Game Theory

Global Games and Coordination

Bank runs are a classic coordination multiplicity: if everyone believes the bank will fail, they withdraw and the bank does fail. Global games show that a small fundamental uncertainty selects exactly one of these equilibria - the one determined by the bank's actual asset quality.

  • **Currency crises:** attacks on the British pound (1992) and Thai baht (1997) are predicted by Morris-Shin with threshold theta* = 1 - c
  • **Bank runs:** Diamond-Dybvig (1983) combined with Morris-Shin: the threshold rule determines when a bank run is objectively justified
  • **IMF interventions:** public announcements of support lower c for attacking speculators, shifting theta* and stabilizing the regime
  • **Crypto de-peg events:** LUNA/UST collapse (2022) - a textbook global game where threshold theta* was reached and triggered unstoppable coordination

Предварительные знания

  • Bayesian games
  • Bayes-Nash equilibrium
  • Iterated elimination of dominated strategies
  • Signaling games

Global games: equilibrium uniqueness

Morris and Shin in 1998 showed that a small informational noise eliminates equilibrium multiplicity in coordination games. The currency attack game under common knowledge of theta has infinitely many equilibria - attack for any theta, do not attack for any theta, and everything in between. Adding a small signal noise selects a unique equilibrium: a threshold rule at theta* = 1 - c. The IMF has used this model since 2002 to assess currency crisis risk. Carlsson and van Damme independently obtained the same result in 1993.

How do global games resolve the multiplicity problem in coordination games?

Carlsson & van Damme (1993), Morris-Shin (1998): in a coordination game with noisy signals theta_i = theta + epsilon_i, as epsilon → 0 iterated deletion of strictly dominated strategies leaves a unique equilibrium. The threshold is determined by risk-dominance, not payoff maximisation.

Coordination and limit analysis

Key distinction between global games and classical coordination games: in the noiseless game with common knowledge of theta, every theta in (0,1) has two equilibria (all attack / none attack). Adding any signal noise removes common knowledge - players no longer know that others know that others know... It is the absence of this higher-order knowledge that selects the unique equilibrium.

What threshold theta* do players follow in a global game with private signals?

With noise epsilon_i ~ U[-eps, eps], player i chooses a threshold: attack if x_i < theta*. The equilibrium condition is u(attack | x_i = theta*) = u(no-attack | x_i = theta*). For symmetric games theta* solves the risk-balance equation; as epsilon → 0 it coincides with Selten risk-dominance.

Applications: bank runs and currency attacks

The Morris-Shin model explains the bank run puzzle: a bank with sound assets (theta > theta*) will not collapse even under panic, if depositors know this. It is uncertainty about the fundamental - not its objective value - that causes instability. This justifies central bank transparency: reducing uncertainty shifts theta* toward weaker fundamentals, stabilizing the regime.

Which real-world phenomena are modelled by global games?

Morris-Shin (1998) applied global games to currency attacks: each trader receives a noisy signal about currency fundamentals and decides whether to attack. The unique threshold equilibrium explains why weakened currencies are attacked "in coordination" without explicit collusion. Analogously: bank runs (Diamond-Dybvig).

Connections to other fields

Global games connect equilibrium theory, macroeconomics, and the theory of financial crises.

  • Signaling Games — Global games generalize signaling to coordination problems with continuous signal noise
  • Bayesian Games and Incomplete Information — Technically global games are Bayesian games with a continuous type space
  • Game Theory in Tech: Pricing and Markets — Bank-run, currency crisis and IPO auction models rest on global games

Итоги

  • Global game: fundamental theta observed with noise x_i = theta + epsilon*xi. As epsilon -> 0, a unique threshold equilibrium is selected
  • Unique threshold theta* = 1 - c: attack if x_i <= theta*, do not attack if x_i > theta*
  • Proof: strict dominance at theta < 0 and theta > 1 anchors IESDS, which collapses the intermediate zone
  • Common knowledge vs. higher-order knowledge: noise breaks common knowledge and removes coordination traps of multiple equilibria
  • Applications: currency attacks, bank runs, stablecoin de-peg events - all described by the same threshold model

Вопросы для размышления

  • Why does the noiseless game with theta as common knowledge have infinitely many equilibria while the game with small noise has exactly one?
  • How do central bank public announcements about defending an exchange rate affect the parameter c and the threshold theta*?
  • How can the global games model help regulators design mechanisms that prevent bank runs?
Global Games and Coordination

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