private knowledge vs. common knowledge carlson and van dam (see econometrica, 1991) and morris and...
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Private Knowledge vs. Common Knowledge
Carlson and Van Dam (see Econometrica, 1991) and Morris and
Shin (see AER, 1998)
Coordination Game
4,40,2
2,03,3
A B
A
B
Two Equilibria: (4,4) and (3,3)
Carlson-Van Dam’s “Global Game”
4+X, 4+X0+X, 2
2 ,0+X3 ,3
A B
A
B
Add X to Action A And view X as a variable
X
-2 3
BB dominates
AAdominates
Both, AA and BBare Equilibria
Global Game:
“Connecting” games together
Private Signal
X
Uniform
XX
i
ii
],[~
Random variable
-2 3
B dominates A
dominates
Both AA and BBare an Equilibrium
iX
If my signal is –2I suspect that the
other player signalis –2 with 50% probability :
thus action B dominates
If my signal is 3I suspect that theother player signal
is 3 with 50% probability :thus action A dominates
-2 3
B dominates
Adominates
iX
X*=1/2
32
1)2(
2
1*)0(
2
1*)4(
2
1 XX X*=1/2
Cutoff X is determinedso that player is
indifferent between playing A or playing B:
A-RDB-RD
A-PD
At X=0, (the original game), BB or AA with 50-50=risk dominance
PD=Pereto dominanceRD=Risk dominance
Morris and Shin
)(*
)(
]1,0[~
fe
f
U
Randomfundamental
FloatingExchange rate
Pegged exchangerate
Speculators’ cost of attack: t>0
Speculator’s gain if peg abandoned: )(* fe
Speculators’ cost of attack: t>0
Speculator’s gain if peg abandoned: )(* fe
Government value of defending peg:
)(
)(),(
),(
),(
0
C
CGov’t cost of defending:
Gov’t observes:
Government indifference:
Government defends if:
-fraction of attackers in the population
C() increases in alphadecreases in theta
Dominance region for attack speculators
),0(C )abandons peg below even if no one
Attacks (.
Attack strategy dominates
tfe )(* )no attack above even if government
abandons peg (.
Attack strategy dominates
No-attack strategy
dominates
PotentialMultiple
Equilibria
Signals
Sequence of steps:1 .Realized;
2 .Speculators observe--
],[~
Ui
ii
small
3 .Government observes and decides on peg--
Abandons if
),(
)(
]1,0[~U
*i
Attack strategy dominates
No-attack strategy
dominates
PotentialMultiple
equilibria
Probability that government abandons the peg
)(1)(1 *i
An indifference at * :
t=prob.gain= )(*)(1 ** fe