Persuasion in Global Games with Application to Stress Testing€¦ · Motivation Model PCP Public...
Transcript of Persuasion in Global Games with Application to Stress Testing€¦ · Motivation Model PCP Public...
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 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 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
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?
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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),...
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
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,+•)
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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
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Beliefs
x = (xi
)i2[0,1] 2 X: beliefs/signal profile with each
x
i
⇠ p(·|q)
i.i.d., given q
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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)]
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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
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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 �
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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 �
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
Perfect Coordination Property [PCP]
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
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
Perfect Coordination Property [PCP]
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”
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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)
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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(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
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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)
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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 �.
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Monotone Tests
θ*θ
π*(0|θ)
1
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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)
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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
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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)
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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)
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 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
)
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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) less aggressive than a
�(n)
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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).
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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) less aggressive than a
�(n).
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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).
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, a�i
(xi
) = 1, all xi
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
Example
Suppose PM informs agents of whether q is extreme or intermediatea
�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, � ispass/fail): a
�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
�(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
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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
]
Motivation Model PCP Public Disclosures Monotone Tests Discriminatory Disclosures Conclusions Extra
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
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
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
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(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