ambiguity and coordination in a global game model of financial crises

7/29/2019 Ambiguity and Coordination in a Global Game Model of Financial Crises

1/23

Ambiguity and Coordination in a GlobalGame Model of Financial Crises

Daniel Laskar

April 10, 2013

Abstract

By using a non-Bayesian approach with ambiguity averse players,we highlight a new channel through which uncertainty can affect crisesin global game models offinancial crises. We show that ambiguity re-duces the amount of coordination perceived by each player. In themodel we consider, which is a two-player global game where creditorsfinance some project, this effect contributes to increasing the probabil-ity of a financial crisis, and therefore provides an additional argumentin favor of transparency.

JEL classification: C72, D81, D82, G01.Keywords: global games, coordination, ambiguity, uncertainty, fi-

nancial crises, transparency.

1 Introduction

Global games have often been used in order to analyse financial crises. In a"global game", the game is not common knowledge and, instead, each playerreceives an imperfect signal on an underlying parameter which characterizes

the game. The rationale of using global games for the analysis offi

nancialcrises is that, by introducing in this way uncertainty on the underlying pa-rameter which represents the value of "fundamentals", one can get rid of themultiplicity of equilibria which would otherwise arise in models offinancialcrises if the value of fundamentals were common knowledge1.

Paris School of Economics-CNRS, France. Address: CEPREMAP, 142 rue duChevaleret, 75013 Paris, France. Ph: 33 (0)1 40 77 84 08. Fax: 33 (0)1 40 77 84 00.E-mail: [email protected]

1 See Carlson and van Damme (1993), which have introduced global games. See alsoMorris and Shin (2003) for a survey on global games. The global game approach has been

1


2/23

In models offinancial crises, multiple equilibria arise because of comple-

mentarities between the decisions of markets participants. Thus, the pos-sibility of a bank run depends on the amount of depositors who withdrawtheir money deposited at the bank; and, in the same way, when a group ofcreditors finance some investment project, a financial crisis, due to an earlyliquidation by creditors, occurs only when an insufficient amount of credi-tors roll over their loans. Also, if we consider a currency crisis, such a crisisrequires that a sufficient amount of speculators attack the currency.

This implies that a market participant will be more willing to take thecorresponding action if (s)he believes that a larger number of market par-ticipants choose the same action, i.e. if (s)he perceives that there is more

coordination betwen his/her action and the actions of other players. As it hasbeen shown in the literature, this may lead to multiple equilibria, at least forsome intermediate range of values of fundamentals. Thus, if each depositorexpects that other depositors keep their money deposited at the bank, thena bank run will not occur. But such a bank run will occur if, on the contrary,each depositor expect other depositors to withdraw their money. Similarly, afailure of the investment project due to an early liquidation by creditors willoccur if each creditor expects that other creditors do not roll over their loans,but there will be no failure if each creditor expects that other creditors rollover their loans. Concerning currency crises, a currency crisis will occur if

everyone expects other participants to attack the currency, but the crisis willnot occur if, on the contrary, each one expects that other participants do notattack the currency. In each case, the corresponding game has an equilibriumwith a financial crisis, and another one without a financial crisis2.

In a global game, instead of assuming that the value of fundamentals iscommon knowledge, it is assumed that each player (market participant) onlyreceives an imperfect signal on the value of fundamentals. It can then beshown that the multiplicity of equilibria disappears: there is a unique equi-librium. Then, comparative statics can easily be done, and one issue whichhas been explored is whether more uncertainty increases the probability ofa financial crisis or not. The answer to such a question has policy impli-cations. For, if more uncertainty can trigger financial crises, then it wouldbe desirable for policymakers to be transparent about the information theyhave; and they should also follow policies aiming at reducing uncertainty.According to the existing literature, the comparative statics exercises done

applied to currency crises by Morris and Shin (1998). Other applications of global gamesto financial crises include the issue of creditors having to roll over their loans to finance aproject (Morris and Shin (2004)), or the issue of bank runs (Goldstein and Pauzner (2005)and Rochet and Vives (2004)).

2 See Diamond and Dybvig ((1983) and Obstfeld (1996) for such analyses.

2


3/23

with these models lead to results which are not very clear-cut. Depending

on the model and on the parameters of the model, greater uncertainty mayeither increase or decrease the probability of a crisis3.

In the present paper, we want to highlight a new channel, which is absentin the existing literature, through which uncertainty can affect crises in globalgame models of financial crises. This channel goes through the amount ofcoordination perceived by each player when (s)he chooses the action whichrequires coordination. Because of the complementarities involved, the morecoordination each player perceives, the greater is his/her willingness to choosethis action. For example, consider the case of creditors who have to decidewhether to roll over their loans for some investment project. If a player

perceives that, when (s)he rolls over his/her loan, more players coordinate,i.e. more players also roll over their loans, then this player has a greaterincentive to roll over his/her loan. As a consequence, this decreases theprobability of a financial crisis.

However, in the existing literature, uncertainty, in some sense, does notaffect the amount of coordination perceived by each player in a global game.This means that, in the literature, when one consider the effect of uncer-tainty on financial crises in global games, there is no channel going throughthis perceived coordination. More precisely, the literature has underlined thefollowing property4. Assume that there is a non informative prior on funda-

mentals and consider the situation where a player receives a signal equal tothe threshold value which makes himself/herself indifferent between the twoactions (s)he can choose5. Then, at the equilibrium, any other player alwayshas a probability 1

2of taking the action which requires coordination. This

means that, in this case, each player always perceives a degree of coordina-tion equal to 1

2. The amount of uncertainty has no effect on this perceived

degree of coordination, which is always equal to 12

.In our analysis, we will show that this property is actually very dependent

on the approach to uncertainty which is taken. The literature on global gamesgenerally uses the standard Bayesian approach of expected utility maximiza-tion. However, such an approach has been challenged. Knight (1921) hadalready made the distinction between a situation of "risk" where there is aknown probability distribution, and a situation of "(Knightian) uncertainty"where no known probability distribution exists. Ellsberg (1961), throughsome experiments, has shown that decision makers exhibit some "aversion to

3 See for example Metz (2003).4 See Morris and Shin (2003)).5 As it is the case in these global game models of crises, we consider a binary action

game where only two actions are possible: depending on the model of crises, they consistin withrawing deposits or not; rolling over the loan or not; attacking the currency or not.

3


4/23

ambiguity", and further experiments confirmed this finding. Decision makers

usually prefer a situation where there is a known probability distribution to asituation where this distribution is unknown. This is why, in the past two orthree decades, some new non-Bayesian approaches to uncertainty, where suchan aversion to ambiguity could be taken into account, have been developped6.

We will therefore use such a non-Bayesian approach to uncertainty andintroduce ambiguity (or "Knightian uncertainty"), with ambiguity averseplayers, in a global game model of financial crises. We will show that theamount of ambiguity will affect the perceived degree of coordination. Whenplayers receive a signal equal to the threshold value (and have an uninfor-mative prior), this perceived degree of coordination will no longer be equal

to12 , as in the standard analysis of the literature on global games. We will

actually show that, in the presence of ambiguity, it is in general less than12

, and that it is a decreasing function of ambiguity. If ambiguity is largeenough it could even become equal to zero. This effect of ambiguity on per-ceived coordination gives a new channel through which uncertainty can affectfinancial crises.

We will consider a two-player global game between two creditors whofinance some investment project. At an interim stage, each player has todecide whether to roll over his/her loan or not. As it will be shown, moreambiguity tends to decrease the degree of coordination of players which is

perceived by each player when s(he) considers rolling over his/her loan. Thiswill make each player less willing to roll over the loan, and therefore makes afinancial crisis more likely. From a policy point of view, our result is in favorof transparency, because more transparency from the public authorities mayreduce the amount of ambiguity that each player has to face.

In the literature, there are actually some analyses which introduce ambi-guity in global games (Kawagoe and Ui (2010) and Ui (2009)). However, inthese analyses, ambiguity has no effect on the perceived degree of coordina-tion between players, which actually remains equal to 1

2, as in the literature

which uses the standard expected utility approach. Therefore, the channelthat we underline, and which goes through the effect of ambiguity on theperceived degree of coordination, does not exist in these analyses. As we willshow, the reason is that they make an implicit assumption which precludes

6 Two classical references are Gilboa and Schmeidler (1989) and Schmeidler (1989).In these non-Bayesian frameworks, the presence of some aversion to ambiguity leads thedecision maker to give more weight to the bad outcomes implied by each decision.

Such a non Bayesian approach has been applied to a large variety of issues in economics.It has been shown that this approach could help explain some stylized facts, and couldallow us to better understand some results which otherwise looked quite paradoxical (seefor example Mukerji and Tallon (2004)).

4


5/23

such a channel. They a priori assume that the probability distributions of

the signals received by all the players are identical. Such an assumptionseems actually restrictive, and in the present paper we will relax this as-sumption. We will therefore allow the probability distributions of the signalsto be different for different players. Then, it will be shown that ambiguityhas an adverse effect on the belief that each player holds on the amount ofcoordination.

The model is presented in Section 2. Section 3 derives the equilibriumof the game. Section 4 considers the effect of more ambiguity. Section 5concludes.

2 Model

As in the global game model of Morris and Shin (2004), a group of credi-tors are financing some investment project through some collaterized debt.At some interim stage, each creditor can either roll over the loan until thematurity of the project, or seize the collateral, which has some given knownvalue . The value of the investment project at maturity will depend onthe state of fundamentals, represented by some real variable . The value offundamentals is not known, and each creditor receives an imperfect signalon . The value of the project at maturity will also be negatively affected

if an insufficient amount of creditors do not roll over the loan. This createssome complementarities between the decisions of creditors to roll over theirloans or not. The model considers the corresponding game between the cred-itors. As in Kawagoe and Ui (2010), we will consider a two-player version ofthe game, and we will introduce some ambiguity on with ambiguity averseplayers.

The actions and payoffs of this two-player game are the following. Eachplayer has two possible actions: to roll over the loan (R), or not to roll overthe loan (N). By choosing N, the player receives the value of the collateral, where satisfies the inequality 0 < < 1. By choosing to roll over the

loan (R), a player gets a payoff equal to 1 when the project succeeds, and apayoffequal to 0 when the project fails. It is assumed that the project alwayssucceeds when the fundamentals are good enough, i.e. when we have > 1;and that the project always fails when the fundamentals are bad enough, i.e.when we have < 0. For intermediate values of the fundamentals, i.e. in thecase 0 1, it is assumed that the investment project succeeds only inthe case where both players roll over the loan. The payoffmatrix in the case0 1 (which is the case where the coordination of players matters) istherefore:

5


6/23

R N

R 1, 1 0, N , 0 ,

If the value of were common knowledge, then, in the case > 1, R wouldbe a dominant strategy; while in the case < 0, N would be a dominantstrategy. In the case 0 1, both (N,N) and (R,R) would be equilibria:(N,N) is a financial crisis equilibrium, where no player rolls over the loan andwhere, consequently, the project fails; and (R,R) is an equilibrium withouta financial crisis, where each player rolls over the loan, and where, as aconsequence, the project succeeds.

In our analysis, however, as in the global game literature, it is assumedthat is not common knowledge. Each player receives a signal xi given by

xi = + i (1)

where i is a random variable.The literature on global games usually assumes that i follows some given

known probability distribution. Then, it can be shown that, even in thecase 0 1, there is a unique equilibrium. This occurs even when theuncertainty becomes small, i.e. when the variance of i goes to zero.

As in Kawagoe and Ui (2010), the assumption that the probability dis-tribution of i is known will be removed and it will be assumed that there

is (Knightian) uncertainty, or "ambiguity", on some parameter underlyingthe distribution of i. As indicated in the Introduction, Kawagoe and Ui(2010) assume that the probability distributions of the signals are the samefor all the players, but, in our analysis, we will relax this assumption. Suchan assumption seems restrictive. Each player may not necessarily get thesame kind of information. Each player may get information from differentsources or may obtain the opinions of different experts, or may ponder themin a different way. This could change not only the signal received but alsothe underlying probability distribution of the signal. In our analysis, we willtherefore assume that the signal received by each player follows a probability

distribution which may not be the same for all the players. As we will see,this will allow ambiguity to have an effect on the perceived coordination ofthe players.

We will assume that the mean of the probability distribution of i isuncertain. Thus player i receives a signal with a possible "bias" i, which isassumed to be uncertain7. We will write

i = i + i (2)

7 In Kawagoe and Ui (2010), the mean is assumed to be equal to zero (as it is usually thecase in the global game literature) and there is ambiguity on the variance of the probability

6


7/23

where i is a zero-mean random variable which has a known probability

distribution, and where the mean i of i depends on i, and therefore isnot necessarily the same for the two players. The random variables 1 and2 are identically distributed, and the variables , 1 and 2 are mutuallyindependent. From (1) and (2) we have

xi = + i + i (3)

There is ambiguity on the value of the mean (or "bias") i. We willassume that each i, i = 1, 2, belongs to the interval

,

and that each

player has a corresponding maxmin criterion of expected utility8. We allow

the distributions of the signals received by the players to be diff

erent, byallowing 1 6= 2.As in Kawagoe and Ui (2010), we assume that is uniformally distributed

on [, 1 + ] ; and that i is uniformally distributed on [, ] . We willassume9 that we have 0 <

2; and, to simplify the presentation10, it will

also be assumed that we have 12

.

distribution (i.e. on the length of the support of the uniform distribution of i). To takethe mean rather than the variance as the ambiguous parameter simplifies the analysis.Cheli and Della Posta (2007) have argued that it might be justified to introduce biasedsignals in global game models.

8 This is a standard criterion in the literature on decision under uncertainty with am-

biguity aversion (see Gilboa and Schmeidler (1989)). This was also the criterion used byUi (2009) and Kawagoe and Ui (2010).

The minimum will be taken with respect to i belonging to

,

, but we could as welltake the minimum with respect to all probability distributions having support includedin

,

. The worst case would be the same because it would lead to a distributionconcentrated at either or .

Note that the range

,

, which enters the maxmin criterion, may be considered asreflecting both the ambiguity of the available information and the aversion to ambiguityof the players (see Gajdos et al. (2004)).

9 In Kawagoe and Ui (2010), the less restrictive assumption 0 < is made. Thiswas necessary to encompass the specific parameter values they took in their experiments.Then, when they developed their theoretical analysis, they had to restrict the possible

values of . Thus, parameter had to belong to the interval38 ,

5

8

. Here, by making theassumption 0 <

2, our analysis can be made without further restraining . Any value

of satisfying 0 < < 1 is allowed.10 Note that this is compatible with the case, often considered, of a small uncertainty.

The case 12

< 2

could be taken into account without any difficulties, and would leadto the same kind of basic qualitative results. However, adding this case would have madethe analysis and the presentation of the results more cumbersome.

7


8/23

3 Equilibrum

As it is usually done in such global games, we will look for equilibria in"switching strategies". When player i follows the switching strategy withswitching point k, s(he) chooses R if xi > k and chooses N if xi k.

From the payoffstructure previously given, the expected payoffof a playerwho chooses N (i.e. who does not roll over the loan) is equal to , while theexpected payoff of a player who chooses R (i.e. who rolls over the loan) isequal to the probability that the investment project succeeds in that case.

Let i,j (k, xi) denote the probability, conditional on having received thesignal xi, that player i holds on the event that the project succeeds when the

other player j follows the switching strategy with switching point k. Thisprobability is therefore given by

i,j(k, xi) = Pri,j[(0 1 and xj > k) or ( > 1) | xi] (4)

Note that this probability depends on both parameters i and j becauseit depends on the probability distributions of both random variables i andj.

As underlined before, each player has some aversion to ambiguity anduses a maxmin criterion. This means that player i chooses action R if andonly if we have11 min

i[,],j[,]

i,

j

(k, xi) > .

3.1 Worst case for each player

We first have the following result:

Proposition 1 We have mini[,],j[,] i,j(k, xi) = ,(k, xi). The

worst case for player i is therefore obtained for i = andj = .

Proof: From (3) , we have = xi i i. A larger bias i shifts theprobability distribution that player i holds on , conditional on having re-

ceived a signal xi, towards lower values of . This reduces (or at most leavesunchanged) the probability of success of the project through two channels.First, this may decrease the probability of being in the case > 1 where theproject is always successful, or this may increase the probability of being inthe case < 0 where the project is never successful. Second, as, from (3) ,we have xj = + j + j, this reduces the probability that the other player

11 As in Morris and Shin (2004), we assume that player i chooses N if (s)he is indifferentbetween N and R. This assumption is actually without any consequences on the analysisand the results.

8


9/23

rolls over the loan, because this decreases the probability of having xj > k.

As a consequence of these two effects, player i considers that the worst caseis obtained when the bias i of her message takes its maximum value .

Now, consider the effect of the bias j of the signal received by the otherplayer. This bias changes the probability distribution (conditional on xi)that player i holds on the signal xj received by the other player. From(3) , we have xj = + j + j , and therefore a smaller bias j shifts thisprobability distribution toward lower values ofxj. This implies that a smallerbias j decreases (or at most leaves unchanged if this probability was alreadyequal to zero) the probability of having xj > k, and therefore decreases theprobability that player j rolls over the loan. This tends to reduce the success

of the investment project. Consequently, the worst case would occur whenthis bias would be the lowest (j = ). QED

Each player has only to consider what happens in the worst case. Asa consequence, Proposition 1 implies that each player behaves as if (s)hebelieves that the bias of his/her own signal is the maximum bias and thatthe bias of the signal of the other player is the minimum bias .

3.2 Equilibrium switching point

Proposition 1 implies that player i (strictly) prefers action R (and therefore

chooses R) if and only if we have,(

k, xi)

> .Let

b(

k) be the value ofxi which makes player i indifferent between R and N, and which therefore

satisfies the equality ,(k, b(k)) = . It can be seen12 that ,(k, xi) isa non-decreasing function of xi, and, in the neighborhood of k, a (strictly)increasing function of xi. Consequently, if we have xi > b(k), then we have,(k, xi) > , and therefore player i chooses R; and if we have xi b(k)then we have ,(k, xi) and therefore player i chooses N. This impliesthat the switching strategy with the switching point b(k) is the best responseof player i to the switching strategy with switching point k of player j .

As a consequence, when both players use switching strategies with switch-ing point k, where k is a solution of the equation b(k) = k, we have an

12 When we consider ,(k, xi), we could make the same kind of analysis as is done

for ,(k, k) in the Appendix. The difference is that point E in the Figures would have

coordinates (xi , 0) instead of(k , 0) . An increase in xi would then shift toward theright the support of the uniform distribution which represents the conditional distributionof the joint variable (, j) (represented by the square ABCD in the Figures). This wouldincrease ,(k, xi), or possibly leave it unchanged in some cases, but it can be seen that

in the case where we have 0 < ,(k, xi) < 1 (which is satisfied in the neighborhood

of k for xi because we have ,(k, k) = with 0 < < 1), this would stricly increase

,(k, xi).

9


10/23

equilibrium of the game. This equation can be written ,(k, k) = . The

following proposition shows that (in general) the solution of this equation isunique, and it explicitly gives its value13.

Proposition 2 Let us consider the ambiguity parameter defined by

12

( ) (5)

1. In the case < , if we have 6= 12

1

2, there is a unique

equilibrium in switching strategies, where the equilibrium switching point k

is

k

= + g( , , ) (6)where the function g( , , ) is given by:

(i) if we have 0 < < 12

1

2, then we have

g( , , ) =

1 2

s1

2 2

2 (7)

(ii) if we have 12

1

2< < 1

, then we have

g( , , ) = 1 + 2vuut2 + 221

2! 2 (8)

(iii) if we have 1 < 1, then we have

g( , , ) = 1 + 2 (9)(and therefore does not depend on in this case)

In the case < , if we have = 12

1

2, then any switching pointk

such that2 k 1 gives an equilibrium in switching strategies.2. In the case

, then, for all (where 0 < < 1), there is a unique

equilibrium in switching strategies, where the equilibrium switching point k

is given by (6) and (9) (and therefore k does not depend on in thiscase).

The proof is given in the Appendix.

13 Following the same kind of argument as in, for example, Morris and Shin (2003), itcould be shown that the switching strategy with switching point k, where k is a solutionof the equation b(k) = k, survives the iterated deletion of interim-dominated strategies.And, when there is a unique solution to this equation, the corresponding switching strategyis the unique strategy which survives the iterated deletion of interim-dominated strategies.

10


11/23

4 Effect of ambiguity on financial crises

4.1 Two channels

Ambiguity has been defined by the couple of values

,

, where .We will say that

0, 0

is more ambiguous than

,

if we have

0, 0

,

, which can be written: 0 and 0 , with at least one strictinequality. Note that this implies 0 > , where parameter is defined by(5) .

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

0, 0 , where0

, 0

is more ambiguous than

,

, and where we have 1 , 2 , 0 1 0 and 0 2 0.In the model, a financial crisis occurs when the investment project fails

because of a problem of early liquidation by creditors. This happens in thecase 0 1 when at least one of the two creditors does not roll over theloan. Therefore, more ambiguity increases the probability of a crisis if andonly if we have k0 > k.

From equation (6) of Proposition 214, we see that ambiguity affects theequilibrium switching point k, and therefore the probability of a crisis,through two channels. The first goes through the highest bias . The reasonis that, in the worst case, we have

i= , and therefore player i acts as

if (s)he believed that the bias of his/her signal was at its maximum value. As we have 0 , this makes k0 k. Thus, as more ambiguity gen-erally raises the highest possible bias, each player interprets less favorablyany message (s)he receives: (s)he infers lower values for the fundamentals ,the conditional distribution of the fundamentals being shifted toward lowervalues. This kind of channel was present in the literature which introducedambiguity in global games15.

The second channel goes through the function g( , , ), which, in cases 1-(i) and 1-(ii) of Proposition 2, depends on the ambiguity parameter defined

14

This equation is valid in the general case where there is a unique equilibrium. In thespecial case given by < and = 1

2

1

2, where, from Proposition 2, we have

a continuum of equilibria for k in the interval [ + 2, + 1 ] , we should add anupward jump from k to + 1 in the case of more ambiguity, or a downward jumpfrom k to + 2 in the case of less ambiguity. These additional jumps would thenreinforce the effect found in the general case where there is a unique equilibrium k.

15 In Kawagoe and Ui (2010), it is not the mean but the variance, or equivalently thelength of the support of the uniform distribution of i which is ambiguous: we have

,

. Then this channel occurs through the maximum (or the minimum , dependingon the value of ), rather than through the maximum mean .

11


12/23

by (5) . In these cases, from (7) or (8) , we have g(,,)

> 0, which makes

k0 > k. Thus, through this second channel, more ambiguity also increasesthe probability of a crisis. This kind of channel was not present in theliterature on the effect of uncertainty in global games. It is actually the newchannel we want to underline in our analysis, and we will discuss it in moredetails in the next subsection.

The two channels go into the same direction. As a consequence, wefind that more ambiguity increases the probability of a financial crisis. Thismakes more ambiguity harmful and, therefore, gives an argument for moretransparency16.

4.2 The role of the perceived coordination of players

The second channel is entirely due to the fact that we have allowed i 6= j ,because it comes from the property that, in the worst case, i and j actuallydiffer and take the opposite extreme values (from Proposition 1 we havei = and j = in the worst case). If, as it is done in the literature, wehad assumed that the distributions of the signals of the two players were thesame, and therefore that we had i = j , this channel would have disappearedand only the channel going through would have remained17.

Thus, each player, in the worst case, substracts to his/her own signal the

maximum bias

but considers that the other player receives a signal withthe minimum bias . As a consequence of such a gap (which is equal16 This result contrasts with what would be the effect of more uncertainty under a

Bayesian approach in the model. Thus, take the Bayesian case where we have = 0,and consider the effect of more uncertainty, in the sense of an increase in the variance 2of the noise i. This is actually equivalent to an increase in parameter of the uniformdistribution followed by each i.

In the Bayesian case = 0, Proposition 2 would give : g(0, , ) =

1 2

1 2

in

the case < 12

; and g(0, , ) = 1 +

2

2 1 1

in the case > 12

(Note that theseexpressions also appears in Kawagoe and Ui (2010) p.6). As a consequence, an increase in increases the probability of a crisis in the case 3

8< < 1

2or 5

8< < 1, but decreases

the probability of a crisis in the case 0 < < 38

or 12

< < 58

. (This corresponds to theresult underlined by Kawagoe and Ui (2010) that the effect of the "quality" of informationdepends on parameter ).

Thus, as it is often the case in global game models of financial crises under a Bayesianapproach (see the Introduction), the sign of the effect of uncertainty (i.e. of "risk") onthe probability of a crisis would depend on the value of some parameter of the model. Incontrast, under the present non-Bayesian approach, more ambiguity always increases theprobability of a crisis.

17 If we had imposed i = j , we can easily see that the worst case would have beenobtained for i = j = . Therefore, ambiguity would have had an effect (only) through, as in the first channel.

12


13/23

to 2), in the worst case each player believes that the other player receives

signals which are on average less favorable (i.e. lower) than the signal (s)hereceives. This lowers the probability that the other player rolls over his/herloan. More ambiguity, which increases this gap, has therefore the effectof decreasing this probability. In other words, more ambiguity decreasesthe amount of coordination that each player expects when (s)he rolls overhis/her loan. This makes each player less inclined to roll over his/her loanand, consequently, increases the probability of a financial crisis.

As we have indicated in the Introduction, the existing literature had un-derlined the following property: when player i receives a signal equal to theequilibrium switching point (xi = k), and if the prior on is non infor-

mative, then the other player has probability12 of rolling over the loan. In

other words, when, in this case, a player rolls over the loan, (s)he perceivesa degree of coordination of 1

2. In our analysis, this property does not hold

anymore because, through the effect just described, ambiguity decreases theprobability that the other player rolls over the loan.

This can be seen formally in the following way. The probability thatplayer j rolls over the loan is given by the probability of having xj > k

.

According to (3) , this is equal to the probability of having j > k j .On the other hand, from (3) , the equality xi = k can be written = ki i. When the prior on is non informative18 this equality gives the

distribution of conditional on xi = k

. Then, replacing by this expressionin the previous inequality, the probability (conditional on xi = k) of havingxj > k

is equal to the probability of having j i > i j . If we hada priori assumed, as in Ui (2009) and Kawagoe and Ui (2010), that theprobability distributions of the signals are the same for the two players,which would imlpy i = j, this probability would be equal to

12

. But, in ouranalysis, player i evaluates this probability by taking the worst case for iand j . Therefore player i has instead to consider the probability of havingj i > , which can be written j i > 2. This probability is equalto 1

2in the usual Bayesian case = 0 of no ambiguity, but it is less than 1

2

when we have > 0, i.e. when there is ambiguity; and this probability is adecreasing function of the ambiguity parameter . In this case, when a playerrolls over the loan, the perceived degree of coordination is less than 1

2; and

more ambiguity decreases this perceived coordination.Note that the effect of more ambiguity going through the perceived coor-

dination of players does not play a role in cases 1-(iii) and 2 of Proposition2, where g( , , ) is given by (9) and does not depend on . The reasons arethe following. First, consider case 2 of Proposition 2. This case arises when

18 As shown in the Appendix this is satisfied when we have i = .

13


14/23

the level of ambiguity is already large enough to make the probability of

having j i > 2 equal to zero. This happens in the case , becausethe values of j i satisfying the inequality j i > 2 would then beoutside the support ofji. In that case any marginal increase in the valueof would be without any effect on this probability, which would remainequal to zero. Second, when the inequality xj > k can be satisfied only forvalues of greater than 1, then the coordination of players does not matterbecause the investment project is always successful. When we have < ,this corresponds to case 1-(iii) of Proposition 219.

5 ConclusionWe have considered a standard two-player global game of coordination be-tween creditors who have to decide whether to roll over their loans or not,in order to finance some investment project. We have introduced some am-biguity, players being assumed to have some aversion to ambiguity. We haveshown that more ambiguity tends to reduce the perceived coordination ofplayers. More ambiguity leads each player to act under the belief that, when(s)he rolls over his/her loan, there is less coordination of the players. Thisreduces the incentive to roll over the loan, and therefore increases the prob-ability of a financial crisis due to early liquidation by creditors.

This effect of ambiguity, going through the belief that each creditor hason the degree of coordination, actually reinforces a channel which was presentin the analyses of the literature which introduced ambiguity in global games.But, this effect, going through the perceived coordination, was absent in theanalyses on global games which can be found in the literature. This occurredeither because this literature used a Bayesian expected utility approach touncertainty, or because, when it used a non-Bayesian approach with someaversion to ambiguity, it did so under the implicit restrictive assumptionthat the probability distributions of the signals received by the players werethe same. By removing this last assumption, we have been able to show

that ambiguity reduces the perceived coordination of players in rolling overtheir loans. This highlights a new channel through which more ambiguity

19 This can easily be seen in the following way. From (3) and j = , the inequalityxj > k

can be written > k j. Using (6) and (9) , this inequality becomes > 1 + 2 + (2 1) j . Consequently, this inequality will be satisfied only for valuesof greater than 1 if and only if we have 2 + (2 1) j 0. This is verified for all jif and only if it is verified for j = . As we have 2 + (2 1) j = 0 for j = and = 1

, then the inequality is satisfied if and only if we have 1

, which gives

the inequalities for of case 1-(iii) of Proposition 2.

14


15/23

increases the probability of a financial crisis.

From a policy point of view, the results we obtain are in favor of moretransparency because more ambiguity increases the probability of a crisis.The central point of our analysis that we have emphasized is that, by reduc-ing ambiguity, more transparency would increase the perceived coordinationbetween creditors, which would help to prevent a financial crisis. In theircommunication to the private sector, the public authorities should thereforetry to make statements which may not be interpreted in an ambiguous way,and should provide information which would reduce the amount of ambiguityon the underlying fundamentals.

We have nonetheless to be careful in interpreting these results because

they may depend on the specific global game model used. Actually, the mainpoint of our analysis is that ambiguity could change the belief that eachplayer holds on the degree of coordination between his/her own action andthe actions of the other players. We have shown that ambiguity tend to de-crease this perceived coordination of players. In the model considered, whichis a model of coordination between creditors who roll over their loans, the co-ordination of players has a beneficial role, because it helps to prevent a crisis;and this is why the negative effect of ambiguity on perceived coordinationcan trigger a crisis and is therefore harmful.

However, the same kind of effect on perceived coordination could on the

contrary make ambiguity benefi

cial if the coordination of players has a harm-ful role and can trigger a crisis. This would be the case in a global game modelof currency attacks. For, in such a model, a player (speculator) is more willingto attack the currency if (s)he believes that there is more coordination withthe other players when attacking the currency. If more ambiguity reducedthe degree of perceived coordination of players (speculators) who attack thecurrency, this would decrease the incentive of each player to attack the cur-rency. Consequently, more ambiguity would, on the contrary, decrease theprobability of a crisis (i.e. of a currency attack). This would make ambigu-ity beneficial and, consequently, transparency harmful20. In further reseach

20

Although the effect on the perceived coordination of players that we underline in thepresent paper was not present in the literature, the existing literature (Kawagoe and Ui(2010)) and Ui (2009)) has shown that more ambiguity has also opposite effects on theprobability of crises in these two different models. As in our analysis, more ambiguityincreases the probability of a crisis in a model of creditors having to choose whether toroll over their loans (or in a bank run model), but decreases the probability of a crisis ina model of currency attacks.

As explained in Ui (2009), the reason is that, in the model of creditors having to rollover their loans or not, the action which has a payoff which depends on the value offundamentals is the action which helps to prevent a crisis ("to roll over the loan"); while,in the currency attack model, the action which has a payoff which depends on the value

15


16/23

it may therefore be worthwhile, both from a theoretical and an empirical

point of view, to further investigate the implications of such opposite andsymmetric effects that ambiguity may have on different types of crises.

Appendix

Proof of Proposition 2

As indicated in Section 3, the equilibrium condition is ,(k, k) = .From (3) and (4) , we have ,(k, k) = Pri=,j=[(0 1 and + j >k ) or ( > 1) | xi = k]. Consider the distribution of conditional onhaving received the signal xi = k, when we have i = . From (3) , the

equality xi = k can be written = k i. Consider a value of k whichbelongs to the interval

2

, + 1 + 2

. As we have assumed

2, this

equality implies 1 + . Consequently, for k

2

, + 1 + 2

, all the possible values of which are compatible with having received thesignal xi, belong to the support of the uniform prior on . This implies thatthe conditional distribution of is a uniform distribution with mean k and support [k , k + ] . As j is independent of i, this impliesthat the conditional distribution of the joint variable (, j) is a uniformdistribution having support [k , k + ] [, ].

Then, we could caIculate ,(k, k) through some integration calculus,but it will be simpler, or at least more illuminating, to do it graphically.In the plane of coordinates and j, the support of the uniform conditionaldistribution of the joint variable (, j) can be represented by a square ABCDcentered on point E of coordinates (k , 0) on the horizontal axis. This isdone in Figures 1, 2 and 3. As we have assumed 1

2, there are only three

possible cases which are represented by these three figures. In Figure 1, thesquare ABCD is entirely in the area where we have 0 1. Figure 2corresponds to the case where the square ABCD intercepts the vertical axis,in which case we can have < 0 or 0 1 (but never > 1). In figure3, where the square ABCD intercepts the vertical line where = 1, we canhave > 1 or 0

1 (but never < 0).

of fundamentals is the action which helps to trigger a crisis ("to attack the currency").As more ambiguity decreases the incentive to choose the action which has a payoffwhichdepends on the level of fundamentals, this explains the opposite results that ambiguityhas in these two models, which are found in Ui (2009) and Kawagoe and Ui (2010).

In our analysis, we have emphasized that an other characteristic of the action can alsoplay a role: it is whether the action has a payoff which depends on the coordination ofplayers or not. As we have indicated, the implied channel going through the perceiveddegree of coordination reinforces the previous channel underlined in the literature, andshould also lead to opposite effects when we go from a model of debt with creditors to amodel of currrency attacks by speculators.

16


17/23

The length of each side of the square ABCD is equal to 2. We have

AB = CD = AC = BD = 2. In these figures we have represented thestraight line (Z) of slope (1) of equation + j = k . We have alsodrawn the straight line parallel to (Z) going through point E, which is adiagonal of the square ABCD. This diagonal would be the same as (Z) inthe special case = where there would be no ambiguity.

The straight line (Z) intercepts the horizontal axis at point F, whichhas coordinates (xi , 0) and is therefore at the right of point E. We haveEF = = 2. Therefore, (Z) intersects the square ABCD in two pointsG and H if and ony if we have < .

In Figure 1, we always have 0 1, which occurs when we have k 1 . (This case is possible because we have assumed

12).

Then, we simply have ,(k, k) = Pri=,j=( + j > k | xi = k). In thecase < , this probability is given by A(GBH)

A(ABCD), where A (F) represents the

area offigure F. We have A (ABCD) = AB2 = 42and A (GBH) = 12

GB2.

As we have GB = AB AG = AB EF = 2 ( ) , we get A (GBH) =2 ( )2 . This gives ,(k, k) = 12

1

2. In the case , where the

straight line (Z) does not intersects the square ABCD, we have ,(k, k) =Pri=,j=( + j > k | xi = k) = 0.

The case represented in Figure 2 occurs when we have k < . In this

case, the area where we have < 0 should be excluded in order to calculate,(k, k). If we have < , then, from Figure 2, this gives ,(k, k) =A(GBH)A(GPR)

42in the case where (as in Figure 2) G is at the left of P,

i.e. when we have AG < AP. We have AG = EF = 2, and AP = SO =SE+EO = (k ) (because, in Figure 2, point E being on the left of pointO, we have k < 0 and therefore EO = (k )). Thus, the inequalityAG < AP gives the condition k < 2. As we have A (GP R) = 1

2GP2

and GP = APAG = (k )2, this gives ,(k, k) = 12

1

2

182

[ (k ) 2]2 . If G is on the right of P, which occurs when we havek

2, then there is no relevant area where we have < 0, and,

as in the first case, we have ,(k, k) =12

1

2

. Finally, as in the first

case, when we have , we always have ,(k, k) = 0.In the third case, depicted in Figure 3, the square ABCD intercepts the

vertical line = 1. This occurs when we have k > 1 . Then allthe area where we have > 1 should be included. If we have < , thismeans that, in Figure 3, to the area of the triangle GBH, we have to addthe area of LNDH. We therefore have ,(k, k) =

A(GBH)+A(LHND)42

. We can

write A (LHND) = A (LIH) + A (IHND). We have A (LIH) = 12

IH2

17


18/23

and A (IHND) = IH.HD. We have HD = EF = 2 and IH = OT

1 = OE + ET 1 = k + 1. Let us define k + 1.we get A (LIH) = 1

22 and A (IHND) = 2. This implies ,(k, k) =

12

1

2+ 182 (

2 + 4) . However this calculus is valid only when G is

on the left of M, as in Figure 3. This occurs when we have AG < AM. As wehave AG = EF = 2 and AM = AB IH = 2 , this inequality can bewritten 2 < 2, or equivalently < 2 ( ) , which can also be writtenk < 1 + 2 ( ) . When this inequality is not verified, i.e. when wehave k 1 + 2 ( ) , then G is on the right of M, and we simplyhave ,(k, k) =

A(MBND)42

. As we have A (MBND) = IH.BD = 2, this

implies ,(k, k) =12. Finally, when we have , then we also have

,(k, k) =A(MBND)

42= 1

2

.

These results can be summarized in the following Proposition:

Proposition 3 (i) In the case k < 2, when we have < , we get

,(k, k) =1

2

1

2 1

82[ (k ) 2]2 (10)

and, when we have , we get ,(k, k) = 0(ii) In the case

2

k

1

, when we have < , we get

,(k, k) =1

2

1

2(11)

and, when we have , we get ,(k, k) = 0(iii) In the case < and1 < k < 1 + 2 ( ) we get

,(k, k) =1

2

1

2+

1

82

2 + 4

(12)

where we have defined k + 1 (13)

(iv) In the case k 1 + 2 ( ) if we have < , and in thecase k > 1 if we have , we get

,(k, k) =1

2

(14)

The equilibrium condition is ,(k, k) = . First, consider the case < . Then, if we look for a solution satisfying the inequality k < 2 of

18


19/23

case (i) of Proposition 3. In the case < d, using (10) , ,(k, k) = gives

a second order equation in the variable (k ). It can easily be shown thatthis equation has solutions if and only if we have 0 < < 1

2

1

2; and,

when this last inequality is satisfied, there is a unique solution satisfying theinequality k < 2, and this is the solution given by (6) and (7) ofProposition 2.

In the same way, we can look for a solution satisfying 1 < k 1 .As we have > 0, the equilibrium condition ,(k, k) = in the case requires k > 1 . Then, this solution is given by = 2, which yieldsthe solution for k given by case 2, and therefore by equations (6) and (9) , ofProposition 2. Note that for all possible values of (which satisfy 0 < < 1)this solution verifies the required inequality k > 1 .

Finally, at the beginning of the Appendix, in order to calculate the prob-ability distribution of, conditional on having xi = k, we have assumed thatk belongs to the interval

2, + 1 +

2

. We therefore have to verify that

the equilibrium values of k we found satisfy this requirement. This can beseen in the following way. In the extreme case k =

2, we would be in

the case of Figure 2 where E would have coordinates

2, 0

. As we have

19


20/23

Figure 1

O

A B

DC

E F

G

H

1

j

(Z)

Figure 1:

assumed 2 , this implies that the square ABCD would be necessarilyentirely contained in the region 0 on the left of the vertical axis. Wewould thus have ,(k, k) = 0 in this case. In the same way, in the other

extreme case k = + 1 + 2

, we would be in the case of Figure 3 where Ewould have coordinates

1 +

2, 0

. As we have assumed 2

, this impliesthat the square ABCD would be necessarily entirely contained in the region 1 on the right of the vertical axis = 1. This would give ,(k, k) = 1 inthis case. But, in Figures 1, 2 and 3, an increase in k has the effect of shiftingthe square ABCD and the straight line (Z) to the right by the same amount.Thus, a greater k decreases the probability of having < 0 or increases the

probability of having > 1 (or possibly could leave them unchanged). Thisimplies that ,(k, k) is a non decreasing function of k (this property couldalso easily be seen from Proposition 3). As a consequence, because we have0 < < 1, the solution of the equation ,(k, k) = necessarily belongs to

the interval

2

, + 1 + 2

. QED

20


21/23

Figure 2

O

A B

C D

E F

H

R

PG

j

Q

S

(Z)

Figure 2:

Figure 3

A B

C D

E

F

G

HI

L

M

N

O

j

1

(Z)

T

Figure 3:

21


22/23

References

[1] Carlsson, H. and E. van Damme (1993), "Global Games and EquilibriumSelection", Econometrica, Vol. 61, No. 5, 989-1018.

[2] Cheli, B. and P. Della Posta (2007), "Self-fulfilling Currency Attackswith Biased Signals, Journal of Policy Modeling, 29, 381-396.

[3] Diamond, D. and P. Dybvig (1983), Bank Runs, Deposit Insurance, andLiquidity", Journal of Political Economy, 91, 401-419.

[4] Ellsberg, D. (1961),"Risk, Ambiguity, and the Savage Axioms", Quar-terly Journal of Economics, 75,.643-669.

[5] Gajdos, T, Tallon, J.M. and J.-C. Vergnaud (2004), "Decision Makingwith Imprecise Probabilistic Information", Journal of MathematicalEconomics, 40, 647-681.

[6] Gilboa, I. and D. Schmeidler ((1989), "Maximin Expected Utility withNon-Unique Prior", Journal Mathematical Economics,18, 141-153.

[7] Goldstein, I. and A. Pauzner (2005), "Demand-Deposit Contracts andthe Probability of Bank Runs", The Journal of Finance, Vol. LX,No. 3, 1293-1327.

[8] Kawagoe, T. and T. Ui (2010), "Global Games and Ambiguous

Information: An experimental Study", Department of ComplexSystems, Future University-Hakodate, and Faculty of Economics,Yokohama National University, September, http://wakame.econ.hit-u.ac.jp/~oui/ggex.pdf

[9] Knight, F.H. (1921), Risk, Uncertainty and Profit, Boston: HoughtonMifflin.

[10] Metz, C.E. (2003), "Private and Public Information in Self-FulfillingCurrency Crises", University of Frankfurt, Finance Department,June, http://128.118.178.162/eps/if/papers/0309/0309006.pdf

[11] Morris, S. and H.Y. Shin (1998), "Unique Equilibrium in a Model ofSelf-fulfilling currency attacks, American Economic Review, 88 (3),587-597.

[12] Morris, S. and H.Y. Shin (2003), "Global Games: Theory and Appli-cations", in Advances in Economics and Econometrics: (Proceedingsof the Eighth World Congress of the Econometric Society), ed. by M.Dewatripont, L.P. Hansen and S.J. Turnovsky, Cambridge UniversityPress.

22


23/23

[13] Morris, S. and H.Y. Shin (2004), "Cordination and the Price of Debt",

European Economic Review, 48, 133-153.

[14] Mukerji, S. and J.-M. Tallon (2004), "An overview of economic appli-cations of David Schmeidlerss models of decision making under un-certainty" in I. Gilboa, (ed), Uncertainty in Economic Theory: Acollection of essays in honor of David Schmeidlers 65th birthday,Routledge Publishers.

[15] Obstfeld, M. (1996), "Models of Currency Crisis with Self-Fulfilling Fea-tures", European Economic Review, 40, 1037-1047.

[16] Rochet, J-C. and X. Vives (2004), "Coordination Failures and the

Lender of Last Resort: Was Bagehot Right after All?", Journal ofthe European Economic Association, December, 2(6), 1116-1147.

[17] Schmeidler, D. (1989), "Subjective Probability and Expected Utilitywithout Additivity", Econometrica, Vol.57, No. 3, May, 571-587.

[18] Ui, T. (2009), "Ambiguity and Risk in Global Games", Fac-ulty of Economics, Yokohama National University, June,http://wakame.econ.hit-u.ac.jp/~oui/ggmp.pdf

23

ambiguity and coordination in a global game model of financial crises

Documents