56
Introduction to Global Games Hyun Song Shin Princeton University Lecture at University of Western Ontario 5th September 2008
Introduction to Global Games
Hyun Song ShinPrinceton University
Lecture at University of Western Ontario5th September 2008
Global Games: A First Look
Model of self-ful�lling currency attack.
Net bene�t to govt. of holding peg is B(+
�;�`), where
� is underlying strength of economy (reserves)
` is proportion of speculators who attack
For concreteness,B (�; `) = � � `
So, peg abandoned if and only if
� < `
) Potential for multiple equilibria
1
� When � < 0, peg fails irrespective of speculators' actions
� When � � 1, peg survives irrespective of speculators' actions
� When 0 < � � 1, the peg is \ripe for attack". Outcome depends onaggregate actions of speculators.
The model can alternatively be interpreted as a \creditor grab race" whereindividual secured creditors can seize assets of a distressed �rm.
Fears that others would seize assets (and hence derail the �rm) justi�esseizing assets oneself.
2
� Elements of model{ Speculators, indexed by [0; 1]{ Two actions: attack, refrain.{ Payo� to `refrain' is zero{ Cost of attack is t, but pro�t from collapse of peg is 1.
The cost of attack can be interpreted as the interest di�erential betweendollars and the target currency.
Payo� to `attack' depends on state �, proportion ` of creditors who attack
v (�; `) =
�1� t if ` > ��t if ` � �
Coordination problem when � 2 (0; 1). There are two equilibria for each �- \all attack", and \all refrain".
3
Global GameNow suppose that � is not common knowledge. Instead, � uniformlydistributed over an interval that includes [0; 1].
Rather than knowing � for sure, speculator i observes noisy signal
xi = � + si
where si uniformly distributed over [�"; "], where " > 0 is small.
Now, strategies must be conditional on the signal xi, rather than �. Hence,strategy is mapping
xi 7�! fAttack, Refraing
Posterior density over � conditional on xi is uniform over
[xi � "; xi + "]
4
As a �rst step, restrict attention to switching strategies.�attack if xi < x
�
refrain if xi � x�
Let \failure point" �� be the threshold value of � where the peg just fails.
� Failure point �� depends on switching point x�
� Switching point x� depends on failure point ��
5
Failure point �� solves � = `. If all follow x�-switching, ` is the proportionwhose signal is below x� when the true state is ��.
` =x� � (�� � ")
2"
So, �� = ` if and only if
�� =x� � (�� � ")
2"
This gives a linear relationship between the switching point x� and thefailure point ��.
x� = (1 + 2") �� � " (Eq 1)
This equation gives �� as a function of x�.
6
Now, consider reasoning of speculators. At switching point x�, speculatoris indi�erent between `attack' and `refrain'.
Pr (peg failsjx�) (1� t) + Pr (peg staysjx�) (�t)= Pr (peg failsjx�)� t= 0
Peg fails i� � < ��. So Pr(� < ��jx�) = t, or
�� � (x� � ")2"
= t
This gives another linear relationship between x� and ��. This time, thisrelationship gives x� as a function of ��.
x� = �� + " (1� 2t) (Eq 2)
7
*x
*θ
1
ε21+
Two linear equations in two unknowns: ��; x�. There is a unique crossingpoint, as long as " > 0, however small. Solving,
��� = 1� tx� = 1� t� " (2t� 1)
8
As a �nal check, we need to show that:�when xi � x�, speculator wants to attackwhen xi > x
�, speculator wants to refrain
Say xi < x�.
Pr (peg failsjxi) =�� � (xi � ")
2"
>�� � (x� � ")
2"
= Pr (peg failsjx�)
and conversely for when xi > x�.
So, switching strategy around x� is an equilibrium. In fact, we've shownthat it's the only equilibrium in switching strategies.
9
As " ! 0, x� ! ��. So, the limit as noise tends to zero still gives us aunique switching equilibrium.
Three questions
� Where did all the other equilibria disappear to?
� Can we say more than simply that we have unique equilibrium whenrestricted to switching strategies?
� What's happening in the limit as "! 0 to give us discontinuity?
10
Dominance Solvability
*0x
*θ
( )** xθ ( )** θxdominanceregion
no attack
11
After two rounds of deletion, we have
*0x
*θ
( )** xθ ( )** θxdominanceregion(iterated)
no attack(after two rounds)
*1x
12
Iterated deletion converges to switching equilibrium threshold
*0x
*θ
( )** xθ ( )** θxdominanceregion(iterated)
no attack(after iterated deletion)
*1x
*∞x
But there is an exactly symmetric argument \from below". Switchingequilibrium is dominance solvable (hence unique equilibrium).
13
Recapping the Story So Far...
In unique equilibrium of currency attack model,
�� = 1� tx� = 1� t� " (2t� 1)
Note that as "! 0, we have x� ! ��.
Fundamental uncertainty (i.e. uncertainty about �) disappears as " ! 0.However, there is still uniqueness of equilibrium. So, there is a discontinuityof the equilibrium correspondence in the limit as "! 0.
Why?
14
Strategic/Fundamental Uncertainty
A speculator's reasoning takes account of:
� uncertainty over true state �
� uncertainty over incidence of attack, `.
The fact that I am indi�erent between attacking and not attacking suggeststhat I hold certain beliefs about the the incidence of attack. What arethese beliefs?
Uncertainty over incidence of attack ` is over actions of others - i.e. it isstrategic uncertainty.
What happens to strategic uncertainty as "! 0?
15
Density Over Incidence of Attack
Consider the following question.
(*) Question. My signal is exactly x�. What is the probability thatproportion z or less of the speculators are attacking the currency?
The answer to this question gives the cumulative distribution function overthe proportion who attack, conditional on x�, evaluated at z. Denote it by
G (zjx�)
If we can obtain G (zjx�), we can di�erentiate it to obtain density overproportion who attack.
16
Two steps to answer question (*).
Step 1. If the true state � is higher than some benchmark level �0, then theproportion of speculators receiving signal lower than x� is z or less. Thisbenchmark state �0 satis�es:
x� � (�0 � ")2"
= z
Or�0 = x
� + "� 2"z
17
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�
z
G (zjx�)
�0 � " �0x�
�0x� x� + "
...................................................................................................... ...................
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Figure 1: Deriving G (zjx�)
18
Step 2. So, the answer to question (*) is given by the probability that thetrue state is higher than �0; conditional on signal x
�. This is,
(x� + ")� �02"
=(x� + ")� (x� + "� 2"z)
2"= z
The cumulative distribution function over the incidence of attack is theidentity function ) density function over the incidence of attack is uniformover [0; 1].
As " ! 0, the uncertainty concerning � dissipates, but the strategicuncertainty is very severe.
19
Back to Currency Attack Model Once More
In unique equilibrium of currency attack model, �� = 1 � t. This nowmakes sense, since density over ` is uniform.
l0 1
0
payoff toattack
t−1
t−1
t−
20
Finite Example
Two players (e.g. Thai importer, Thai property developer)
Sell Baht Hold BahtSell Baht �w;�w 0;�2wHold Baht �2w; 0 r; r
(w > 0; r > 0)
� Both (Sell, Sell) and (Hold, Hold) are equilibria.
� (Sell, Sell) risk-dominant if w > r:
� (Hold, Hold) risk-dominant if w < r:
21
Fundamentals, equally likely ex ante.
� � � � � � � � � � � � �0 1 2 3 � � � � � � N � 1 N N + 1
Fundamental 0 is \extremely bad". The game at 0 is
Sell Baht Hold BahtSell Baht �w;�w 0;�2wHold Baht �2w; 0 �w;�w
Fundamental N + 1 is \extremely good". The game at N + 1 is
Sell Baht Hold BahtSell Baht 0; 0 0; rHold Baht r; 0 r; r
22
Information Structure
fundamental 0 1 2 3 � � � N � 1 N N + 1� � � � � � � � � �#& #& #& #& � � � #& #& #� � � � � � � � � �
signal 0 1 2 3 � � � N � 1 N N + 1
Pair (i; j): fundamental i, signal j
Prob (i; j) =
�12Prob (i) if i = j or j = i+ 10 otherwise
unless i = N + 1, when
Prob (N + 1; N + 1) = Prob (N + 1)
23
Case 1 (w < r)
\Hold" is risk-dominant
Player 1's signal has no noise, but player 2's signal is noisy
0 1 2 3 � � � N � 1 N N + 1� � � � � � � � � � Player 1#& #& #& #& � � � #& #& #� � � � � � � � � � Player 20 1 2 3 � � � N � 1 N N + 1
Solution concept is iterated deletion of strictly interim dominated strategies.
24
Case 2 (w > r)
\Sell" is risk-dominant, and same information structure as case 1.
0 1 2 3 � � � N � 1 N N + 1� � � � � � � � � � Player 1#& #& #& #& � � � #& #& #� � � � � � � � � � Player 20 1 2 3 � � � N � 1 N N + 1
Solution concept is iterated deletion of strictly interim dominated strategies.
25
State Space
N + 1 � �N � �
Signal ... � � � ...3 � �2 � �1 � �0 �
0 1 2 3 � � � N N + 1Fundamental
26
Event GG = f(i; j)jmiddle game is played at (i; j)g
N + 1 � �N � �
Signal ... � � � ...3 � �2 � �1 � �0 �
0 1 2 3 � � � N N + 1Fundamental
27
Event K2 (G)
K2 (G) = f(i; j)j 2 knows G at (i; j)g
N + 1 � �N � �
Signal ... � � � ...3 � �2 � �1 � �0 �
0 1 2 3 � � � N N + 1Fundamental
28
Event K1K2 (G)
K1K2 (G) = f(i; j)j 1 knows K2 (G) at (i; j)g
N + 1 � �N � �
Signal ... � � � ...3 � �2 � �1 � �0 �
0 1 2 3 � � � N N + 1Fundamental
29
Event E is common knowledge at ! if
! 2\i
Ki (E)
and
! 2\i
Ki
\i
Ki (E)
!and
! 2\i
Ki
\i
Ki
\i
Ki (E)
!!...
Event G is never common knowledge.
30
Public Good Contribution Gameaka Larry Summers Game
Continuum of players indexed by [0; 1]
Action set fContribute, Opt outg
� is proportion who contribute, but there is critical threshold �̂ necessaryto produce public good
Common cost of contribution c
Payo� to Contribute is �1� c if � � �̂�c if � < �̂
Payo� to Opt out is zero.
31
Intuitions
In the benchmark case where c is common knowledge, there are twoequilibria for each c 2 (0; 1).
\all contribute" and \all opt out" are both equilibria for c 2 (0; 1). Bothequilibria are strict (trembling hands will not a�ect them).
Ostensibly, c is common knowledge, but behavior (say, in experiments)betrays lack of con�dence in this proposition.
Larry Summers's thought experiment
\Reverberant doubt"
32
Intuitition suggests we can draw a (possibly fuzzy) boundary betweensuccessful provision and failure.
1 failure
c
success0 �̂ 1
Boundary should be downward sloping.
Suggestive of switching strategy around c�, but where c� depends on �̂.
33
\Private Values" Global Game
Depart from benchmark model by introducing small heterogeneity in thecosts of provision (hence \private values" global game).
There are two components in the cost of provision - a population-wide term�, and idiosyncratic term si. Player only knows his own cost, and not �and si separately.
ci = � + si
�, uniform
si � U [�"; "], i.i.d.
34
Look for equilibrium in switching strategies around common c�.�contribute if ci � c�not contribute if ci > c
�
Denote G (�̂jc�) = Pr (� < �̂jc�). Expected payo� to contribute is
�c�G (�̂jc�) + (1� c�) (1�G (�̂jc�))= (1� c�)�G (�̂jc�)= 0
So G (�̂jc�) = 1 � c�. We can solve for switching equilibrium if we knowG (�̂jc�).
35
Suppose all players use switching strategy around c�:
Question: \My cost is c�. What is the probability that � is less than z?"In other words, what is G (zjc�)?
Answer: G (zjc�) = z.
) Density over � is uniform. As " ! 0, density over � stays uniform.Then
1� c� = G (�̂jc�)= �̂
So, we can solve for switching point c� as function of �̂.
c� = 1� �̂
36
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c�
�̂00 1
1
1=3
2=3
Figure 2: c� as function of �̂
37
In Larry Summers's thought experiment, investor stakes 10 dollars, andeither gets 11 dollars or loses the whole stake.
) normalized cost: c = 10=11
) �̂ = 2=3 is too stringent.
Prediction is that there will be failure to provide public good. In general,
� There is large region of ine�ency: the whole upper triangle is regionwhere successful provision is feasible, but does not happen.
� Presumption (common in corporate �nance) that outcomes are alwayse�cient at the ex post stage may be too strong
{ bargaining may not be a good modelling tool{ gets worse as decisions become decentralized
38
Uniform Density - How Restrictive?
A Gaussian example
ci = � + si
� � N (y; 1=�)
si � N (0; 1=�)
Question (*) once again. My cost is exactly c�. What is the probabilitythat proportion z or less contribute?
Let �0 be the such that, when � = �0 the proportion of players with costlower than c� is exactly z.
��p
� (c� � �0)�= z
39
or
�0 = c� � �
�1 (z)p�
Next, what is the probability that � lies above �0 conditional on my costbeing exactly c�? Answer gives G (zjc�)
G (zjc�) = 1� ��p
�+ ���0 � �y+�c
�
�+�
��= �
�p�+ �
��y+�c�
�+� � �0��
= ��p
�+ ���y+�c�
�+� � c� + �
�1(z)p�
��= �
��p�+�
(y � c�) +q
�+�� �
�1 (z)
�6= z
40
So, density is not uniform in the Gaussian case. But note that
G (zjc�)! z as � !1
In the limit as heterogeneity disappears, density over proportion whocontribute is uniform. Uniform density over proportion who contributeis a robust result.
Logistic approximation: � (x) ' 11 + e�mx
G (zjc�) ' 1
1 +�1�zz
�q�+�� exp
�m�p�+�
(c� � y)�
41
0
0.2
0.4
0.6
0.8
1
G(z|c*)
0.2 0.4 0.6 0.8 1z
� = 1; � = 3, y = 0:2 (dark), y = 0:8 (faint)
42
Densities
0
0.2
0.4
0.6
0.8
1
1.2
g(k|c*)
0 1k
� = 1; � = 3, y = 0:2 (dark), y = 0:8 (faint)
43
Solve c� as a function of �̂ from indi�erence condition: 1� c� = G (�̂jc�)
�̂ ' 11+�
c�1�c� exp
�m�p�+�
(y�c�)��r �
�+�
44
0
0.2
0.4
0.6
0.8
1
c*
0.2 0.4 0.6 0.8 1k^
Plot of c� (�̂) for y = 0:2 (dark),y = 0:8 (faint)
45
Liquidity Black Holes
1987 crash, 1998 crisis...
� Not merely rapid adjustment of prices
� Feedback process
� Endogenous shortening of horizons: agency problems, bankruptcy andmargin constraints
� Severity depends on residual demand curve
46
Global Game Model
� Asset trades at date 1 and date 2
� Liquidation value is v + z
{ v common knowledge at date 1{ z � N
�0; �2
�� Two groups of traders:
{ Risk-neutral traders, with loss limits fqig{ Risk averse, long-horizon traders (exponential utility)
) residual demand curve for asset where s is proportion of risk-neutraltraders who sell
p = v � cs
47
Loss limit for trader iqi = � + �i
where� � U
��; ���; �i � U [�"; "]
Strategy is mapping(v; qi) 7�! fhold, sellg
Three cases:v � c � qi Hold is dominantv < qi Sell is dominantqi � v < qi + c Intermediate region
48
Payo�s
Payo� when loss limit is breached is zero.
If trader i sells when aggregate sale is s, his place in queue is uniformlydistributed over [0; s].
Let ŝi be largest aggregate sale that does not breach trader i's loss limit.That is, ŝi is de�ned as
qi = v � cŝior
ŝi =v � qic
Probability of no breach in loss limit
ŝi=s
49
Expected payo� to sell is
w (s) =
8>>>:v � 12cs if s � ŝi
ŝis
�v � 12cŝi
�if s > ŝi
Expected payo� to hold is
u (s) =
�v if s � ŝi0 if s > ŝi
Payo� di�erence u (s) � w (s) is non-monotonic in s. Hence, strategiccomplementarity fails, and dominance solvability cannot be guaranteed.
50
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. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
s
v
0
qi
1bsi
p = v � cs
u(s)
w(s)
Figure 3: Payo�s
51
Solving for Switching Equilibrium
Trader i uses switching strategy
(v; qi) 7�!�hold if v � v� (qi)sell if v < v� (qi)
At switching point v� (qi), the density over aggregate sale s is uniform overthe unit interval [0; 1].
Solve for switching point from
Z 10
[u (s)� w (s)] ds = 0
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s
v� (qi)
0 1
qi
bsi
u(s)
w(s)
A
B
Figure 4: Expected Payo�s in Equilibrium
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Substituting out u (s), w (s) and noting ŝ = (v � q�) =c,
12c
Z v�q�c
0
sds =(v�q�)(v+q�)
2cs
Z 1v�q�c
1
sds
or
v � q� = 2 (v + q�) log cv � q�
Theorem 1. There is an equilibrium in threshold strategies where thethreshold v� (qi) for trader i is given by the unique value of v that solves
v � qi = c exp�
qi � v2 (v + qi)
�
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There is no other threshold equilibrium.
c0 1 2 3 4 5
v
0
1
2
3
4
5
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v � c = qi
v = qi
v�hold dominant
sell dominant
Figure 5: v� as a function of c. qi = 1
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