Lecture 3: Bank Runs

Multiple Equilibria, Global Games, and Anti-Run Policies

Rajesh Narayanan

LSU Business - Department of Finance

February 2026

1 Overview

1.1 Learning Objectives

By the end of this lecture, you should be able to:

1. Explain why the DD banking contract is vulnerable to self-fulfilling runs
2. Derive the tipping point and characterize the three equilibria
3. Distinguish fundamental-based runs from panic-based runs
4. Describe how Goldstein-Pauzner (2005) resolves the multiple equilibria problem
5. Explain the key comparative static: liquidity mismatch → fragility
6. Evaluate anti-run policies (suspension, deposit insurance, LOLR)

1.2 Roadmap

Part 1: The Dark Side of Liquidity Creation

• Recap of DD, setting up the run problem, good and bad equilibria

Part 2: Strategic Complementarities

• Payoff structure, the tipping point, three equilibria, what triggers a run

Part 3: Fundamental vs. Panic-Based Runs

• Information-based models, the game-theoretic distinction, reconciling the evidence

Part 4: The Global Games Approach (Goldstein-Pauzner 2005)

• Private signals, uniqueness via iterated dominance, three regions, key comparative static

Part 5: Anti-Run Policies

• Suspension, deposit insurance, lender of last resort, capital requirements

2 Part 1: The Dark Side of Liquidity Creation

2.1 Recap: The DD Banking Contract

From Lecture 2:

• 100 depositors, each deposits 1 at \(t=0\)
• Illiquid asset: \(r_1 = 1\), \(R = 2\)
• Fraction \(\lambda = 1/4\) impatient
• Optimal contract: \(c_1^* = 1.28\), \(c_2^* = 1.813\)

The Central Paradox

The same features that make banking valuable also make it fragile:

• Liquid deposits → depositors can withdraw at any time
• Illiquid assets → bank cannot meet all withdrawals simultaneously
• Sequential service → early withdrawers paid in full, late withdrawers may get nothing

2.2 The Good Equilibrium

Suppose all depositors expect only impatient types will withdraw:

• Fraction \(f = \lambda = 1/4\) withdraw at \(t=1\)
• Total payout: \(25 \times 1.28 = 32\)
• Remaining assets at \(t=2\): \((100 - 32) \times 2 = 136\)
• Per remaining depositor: \(136/75 = 1.813\)

Since \(c_2^* = 1.813 > 1.28 = c_1^*\), patient depositors prefer to wait.

Expectation confirmed → Nash equilibrium.

2.3 The Bad Equilibrium: A Bank Run

Now suppose all depositors expect everyone will withdraw at \(t=1\):

• Total demanded: \(100 \times 1.28 = 128\)
• Total assets available at \(t=1\): \(100\)
• Bank runs out of assets before paying everyone

The bank pays at most \(\lfloor 100/1.28 \rfloor = 78\) depositors. Depositors 79–100 receive nothing.

Self-Fulfilling Prophecy

A patient depositor who believes everyone else will withdraw faces:

• Withdraw now: Get \(c_1^* = 1.28\) (if early enough)
• Wait until \(t=2\): Get nothing

Withdrawing is rational. But when all patient depositors reason this way, the bank is liquidated. The fear of a run causes the run.

3 Part 2: Strategic Complementarities

3.1 The Payoff Structure

Let \(\hat{\lambda}\) = fraction who actually withdraw at \(t=1\) (including impatient + any patient who run).

Early withdrawer’s payoff: \[ \text{Early payoff} = \min\left(c_1^*, \frac{1}{\hat{\lambda}}\right) \]

Late withdrawer’s payoff: \[ \text{Late payoff} = \max\left(\frac{(1 - \hat{\lambda} c_1^*)R}{1 - \hat{\lambda}}, \; 0\right) \]

Diamond’s Numerical Example

With \(c_1^* = 1.28\), \(R = 2\):

• \(\hat{\lambda} = 0.25\): Late payoff = \(1.813\) ✓
• \(\hat{\lambda} = 0.50\): Late payoff = \(1.44\)
• \(\hat{\lambda} = 0.75\): Late payoff = \(0.32\)
• \(\hat{\lambda} \geq 1/c_1^* = 0.78\): Late payoff = \(0\) (insolvent)

3.2 Strategic Complementarities: The Diagram

Adapted from Tirole (2006), Figure 12.2.

3.3 Reading the Diagram

• Early withdrawer (blue): Flat at \(c_1^*\) while bank is solvent (\(\hat{\lambda} < 1/c_1^*\)), then declines to 1

• Late withdrawer (red): Starts at \(c_2^*\) at \(\hat{\lambda} = \lambda\), drops to zero at \(\hat{\lambda} = 1/c_1^*\)

• “No run” region (left of crossing): Late payoff \(> c_1^*\) → patient depositors wait

• “Run” region (right of crossing): Early payoff \(>\) late payoff → patient depositors withdraw

Strategic Complementarities

My incentive to run increases when more others run:

1. Late withdrawer’s payoff falls (fewer assets remain)
2. Early withdrawer’s payoff stays constant (first-come, first-served)
3. At some point, withdrawing dominates waiting

3.4 The Tipping Point

The curves cross when early and late payoffs are equal:

\[ c_1^* = \frac{(1 - \hat{\lambda} c_1^*)R}{1 - \hat{\lambda}} \]

Solving:

\[ \hat{\lambda}^* = \frac{R - c_1^*}{c_1^*(R - 1)} \]

Diamond’s Numerical Example

With \(c_1^* = 1.28\) and \(R = 2\):

\[ \hat{\lambda}^* = \frac{2 - 1.28}{1.28(2 - 1)} = \frac{0.72}{1.28} = 0.5625 \]

Run becomes self-fulfilling when more than 56.25% of depositors are expected to withdraw — roughly 42% of patient depositors.

3.5 Three Equilibria

1. Good equilibrium (\(\hat{\lambda} = \lambda = 0.25\)): Only impatient withdraw. Patient depositors wait because \(c_2^* = 1.813 > 1.28 = c_1^*\).

2. Unstable equilibrium (\(\hat{\lambda} = \hat{\lambda}^* = 0.5625\)): Patient depositors indifferent. Any perturbation pushes to a stable equilibrium.

3. Run equilibrium (\(\hat{\lambda} = 1\)): Everyone withdraws. Bank is liquidated. Everyone gets less than in the good equilibrium.

Why Two Stable Equilibria?

Starting from the good equilibrium, a small increase in \(\hat{\lambda}\) does not change incentives. It takes a coordinated shift in beliefs past the tipping point of 0.5625.

Diamond (2007): initiating a run requires “something that all (or nearly all) depositors see (and believe that others see).”

3.6 What Triggers a Run?

DD do not provide a theory of equilibrium selection. Runs can be triggered by publicly observable signals:

• Sunspots: Arbitrary events that serve as coordination devices

• News about fundamentals: A report that the bank’s assets have declined

• Observable queues: Diamond (2007): “it would make sense for a bank to have a large lobby”

The Gorton Critique

The sunspot approach makes the model “empirically vacuous and untestable” (Gorton, 1988). If runs are driven by arbitrary signals, the model cannot predict:

• When runs occur
• Which banks are vulnerable
• How policy affects run probability

4 Part 3: Fundamental vs. Panic-Based Runs

4.1 Two Views of Bank Runs

Panic-based runs (DD)

• Bank is solvent but illiquid
• Withdrawals driven by coordination failure
• Beliefs about other depositors are important
• Runs are inefficient (destroy viable institutions)

Fundamental-based runs

• Bank is insolvent (or likely to be)
• Withdrawals are informationally driven
• Beliefs about other depositors not important
• Runs may be efficient (prevent throwing good money after bad)

The Game-Theoretic Distinction

• Fundamental run: Withdrawing is a dominant strategy — optimal regardless of what others do
• Panic run: Withdrawing is only optimal if enough others withdraw — a coordination game

4.2 Key Fundamental-Based Models

Chari and Jagannathan (1988)

• Some depositors observe signal about asset quality
• Uninformed depositors observe the withdrawal queue and try to infer
• Long queue = bad news or many legitimate liquidity needs → inference problem
• Runs driven by information extraction, not coordination failure

Jacklin and Bhattacharya (1988)

• Depositors receive private information about asset returns
• Those with bad news withdraw early to avoid losses
• Runs serve a disciplining function — penalize poor investment decisions

Allen and Gale (1998)

• Runs can be part of optimal risk-sharing
• When economy is in bad state, run forces liquidation → transfers resources to those who need them now
• Runs as an efficient response to aggregate shocks

4.3 Reconciling the Evidence

Gorton (1988): Historical U.S. run episodes consistently preceded by deteriorating fundamentals.

Initially interpreted as evidence against the panic view.

Fundamentals and Panics Are Not Mutually Exclusive (Goldstein, 2013)

A prerequisite for panic-based runs is weaker fundamentals. Strategic complementarities amplify the effect of bad news.

The key question: does coordination failure make the response to fundamentals larger than it would be without strategic complementarities?

This is exactly what Goldstein and Pauzner (2005) formalize.

5 Part 4: The Global Games Approach

5.1 The Problem with Common Knowledge

In the standard DD model, depositors have common knowledge of bank fundamentals:

• Everyone knows the bank is solvent
• Everyone knows that everyone knows
• Everyone knows that everyone knows that everyone knows…
• → Both equilibria are sustainable

This means DD cannot answer:

• Which banks are more likely to experience runs?
• What bank characteristics make runs more likely?
• How does policy affect the probability of a run?

Goldstein and Pauzner (2005): Introduce incomplete information into DD using the global games methodology (Carlsson and van Damme, 1993; Morris and Shin, 1998).

5.2 GP Model Setup

Bank fundamentals \(\theta\): Asset return depends on \(\theta \sim [\theta_L, \theta_H]\). Higher \(\theta\) = better asset quality.

Late withdrawer’s payoff depends on both \(\hat{\lambda}\) and \(\theta\): \[ \text{Late payoff}(\hat{\lambda}, \theta) = \max\left(\frac{(1 - \hat{\lambda} c_1^*) R(\theta)}{1 - \hat{\lambda}}, \; 0\right) \]

Private signals: Each depositor \(i\) observes: \[ x_i = \theta + \epsilon_i \] where \(\epsilon_i\) independent with small variance \(\sigma^2\).

Dominance regions:

• \(\theta < \underline{\theta}\): Bank insolvent even if only impatient withdraw → withdrawing is dominant strategy
• \(\theta > \overline{\theta}\): Bank so strong that waiting always dominates → dominant strategy to wait
• \(\theta \in [\underline{\theta}, \overline{\theta}]\): DD multiple-equilibria region

5.3 How Uniqueness Emerges

With private signals, the game is no longer one of common knowledge:

Step 1: Dominance at the extremes. Very low signal → bank almost certainly insolvent → withdraw. Very high signal → bank is safe → wait.

Step 2: Iterated elimination. Signal just above \(\underline{\theta}\) → knows slightly-lower-signal depositors withdraw → expects some withdrawals → strategic complementarities → she also withdraws.

Step 3: Contagion propagates upward. A depositor slightly higher knows Step 2 depositors withdraw. This further increases expected withdrawals. Reasoning cascades upward.

Step 4: The cascade stops. At critical signal \(x^*\), depositors above find it optimal to wait despite some withdrawals. This pins down the unique threshold.

The Role of Higher-Order Beliefs

The key is the destruction of common knowledge. With private signals, a depositor with middling signal \(x_i\) is uncertain about what others believe about what others believe. This higher-order uncertainty unravels the multiplicity.

5.4 The Threshold and Three Regions

As \(\sigma \to 0\), a critical fundamental \(\theta^*\) determines the bank’s fate. \(\theta^*\) lies within \([\underline{\theta}, \overline{\theta}]\) and splits the DD multiple-equilibria region:

1. \(\theta < \underline{\theta}\) (Insolvent): Fundamental run — bank fails regardless of coordination
2. \(\underline{\theta} < \theta < \theta^*\) (Panic): Bank is solvent but fails due to coordination failure
3. \(\theta > \theta^*\) (No run): Fundamentals strong enough to sustain confidence

5.5 Fundamentals and Panics Coexist

The Resolution

Both forces operate simultaneously:

• Fundamentals determine which region the bank falls into
• Panic matters because in \([\underline{\theta}, \theta^*]\), the bank fails only because of coordination failure

The probability of a run is \(\Pr(\theta < \theta^*)\) — well-defined and depends on bank characteristics.

This makes the model empirically testable.

5.6 The Key Comparative Static

Greater Liquidity Mismatch → Larger Panic Region

Higher \(c_1^*\) (more generous demand deposit contract) → higher \(\theta^*\):

• Panic region \([\underline{\theta}, \theta^*]\) is wider
• Probability of a run is higher
• Bank is more fragile

Intuition: When the bank promises more to early withdrawers, remaining assets for late withdrawers are smaller if others withdraw. This strengthens strategic complementarities → larger panic region.

Testable prediction: Banks with greater liquidity mismatch should exhibit stronger sensitivity of deposit outflows to bad performance.

5.7 Detecting Panic in the Data

Challenge: Fundamental and panic-based runs look similar ex post — both involve large outflows after bad news.

GP’s identification strategy:

If runs are purely fundamental:

• Response to bad news depends only on severity
• Bank’s liability structure is irrelevant
• Depositors withdraw because assets are bad

If panic dynamics are present:

• Response is amplified by liquidity mismatch
• Banks with greater maturity transformation show stronger sensitivity
• Strategic complementarities are stronger

Chen, Goldstein, Huang, and Vashishtha (2022)

Test whether the interaction of performance shocks and liquidity mismatch predicts deposit outflows. Finding: it does — and operates through uninsured deposits, not insured ones. Evidence that the panic channel is empirically relevant.

6 Part 5: Anti-Run Policies

6.1 The Policy Challenge

Why the Nature of Runs Matters for Policy

• Panic-based runs: Credible government guarantee is sufficient. Deposit insurance eliminates the bad equilibrium at no cost (off the equilibrium path).

• Fundamental-based runs: Government support = bailout. Ex ante policies (capital requirements, supervision) are more appropriate. Runs serve a disciplining function.

• Both forces operate (GP): Policy must address the panic region (liquidity backstops) and the insolvency region (prudential regulation).

6.2 Suspension of Convertibility

The Mechanism

Bank announces: “We suspend withdrawals once fraction \(\lambda\) of deposits have been withdrawn.”

Then patient depositors know: \[ r_2(\lambda) = \frac{(1 - \lambda c_1^*)R}{1 - \lambda} = c_2^* > c_1^* \]

Waiting always dominates → no patient depositor runs. Run equilibrium eliminated.

Historical practice: U.S. banks suspended convertibility 8 times between 1814 and 1907.

Limitations:

1. Aggregate uncertainty: If \(\lambda\) fluctuates, bank can’t set threshold perfectly
2. Moral hazard: Poorly managed banks can suspend to shield from discipline
3. Political unpopularity: Depositors with genuine needs are locked out

6.3 Deposit Insurance

The Mechanism

Government guarantees every depositor receives the contractual amount — \(c_1^*\) for early, \(c_2^*\) for late — regardless of withdrawals.

If credible, no reason to run: patient depositors get \(c_2^* > c_1^*\) no matter what. Run equilibrium eliminated.

Why the government? Diamond (2007): requires taxation authority — a private insurer might itself face a run.

Advantages over suspension: Handles aggregate uncertainty, no disruption, commitment via law.

Costs:

1. Moral hazard: Insured depositors don’t monitor bank risk-taking
2. Fiscal cost: Insurance fund (and taxpayers) bear losses
3. Coverage limits: Uninsured deposits remain vulnerable (SVB 2023)

GP perspective: Deposit insurance reduces \(\theta^*\) → shrinks panic region. Davila and Goldstein (2023): optimal insurance balances smaller panic region against moral hazard. Unlimited insurance is not optimal.

6.4 Lender of Last Resort

Central bank stands ready to lend to solvent but illiquid banks against good collateral.

Bagehot (1873): “Lend freely, at a high rate, on good collateral.”

Limitation: A private credit line can protect one bank, but not the entire system. Only the central bank (money creation authority) can provide system-wide liquidity (Tirole, 2006).

The Bagehot Boundary (GP Framework)

• Panic region (\(\underline{\theta} < \theta < \theta^*\)): Bank is solvent but illiquid → lend
• Insolvency region (\(\theta < \underline{\theta}\)): Genuine asset impairment → do not lend
• The difficulty: distinguishing the two in real time

6.5 Capital Requirements and Transparency

Capital requirements:

• Improve loss absorption → raise \(\underline{\theta}\) (expand solvency region)
• Reduce leverage → may reduce \(\theta^*\) (shrink panic region)

Transparency has ambiguous effects:

• Better information (smaller \(\sigma\)) makes the threshold sharper
• But more precise public information can increase fragility by coordinating depositors on negative signals (Morris and Shin, 2002)
• Private signals reduce scope for coordination → may reduce fragility

The Transparency Paradox

More information is not always better. Public disclosures can serve as coordination devices that trigger the very panic they reveal.

7 Summary

7.1 Key Takeaways

Main Insights

1. Runs are inherent to liquidity creation: The same maturity transformation that creates value also creates vulnerability to self-fulfilling panics (DD 1983)

2. Multiple equilibria: Good equilibrium (only impatient withdraw) and run equilibrium (everyone withdraws). Tipping point: \(\hat{\lambda}^* = 0.5625\)

3. Fundamental vs. panic: Fundamental runs = dominant strategy (bad assets); panic runs = coordination failure (solvent but illiquid). In practice, both forces operate

4. Global games resolution (GP 2005): Private information → unique equilibrium with threshold \(\theta^*\). Greater liquidity mismatch → larger panic region → more fragility

5. Anti-run policies: Suspension (works but limited), deposit insurance (shrinks panic region but creates moral hazard), LOLR (the Bagehot boundary challenge), capital requirements (expand solvency region)

7.2 References

7.2.1 Primary Sources

• Diamond, D. W., & Dybvig, P. H. (1983). Bank runs, deposit insurance, and liquidity. Journal of Political Economy, 91(3), 401-419.

• Diamond, D. W. (2007). Banks and liquidity creation: A simple exposition of the Diamond-Dybvig model. FRB Richmond Economic Quarterly, 93(2), 189-200.

• Tirole, J. (2006). The Theory of Corporate Finance. Princeton University Press. Ch. 12.3.

7.2.2 Fundamental vs. Panic-Based Runs

• Chari, V. V., & Jagannathan, R. (1988). Banking panics, information, and rational expectations equilibrium. Journal of Finance, 43(3), 749-761.

• Allen, F., & Gale, D. (1998). Optimal financial crises. Journal of Finance, 53(4), 1245-1284.

• Gorton, G. (1988). Banking panics and business cycles. Oxford Economic Papers, 40(4), 751-781.

7.2.3 Global Games and Policy

• Goldstein, I., & Pauzner, A. (2005). Demand deposit contracts and the probability of bank runs. Journal of Finance, 60(3), 1293-1328.

• Carlsson, H., & van Damme, E. (1993). Global games and equilibrium selection. Econometrica, 61(5), 989-1018.

• Davila, E., & Goldstein, I. (2023). Optimal deposit insurance. Journal of Political Economy, 131(7), 1676-1730.

• Chen, Q., Goldstein, I., Huang, Z., & Vashishtha, R. (2022). Liquidity transformation and fragility in the US banking sector. Journal of Finance, 77(4), 2001-2057.