Coordination, incomplete information and crises'

1
Coordination, incomplete information and crises.

• The contagion argument, currency crises,bubbles
  and crashes.

2
The contagion argument

• Does incomplete information provide the Graal ?

3
A normal form game complete information.

• The game Carlsson-Van Damme (1993)
• Analysis
• If is ? known for sure.
• ? gt1 a1, a2, dominant, ? lt0 b1, b2, dominant.
• Intermediate
• 2 nash equilibria in pure strategies, 1 in mixed.
• Note ? gt(1/2)
• If 2 plays a2 with probability ½ at least, 1s BR
  is a1
• If ? lt1/2 replace a by b, 1 by 2
• Outcome.
• ? gt(1/2) (resp. lt) (a1, a2) (resp. (b1,b2)) is
  the risk-dominant Nash equilibrium

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Incomplete information the game made
 global 

• Assume s(i)?ev(i),
• e scaling factor, v(i) noise, mean zero, v(.)
  iid, sym. density
• e going to zero almost complete information.
• The idea
• Uncertainty on pay-offs is a general phenomenon.
• Here incomplete information on payoffs.
• Consequences discontinuities, higher order
  beliefs, strategies..
• ? is no longer CK (as it is assumed in the
  non-noisy version)
• Note, if v is uniform on -e,e,
• if i observes s, he knows ? ?s-e,se.
• He knows that j has observed s(j) ? s-2e,s2e,
• He knows that j knows) that ? ? s-3e,s3e,
• He knows that j knows that he has observed s(i)
  ?s-4e,s4e.
• Strategies have to be defined as functions of
  s(i), not ?.

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The global game the contagion argument.

• Starting point.
• Strategies have to be defined as functions of
  s(i), not ?.
• The  eductive  anchor to the argument e
  small, v(i) finite support.
• For s(i) gtgt1, (?gt1), then (a,a) is played
• For s(i)ltlt0, then (b,b) is played.
• The contagion (infection) argument.
• If you believe that your opponent plays a for s
  Sgt ½ ?.
• When you receive S, you believe that the
  probability of your opponent receiving more than
  S is ½. ..(symmetry)
• Then the probability he will play a is greater
  than ½.
• It is (almost) sure that ?gt1/2. (e small enough)
• You play a for S S-?, ? not too small.
• Remark on the size of the noise..

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The global game incomplete inf.outcome-1.

• The contagion (infection) argument.
• If you believe that your opponent plays a for s
  Sgt ½
• You play a for S S-?, ? not too small.
• Iterate
• The result.
• When the noise is small, then, the Nash
  equilibrium strategy stipulates play a if s
  gt(1/2) .?, play b if slt(1/2) -. ?,
• The equilibrium strategy is the outcome of the
  iterated elimination of dominated strategies.
• Salient features of the outcome.
• The outcome tends to be, either (a,a) or (b,b)
• Unless ? is too close to 1/2

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The global game incomplete inf.outcome - 2

• Salient features of the outcome.
• At the limit, uniqueness, the risk dominant
  equilibrium is selected.
• Even when Pareto-dominated
• The limit equilibrium is dominant solvable
• The graal !
• What matters
• The  anchor  contagion has to start..
• Specific characteristics of the noise, no..
• Small noise, yes.
• Specific structure of the 2.2 game, no
  (Carlsson-Van Damme)
• Two agents, no..see next
• Strategic complementarities..

8
Global game insights into contagion arguments

• The context
• a continuum of agents, i ? 0,1.
• Signals s(i) ?v(i), v(i), iid, mean zero,
• ?, density, F cumulative, sym. (normal ?(n,0,?))
• ? uniformly distributed on the real line !
• Hint results valid for a given law of ? and
  small noise.
• Beliefs
• If I receive s, I believe that ?s-v(i),
• Hence a posteriori on ? normal of mean s and
  precision (1/?2),.
• Beliefs on the signal received by j
• compose the laws, a priori on ?, noise on s from
  ?.
• Beliefs on signals for j Normal of mean s and
  stand dev.(?2 ?)
• Higher order beliefs..
• The beliefs on signals received by others, ..
• conditional on my signal, ..
• Is a basic coordinating device

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More on beliefs and the contagion argument.

• Context
• Signals s(i) ?v(i), v(i), iid, mean zero,
• ?, density, F cumulative, sym. (normal ?(n,0,?))
• ? uniformly distributed on the real line ! (small
  noise).
• Simple beliefs.
• If I receive s, I believe that ?s-v(i),
• Beliefs conditional on signal s.
• Expected mass of people who have received lt s
  ½.
• Beliefs on ranks conditional on signal s.
• I receive s Probability E less than t have
  received lts
• If ?, F(s- ?) observe lts, then the event E occurs
  iif ? ? s-F-1(t),
• whose probability is Fa(?) 1-Fs- s-F-1(t)t
  !!
• t is a random variable of uniform distribution on
  (0,1).
• Expectation t ½.
• Laplace ???

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More on beliefs and the contagion argument.

• Large noise.
• ? normal of mean y (known) and precision ?.
• s ?v(i), v(i) i.i.d and normal, precision,
  ?(1/?2), inv/ var.
• Observe s,
• Beliefs
• on ?
• Mean ?y ?s/(? ?), precision (? ?)
• on the distribution of other players signals.
• Mean ?y ?s/(? ? )
• variance ?2 (12/ (? ?2 ))(11/( ? ?2 )).
• Flatter cumulative..
• with uncertainty about ?
• With small ?.

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Currency crises

• Predictable or unpredictable ?
• Morris-Shin (1998)

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The Model Currency and Government.

• The currency.
• Fixed exchange rate e
•  fundamentals 
• ??m, M, strength
• The Government.
• Optimal defense strategy for the  Centre 
• If ? ltm, devalue.
• If ? gtM, figth
• If m? M, fight / prop.of speculators attacking
  lt a(?), a increasing.
• ? obs. by the Center

• The Govts optimal reaction.

a(?)
m
M
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Complete information multiplicity.

• The Government.
•  fundamentals  m, M, value e.
• Optimal defense strategy for the  Centre 
  a(?)
• The agents.
• If attack, individual cost C,
• Gain if devaluation e-f(?) gt0,
• increasing with the  overvaluation .
• grows when ? decreases.
• If m M, no attack.
• Perfect observability of the fundamentals.
• Outside m, M, unique equilibrium
• Inside two equilibria
• everybody attacks,
• nobody attacks
• Multiplicity unpredictability ? , sunspot ?

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What happens with incomplete information ?

• Assumption
• Signal s(i) ? e, (e uniform, support ? ,?)
  or e  small .
• Again, ? is not CK !
• Strategies
• Strategies are not actions 0-1, but
  actions/signal.
• Trigger strategies.
• Consequences
• There is a unique equilibrium in trigger strategy
  
• ss, attack, s gts, do not attack.
• It is the unique equilibrium, whatsoever.
• It is Strongly Rational (SR),
• (Eductively Stable, Dominant Solvable)
  equilibrium.

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Insights into a trigger strategy equilibrium
Prop/ S, ?

• Assumptions.
• Small uniform noise, ?-?, ??.
• Trigger iif s?s, attack.
• Probability of devaluation/ trigger s
• ? uniform on s-?, s?,
• s uniform on ? -?, ? ?,
• Proportion/slts/ ?
• ½ -(½?)(?-s)
• attacks succeeds iif
• ½ -(½?)(?-s)gta(s..)
• (?-s)gt ?(1-2a(s)),
• An event of prob. (1-a(s..))
• Necessary condition for equ.
• Proba. Success (1-a(s..),
• gain (ef(s. .)C
• Equilibrium.
• Slts, attack, sgts, no attack
• ?lt ?, devaluation succeeds.
• s-? ?,

• Trigger strategy s

s
s?
s2 ?
m
?
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Symmetric equilibrium in trigger strategies.

• A symmetric trigger strategy equilibrium.
• Find s such that iif everybody attacks iif slts,
  the best response is to attack iif slts.
• If I receive s, the probability that t /
  receive less than s is t.
• As if I were agnostic on whom is acting !
• Flat distribution (Laplacian)
• I am indifferent/ attacking in s, if appr.
• Prob t gta(s)(1-a(s)), gain (e-f(?)-t)
• (1-a(s))((e-f(s))t (small noise, ?s),
• Existence and uniqueness for small noise..
• Now,
• No other non symmetric trigger strategy
  equilibrium.
• No other equilibrium.
• SREE.

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Another approach Best resp./trigger strat.
2
3
1
4

• Induction argument
• Evbd obs s lt s attacks
• BR trigger strategy S(s)
• Best response
• 1, attacks,
• 2,3 Expected prop./ attack.people, expect.
  success and expect. gain
• decr. when s increases.
• S(s) BR trigger strategy
• Properties inspect and think.
• S(s) increasing.
• dS/dslt1 ?uniqueness.
• Small noise independant..?

s
S(s)
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The unique equilibrium is a SREE

• The induction argument.
• sltm -?, everybody attacks
• BR(m - ?)gtm- ?.
• Everybody knows that
• attack if BR(m - ?) ? s.
• Attack if BR(BR(m - ?)) ? s.
• Go on,
• and then from the top..
• It is CK
• that the trigger strategies s are played.
• The Graal !

BR(m-?)
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Conclusions.

• Descriptive value.
• Devaluation occurs for some treshold.
• More satisfactory than multiplicity..
• Not necessarily the only alternative
  explanation..
• Can be embedded in a dynamic framework.
• Policy relevance.
• Common sense
• Recommendations a(?, R) increasing in R,
  reserves.
• Increase reserves !

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Back on the global games problematics.

• Global games at intersection of
• a fundamentalist view of game theory
• agents characterized by beliefs, beliefs on the
  beliefs of others, beliefs on beliefs (Harsanyi,
  Mertens-Zamir,).
• an operational concern for Nash equilibrium
  selection,
• Robustness (Harsanyi, Kohlberg-Mertens).
• A concern for uniqueness, dominant solvability
  (SREE).
• Concern for actual economic problems.
• Indeed, the research has delivered
• A story that involves higher order beliefs,
• But on fundamentals onlly
• An attractive reflection on selection criterion,
• Carlsson-Van Damme.
• the notion of p-common beliefs, etc.
• The Graal in some contexts
• But with strategic complementarities,
• Not substitutabilities
• Interesting applications.

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Directions.

• The fundamentalist view of game theory.
• Weinstein-Yilmitz (2007
• Morris-Shin (2007)
• One-dimensional models.
• Away from small noise,
• Public information
• Different structure of the problem ..
• Other directions.
• Dynamics,
• Variants of the views of a crisis.
• Experiments.

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Going further.

• Other applications with one-dimensional
  uncertainty, strategic complementarities and
  large economies.

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A search problem with inc. inf.à la Chamley..
.
Mass a

• Remind
• the model/complete info.
• Population agents a
• Individual Costs c, F(c.),
• Profitability ?da
• Strategic complementarities
• Equilibria
• Three or
• One ?
• How flat is the distribution

C
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The model the role of inc. inf.

• Assumptions
• Agents know their cost but not the exact
  distribution.
• The mean of the distribution m drawn at random
  f(m)
• The cost c drawn at random g(c,m) cumulative
  G
• c is more risky
• A posteriori mp m, c, with standard
  assumptions, variancegtgt
• Consequences
• The cost distribution is flatter
• Example g close to a Dirac measure ___ belief 
  s ½
• Uniqueness is more likely.

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Incomplete information on distribution of costs

• Large noise.
• The mean of the distribution ? drawn at random
• ? normal of mean y (known) and precision ?.
• Then the cost c drawn at random
• c ?v(i), v(i) i.i.d and normal, precision,
  ?(1/?2),
• Everybody observes its own c,
• Beliefs on ?
• Mean ?y ?s/(? ?), precision (? ?)
• Beliefs on the distribution of other players
  signals.
• Mean ?y ?s/(? ? )
• variance ?2 (12/ (? ?2 ))(11/( ? ?2 )).
  Laplace wrong ?
• Flatter cumulative..
• with uncertainty about ?
• With small ?.

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The role of public and private information.

• A priori and a posteriori on ?
• Triggers the distribution of signals..
• Reliable public information shapes my view of the
  signal distribution
• Particular cases.
• g Dirac, whatever the signal, ½ smaller signal, ½
  higher signals..

g(c / ?)
f(?)
?
?y ?s/(? ?),
s
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Solving the incomplete information cas

• The relevant concept.
• V(c)expected proportion of people who have
  received a signal less than c when I have
  received c.
• W(c,c) expected proportion of joining people,
  given that everybody who has received less than c
  joins, given that I observe c
• W(c,c)lt V(c), if cgtc, decreases in c.
• The analysis
• Join iif sltc, c/ V(c)1 is a trigger strategy
  equilibrium.
• Multiplicity or uniqueness depends on the
  flatness of V(c), easy to analyse from the above
  formula on the distribution of signals.
• Case where the conditional distribution of costs
  is Dirac unique equilibrium, note the meaning
  of the equilibrium in terms of observable
  situations (the outcome is (almost) half time,
  everybody joins, half time nobody joins

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The inc. Inform. equil. If unique is a SREE

• Uniqueness
• Trigger strategies..
• If c lt c(0), join.
• Unique equilibrium.
• Strong Rationality
• Dominant solvable,  eductively stable.. 
• Induction c ltc(0) join
• In c, belief on the nunber of joining members
  less optimistic___c
• Convergence obtains but more slowly
• The Graal, again !

V(c)Prob cltc/c
C
C  
C
C(0)
c