Dealing with Multiple Equilibria: Global Games

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Dealing with Multiple Equilibria Global Games
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Attempts to Deal with Multiple Equilibria

• When there are multiple equilibria, how is an
  equilibriun selected?
• Implicit in second generation models and
  financial panics approach sunspots
• Morris and Shin adding noisy information can
  change results dramatically

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The Morris-Shin Model

• Fundamental ? distributed U0,1
• Exchange rate (foreign currency per domestic
  currency) if floating an increasing function
  f(?)
• The government attempts to peg at e gt f(1)

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Speculators

• There is a continuum of speculators that can
  attack the currency, by short selling one peso at
  cost t
• An attacking speculator gains e-f(?) if the peg
  is abandoned, pays t regardless
• Otherwise, payoff is zero

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Government

• Government observes the number of attacking
  speculators (a), and chooses to defend the
  currency or abandon the peg.
• Defending the currency has value v and costs
  c(a,?)
• If the peg is abandoned, government gets zero

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Assumptions

• c(.,.) increases in a, decreases with ?
• c(0,0) gt v (fundamentals may be so bad the
  government will abandon peg regardless of
  speculation)
• c(1,1) gt v (an all attack forces collapse of peg)
• e-f(1) lt t (if fundamentals are strong enough,
  it is not profitable for speculators to attack)

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Two critical values of ?

• Define L to satisfy
• c(0, L) v
• ? If ? lt L, peg is always abandoned
• H is defined to satisfy
• e - f(H) t 0
• ? If ? gt H, attacking is not profitable

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• A critical assumption is that L lt H. Then there
  are three regions for ?
• If ? L, the currency is unstable
• If ? H, the peg is stable
• If L lt ? lt H, the currency is ripe for attack
  (multiple equilibria region)

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Introducing Noise

• Suppose that speculators do not observe ?, but
  observe a signal x
• x is distributed uniform ? e, ? e
• A (mixed) strategy is a function p(x)
  probability that after observing x an speculator
  attacks fraction of speculators that observe x
  and attack

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Analysis

• Let a(?) be the minimum attack size such that
  government devalues if fundamentals are ?
• a(?) 0 for ? lt L, otherwise is the value of a
  that solves c(a,?) v
• Next, calculate expected payoffs
• Let s(?, p) size of attack if fundamental is ?
  and strategy is p

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• Let A(p) set of fundamentals such that peg is
  abandoned ? s(?, p) a(?)
• h(?,p) realized payoff from attacking when
  strategy is p and fundamental is ?
• e - f(?) t if ? is in A(p)
• - t if not

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u(x,p) expected payoff from attacking, after
observing signal x, when strategy profile is p
? p is an equilibrium if p(x) 1 whenever
u(x,p)gt0, and p(x) 0 whenever u(x,p) 0
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Main Theorem

• There is a unique equilibrium. In it, the
  government abandons the peg if and only if
  fundamentals are below a value ? . An speculator
  attacks if and only if he receives a signal below
  some x.

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• Intuition if x is close to L, speculator knows
  that speculators to the left will attack, and
  be successful, so it pays to attack. If x is
  close to H, in contrast, know that people to the
  right will not attack. One can continue this
  reasoning until one finds a marginal speculator
  that knows that everybody to his left attacks,
  and everyone to his right does not
• Fundamentals are not common knowledge

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• In the limit as e goes to zero, ? tends to the
  solution of t (e - f(? ))/2
• So uniqueness prevails even if informational
  noise is arbitrarily small

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Some comparative statics

• A marginal increase in t reduces ? by 2/f(? )
• argument for throwing sand in the wheels of
  finance
• if speculative attacks lead to large effects,
  increasing transactions costs have a small impact
  

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Comments

• This is an elegant model, useful for other
  purposes
• Analysis is compelling, setup relatively
  realistic, suggests one way to deal with the
  multiple equilibria conundrum

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• However, it substitutes one source of uncertainty
  for another the question of how an equilibrium
  is selected when there are many of them is
  exchanged for the question of what information
  people observe

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• Model is very stylized, perhaps too stylized to
  be useful in practice
• More importantly, a number of auxiliary
  assumptions are critical in generating the
  uniqueness result