Persuasion in Global Games with Application to Stress Testing€¦ · Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra Public Disclosures

Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra

Persuasion in Global Gameswith Application to Stress Testing

Nicolas Inostroza Alessandro Pavan

Northwestern University

May 1, 2018


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 ofnonperforming 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


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?


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), Guoand 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),...


Plan

Model

Perfect coordination property

Public disclosures

Pass/Fail Policies

Monotone rules

Benefits of discriminatory disclosures

Extensions and conclusions


Global Games of Regime Change

Policy maker (PM)

Agents i 2 [0,1]

Actions

a

i

=

(

1 (attack)

0 (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> q

0 if A q

“fundamentals” q parametrize amount of performing loans

Supermodular game w. dominance regions: (�•,0) and [1,+•)


Global Games of Regime Change

PM’s payoff

U

P(q ,A) =

(

W if r = 0L<W if r = 1.

Agents’ payoff from attacking (safe action) normalized to zero

Agents’ payoff from not attacking

u(q ,A) =

(

g if r = 0b if r = 1

withg > 0 > b


Beliefs

x = (xi

)i2[0,1] 2 X: beliefs/signal profile with each

x

i

⇠ p(·|q)

i.i.d., given q


Disclosure Policies

m : [0,1]! S message function

m

i

2 S : information disclosed to i

M(S): set of all possible message functions with range S

Disclosure policy �= (S ,p)

with p :⇥!�(M(S))

Non discriminatory disclosures: m

i

=m

j

all i , j 2 [0,1], q 2⇥,m 2 supp[p(q)]


Timing

1 PM announces �= (S ,p) and commits to it

2 (q ,x) realized

3 m drawn from p(q)2�(M(S))

4 Information m

i

disclosed to agent i 2 [0,1]

5 Agents simultaneously choose whether or not to attack

6 Regime outcome and payoffs


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 deletionof interim strictly dominated strategies (IDISDS)

a

� ⌘ (a�i

)i2[0,1]: MARP consistent with �


Perfect Coordination Property [PCP]

Definition

�= {S ,p} satisfies PCP if, for any (q ,x), any message functionm 2 supp(p(q)), any i , j 2 [0,1], a�

i

(xi

,mi

) = a

�j

(xj

,mj

), wherea

� ⌘ (a�i

)i2[0,1] is MARP consistent with �



Theorem

Given any (regular) �, there exists (regular) �⇤ satisfying PCP andyielding PM a payoff weakly higher than �.

Regularity: regime outcome under MARP measurable wrp PM’sinformation



Policy �⇤ = (S⇤,p⇤) removes any strategic uncertainty

It preserves (and, in some cases, enhances) heterogeneity instructural uncertainty

Under �⇤, agents know actions all other agents take but not whatbeliefs rationalize such actions

Inability to predict beliefs that rationalize other agents’ actionsessential to minimization of risk of regime change

“Right” form of transparency

conformism in beliefs about mkt response...not in beliefs about “fundamentals”


PCP: Proof sketch

Let r(w;a�) 2 {0,1} be regime outcome at w ⌘ (q ,x,m) whenagents play according to a

�

Let �⇤ = {S⇤,p⇤} be s.t. S

⇤ = S⇥{0,1} and

p

⇤((m, r(w;a�))|q) = p(m|q), all (q ,m) s.t. p(m|q)> 0

Key step: r(w;a�) = 0 ) MARP under �⇤ less aggressive thanMARP under �, i.e.,

a

�i

(xi

,mi

) = 0 ) a

�⇤i

(xi

,(mi

,0)) = 0, 8i , 8(xi

,mi

)

Size of attack under �⇤ smaller than under �) r(w;a�⇤) = 0

That �⇤ weakly improves upon � follows fromprobability of regime change under �⇤ same as under � (all q)size of attack when r = 0 smaller under �⇤ (relevant for moregeneral payoffs).

(formal proof)


PCP: Lesson

Optimal policy combines:

public Pass/Fail announcement

eliminate strategic uncertainty

additional (possibly discriminatory) disclosures


PCP – General

Optimality of PCP extends to economies in which

regime outcome determined by more general rule R(q ,A)

PM’s payoff

U

P(q ,A) =

(

W (q ,A) if r = 0L(q) if r = 1

with (a) W

A

(q ,A) 0; (b) W (q ,A)�L(q)� 0 if R(q ,A)> 0finitely many agents with asymmetric payoffs u

i

(q ,A)

arbitrary collection of beliefs

⇤i

(xi

) 2�(⇥⇥X)

level-K sophisticationPM has imperfect information about q and agents’ beliefsKey assumptions:

supermodularity of gamemeasurability of regime outcome under MARP wrt PM’s info


Public Disclosures

Designer constrained to non-discriminatory policiesp:⇥!�(M(S)) s.t. m

i

=m

j

all m 2 supp(p(q)), all q .

Optimality of PCP extends to such environments

Theorem

Suppose p(x |q) log-supermodular. Given any non-discriminatory policy�, there exists binary (non-discriminatory) policy �⇤ = (S⇤,p⇤) inwhich S

⇤ = {0,1} satisfying PCP and yielding higher payoff than �.

Optimal non-discriminatory policy: stochastic pass/fail test

Log-SM of p(x |q) ) MLRP (co-movement between state q andbelies x)MARP in threshold strategies: signals other than regime outcomecan be dropped (averaging over m) without affecting incentives

(Example)


Monotone Tests

Condition M:

1 �P(q)⌘W (q ,0)�L(q) non-decreasing2 b(q ,P(x |q)) and p(x |q) log-SM.3 For any x ,

Y (q ;x)⌘ �P(q)

p(x |q)|b(q ,P(x |q))|

nondecreasing in q over [q , q(x)], with q(x) s.t.R(q(x),P(x |q(x))) = 0

Theorem

Suppose Condition M holds.

Given any non-discriminatory �, there exists monotone

non-discriminatory policy �⇤ = ({0,1},p⇤) yielding higher payoff than �.


Monotone Tests

θ*θ

π*(0|θ)

1


Monotone Tests

Optimality of monotone policies not guaranteed even in “canonical”case where W ,L,b,g constant and x

i

= q +se

i

with e

i

⇠ N(0,1)Counter-example

Single receiver (same prior as PM): supermodularity of payoffssuffices (here: monotonicity of �P(q))(Mensch, 2016)


Benefits of Discriminatory Disclosures

In general, optimal stress tests involve

public pass/fail announcementsdiscriminatory disclosures (DD)

Benefits of DD: Divide-and-Conquer

some agents find it dominant not to attackfraction of agents for whom not attacking dominant not CKiteratively dominant for all not to attack (when s = 0)

DD can outperform non-discriminatory ones even when prior beliefsare homogenous


Optimality of non discrimination

Conditions for optimal policy to be non discriminatory

upside risk “dominating” downside risk

sensitivity of payoffs to q 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 ofperforming loans

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

Less precise private info ! mean-preserving-spread in cross-sectionof beliefs ! smaller attack

(NDD)


Extensions and Conclusions

Information design in coordination games with heterogeneouslyinformed agents

Application: Stress Test Design

Perfect coordination property (“right” notion of transparency)Pass/Fail testsmonotone rulesbenefits 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)


THANKS!


PCP Proof

After receiving m

+i

⌘ (mi

,0), agents use Bayes’ rule to updatebeliefs about w ⌘ (q ,x,m):

∂⇤�+i

(w|xi

,(mi

,0)) =1{r(w;a�) = 0}

p

�i

(0|xi

,mi

)∂⇤�

i

(w|xi

,mi

)

wherep

�i

(0|xi

,mi

)⌘Z

{w:r(w;a�)=0}d⇤�

i

(w|xi

,mi

)


PCP Proof

Let a�(n), a�+

(n) be most aggressive profile surviving n round of IDISDSunder � and �+, respectively.

Definition

Strategy profile a

�+

(n) less aggressive than a

�(n) iff, for any i 2 [0,1],

a

�(n),i (xi ,mi

) = 0 ) a

�+

(n),i (xi ,(mi

,0)) = 0

Lemma

For any n, a�+


�(n)


PCP Proof

Proof by induction

Let a�0 = a

�+0 be strategy profile where all agents attack regardless

of their (endogenous and exogenous) information

Suppose that a�+

(n�1) less aggressive than a

�(n�1)

Note that r(w|a�) = 1 ) r(w|a�(n�1)) = 1(a�(n�1) more aggressive than a

� = a

�•)

Hence, r(w;a�) = 0 “removes” from support of agents’ beliefs states(q ,x,m) for which regime change occurs under a�(n�1).


PCP Proof

Becausepayoffs in case of regime change are negativer(w;a�) = 0 removes from support of agents’s beliefs states atwhich regime change occurs also under a�(n�1)

payoff from not attacking under �+ when agents follow a

�(n�1)

U

�+i

(xi

,(mi

,0);a�(n�1)) =R

w

u(q ,A(w;a�(n�1)))1{r(w;a�)=0}d⇤�i

(w|xi

,mi

)

p

�i

(0|xi

,mi

)

>R

w

u(q ,A(w;a�(n�1)))d⇤�i

(w|xi

,mi

)

p

�i

(0|xi

,mi

)

=U

�i

(xi

,mi

;a�(n�1))

p

�i

(0|xi

,mi

)

Hence, U�i

(xi

,mi

;a�(n�1))> 0 ) U

�+i

(xi

,(mi

,0);a�(n�1))> 0


PCP Proof

That a�+

(n�1) less aggressive than a

�(n�1) along with supermodularity

of game implies that

U

�+i

(xi

,(mi

,0);a�(n�1))> 0 ) U

�+i

(xi

,(mi

,0);a�+

(n�1))> 0

As a consequence,

a

�(n),i (xi ,mi

) = 0 ) a

�+

(n),i (xi ,(mi

,0)) = 0

This means that a�+


�(n).


PCP Proof

Above lemma implies MARP under �+, a�+ ⌘ a

�+

(•), less aggressivethan MARP under �, a� ⌘ a

�(•)

In turn, this implies that r(w;a�) = 0 makes it common certaintythat r(w;a�

+) = 0

Hence, no agent attacks after hearing r(w;a�) = 0

Similarly, r(w;a�) = 1 makes it common certainty that q < 1.Under MARP, all agents attack when hearing that r(w;a�) = 1

That �+ weakly improves upon � follows from

probability of regime change under �+ same as under � (all q)size of attack when r = 0 smaller under �+

(relevant for more general payoffs).


Example

Assume g =�b

Attacking rationalizable iff Pr(r = 1)� 1/2


Example

No disclosure: under MARP, a�i

(xi

) = 1, all xi


Example

Suppose PM informs agents of whether q is extreme or intermediatea

�i

(xi

,s) = 0, all (xi

,s)


Example

If, instead, PM only recommend not to attack (equivalently, � ispass/fail): a

�i

(xi

,0) = 1 for all xi

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


Counter-example

Suppose

�(q) =W �L

x

i

= q +se

i

, with e

i

⇠ N (0,1)g = 1� c and b =�c with c > 1/2

Regime change occurs iff q < q

#s

Marginal agent xs

= q

#s

+s��1(q#s

)

lim

s!0+q

#s

= c = q

MS > 1/2Hence, x

s

> q

#s

> q

infs

for small s , where q

infs

is regime thresholdunder best mon. policy.Therefore

Y (q ;xs

)⌘ �P(q)

p(xs

|q)|b(q ,P(xs

|q))| =W �L

f( xs

�q

s

) · c

strictly decreasing over [q ,q infs

], which implies best mon policy

can be improved upon by non-monotone policy


Optimality of NDD

Suppose PM can engineer any public disclosure but constrained toGaussian private communications

F improper uniform over Rexogenous signals x

i

= q +s

h

h

i

, with h

i

⇠ N (0,1)

e

m

i

= q +s

x

x

i

, with x

i

⇠ N (0,1)

agents’ payoffs depend only on q : g(q) and b(q)

Info disclosed to i : m

i

= (s, emi

)

Information contained in (xi

, emi

) summarized by

z

i

⌘s

2x

x

i

+s

2h

m

i

s

2h

+s

2x

PM’s choice of discriminatory part of her policy parametrized bys

z

2 (0,sh

]


Optimality of NDD

MARP a

�i

(xi

,(s,mi

)) = 1{zi

z(s)}

PCP: z(0) =�•, and z(1) = +•.

Letz

⇤s

z

(q) = q +s

z

��1(q),

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

⇤s

z

(q), regime change occurs iff q q

Lety(q , q ,s

z

) = U(z⇤s

z

(q),{q > q}|z⇤s

z

(q))

Defineq

inf

s

z

⌘ infn

q : y(q , q ,sz

)> 0all q

o

.

For any q > q

inf

s

z

, unique rationalizable profile has no agentattacking after s publicly reveals that q � q


Optimality of NDD


Optimality of NDD

Proposition

Lets

⇤z

⌘ argmin

s

z

2(0,sh

]qinf

s

z

Optimal (Gaussian) policy combines public disclosure of whether or notq < q

inf

s

⇤z

, with Gaussian private messages of precision

s

�2x

= [s2h

� (s⇤z

)2]/(s⇤z

)2s

2h

Precision s

�2x

guarantees that sufficient statistics has precision 1/s

⇤z


Optimality of NDD


Optimality of NDD


Optimality of NDD


Optimality of NDD

Suppose agents’ payoffs given by b(q) and g(q)

Let q

#s

z

and z

#s

z

denote regime threshold and “marginal agent” underoptimal policy when information has precision s

�2z

.

Proposition

Suppose that, for any s

z

2 [0,sh

],

E[g 0(q)(q �q

#s

z

)|z#s

z

,q � q

#s

z

]

E[g(q)|z#s

z

,q � q

#s

z

]>

E[b0(q)(q �q

#s

z

)|z#s

z

,q 2 (q inf

s

z

,q#s

z

)]

E[|b(q)||z#s

z

,q 2 (q inf

s

z

,q#s

z

)]

Optimal (Gaussian) policy is non-discriminatory.

Condition says upside risk “dominates” downside risk

Documents

Persuasion in Global Games with Application to Stress Testing€¦ · Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra Public Disclosures