Measured Games · Paper 11 of 07 · Macro-finance

Bank runs

After D. W. Diamond and P. H. Dybvig (1983); uniqueness by H. Carlsson and E. van Damme (1993) and S. Morris and H. S. Shin (1998)

Abstract

A depositor prefers to withdraw only if enough others withdraw, which under common knowledge of the bank's condition supports both a run and no run as equilibria and leaves the theory unable to say which occurs. Giving each depositor a slightly noisy private signal instead removes the multiplicity: play is uniquely determined by a threshold on the signal, and the run occurs precisely when fundamentals fall below a level the model can name.

Keywordsglobal games · equilibrium selection · strategic uncertainty · threshold strategy

1Setup

Setup

A bank holds illiquid assets against demandable deposits. If few depositors withdraw, the bank meets them and everyone who stayed is paid in full. If enough withdraw, the bank is forced to liquidate at a loss and those who stayed lose. Each depositor therefore wants to withdraw exactly when they expect enough others to.

Written that way the game has two equilibria for a wide range of bank conditions. Everyone running is self-justifying, and nobody running is equally self-justifying. The theory says a run is possible and says nothing about when one happens, which is a weak thing for a theory about crises to say.

2The model

The model

# fundamental θ measures how much withdrawal the bank can absorb # depositor withdraws (cost c, certain) or stays (pays 1 if the bank survives) bank fails ⇔ fraction withdrawing ℓ > θ # complete information: for 0 < θ < 1 both are equilibria everyone runs → ℓ = 1 > θ → failure → running was right nobody runs → ℓ = 0 < θ → survival → staying was right # global game: each depositor sees a private signal instead xi = θ + εi, ε ~ U(-η, η) # the unique surviving strategy is a threshold on the signal run ⇔ xi < θ* with θ* = c

The step that removes the multiplicity is not the noise itself but what the noise does to beliefs. A depositor at the threshold has no idea where they sit in the queue of signals, and treats the proportion of others running as uniformly distributed. That single belief pins the indifference condition, and iterated deletion of dominated strategies then removes every equilibrium but one.

Figure 1Threshold play under private information

This figure needs JavaScript. The panels below are live simulations rather than images, so with scripting turned off they stay empty and the controls do nothing. Everything the figure demonstrates is also stated in the surrounding text: the model is given in full in section 2 and the result in section 3.

Depositors ordered by private signal · filled markers withdrew

Probability of failure against fundamentals · measured over many draws

Withdrew Stayed Measured failure Threshold θ*

Charts are focusable. With a chart focused, use the left and right arrow keys to read values along the horizontal axis.

Fundamental θ

Private signal noise η

Cost of withdrawing c

Depositors

Threshold θ*—

Withdrawing—

Outcome—

Predicted—

Failure rate at θ—

Draws0

Figure 1. Each depositor sees only its own noisy signal and applies the threshold rule, so the proportion withdrawing is counted rather than assumed. Sweep θ runs the whole range and measures how often the bank fails: the curve is a sharp switch at θ* rather than the wide band of indeterminacy that complete information leaves. Toggle the multiplicity band to see how much of the range the common-knowledge model refuses to resolve, and shrink the noise to watch the switch sharpen.

3Results

Results

Diamond and Dybvig (1983) built the model that makes runs coherent: demand deposits provide liquidity insurance, and the same contract that provides it admits a run equilibrium alongside the good one. The multiplicity is the achievement and the problem. It explains why runs can happen without any deterioration in fundamentals, and it leaves the timing entirely to an unmodelled sunspot.

Carlsson and van Damme (1993) showed that adding a small amount of private noise to a coordination game selects a unique equilibrium, and Morris and Shin (1998) applied the argument to currency attacks. The result is stronger than it first sounds: uniqueness survives as the noise goes to zero, so the limit of the noisy game is not the complete-information game. An arbitrarily small amount of strategic uncertainty is enough to pin down behaviour that common knowledge leaves undetermined.

What the model buys is a prediction with content. Instead of saying a run may occur anywhere in a wide band, it names a threshold in fundamentals and says the run occurs below it. That is testable, and it makes the comparative statics meaningful: anything that lowers the cost of withdrawing raises the threshold and widens the region in which the bank fails.

4Evidence

Evidence

Iyer and Puri (2012) used depositor-level data from a run on an Indian bank and found the run was not indiscriminate. Depositors with longer relationships and with loans from the bank were markedly less likely to withdraw, deposit insurance reduced but did not eliminate running, and social networks propagated withdrawal. That is a run with structure, not a panic.

Iyer, Puri and Ryan (2016) followed with evidence that both fundamentals and strategic complementarity matter, which is the combination the global-games account predicts rather than either the pure-panic or pure-fundamentals story alone.

For the currency-attack version, the Morris and Shin framework has been applied extensively to speculative attacks and to the 2007 to 2009 wholesale funding runs, where short-term creditors declining to roll over played the depositor role. The empirical work there is more contested, and Heinemann, Nagel and Ockenfels (2004) is the cleaner test: a laboratory implementation of the global game whose subjects used threshold strategies close to those the theory predicts.

5Limitations

Limitations

The uniqueness result depends on assumptions that are easy to state and hard to verify. It requires private information to be sufficiently precise relative to public information; add a strong enough public signal and multiplicity returns. It also relies on the fundamental having dominance regions at both ends, states so good that nobody runs and so bad that everyone does, which is a modelling choice rather than an observed fact.

Real depositors do not compute threshold strategies. The laboratory evidence suggests behaviour close to threshold play in stylised settings, which is encouraging, but the step from an experiment to a queue outside a branch is large.

Finally, the model treats the bank's condition as exogenous. In an actual crisis the run and the fundamentals determine each other, since forced liquidation damages the balance sheet that depositors are trying to assess. Read the threshold as a sharp statement about a simplified world rather than a number anyone could compute for a real bank on a real morning.

References

Carlsson, H., & van Damme, E. (1993). Global games and equilibrium selection. Econometrica, 61(5), 989–1018. https://doi.org/10.2307/2951491
Diamond, D. W., & Dybvig, P. H. (1983). Bank runs, deposit insurance, and liquidity. Journal of Political Economy, 91(3), 401–419. https://doi.org/10.1086/261155
Heinemann, F., Nagel, R., & Ockenfels, P. (2004). The theory of global games on test: Experimental analysis of coordination games with public and private information. Econometrica, 72(5), 1583–1599. https://doi.org/10.1111/j.1468-0262.2004.00544.x
Iyer, R., & Puri, M. (2012). Understanding bank runs: The importance of depositor-bank relationships and networks. American Economic Review, 102(4), 1414–1445. https://doi.org/10.1257/aer.102.4.1414
Iyer, R., Puri, M., & Ryan, N. (2016). A tale of two runs: Depositor responses to bank solvency risk. The Journal of Finance, 71(6), 2687–2726. https://doi.org/10.1111/jofi.12424
Morris, S., & Shin, H. S. (1998). Unique equilibrium in a model of self-fulfilling currency attacks. The American Economic Review, 88(3), 587–597.

Verification Every reference on this page has been checked against Crossref, so the works cited are real and correctly identified. The claims this page makes about what those sources report have not yet been independently checked. Of the nine papers audited so far, eight required a correction, so treat the specifics here as unconfirmed.