Ambiguity and Coordination in a Global

Game Model of Financial Crises

Daniel Laskar (Paris School of Economics-CNRS)

Conference Birbeck June 21-22 2013

Models of financial crises: problem of coordination between the partic-ipants in the financial markets, because of complementarities between theiractions. This may lead to multiple equilibria of the corresponding game, forsome intermediate range of the levels of "fundamentals": currency crises (coor-dination between speculators); liquidity crises (coordination between creditors);bank runs (coordination between depositors).

"Global games" (Carlsson and van Damme (1993)): instead of assuming thatthe level of fundamentals is perfect knowledge, players only receive an imper-fect signal. Then, the equilibrium is always unique, even if the uncertaintyof the signal becomes small.

This has been applied to financial crises: exchange rate crises (Morris and Shin(1998)); bank runs (Goldstein and Pauzner (2005), Rochet and Vives (2004));debt and creditors (Morris and Shin (2004)).

Comparative statics. One issue which can be considered: effect of uncertaintyon the occurrence of the crisis. In the literature, no clear-cut results: thismay depend on the model and, in a model, on the value of some parameter ofthe model.

Usually, this literature uses a standard expected utility approach to uncertainty.But, this rules out ambiguity aversion (Ellsberg (1961)): decision makers oftenprefer situations with known probabilities to situation with unknown probabili-ies. There are more recent approaches to uncertainty which introduce ambigu-ity (Gilboa and Schmeidler (1989)and Schmeidler (1989)): Ambiguity aversionwould imply that, for each decison taken, the decision maker is "pessimistic"and therefore gives more weight to the bad states of the world.

I will use such an approach and I will higlight a new channel through whichuncertainty can affect the occurence of a financial crisis: I will under-line that more ambiguity reduces the amount of perceived coordinationbetween market participants (players). First, I will consider a model of debtfinancing with several creditors.

There are a few analyses of ambiguity in global games, with applications tomodels of financial crises (Ui (2009) and Kawagoe and Ui (2010). But thischannnel through perceived coordination was not present because of their im-plicit assumption that the distrbution of the signals is the same for all theplayers.

Model

Morris and Shin (2004). A group of creditors financing a project through acollaterized debt have to decide whether to roll over their loans or not. Twoplayer version, as in Kawagoe and Ui (2010).

Two actions: R (to roll over the loan) or N (to not roll over the loan).

Payoffs:

- If a player plays N, he gets the value of the collateral λ, 0 < λ < 1;

- If a player plays R, he gets 0 if the project fails, and gets 1 if the projectsucceeds. The project always succeeds if "fundamentals" θ are strong enough(θ > 1), and always fails if fundamentals are weak enough (θ < 0).

In the intermediate case 0 ≤ θ ≤ 1, the project succeeds if both creditors playR, but fails otherwise:

R NR 1, 1 0, λN λ, 0 λ, λ

If θ were common knowledge, there would be two Nash equilibria: one withouta financial crisis (R,R); and one with a financial crisis (N,N), due to a lack ofcoordination between creditors.

But each player only receives an imperfect signal on θ :

xi = θ + ξi (1)

Furthemore, it is assumed that the distribution of ξi is not known: its meanµi is ambiguous:

xi = θ + µi + εi (2)

The signal can therefore be biased (Cheli and Della Posta (2007)), and thereis ambiguity on the value of the mean (or "bias") µi. We have µi ∈

hµ, µ

i,

and each player has a corresponding maxmin criterion of expected utility.We allow the distributions of the signals received by the players to bedifferent, by allowing µ1 6= µ2.

Define the ambiguity parameter

η ≡ 1

2(µ− µ) (3)

θ is uniformally distributed on [−δ, 1 + δ] ; and εi is uniformally distributed on[−γ, γ] . It is assumed that we have 0 < γ ≤ δ

2 (which will make the prioron θ non informative when we consider the equilibrium condition) and (tosimplify the analysis) γ ≤ 1

2.

Switching strategy sk with switching point k: play R if xi > k, and play Nif xi ≤ k

We look for equilibria in switching strategies.

Proposition 1 When player j follows a switching strategy, then, the worst casefor player i is obtained for µi = µ and µj = µ.

This implies E³xj | xi

´= xi − 2η. Therefore player i expects that the

other player j receives a signal which is lower (i.e. worse) than his/hersignal by an amount which increases with the ambiguity parameter η.

Proposition 2 1. In the case η < γ, if we have λ 6= 12

³1− η

γ

´2, there is

a unique equilibrium in switching strategies, where the equilibrium switchingpoint k∗ is

k∗ = µ+ g(η, γ, λ) (4)

where the function g(η, γ, λ) is given by:

(i) if we have 0 < λ < 12

³1− η

γ

´2, then we have

g(η, γ, λ) = γ

⎛⎜⎝1− 2vuutÃ1− η

γ

!2− 2λ

⎞⎟⎠− 2η (5)

(ii) if we have 12³1− η

γ

´2< λ < 1− η

γ, then we have

g(η, γ, λ) = 1− γ + 2

vuuutη2 + γ2

⎛⎝2λ− Ã1− η

γ

!2⎞⎠− 2η (6)

(iii) if we have 1− ηγ ≤ λ < 1, then we have

g(η, γ, λ) = 1− γ + 2λγ (7)

(and therefore does not depend on η in this case)

In the case η < γ, if we have λ = 12

³1− η

γ

´2, then any switching point k∗

such that γ−2η ≤ k∗−µ ≤ 1−γ gives an equilibrium in switching strategies.

2. In the case η ≥ γ, then, for all λ (where 0 < λ < 1), there is a uniqueequilibrium in switching strategies, where the equilibrium switching point k∗

is given by (12) and (7) (and therefore k∗ − µ does not depend on η in thiscase).

Effects on the probality of a crisis

A financial crisis occurs when the investment project fails because of a problemof early liquidation by creditors. This happens in the case 0 ≤ θ ≤ 1 whenat least one of the two creditors does not roll over the loan. Therefore, moreambiguity increases the probability of a crisis if and only if we have k0∗ > k∗.

The equilibrium switching point k∗ is an increasing function of λ. It may beincreasing or decreasing with γ (i.e. with the variance of the shocks (εi)),depending on the values of the other parameters of the model.

Effect of ambiguity:

³µ0, µ0

´is more ambiguous than

³µ, µ

´if we have

hµ0, µ0

i⊃hµ, µ

i, which

can be written: µ0 ≤ µ and µ0 ≥ µ, with at least one strict inequality. Notethat this implies η0 > η

Let µ1 and µ2 be the true given biases of the signals received by player 1and player 2, respectively. Then, we will compare

³µ, µ

´to³µ0, µ0

´, where³

µ0, µ0´is more ambiguous than

³µ, µ

´, and where we have µ ≤ µ1 ≤ µ, µ

≤ µ2 ≤ µ, µ0 ≤ µ1 ≤ µ0 and µ0 ≤ µ2 ≤ µ0.

Two channels

k∗ = µ+ g(η, γ, λ) (8)

The first channel goes through the highest bias µ. The reason is that, in theworst case, we have µi = µ, As we have µ0 ≥ µ, this makes k0∗ ≥ k∗. Thus, asmore ambiguity generally raises the highest possible bias, each player interpretsless favorably any message received: a lower values for the fundamentals θ isinferred.

The second channel goes through the function g(η, γ, λ), which, in cases 1-(i)and 1-(ii) of Proposition 2, depends on the ambiguity parameter η. In thesecases, we have ∂g(η,γ,λ)

∂η > 0, which makes k0∗ > k∗. Thus, through thissecond channel, more ambiguity also increases the probability of a crisis.

The two channels go into the same direction. As a consequence, we findthat more ambiguity increases the probability of a financial crisis. Thismakes more ambiguity harmful and, therefore, gives an argument for moretransparency.

The second channel is the channel I want to emphasize. It is due tohaving, in the worst case, µi = µ and µj = µ. Therefore this channel woulddisappear in the case without ambiguity (µ = µ and therefore η = 0), but alsounder ambiguity (η > 0), if we had assumed (as it is done in the literature onglobal games with ambiguity ) that the probability distributions of the signalswere the same for all the players, which would have imposed the constraintµi = µj. This is why this channel was not present in the literature.

This channel comes from having each player expect (in the worst case) thatthe other player receives a signal which is worse than his/her signal, and wouldtherefore have a lower probability to play R. Therefore, more ambiguity makeseach player expects that, when R is played, there is a lower amount ofcoordination with the other player. This makes each player less willing toplay R. This raises k∗ and therefore increases the probability of a crisis.

Figure 1

O

A B

DC

E F

G

H1 θ

ε j

(Z)

Figure 1:

Figure 2

O

A B

C D

E FH

R

PG

θ

ε j

Q

S

(Z)

Figure 2:

Figure 3

A B

C D

E

F

G

HI

L

M

N

O θ

ε j

1

(Z)

T

Figure 3:

There are two reasons why ambiguity increases the probability of a crisis:

(i) The payoff to action R depends on θ while the payoff to N does notdepend on θ. More ambiguity increases the incentive to choose the action (N)which does not depend on θ.

(ii) The payoff to action R increases with the coordination of playerswhile the payoff to N does not depend on the coordination of players.More ambiguity decreases the amount of perceived coordination, which reducesthe incentive to choose the action (R) which depends on coordination.

Two remarks on that:

1. Both effects are of the same sign in the model considered, but this may notbe necessarily so in other games.

2. In an exchange rate crisis model, where players are speculators, these twoeffects would also be of the same sign, but this sign would be opposite to thesign it has in the present model of a liquidity crisis in rolling over a debt. Thereason is that it would be the payoff to the action "to attack the currency"which would depend both on the state of fundamentals and on the coordinationof players, while the payoff to the action "not to attack the currency" wouldnot depend on these variables. For the two previous reasons (and, in particular,because it reduces the perceived coordination between speculators), moreambiguity would therefore give more incentive to play the action "not to attackthe currency". As a consequence, more ambiguity would on the contraryalways decrease the probability of a crisis. Thus, for exchange crises, thiswould tend to make more transparency harmful.

Complementary paper: Daniel Laskar (2013), "Ambiguity, Pessimism, Opti-mism and Financial Crises in a Simple Global Game Model":

- same kind of analysis in a simpler linear global game model with a con-timuum of players

ui(θ, l, ai) = u(θ, l, ai) = θ − 1 + l if ai = C

0 if ai = N(9)

where l is the proportion of players who take action C (action "C" has a payoffwhich increases with l, and therefore which benefits from "coordination", andcorrespond to rolling over the loan (R) in the previous analysis

- introduce both optimism (preference for ambiguity ) and pessimism(ambiguity aversion):

Ωα,i

³k, xi,ai

´= α min

M∈MπM (k, xi, ai) + (1− α) max

M∈MπM (k, xi, ai)

(10)

where α (0 ≤ α ≤ 1) is the degree of pessimism (and 1 − α the degree ofoptimism)

- explicitly compare two models which may represent two types of financialcrises: a liquidity crisis model with creditors as players (model above);and an exchange rate crisis model with speculators as players:

u0i(θ0, l0, a0i) = u0(θ0, l0, a0i) =

l0 − θ0 if a0i = C0

0 if a0i = N 0 (11)

where now C’ represents the action "to attack the currency"

- consider the effect of shifts in "market sentiment" (Cheli and Della Posta(2007)), corresponding to shifts in the biases µi of the signals.

Equilibrium for the first model: switching strategy with switching point k∗

k∗ =1

2+ µ0 + (2α− 1)

∙η + ψ (2η)− 1

2

¸(12)

where µ0 ≡ 12

³µ+ µ

´; η ≡ 1

2(µ−µ); and ψ (.) is the cumulative distributionfunction of the distribution of εi − εj, i 6= j.

Case of pure pessimism α = 1 (first paper)

k∗ = µ+ ψ (2η) (13)

El∗ can be taken as an inverse index of financial difficulties:

El∗ =Z 10Pr³θ + εj > k∗ − µj

´dj (14)

Results:

- Ambiguity has opposite effects according to whether pessimism or optimismdominates. An increase in η reduces the expected coordination when pes-simism dominates

³α > 1

2

´, but it increases the expected coordination

when optimism dominates³α < 1

2

´.

- More pessimism decreases the expected coordination.

- If we want to know the effect on the equilibrium switching point, and onfinancial difficulties, we can go from one model to the other by simply re-placing pessimism by optimism (i.e. α by 1−α), and vice versa. A modelof liquidity crises with pessimistic creditors is similar to a model of exchangerate crises with optimistic speculators, and vice versa. As a consequence, moreambiguity has opposite effects on financial difficulties in the two models offinancial crises, with opposite conclusions about transparency.

- We can obtain real effects of shifts in market sentiment in a rationalway: ambiguity gives some room of maneuver to have shifts in market sen-timent without equal corresponding shifts in expectations. We only have tosatisfy the constraints µi ∈

hµ, µ

ifor all i.