Inside Outside Information

Transcription

1 Inside Outside Information Daniel Quigley and Ansgar Walther Presentation by: Gunjita Gupta, Yijun Hao, Verena Wiedemann, Le Wu

2 Agenda Introduction Binary Model General Sender-Receiver Game Fragility of Information Better Outside Information can Reduce Total Information Disclosures and the Shape of Payoffs Applications Conclusion and Discussion

3 Introduction - Motivation Research question: Do better public signals crowd out the disclosure of inside information? Motivation: Study the impact of outside information and where these can create frictions Related literature and models: Communication with self-interested experts ÞContext of verifiable communication and unravelling equilibria

4 Introduction - Setting Inside information can be disclosed by self-interested insiders Uninformed agents also have access to outside information (e.g. online reviews and policy-maker announcements) Then insiders incentives to disclose information depends on availability and quality of outside information available Substantial cost of disclosure for Sender Uncertainty about whether Sender has information to disclose Sender discloses information before outside information is received

5 Introduction - Setting What is new? Disclosure costs and uncertainty favour opacity and lead to discontinuities in equilibrium play Outside information is first-order determinant of incentives to communicate Where does it matter? Political contests with several candidates as Senders, voters as Receivers and outside information through media coverage Financial panics with financial institutions are Senders, investors as Receivers and outside information by policy makers such as central banks

6 Binary Model - Setup Informed Agent: Sender offers an indivisible good at a market price p Uninformed Agent: Receiver take Binary Action (can either buy the good or not) Quality of good: θ with commonly known distribution and smooth density f(θ) Timing: ØSender privately observes θ, and send a message m {θ, } m = θ verifiable but with cost c + m = no information and costless ØReceiver observes m and an outside signal s = θ + kε, k measures noise andε [ 1,1]with smooth log-concave density g(ε) and cumulative distribution G(ε) ØReceiver decides whether or not to buy good

7 Binary Model - Setup Assumptions: E[θ] < p: without information, not worth buying good p > c: profits from sale cover the costs of disclosure High realizations of s are good news about θ: if a higher s is received, θ more likely to be high Payoffs: Receiver L s Payoff=θ p If receiver buys good, + Sender L s Payoff = p c 1 STU Receiver L s Payoff=0 If receiver does not buy good, + Sender L s Payoff = c 1 STU

8 Binary Model - Equilibrium Under full information, Receiver would buy good iff θ p For Bad Senders, θ < p, disclosing true θ never leads to a sale, but incurs cost c dominant strategy: stay quiet m = For Good Senders, θ p, find Transparent Equilibrium, i.e. all good types disclose ØIf sender quiet, receiver assume θ < p unless s proves opposite ØReceiver buys good only if s p + k ØSender discloses if θ p and M(θ) c, where θ p M θ p Pr s < p + k θ = p G 1 k M θ is the maximal punishment for staying quiet, M θ _ = c

9 Binary Model - Equilibrium If k is above a critical level k` Transparent Equilibrium exists M(θ) c for all θ p M θ _ = c k U bcd k` ecf gh i j If k is below a critical level k` Reverse Unraveling arises as follow If k < k`, a set of best types, θ (θ _, θ ], have dominant strategy to stay quiet No news is now ambiguous news Receiver values silence more favorably Silence is amplified

10 General Sender-Receiver Game - Setup Receiver needs to choose an action a A R The state of the world: θ Θ = {θ e,, θ q } R, θ q > θ qce > > θ e Sender s payoff: v(a) Receiver s payoff: u a, θ, a and θ are strategic complementarities, i.e. Receiver chooses higher a when optimistic about θ μ 0 (θ): prior distribution of θ π(s θ): conditional distribution of s given θ

11 General Sender-Receiver Game - Equilibrium In Perfect Bayesian Equilibria: Sender sets m to maximize v(a) and Receiver sets a to maximize u a, θ On equilibrium path, Receiver s posterior beliefs on θ follows Bayes law Off equilibrium path, Receiver places zero probability on type θ if she observes a signal such that π(s θ ) = 0 Receiver s Best Response if knows θ with certainty a (θ) = argmax y z u(a, θ) Receiver s Best response if sender stays quiet and outside information is s α(s) argmax E μ [u(a, θ) s, m = ] Receiver with skeptical beliefs assumes Sender to be the worst type given s, θ(s), if m =

12 General Sender-Receiver Game - Equilibrium Sender s payoff when he is taken to be type θ for certain V(θ) = v(a (θ)) Sender s payoff if stays quiet and faces a skeptical receiver who assumes that E[V(θ(s)) θ] Sender s net payoff fromverifiable disclosure N(θ) V(θ) E[v(α(s)) θ] Sender s maximal punishment for staying quiet M(θ) = V(θ) E[V(θ(s)) θ] Sender discloses if N(θ) c, i.e. net payoff from disclosure is more than cost Sender with type θ has a dominant strategy to stay quiet iff M(θ) < c Transparent equilibrium exists iff c S UƒU h M(θ) c 0

13 General Sender-Receiver Game - Benchmark Benchmark: No outside information Disclosure strategies monotone increasing qualitatively similar to transparent equilibrium All types valued equally if quiet high types gain most from disclosure Unraveling result if c is not prohibitively high We then consider the presence of outside information

14 Fragility of Information More Setup We consider how equilibrium disclosure changes as the distribution of s varies from pure noise to full information in a continuous fashion. π s θ; t : conditional distribution of s given θ as a function of t Π t = (π s θ; t )ˆ,U : stochastic matrix that defines t-dependent structure of s pure noise if π s θ; t is independent of θ Π t is + fully revealing if for each s, only one θ s. t. π s θ; t > 0

15 Fragility of Information Proposition 1 Proposition 1: For any payoffs {u, v} and any prior μ _, assume that c is sufficiently small to ensure that there is a transparent equilibrium when public signals are pure noise. Then there exists a continuous path of signal structures Π(t) for t [0, 1] with the following properties: Π(0) is pure noise, while Π(1) is fully revealing, and There exists a critical point t 0, 1 such that, when Receiver observes the signal induced by Π(t), there is a transparent equilibrium for t t, while full opacity is the unique equilibrium for t > t

16 Fragility of Information Binary Model B θ L : highest type of Sender who prefers to disclose when Receiver expects disclosures from θ (p, θ L ), increase in θ L due to Strategic complementarities Transparent Equilibrium Imprecise s Precise s Full Opacity

17 Fragility of Information Binary Model Transparent Equilibrium k = k` k = k` ε

18 Fragility of Information General Model

19 Fragility of Information Caveat Caveat: The sequence of signals that leads to a discontinuity does not have full support, that is, the set of possible signals is different for different types If full support is enforced, cannot construct a threshold t where M θ q while M θ q < M θ However, deviations from full support are empirically reasonable in many settings, e.g. very favorable news tend to rule out very bad outcomes and vice versa = c

20 Better Outside Information can Reduce Total Information Rank Information Blackwell (1953): Signal s L is more informative about θ than s if s is a garbled signal of s L Or put differently, Bayesian decision-maker always weakly prefers s L to s We need it to show (only using intuition here) that better outside signals can always decrease informativeness as long as there is some disclosure in equilibrium Before: Fully Revealing Equilibrium v.s. Fully Opaque Equilibrium Here: Generalization

21 Better Outside Information can Reduce Total Information Proposition 2 Proposition 2: Suppose that, when outside information is s, there is an equilibrium E in which Sender makes a disclosure m = θ with strictly positive probability. Then there exists an outside signal s L such that s L is more informative than s in the sense of Blackwell, and In the game where outside information is s L, there is an equilibrium E L in which Receiver is less informed than in E in the sense of Blackwell

22 Better Outside Information can Reduce Total Information - Intuition Another equilibrium E L exists where one Sender switches from full disclosure to not revealing any information and everyone else continues to play there equilibrium strategy Overall this means less information available to the Receiver as she does not learn about type who changed his behaviour

23 Better Outside Information can Reduce Total Information - Intuition How do they get there? In equilibrium E with outside information s the Sender strictly prefers to reveal his type. With probability close to 1, information bundle s L is drawn that perfectly reveals one sender s type. Therefore he expects a payoff close to full disclosure where he saves the cost of disclosure by not revealing any information. Everyone else continues to strictly prefer their equilibrium strategy fromequilibrium E.

24 Disclosures and the shape of payoffs More Setup A complementary analysis of prior results: Given a distribution of outside info, when does a sender have incentive to stay quiet in the equilibrium? Relate receiver s utility to some increasing function X(θ) (X ) = X θ e X(θ ) for the increment of receiver s action if receiver learns that sender s type increases from θ to the next-best type θ e

25 Disclosures and the shape of payoffs Concavity and Convexity Concavity = min ( ) ( h) Convexity = min ( h) ( ) When concavity > 1, payoffs are concave in the sense that marginal value of being perceived as a better type diminishes as sender s type imrpoves When convexity > 1, payoffs are convex in the sense that marginal value of being perceived as a better type increases as sender s type imrpoves

26 Disclosures and the shape of payoffs Downside and Upside Risk We can write payoff into an equation in which net payoff from disclosure is equal to the sum of two components: Downside risk : ce X Q Te qce ; Q =probability of sender s type<θ for j < i Upside risk : T X (1 Q )(Sender s type >θ for j > i) The sender discloses if the downside exceeds the upside more than c

27 Disclosures and the shape of payoffs Proposition 3 Proposition 3: If the concavity of payoffs is sufficiently large, then for any level of disclosure cost c > c _, all equilibria are non-monotonic or fully opaque. Conversely, if convexity is sufficiently large, then there are no non-monotonic equilibria. The strongest type of sender stay quiet in any equilibrium if the payoff is concave enough If payoffs are concave enough, incentives lie on the downside, but Q drops as type improves, incentive becomes weaker, leads to reverse unravelling, nonmonotone or fully opaque equilibrium Analogous in the context of convexity

28 Disclosures and the shape of payoffs Iterated deletion of dominated Define a procedure for iterated deletion of strictly dominated non-monotone disclosure (DNMD): Θ : the set of types staying quiet after n steps of iteration. Θ = {θ e, θ,, θ } and θ is the highest among these types. Θ ce Θ and this set converges to a set Θ Θ is non-empty b/c the worst type has the strictly dominant strategy to stay quiet

29 Disclosures and the shape of payoffs Proposition 4 Proposition 4: If no type θ Θ prefers to disclose when receiver expects the cutoff strategy Pr(m = θ θ) = 1( θ Θ ), then a monotone increasing equilibrium exists. Conversely, if some type θ Θ prefers to disclose when receiver expects the fully opaque strategy Pr(m = θ θ) = 1, then all equilibria are non-monotone A simple test for identifying whether monotone equilibria survive the introduction of outside info Generalize our intuition built up in the concavity argument in proposition 3

30 Applications Financial Panics Question: how much outside info should a policy maker release into a potentially panicked banking system? Key: Hirshleifer effect: opacity provides implicit insurance (the silence of strong strictly improves the utility of weak) Answer: Policy makers should ensure a minimum level of transparency as reducing noise of outside info enhances implicit insurance (efforts strong banks made to distinguish themselves are crowded out)

31 Applications Political Contest A game with two competing senders (candidates) and a mass of receivers (voters) Result: 1) Welfare is a convex function of voters expectation of θ, thus either releasing no info at all or eliminating all noise in the signal. 2) Inside disclosure happens only when θ is small, all landslide wins are associates with no inside info being revealed.

32 Applications Corporate Disclosure A firm wishing to raise funds from investors by issuing bonds or shares Bonds give investor a concave claim Shares give investors a convex claim Result: 1) Disclosure comes from high-quality firms mainly if they are selling shares 2) Disclosure comes from intermediate-quality firms mainly if they are selling bonds 3) Disclosure comes from firms with favorable subsequent realizations of outside info if selling shares 4) Disclosure comes from firms with intermediate signals like mediocre credit ratings if selling bonds

33 Conclusion and Discussion- Conclusion Existence of outside info alters insiders incentive to disclose Reverse unravelling Key result: 1) Fragility of information (amplification effect) 2) Improved outside info could generate discontinuous drop in inside disclosure 3) Tendency of disclosure depends on the shape of sender s disclosure function Key implication - Problem of selling information: Receiver who purchases outside info could end up with worse information overall in equilibrium

34 Conclusion and Discussion- Discussion Natural extension of topic: The role of information market for overall information and market efficiency Timing of model is very good for selected examples: Information by insiders will no longer be of any use of not be perceived very credible once public outside information has been received Various Best Responses by Receivers: There could be multiple Senders who value information differently, e.g. Trump voters who think the New York Times is too liberal and spreading lies anyways, or investors who think the local Central Bank is not independent or has a specific interest in keeping banks that are relevant to the system alive. Thus, not everyone will have same best-response function based on inside and outside information revealed