Persuasion in Global Games with Application to Stress Testing

Transcription

1 Persuasion in Global Games with Application to Stress Testing Nicolas Inostroza Alessandro Pavan Northwestern University July 22, 2017

2 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

3 Questions Structure of optimal stress tests? Should all information collected be passed on to mkt? Should government be specific about recapitalization asked to the bank? Benefits to discriminatory disclosures? Optimality of simple policies (e.g., monotone deterministic pass/fail tests)? Properties of persuasion in global games?

4 Related literature Information design: Myerson (1986), Aumann and Maschler (1995), Calzolari and Pavan (2006,a,b), Kamenica and Gentzkow (2011),... Ely (2016)...Bergemann and Morris (2017)... Persuasion in Games: Alonso and Camara (2013), Barhi and Guo (2016), Mathevet, Perego, Taneva (2016), Taneva (2016)...,Doval and Ely (2017)... Persuasion with ex-ante heterogenous receivers: Bergemann and Morris (2016), Kolotilin et al (2016), Laclau and Renou (2017), Chan et al (2016), 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), Bouvard et al. (2015), Goldstein and Huang (2016)... Global Games w. Endogenous Info: Angeletos, Hellwig and Pavan (2006, 2007), Angeletos and Pavan (2013), Edmond (2013), Iachan and Nenov (2015), Denti (2016),...

5 Plan Model Perfect coordination property Public disclosures Monotone disclosures Discriminatory disclosures Extensions and conclusions

6 Global Games of Regime Change Policy maker (PM) Agents i [0,1] Actions a i = { 1 (attack) 0 (not attack) A [0,1] : aggregate attack Regime outcome: r {0,1}, with r = 1 in case of regime change (e.g., default) Regime rule { 1 if A > θ r = 0 if A θ fundamentals θ parametrize amount of performing loans Dominance regions: (, 0) and [1, + )

7 Global Games of Regime Change PM s payoff U P (θ,a) = { W if r = 0 L < W if r = 1. Agents payoff from attacking normalized to zero Agents payoff from not attacking { g if r = 0 u(θ,a) = b if r = 1 with g > 0 > b

8 Beliefs x = {x i } i [0,1] X : beliefs/signal profile with each i.i.d., given θ. x i p( θ)

9 Disclosure Policies m : [0,1] S message function m i 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))

10 Timing 1 PM announces Γ = (S,π) and commits to it 2 (θ,x) realized 3 m drawn from π(θ) (M(S)) 4 Agents receive information m i 5 Agents simultaneously choose whether or not to attack 6 Regime outcome and payoffs

11 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 of interim strictly dominated strategies (IDISDS) a Γ (a Γ i ) i [0,1]: MARP consistent with Γ

12 Perfect Coordination Property [PCP] Definition Γ = {S,π} satisfies PCP if, for any (θ,x), any message function m supp(π(θ)), any i,j [0,1], a Γ i (x i,m i ) = a Γ j (x j,m j ), where a Γ (a Γ i ) i [0,1] is MARP consistent with Γ

13 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 (here θ)

14 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 what all other agents do but not why they do what they do Inability to predict specific beliefs that rationalize other agents actions essential to minimization of risk of regime change However, no value from mis-coordinating the mkt

15 PCP: Proof sketch Let r(θ,m;a) {0,1} be regime outcome at (θ,m) when agents play according to a Let Γ + = {S +,π + } be s.t. S + = S {0,1} and π + ((m,r(θ,m;a Γ )) θ) = π(m θ), all (θ,m) s.t. π(m θ) > 0 Key step: MARP under Γ + less aggressive than MARP under Γ +, i.e., a Γ i (x i,m i ) = 0 a Γ+ i (x i,(m i,0)) = 0, i, (x i,m i ) 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 general payoffs). (formal proof)

16 PCP General Optimality of PCP extends to economies in which regime outcome determined by more general rule R(θ, A) PM s payoff { U P W (θ,a) if r = 0 (θ,a) = L(θ) if r = 1 with (a) W A (θ,a) 0; (b) W (θ,a) L(θ,A) > 0 if R(θ,A) > 0 finitely many agents with asymmetric (but supermodular) payoffs u i (θ,a) arbitrary beliefs Λ i (x i ) (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 of any feasible Γ wrt PM s info

17 Public Disclosures Designer constrained to policies π:θ (M(S)) s.t. m i = m j all m supp(π(θ)), all θ. Theorem Suppose p(x θ) is log-supermodular. Gven 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) guarantees MARP is in threshold strategies: signals other than regime outcome can be dropped without affecting incentives (Example)

18 Monotone Disclosures Let P (θ) W (θ,0) L(θ) Condition M: b(θ,p(x θ)) and p(x θ) log-sm. Furthermore, for any x, P (θ) Y (θ;x) p(x θ) b(θ,p(x θ)) nondecreasing in θ over [θ, ˆθ(x)], with ˆθ(x) s.t. R(ˆθ(x),P(x ˆθ(x))) = 0 Theorem Suppose P (θ) nondecreasing and Condition M holds. Given any non-discriminatory Γ, there exists monotone non-discriminatory policy Γ = ({0,1},π ) yielding higher payoff than Γ.

19 Monotone Disclosures

20 Monotone Disclosures Optimality of monotone policies not guaranteed even in standard case where W,L,b,g constant and x i = θ + σε i with ε i N(0,1) (contrast to: Goldstein and Huang, 2016) (Counter-example) Single receiver (same prior as PM): supermodularity of payoffs suffices (here: monotonicity of P (θ)) (Mensch, 2016) Multiple receivers with heterogeneous info: Condition M guarantees swapping prob. regime change from high to low θ improves PM s payoff while preserving agents incentives Difference wrt literature on (noisy) signaling in global games (Angeletos, Hellwig, Pavan (2007), Edmond(2013)).

21 Benefits of Discriminatory Disclosures In general, optimal stress test involves public pass/fail announcement discriminatory disclosures (DD) Benefits of DD do NOT come from possibility of tailoring information to agents prior beliefs Suppose agents exogenous information is homogenous For any ˆθ s.t. all agents attack under MARP u(θ,1)df (θ θ > ˆθ) 0, Suppose PM sends to each i private signal m i = θ + σξ i, with ξ i drawn from smooth distribution over R (e.g., standard Normal), independently across i, and independent from θ In addition PM announces whether θ is above or below cut-off ˆθ

22 Benefits of Discriminatory Disclosures Let θ MS be regime threshold under Laplacian actions, defined by 1 u(θ MS,l)dl = 0 and ˆθ = inf{ˆθ R s.t. 0 u(θ,1)df (θ θ > ˆθ) > 0} regime threshold under optimal non-discriminatory policy. Proposition Assume no heterogeneity in exogenous structural uncertainty. If θ MS < ˆθ, then DD strictly improve upon NDD 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)

23 Optimality of non discrimination Conditions for optimal policy to be non discriminatory: upside risk dominating downside risk Sensitivity of payoffs to θ higher when regime survives than when it goes down Condition holds, e.g., when investors are equity holders (junior/subordinated claims) - 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 of beliefs smaller attack (NDD)

24 Extensions and Conclusions Information design in coordination settings Applications Extension 1: PM uncertain about prior beliefs (robust-undominated design) Extension 2: timing of optimal disclosures Extension 3: Screening of banks balance sheets

25 THANKS!

26 PCP Proof After receiving m i + (m i,0), agent uses Bayes rule to update beliefs about (θ,m): i (θ,m x i,(m i,0)) = 1{r(θ,m;aΓ ) = 0} πi Γ(0 φ Λ Γ i i,m i ) (θ,m x i,m i ) Λ Γ+ where πi Γ (0 x i,m i ) dλ Γ {(θ,m):r(θ,m;a Γ i (θ,m x i,m i ) )=0}

27 PCP Proof Let a(n) Γ, aγ+ (n) be most aggressive profile surviving n round of IDISDS under Γ and Γ +, respectively. Definition Strategy profile a(n) Γ+ less aggressive than aγ (n) iff, for any i [0,1], a Γ (n),i (x i,m i ) = 0 a Γ+ (n),i (x i,(m i,0)) = 0 Lemma For any n, a Γ+ (n) less aggressive than aγ (n)

28 PCP Proof Proof by induction Let a Γ 0 = aγ+ 0 be strategy profile where all agents attack regardless of their (endogenous and exogenous) information Suppose that a Γ+ (n 1) is less aggressive than aγ (n 1) Note that r(θ,m a Γ ) = 1 r(θ,m a Γ (n 1) ) = 1 (a Γ (n 1) less aggressive than aγ = a Γ ) Hence, r(θ,m;a Γ ) = 0 removes from support of agents beliefs states (θ,m) for which regime change occurs under a Γ (n 1).

29 PCP Proof Because payoffs in case of regime change are negative r(θ,m;a Γ ) = 0 removes from support of agents s beliefs states at which regime change occurs also under a Γ (n 1) payoff from not attacking under Γ + when agents follow a Γ (n 1) Ui Γ+ (x i,(m i,0);a(n 1) Γ ) = (ω,m) u(θ,a(θ,m;aγ (n 1) ))1{r(θ,m;aΓ )=0}dΛ Γ i (θ,m x i,m i ) πi Γ(0 φ i,m i ) > = (ω,m) u(θ,a(θ,m;aγ (n 1) ))dλγ i ((θ,m) x i,m i ) π Γ i (0 φ i,m i ) U Γ i (x i,m i ;a Γ (n 1) ) π Γ i (0 φ i,m i ) Hence, U Γ i (x i,m i ;a Γ (n 1) ) > 0 UΓ+ i (x i,(m i,0);a Γ (n 1) ) > 0

30 PCP Proof That a(n 1) Γ+ is less aggressive than aγ (n 1) along with supermodularity of the game implies that Ui Γ+ (x i,m i ;a(n 1) Γ ) > 0 UΓ+ i (x i,(m i,0);a(n 1) Γ+ ) > 0 As a consequence, a Γ (n),i (x i,m i ) = 0 a Γ (n),i (x i,(m i,0)) = 0 which implies that a Γ+ (n) is less aggressive than aγ (n).

31 PCP Proof Above lemma implies MARP under Γ +, a Γ+ a( ) Γ+, is less aggressive than MARP under Γ, a Γ ā( ) Γ In turn, this implies that r(θ,m;a Γ ) = 0 makes it common certainty that r(θ,m;a Γ+ ) = 0 Hence, no agent attacks after hearing r(θ,m;a Γ ) = 0 Similarly, r(θ,m;a Γ ) = 1 makes it common certainty that θ < 1. Hence, under MARP, all agents attack when hearing that r(θ,m;a Γ ) = 1 That Γ + weakly improves upon Γ then 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).

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

33 Example No disclosure implies that, under MARP, a Γ i (x i) = 1, all x i

34 Example Suppose agents are informed whether θ is extreme or intermediate a Γ i (x i,s) = 0, all (x i,s)

35 Example Now suppose Γ only recommends not to attack. Then, a Γ i (x i,0) = 1 for all x i Suboptimality of P/F policies (+ failure of RP)

36 Counter-example Suppose (θ) = W L x i = θ + σε i, with ε i N (0,1) g = 1 c and b = c with c > 1/2 Observe that θ MS = c Regime change occurs iff θ < θ σ # Marginal agent x σ = θ σ # + σφ 1 (θ σ # ) lim σ 0 θ σ # = θ MS > 1/2 Hence, x σ > θ σ # > θσ inf for small σ, where θσ inf is regime threshold under best mon. policy. Therefore Y (θ;x σ ) P (θ) p(x σ θ) b(θ,p(x σ θ)) = W L φ( x σ θ σ ) c strictly decreasing over [θ,θσ inf ], which implies best mon policy can be improved upon by a non-monotone policy

37 Optimality of NDD Suppose PM can engineer any public disclosure but constrained to Gaussian private communications F improper uniform over R exogenous signals x i = θ + σ η η i, with η i N (0,1) m i = θ + σ ξ ξ i, with ξ i N (0,1) agents payoffs depend only on θ : ḡ(θ) and b(θ) Info disclosed to i: m i = (s, m i ) Information contained in (x i, m i ) summarized by z i σ 2 ξ x i + σ 2 η m i σ 2 η + σ 2 ξ PM s choice of discriminatory part of her policy parametrized by σ z (0,σ η ]

38 Optimality of NDD MARP a Γ i (x i,(s, m i )) = 1{z i z(s)} PCP: z(0) =, and z(1) = +. Let z σ z (θ) = θ + σ z Φ 1 (θ), denote marginal agent s.t., when agents follow cut-off strategies with cut-off z σ z (θ), regime change occurs iff θ θ Let Define ψ(θ, ˆθ,σ z ) = U(z σ z (θ),{ θ > ˆθ} z σ z (θ)) } θσ inf z inf{ˆθ : ψ(θ, ˆθ,σ z ) > 0 all θ. For any ˆθ > θ inf σ z, unique rationalizable profile has no agent attacking after s publicly reveals that θ ˆθ

39 Optimality of NDD

40 Optimality of NDD Proposition Let σ z argmin σz (0,σ η ]θ inf σ z Optimal (Gaussian) policy combines public disclosure of whether or not θ < θσ inf z, with Gaussian private messages of precision σ 2 ξ = [σ 2 η (σ z ) 2 ]/(σ z ) 2 σ 2 η Precision σ 2 ξ guarantees that sufficient statistics has precision 1/σ z

41 Optimality of NDD

42 Optimality of NDD

43 Optimality of NDD

44 Optimality of NDD Let θ # σ z and z # σ z denote regime threshold and marginal agent under optimal policy when information has precision σ 2 z. Proposition Suppose that, for any σ z [0,σ η ], E[ḡ (θ)(θ θ # σ z ) z # σ z,θ θ # σ z ] E[ḡ(θ) z # σ z,θ θ # σ z ] Optimal Gaussian policy is non-discriminatory. > E[ b (θ)(θ θ σ # z ) z σ # z,θ (θσ inf z,θ σ # z )] E[ b(θ) z σ # z,θ (θσ inf z,θ σ # z )] Condition says upside risk dominates downside risk