Morris–Shin Global Games Model

The Morris–Shin Global Games Model is a game-theoretic framework in macroeconomics, international finance, and business and economics theory, developed by economists Stephen Morris and Hyun Song Shin in their landmark 1998 American Economic Review article "Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks" and generalized in their 2003 survey "Global Games: Theory and Applications." Extending the two-player global games incomplete-information perturbation method of Hans Carlsson and Eric van Damme (1993, Econometrica) to a continuum of heterogeneous market speculators, Morris and Shin resolved a longstanding dilemma in macroeconomic modeling: why coordination games with strategic complementarities—such as second-generation currency-peg attacks (Obstfeld, 1996), commercial bank runs (Diamond–Dybvig model, 1983), and sovereign debt rollovers—exhibit multiple self-fulfilling Nash equilibria when the underlying economic fundamental θ is common knowledge, preventing comparative-statics predictions or policy evaluation. Morris and Shin proved that if agents instead observe the true fundamental θ with an arbitrarily small amount of idiosyncratic private observation noise (x_i = θ + ε_i with variance σ² → 0), the absence of common knowledge triggers a contagious wave of iterated deletion of dominated strategies anchored in the upper and lower dominance regions, collapsing the continuum of self-fulfilling equilibria into a unique switching-threshold Bayesian Nash equilibrium (x*, θ*).

First-Generation vs. Second-Generation Currency Crisis Models

In international macroeconomics, theories of balance-of-payments and exchange-rate crises evolved through two classic generations before Morris and Shin (1998):

1. First-Generation Models (Krugman, 1979; Flood and Garber, 1984): A government runs fiscal deficits financed by domestic credit expansion while attempting to peg its nominal exchange rate using finite foreign-exchange reserves. Speculators foresee reserve exhaustion and launch a uniquely timed attack when the shadow floating rate equals the peg, but such models could not explain crises in countries with sound fiscal balances.
2. Second-Generation Models (Obstfeld, 1986, 1996): Maurice Obstfeld modeled the central bank as a policymaker weighing the benefit of defending an exchange-rate peg against the unemployment and interest-rate cost of fending off speculators. Because defense costs rise with the fraction l ∈ [0, 1] of attacking speculators, speculators' actions are strategic complements: an individual speculator wants to short the currency if and only if enough other speculators also attack.

Tripartition of Fundamentals Under Common Knowledge (θ̲, θ̄)

Consider a continuum of risk-neutral speculators i ∈ [0, 1], each choosing to Attack (a_i = 1) at transaction cost t ∈ (0, 1) or Not Attack (a_i = 0) (payoff 0). Let θ ∈ ℝ index the macroeconomic fundamental and l = ∫₀¹ a_i di ∈ [0, 1] denote the aggregate attack mass. Devaluation yields gross gain 1 (net profit 1 − t > 0), while a successful defense costs −t < 0. Under the linear defense threshold a(θ) = θ, the peg collapses if and only if l ≥ θ:

When θ is common knowledge, the parameter space ℝ partitions into three zones:

1. Lower Dominance Region (θ ≤ θ̲ = 0): The central bank devalues even if l = 0; attacking (a_i = 1) is dominant (unique Nash equilibrium l* = 1).
2. Upper Dominance Region (θ > θ̄ = 1): The peg survives even if l = 1; not attacking (a_i = 0) is dominant (unique Nash equilibrium l* = 0).
3. Intermediate Multiple-Equilibrium Zone (θ ∈ (0, 1]): Both self-fulfilling attack (l* = 1) and no-attack (l* = 0) are pure-strategy Nash equilibria, preventing probability or comparative-statics predictions.

Breaking Common Knowledge via Noisy Private Signals (x_i = θ + ε_i)

Following Carlsson and van Damme (1993), Morris and Shin (1998) relaxed the assumption that all speculators know θ and know that everyone else knows θ to infinite higher-order beliefs. Instead, suppose that nature draws θ from an improper uniform prior on ℝ (or a normal public prior N(y, 1/α)), and each speculator i ∈ [0, 1] privately observes a noisy private signal x_i:

where the idiosyncratic observation errors ε_i are independent and identically distributed across speculators according to a continuous symmetric distribution (such as normal N(0, σ²) with precision β = 1/σ²).

When speculator i observes signal x_i, speculator i faces strategic uncertainty about what signals x_j other speculators j ≠ i have observed. Even if θ ∈ (0, 1), speculator i knows that some other speculators received lower signals x_j < x_i, who in turn worry about speculators who received even lower signals x_k < x_j reaching down into the lower dominance region θ ≤ 0.

Step-by-Step Derivation of the Unique Switching Equilibrium

Assume an improper uniform prior on θ and normal idiosyncratic signal errors ε_i ~ N(0, σ²) with cumulative standard normal distribution Φ(·). Consider a symmetric switching-threshold strategy around a cutoff signal x*, under which every speculator j attacks (a_j = 1) if and only if x_j < x*.

1. Aggregate Attack Mass l(θ, x*) in State θ: Given the true fundamental θ, the fraction of speculators who receive signals x_j = θ + ε_j < x* and therefore attack the currency is deterministic by the law of large numbers:

Notice that l(θ, x*) is decreasing in the true fundamental θ: stronger fundamentals lead fewer speculators to receive pessimistic signals below x*.

1. Unique Fundamental Cutoff State θ*(x*): Since the central bank devalues if and only if aggregate attack mass satisfies l(θ, x*) ≥ θ, there is a unique fundamental cutoff state θ* at which the central bank is marginally indifferent between defending and abandoning the peg:

For all true fundamentals θ ≤ θ*, the attack mass l(θ, x*) ≥ θ* ≥ θ succeeds and the currency collapses; for all θ > θ*, the attack mass l(θ, x*) < θ* < θ fails and the peg survives.

1. Indifference Condition of the Marginal Speculator Observing Signal x*: Conditional on observing x_i = x* under a uniform prior, speculator i's posterior belief about θ is normally distributed with mean x* and variance σ²: θ | x_i = x* ~ N(x*, σ²). Therefore, the marginal speculator's subjective posterior probability that the currency collapses (θ ≤ θ*) is Φ((θ* − x*)/σ). For the marginal speculator at x_i = x* to be indifferent between attacking and not attacking, expected profit must equal zero:

Exact Threshold Formulas for (θ*, x*)

Using the symmetry of the standard normal distribution Φ(−z) = 1 − Φ(z), the marginal indifference condition Φ((θ* − x*)/σ) = t implies Φ((x* − θ*)/σ) = 1 − t. Substituting this into the Regime-Failure Condition Φ((x* − θ*)/σ) = θ* determines the unique equilibrium fundamental threshold θ* and signal threshold x*:

As the private observation noise vanishes (σ → 0), both the signal cutoff x* and the fundamental cutoff θ* converge to the same unique number inside (0, 1):

Theorem (Unique Equilibrium in the Morris–Shin Global Game — 1998, 2003): In the limit of vanishingly small idiosyncratic private signal noise (σ → 0), iterated deletion of dominated strategies eliminates all equilibria except the unique switching equilibrium at θ* = 1 − t: the currency peg collapses if and only if θ ≤ 1 − t and survives if and only if θ > 1 − t.

Strategic Uncertainty and Uniform Belief Over Attack Mass l

Why does the unique threshold equal θ* = 1 − t independent of the noise distribution? By the probability integral transform (Laplace's Principle of Indifference over rank order), for the marginal speculator whose own signal equals the threshold x_i = x*, the subjective posterior distribution over the fraction l ∈ [0, 1] of other speculators who received signals lower than x* is Uniform on [0, 1]. Because l ~ Uniform[0, 1], the probability that l ≥ θ* is 1 − θ*, and equating expected profit (1 − θ*)(1 − t) − θ*t = 0 gives θ* = 1 − t.

Bayesian Updating with Public Signal y (Precision α) and Private Signal x_i (Precision β)

Suppose speculators observe both a common public signal y = θ + η with precision α = 1/σ_y² and an idiosyncratic private signal x_i = θ + ε_i with precision β = 1/σ_x². Bayesian updating yields a normal posterior distribution for the true fundamental θ:

Substituting this weighted posterior mean into the marginal speculator's indifference condition at x_i = x*(θ*) produces the general Morris–Shin equilibrium equation for θ*:

Differentiating this locus with respect to θ* yields the Morris–Shin Uniqueness Criterion:

• When Private Information Is Sufficiently Precise (β ≥ α² / (2π)): The slope of the marginal speculator's posterior belief with respect to θ* is monotone, ensuring a unique global game equilibrium θ*. As β → ∞ (for any fixed public precision α), α/√β → 0 ≤ √(2π), so vanishing private noise always guarantees uniqueness.
• Focal Coordination on Public Announcements (Morris and Shin, 2002): Because every speculator knows that every other speculator observed the same public announcement y, agents place disproportionate weight on y as a coordination focal point. In "Social Value of Public Information" (2002), Morris and Shin showed that when coordination motives are strong, releasing noisy public signals can increase aggregate volatility by causing market participants to overreact collectively to the public signal error η.

Bank Runs (Goldstein–Pauzner, 2005) and Debt Rollover Risk

The Morris–Shin global games methodology became the standard equilibrium-selection engine across financial economics:

1. Endogenous Probability of Bank Runs (Goldstein and Pauzner, 2005): In the original Diamond–Dybvig model (1983), because the probability of a sunspot bank run was indeterminate, economists could not solve for the optimal demand-deposit interest rate r₁ trading off liquidity insurance against bank-run risk. Itay Goldstein and Ady Pauzner (2005) applied global games to Diamond–Dybvig, deriving a unique bank-run threshold θ*(r₁) that increases with the short-term deposit payment r₁.
2. Corporate Debt Rollover and Credit Risk (Morris and Shin, 2004): When short-term institutional creditors decide whether to roll over wholesale funding to a leveraged borrower (compare Calomiris–Kahn model and Diamond–Rajan model), global games pin down the illiquidity threshold below which coordination fear triggers a preemptive creditor run.

What is a "global game" and how did Morris and Shin adapt it to macroeconomics?

A global game—originally introduced for two-player 2 × 2 games by Hans Carlsson and Eric van Damme (1993)—is an incomplete-information game in which the payoff structure is drawn from a family of games indexed by an underlying state θ, and each player observes θ with a small amount of private noise x_i = θ + ε_i. Stephen Morris and Hyun Song Shin (1998) extended global games to a continuum of speculators in macroeconomic crisis models, proving that vanishingly small private noise eliminates sunspot multiplicity and selects a unique switching-threshold equilibrium θ*.

Why does arbitrarily small private noise (σ → 0) eliminate multiple equilibria when common knowledge (σ = 0) has multiple equilibria?

Under common knowledge (σ = 0), every speculator knows θ, knows that everyone else knows θ, and can coordinate on either "everyone attacks" (l = 1) or "nobody attacks" (l = 0) whenever θ ∈ (0, 1). Under noisy private signals x_i = θ + ε_i, an agent observing x_i slightly above the lower dominance threshold θ̲ = 0 knows that half of the population observed x_j < x_i, and that agents with x_j < 0 have a dominant strategy to attack. This initiates a chain of iterated deletion of dominated strategies from both dominance regions (θ < 0 and θ > 1) that meets at the single unique threshold θ* = 1 − t.

What is the Morris–Shin uniqueness condition α / √β ≤ √(2π)?

When speculators observe both a common public signal y with precision α = 1/σ_y² and an idiosyncratic private signal x_i with precision β = 1/σ_x², a unique equilibrium exists across all public realizations y if and only if α / √β ≤ √(2π). Because α grows linearly while √β enters as a square root, as private information becomes increasingly accurate (β → ∞ for any fixed public precision α), α / √β → 0, guaranteeing a unique equilibrium.

How can central bank transparency or public announcements sometimes increase market volatility in the Morris–Shin framework?

In games with strategic complementarities (where agents want to match the average action of other market participants, as in Keynes's beauty contest), public announcements y serve as a focal point because every trader knows that every other trader also saw y. Consequently, traders place extra weight on the public signal y relative to its statistical accuracy α, which can cause the market to coordinate on the public signal's noise realization η.

1. Morris, S., & Shin, H. S. (1998). "Unique Equilibrium in a Model of Self-Fulfilling Currency Attacks." American Economic Review, 88(3), 587–597. https://doi.org/10.1257/aer.88.3.587
2. Carlsson, H., & van Damme, E. (1993). "Global Games and Equilibrium Selection." Econometrica, 61(5), 989–1018. https://doi.org/10.2307/2951491
3. Obstfeld, M. (1996). "Models of Currency Crises with Self-Fulfilling Features." European Economic Review, 40(3–5), 1037–1047. https://doi.org/10.1016/0014-2921(95)00111-500111-5)
4. Morris, S., & Shin, H. S. (2002). "Social Value of Public Information." American Economic Review, 92(5), 1521–1534. https://doi.org/10.1257/000282802762024610
5. Goldstein, I., & Pauzner, A. (2005). "Demand-Deposit Contracts and the Probability of Bank Runs." The Journal of Finance, 60(3), 1293–1327. https://doi.org/10.1111/j.1540-6261.2005.00762.x