Persuasion in Global Games with Application to Stress Testing - - PowerPoint PPT Presentation

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Persuasion in Global Games with Application to Stress Testing

Nicolas Inostroza Alessandro Pavan

Northwestern University

May 1, 2018

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Motivation

Coordination: central to many socio-economic environments Damages to society of mis-coordination can be severe Monte dei Paschi di Siena (MPS) creditors with heterogenous beliefs about size of nonperforming loans default by MPS: major crisis in Eurozone (and beyond) Government intervention limited by legislation passed in 2015 Persuasion (stress test design): instrument of last resort

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Questions

Structure of optimal stress tests? What information should be passed on to mkt? “Right” notion of transparency? Optimality of pass/fail policies monotone rules Benefits to discriminatory disclosures? Properties of persuasion in global games?

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Related literature

Persuasion and Information design: Myerson (1986), Aumann and Maschler (1995), Calzolari and Pavan (2006,a,b), Glazer and Rubinstein (2004, 2012), Rayo and Segal (2010), Kamenica and Gentzkow (2011), Ely (2016), Bergemann and Morris (2017), Lipnowski and Mathevet (2017), Mathevet, Pearce, Stacchetti (2017),... Persuasion in Games: Alonso and Camara (2013), Barhi and Guo (2016), Taneva (2016), Mathevet, Perego, Taneva (2017)... Persuasion with ex-ante heterogenous receivers: Bergemann and Morris (2016), Kolotilin et al (2016), Laclau and Renou (2017), Chan et al (2016), Basak and Zhou (2017), Che and Horner (2017), Doval and Ely (2017), Guo and Shmaya (2017)... Discrimination and “Divide and Conquer”: Segal (2006), Wang (2015), Yamashita (2016)... Financial Regulation and Stress Test Design: Goldstein and Leitner (2015), Goldstein and Sapra (2014), Alvarez and Barlevy (2017), Bouvard et al. (2015), Goldstein and Huang (2016), Williams (2017),... Global Games w. Endogenous Info: Angeletos, Hellwig and Pavan (2006, 2007), Angeletos and Pavan (2013), Edmond (2013), Iachan and Nenov (2015), Denti (2016), Yang (2016), Morris and Yang (2017),...

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Plan

Model Perfect coordination property Public disclosures

Pass/Fail Policies Monotone rules

Benefits of discriminatory disclosures Extensions and conclusions

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Global Games of Regime Change

Policy maker (PM) Agents i 2 [0,1] Actions ai = ( 1 (attack) (not attack)

A 2 [0,1] : aggregate attack

Regime outcome: r 2 {0,1}, with r = 1 in case of regime change (e.g., default) Regime rule r = ( 1 if A > θ if A  θ “fundamentals” θ parametrize amount of performing loans Supermodular game w. dominance regions: (∞,0) and [1,+∞)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Global Games of Regime Change

PM’s payoff UP(θ,A) = ( W if r = 0 L < W if r = 1. Agents’ payoff from attacking (safe action) normalized to zero Agents’ payoff from not attacking u(θ,A) = ( g if r = 0 b if r = 1 with g > 0 > b

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Beliefs

x = (xi)i2[0,1] 2 X: beliefs/signal profile with each ˜ xi ⇠ p(·|θ) i.i.d., given θ

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Disclosure Policies

m : [0,1] ! S message function mi 2 S: information disclosed to i M(S): set of all possible message functions with range S Disclosure policy Γ = (S,π) with π : Θ ! ∆(M(S)) Non discriminatory disclosures: mi = mj all i,j 2 [0,1], θ 2 Θ, m 2 supp[π(θ)]

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Timing

1 PM announces Γ = (S,π) and commits to it 2 (θ,x) realized 3 m drawn from π(θ)2 ∆(M(S)) 4 Information mi disclosed to agent i 2 [0,1] 5 Agents simultaneously choose whether or not to attack 6 Regime outcome and payoffs

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Solution Concept: MARP

Robust/adversarial approach PM can not select agents’ strategy profile Most Aggressive Rationalizable Profile (MARP): minimizes PM’s payoff across all profiles surviving iterated deletion

• f interim strictly dominated strategies (IDISDS)

aΓ ⌘ (aΓ

i )i2[0,1]: MARP consistent with Γ

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Perfect Coordination Property [PCP]

Definition

Γ = {S,π} satisfies PCP if, for any (θ,x), any message function m 2 supp(π(θ)), any i,j 2 [0,1], aΓ

i (xi,mi) = aΓ j (xj,mj), where

aΓ ⌘ (aΓ

i )i2[0,1] is MARP consistent with Γ

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Perfect Coordination Property [PCP]

Theorem

Given any (regular) Γ, there exists (regular) Γ⇤ satisfying PCP and yielding PM a payoff weakly higher than Γ. Regularity: regime outcome under MARP measurable wrp PM’s information

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Perfect Coordination Property [PCP]

Policy Γ⇤ = (S⇤,π⇤) removes any strategic uncertainty It preserves (and, in some cases, enhances) heterogeneity in structural uncertainty Under Γ⇤, agents know actions all other agents take but not what beliefs rationalize such actions Inability to predict beliefs that rationalize other agents’ actions essential to minimization of risk of regime change “Right” form of transparency conformism in beliefs about mkt response ...not in beliefs about “fundamentals”

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP: Proof sketch

Let r(ω;aΓ) 2 {0,1} be regime outcome at ω ⌘ (θ,x,m) when agents play according to aΓ Let Γ⇤ = {S⇤,π⇤} be s.t. S⇤ = S ⇥{0,1} and π⇤((m,r(ω;aΓ))|θ) = π(m|θ), all (θ,m) s.t. π(m|θ) > 0 Key step: r(ω;aΓ) = 0 ) MARP under Γ⇤ less aggressive than MARP under Γ, i.e., aΓ

i (xi,mi) = 0 ) aΓ⇤ i (xi,(mi,0)) = 0,

8i, 8(xi,mi) Size of attack under Γ⇤ smaller than under Γ ) r(ω;aΓ⇤) = 0 That Γ⇤ weakly improves upon Γ follows from probability of regime change under Γ⇤ same as under Γ (all θ) size of attack when r = 0 smaller under Γ⇤ (relevant for more general payoffs).

(formal proof)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP: Lesson

Optimal policy combines: public Pass/Fail announcement eliminate strategic uncertainty additional (possibly discriminatory) disclosures

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP – General

Optimality of PCP extends to economies in which regime outcome determined by more general rule R(θ,A) PM’s payoff UP(θ,A) = ( W (θ,A) if r = 0 L(θ) if r = 1 with (a) WA(θ,A)  0; (b) W (θ,A)L(θ) 0 if R(θ,A) > 0 finitely many agents with asymmetric payoffs ui(θ,A) arbitrary collection of beliefs Λi(xi) 2 ∆(Θ⇥X) level-K sophistication PM has imperfect information about θ and agents’ beliefs Key assumptions: supermodularity of game measurability of regime outcome under MARP wrt PM’s info

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Public Disclosures

Designer constrained to non-discriminatory policies π:Θ ! ∆(M(S)) s.t. mi = mj all m 2 supp(π(θ)), all θ. Optimality of PCP extends to such environments

Theorem

Suppose p(x|θ) log-supermodular. Given any non-discriminatory policy Γ, there exists binary (non-discriminatory) policy Γ⇤ = (S⇤,π⇤) in which S⇤ = {0,1} satisfying PCP and yielding higher payoff than Γ. Optimal non-discriminatory policy: stochastic pass/fail test Log-SM of p(x|θ) ) MLRP (co-movement between state θ and belies x) MARP in threshold strategies: signals other than regime outcome can be dropped (averaging over m) without affecting incentives

(Example)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Monotone Tests

Condition M:

1 ∆P(θ) ⌘ W (θ,0)L(θ) non-decreasing 2 b(θ,P(x|θ)) and p(x|θ) log-SM. 3 For any x,

Y (θ;x) ⌘ ∆P(θ) p(x|θ)|b(θ,P(x|θ))| nondecreasing in θ over [θ, ˆ θ(x)], with ˆ θ(x) s.t. R(ˆ θ(x),P(x|ˆ θ(x))) = 0

Theorem

Suppose Condition M holds. Given any non-discriminatory Γ, there exists monotone non-discriminatory policy Γ⇤ = ({0,1},π⇤) yielding higher payoff than Γ.

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Monotone Tests

θ* θ π*(0|θ) 1

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Monotone Tests

Optimality of monotone policies not guaranteed even in “canonical” case where W ,L,b,g constant and xi = θ +σεi with εi ⇠ N(0,1) Counter-example Single receiver (same prior as PM): supermodularity of payoffs suffices (here: monotonicity of ∆P(θ)) (Mensch, 2016)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Benefits of Discriminatory Disclosures

In general, optimal stress tests involve public pass/fail announcements discriminatory disclosures (DD) Benefits of DD: Divide-and-Conquer some agents find it dominant not to attack fraction of agents for whom not attacking dominant not CK iteratively dominant for all not to attack (when s = 0) DD can outperform non-discriminatory ones even when prior beliefs are homogenous

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of non discrimination

Conditions for optimal policy to be non discriminatory upside risk “dominating” downside risk sensitivity of payoffs to θ higher when r = 0 than when r = 1 Condition holds, e.g., under equity claims (junior/subordinated)

• under default, liquidation value little sensitive to amount of

performing loans

• when bank survives, value of claims reflects long-term profitability

Less precise private info ! mean-preserving-spread in cross-section

• f beliefs ! smaller attack

(NDD)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Extensions and Conclusions

Information design in coordination games with heterogeneously informed agents Application: Stress Test Design Perfect coordination property (“right” notion of transparency) Pass/Fail tests monotone rules benefits to discrimination (type of securities) Extension 1: PM uncertain about prior beliefs robust-undominated design Extension 2: timing of optimal disclosures Extension 3: Screening of banks’ balance sheets (MD+persuasion) (Calzolari Pavan, 2006, Dworczak, 2017)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

THANKS!

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP Proof

After receiving m+

i ⌘ (mi,0), agents use Bayes’ rule to update

beliefs about ω ⌘ (θ,x,m): ∂ΛΓ+

i

(ω|xi,(mi,0)) = 1{r(ω;aΓ) = 0} πΓ

i (0|xi,mi)

∂ΛΓ

i (ω|xi,mi)

where πΓ

i (0|xi,mi) ⌘

Z

{ω:r(ω;aΓ)=0} dΛΓ i (ω|xi,mi)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP Proof

Let aΓ

(n), aΓ+ (n) be most aggressive profile surviving n round of IDISDS

under Γ and Γ+, respectively.

Definition

Strategy profile aΓ+

(n) less aggressive than aΓ (n) iff, for any i 2 [0,1],

aΓ

(n),i(xi,mi) = 0 ) aΓ+ (n),i(xi,(mi,0)) = 0

Lemma

For any n, aΓ+

(n) less aggressive than aΓ (n)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP Proof

Proof by induction Let aΓ

0 = aΓ+

be strategy profile where all agents attack regardless

• f their (endogenous and exogenous) information

Suppose that aΓ+

(n1) less aggressive than aΓ (n1)

Note that r(ω|aΓ) = 1 ) r(ω|aΓ

(n1)) = 1

(aΓ

(n1) more aggressive than aΓ = aΓ ∞)

Hence, r(ω;aΓ) = 0 “removes” from support of agents’ beliefs states (θ,x,m) for which regime change occurs under aΓ

(n1).

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP Proof

Because payoffs in case of regime change are negative r(ω;aΓ) = 0 removes from support of agents’s beliefs states at which regime change occurs also under aΓ

(n1)

payoff from not attacking under Γ+ when agents follow aΓ

(n1)

UΓ+

i

(xi,(mi,0);aΓ

(n1)) = R

ω u(θ,A(ω;aΓ (n1)))1{r(ω;aΓ)=0}dΛΓ i (ω|xi,mi )

πΓ

i (0|xi,mi )

>

R

ω u(θ,A(ω;aΓ (n1)))dΛΓ i (ω|xi,mi )

πΓ

i (0|xi,mi )

=

UΓ

i (xi,mi ;aΓ (n1))

πΓ

i (0|xi,mi )

Hence, UΓ

i (xi,mi;aΓ (n1)) > 0 ) UΓ+ i

(xi,(mi,0);aΓ

(n1)) > 0

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP Proof

That aΓ+

(n1) less aggressive than aΓ (n1) along with supermodularity

• f game implies that

UΓ+

i

(xi,(mi,0);aΓ

(n1)) > 0 ) UΓ+ i

(xi,(mi,0);aΓ+

(n1)) > 0

As a consequence, aΓ

(n),i(xi,mi) = 0 ) aΓ+ (n),i(xi,(mi,0)) = 0

This means that aΓ+

(n) less aggressive than aΓ (n).

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

PCP Proof

Above lemma implies MARP under Γ+, aΓ+ ⌘ aΓ+

(∞), less aggressive

than MARP under Γ, aΓ ⌘ aΓ

(∞)

In turn, this implies that r(ω;aΓ) = 0 makes it common certainty that r(ω;aΓ+) = 0 Hence, no agent attacks after hearing r(ω;aΓ) = 0 Similarly, r(ω;aΓ) = 1 makes it common certainty that θ < 1. Under MARP, all agents attack when hearing that r(ω;aΓ) = 1 That Γ+ weakly improves upon Γ follows from probability of regime change under Γ+ same as under Γ (all θ) size of attack when r = 0 smaller under Γ+ (relevant for more general payoffs).

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Example

Assume g = b Attacking rationalizable iff Pr(r = 1) 1/2

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Example

No disclosure: under MARP, aΓ

i (xi) = 1, all xi

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Example

Suppose PM informs agents of whether θ is extreme or intermediate aΓ

i (xi,s) = 0, all (xi,s)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Example

If, instead, PM only recommend not to attack (equivalently, Γ is pass/fail): aΓ

i (xi,0) = 1 for all xi

Suboptimality of P/F policies (+ failure of RP)

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Counter-example

Suppose ∆(θ) = W L xi = θ +σεi, with εi ⇠ N (0,1) g = 1c and b = c with c > 1/2 Regime change occurs iff θ < θ #

σ

Marginal agent xσ = θ #

σ +σΦ1(θ # σ )

limσ!0+θ #

σ = c = θ MS > 1/2

Hence, xσ > θ #

σ > θ inf σ

for small σ, where θ inf

σ

is regime threshold under best mon. policy. Therefore Y (θ;xσ) ⌘ ∆P(θ) p(xσ|θ)|b(θ,P(xσ|θ))| = W L φ( xσ θ

σ

)·c strictly decreasing over [θ,θ inf

σ ], which implies best mon policy

can be improved upon by non-monotone policy

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of NDD

Suppose PM can engineer any public disclosure but constrained to Gaussian private communications F improper uniform over R exogenous signals xi = θ +σηηi, with ηi ⇠ N (0,1) e mi = θ +σξξi, with ξi ⇠ N (0,1) agents’ payoffs depend only on θ : ¯ g(θ) and ¯ b(θ) Info disclosed to i: mi = (s, e mi) Information contained in (xi, e mi) summarized by zi ⌘ σ 2

ξ xi +σ 2 η ˜

mi σ 2

η +σ 2 ξ

PM’s choice of discriminatory part of her policy parametrized by σz 2 (0,ση]

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of NDD

MARP aΓ

i (xi,(s, ˜

mi)) = 1{zi  ¯ z(s)} PCP: ¯ z(0) = ∞, and ¯ z(1) = +∞. Let z⇤

σz (θ) = θ +σzΦ1(θ),

denote “marginal agent” s.t., when agents follow cut-off strategies with cut-off z⇤

σz (θ), regime change occurs iff ˜

θ  θ Let ψ(θ, ˆ θ,σz) = U(z⇤

σz (θ),{˜

θ > ˆ θ}|z⇤

σz (θ))

Define θ inf

σz ⌘ inf

n ˆ θ : ψ(θ, ˆ θ,σz) > 0all θ

• .

For any ˆ θ > θ inf

σz , unique rationalizable profile has no agent

attacking after s publicly reveals that θ ˆ θ

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of NDD

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of NDD

Proposition

Let σ⇤

z ⌘ argminσz2(0,ση]θ inf σz

Optimal (Gaussian) policy combines public disclosure of whether or not θ < θ inf

σ⇤

z , with Gaussian private messages of precision

σ2

ξ

= [σ 2

η (σ⇤ z )2]/(σ⇤ z )2σ 2 η

Precision σ2

ξ

guarantees that sufficient statistics has precision 1/σ⇤

z

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of NDD

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of NDD

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of NDD

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Optimality of NDD

Suppose agents’ payoffs given by ¯ b(θ) and ¯ g(θ) Let θ #

σz and z# σz denote regime threshold and “marginal agent” under

• ptimal policy when information has precision σ2

z .

Proposition

Suppose that, for any σz 2 [0,ση], E[¯ g0(θ)(θ θ #

σz )|z# σz ,θ θ # σz ]

E[¯ g(θ)|z#

σz ,θ θ # σz ]

> E[¯ b0(θ)(θ θ #

σz )|z# σz ,θ 2 (θ inf σz ,θ # σz )]

E[|¯ b(θ)||z#

σz ,θ 2 (θ inf σz ,θ # σz )]

Optimal (Gaussian) policy is non-discriminatory. Condition says upside risk “dominates” downside risk