (high c S ) or the probability that S observes no informative signal in period 2 is high (high p 2 ), R should ask for a report in every period. Frequent updating exploits S s willingness to reveal bad news early on for fear that concealment (combined with no more information in the future) will lead to acceptance. Important: the only opportunity for S to reveal s 1 is in period 1. Alternative interpretation: different senders observe different aspects and only one sender is consulted in each period.

69 Summary High c R (nondisclosure leads to rejection): equilibrium outcome is same regardless of frequency of update. Low c R : If divergence of interest is sufficiently low or prob. of the sender observing an informative signal in a later period is sufficiently low, frequent updating is beneficial. If divergence of interest is relatively high and prob. of the sender observing an informative signal in a later is relatively high, should ask for only a summary report.

70 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).

71 Extensions and variations Multiple senders who observe signals of different aspects: Is it better if they report sequentially or if they make a joint report? If sequential reporting is preferable, how to order them? What happens if there is full commitment? (Hart, Kremer and Perry (2016); Deb, Pai and Said (2017) )