1 Introduction
Running is considered a healthy activity, the enjoyment of which often grows when performed with others.
1 These benefits however do not apply to running on a bank. On the contrary, the depositors’ dash to the bank to withdraw money on a large scale can be detrimental – not only to themselves and their bank but also to the health of the financial system as a whole.
Up until relatively recently, high-income countries with sophisticated financial markets were believed to be safe from bank runs. That notion started changing dramatically in 2007 with the runs on Countrywide Financial (the largest US mortgage lender the year before) and Northern Rock (the UK’s first bank run in 150 years). These episodes are often seen as important contributors to the Global Financial Crisis (GFC) (Shin
2009).
Many bank runs occurred during the GFC; even in advanced countries such as the United States (Bear Stearns, Washington Mutual, Wachovia), Canada (Home Capital Group), Iceland (Kaupthing, Landsbanki), the Netherlands (DSB Bank) and Spain (Bankia). Run-like phenomena have also been observed in other segments of the financial system, namely the repo market (Gorton and Metrick
2012), bank lending (Ivashina and Scharfstein
2010) and money market funds (Baba et al
2009). All these developments have rekindled interest by researchers and regulators, primarily because bank runs have the potential to evolve into major financial crises requiring costly policy interventions (Caprio and Klingebiel
1999; Laeven and Valencia
2013; Claessens et al
2014). The Covid-19 pandemic provided some run-like phenomena of its own, for example on toilet paper.
Late 2022 and early 2023 provided a reminder that the danger of bank runs has not gone away; if anything, it has intensified. We have observed runs in the cryptocurrency markets (see e.g. WSJ (
2022) aptly titled “Crypto Has Reinvented Bank Runs”). More importantly, three U.S. banks experienced runs as the final version of the paper was prepared. In one of them, the depositors and investors of the Silicon Valley Bank withdrew around USD 42 billion in just one day. The fact that this was the second largest collapse in the history of the U.S. banking system and that it necessitated a complex policy response speaks volumes of the havoc that bank runs may create.
All these developments underscore the importance of shedding light on the psychological and strategic aspects of bank runs. While some researchers have considered bank runs driven by fundamentals (Gorton
1988; Calomiris and Mason
2003; Goldstein and Pauzner
2005), there is substantial empirical evidence that coordination problems may play a vital role (Davison and Ramirez
2014; De Graeve and Karas
2014). Bank runs may thus be self-fulfilling phenomena. The characterization of the recent Silicon Valley Bank collapse as "the first social media bank run" (YahooFinance
2023) is consistent with this conclusion.
The contribution of this paper is to provide insights about self-fulfilling bank runs, and to consider the effects of some policy recommendations about how to minimize these costly episodes. Our game-theoretic framework of Stochastic Leadership enriches the strategic interactions amongst depositors. It relaxes the widely-used (yet unrealistic) timing assumption that depositors only make a one-off withdrawal decision and are unable to change it. Mimicking the real world, our depositors are able to revise their initial decision with some (heterogeneous) probability, based on observing what others have done. In principle, they can run out of a bank run, or run into it, once they see whether other depositors have run on the bank.
To better understand the finer details of the Stochastic Leadership framework, let us frame it within the bank-run literature, initiated by Diamond and Dybvig (
1983). Studies have used a wide range of approaches and models. Despite this, the strategic aspect of a bank-run situation has always been captured as a coordination game, namely Pure Coordination or Stag Hunt (Bryant
1994; Arifovic et al
2013; Peia and Vranceanu
2019; Shell and Zhang
2018). Each depositor has the option of leaving the money in the bank (action
L) or withdrawing it before maturity (action
W). The incentives and payoffs are such that each depositor would like to do what other depositors are doing. If others run on the bank, one wants to run as well to withdraw their money before the bank goes bankrupt. If others leave their money in the bank, each wants to do so too in order to collect interest on the deposit at maturity.
Given this structure, both the Pure Coordination and the Stag Hunt games feature two Pareto-ranked pure-strategy Nash equilibria. In the payoff-dominant equilibrium (denoted
\(\mathcal {L}\)) all depositors leave their money in the bank so business as usual prevails. In contrast, in the payoff-inferior equilibrium (denoted
\(\mathcal {W}\)) all depositors withdraw their money and a bank run occurs. In the commonly examined simultaneous-move game, standard tools cannot uniquely select between these pure-strategy equilibria
\(\mathcal {L}\) and
\(\mathcal {W}\).
2 The existence of inferior mixed-strategy equilibria further amplifies the equilibrium selection problem and the danger of a bank run occurring. In the Pure Coordination game, it is less of a threat because the Schelling (
1960) focal argument selects the efficient equilibrium
\(\mathcal {L}\). However, in the Stag Hunt game the bank-run equilibrium
\(\mathcal {W}\) is risk dominant and thus commonly considered to be the most likely outcome (Cooper et al
1990,
1992; Harsanyi
1995).
Such equilibrium selection problems of the simultaneous move setup steered recent theoretic and experimental bank-run literature into an investigation of sequential environments (Kinateder and Kiss
2014; Kiss et al
2014; Davis and Reilly
2016).
3 Under the Stackelberg leadership timing, the depositors decide one after another upon observing the full history of previous decisions. In such a sequential setup, the payoff-dominant equilibrium
\(\mathcal {L}\) is uniquely selected, and the threat of bank runs is eliminated. The outcomes of the Stackelberg and simultaneous-move setups are hence in stark contrast, and their conflicting predictions are one of the key reasons for our investigation of self-fulfilling bank runs within a more general framework.
Our Stochastic Leadership framework nests the conventional simultaneous and Stackelberg timing structures as two special cases, while also capturing everything in between them. The game starts with the conventional simultaneous move, but each depositor j has some ex-ante exogenous probability \(1-\theta _{j}\) of being able to change their initial action. Intuitively, depositors may change their minds about withdrawing their money during a banking frenzy, i.e. run out of a bank run, or run into it at a later stage. This revision probability is known to all depositors in advance. The focus on self-fulfilling runs is captured by the assumption that the depositors’ decision to change the initial action is based solely upon observing the behavior of others; the fundamentals of the bank are known to be solid and do not provide any reason for withdrawing money prematurely.
The complementary probability
\(\theta _{j} \in \left[ 0, 1 \right] \) can be interpreted as depositor
j’s degree of rigidity. This is analogous to the Calvo (
1983) timing utilized in macroeconomics. However, in his influential setup the same revision probability applies to all agents, whereas in our framework it is heterogeneous (depositor-specific). It captures the fact that some depositors are rigid (have a high
\(\theta _{j}\)) whereas others are flexible (have a low
\(\theta _{j}\)).
There are a number of features of the world that can motivate probabilistic changes of mind and depositors’ rigidity in the bank-run context. First, the literature provides ample evidence that depositors gather information and react to it. The information linkages happen through social networks including families and neighborhoods (Kelly and O Grada
2000; Iyer and Puri
2012; Atmaca et al
2017), or through mere observation of the behaviour of strangers (Starr and Yilmaz
2007; Davison and Ramirez
2014). As an example, Starr and Yilmaz (
2007) analyze a 2001 bank-run episode in Turkey that lasted for several months. The authors show that many depositors did not rush to the bank from the outset of the turmoil, only withdrawing their money upon seeing others do that. As another indication of the depositors’ change of mind, the authors document that many began redepositing their money after a period.
Second, heterogeneous rigidity
\(\theta _{j}\) can capture important differences between term and demand deposits (Niinimäki
2002), whereby the former are more rigid. Term depositors face greater withdrawal constraints, making them less likely to be able to change their minds. Third,
\(\theta _{j}\) can reflect heterogeneity across depositors in their education, financial sophistication or wealth. As Kiss (
2018) shows, these personal characteristics can often explain whether depositors collect information on the bank’s fundamentals and therefore their behavior when a rumor spreads. Osili and Paulson (
2014) and Fungáčová et al (
2021) show how experience of banking crises shapes depositors’ trust towards banks (and potentially their rigidity).
Fourth, the theory of rational inattention (Huang and Liu
2007; Sims
2010; Caplin and Dean
2015; Matějka and McKay
2015; Bartoš et al
2016) sheds light on why agents may not respond to various developments. Essentially, they attempt to avoid the costs of information acquisition and processing. The implication is that some depositors may simply find it optimal to wait and react to the observed decisions of other (informed) depositors rather than engage in a costly information search and processing themselves.
4 Fifth, the depositors’ rigidity may be driven by emotional and transactional switching costs (for formal modeling see Guin et al (
2015)). Similarly,
\(\theta _{j}\) may also capture the nature of the bank-depositor relationship, see e.g. Guin et al (
2015), Iyer et al (
2016). Stronger relationships can be viewed as a form of rigidity, because they reduce the probability of a deposit withdrawal.
Given these diverse influences from the real world on the rigidity of depositors, we postulate \(\theta _{j}\) as exogenous in this study. But it should be kept in mind that it can be endogenized in various ways, depending on which of the above features is considered the most relevant.
We first obtain some general results for our n-depositor bank-run game without Stochastic Leadership. We derive the classes of games that may arise under various parameter values (the bank’s reserve ratio and investment return on deposits). In addition to the Pure Coordination game and the Stag Hunt, a Deadlock game may also occur under some circumstances.
We then move from the normal-form game to the extensive-form game to explore the effect of Stochastic Leadership. Our analysis demonstrates that if there is sufficient heterogeneity in the revision opportunities across the depositors, occurrence of self-fulfilling bank runs is reduced or fully eliminated - even without a deposit insurance scheme. Under some circumstances (which we derive explicitly for the two-depositor and three-depositor cases) we obtain the
No-run region. It is the set of parameters (namely payoffs and revision probabilities) under which the efficient
\(\mathcal {L}\) is the unique equilibrium outcome. In the No-run region all depositors leave their money at the bank - both initially and in their revision, i.e. the risk-dominant as well as the mixed-strategy equilibria of the normal-form game are eliminated from the set of equilibria.
5
Intuitively, the rigid depositors (e.g. owners of term deposits) act as Stochastic leaders in the game. Conversely, flexible depositors (e.g., owners of demand deposits) act as Stochastic followers.
6 To avoid the bank run, members of each group need the other group to provide the right incentives to them. The mechanics behind this are analogous to Stackelberg leadership. Using backwards induction, if the Stochastic followers have a sufficiently low rigidity
\(\theta _{j}\), they are
flexible in running out of the run at the revision stage. This incentivizes the Stochastic leaders to leave their money in the bank (both initially and in their revision). Conversely, if the Stochastic leaders have a sufficiently high rigidity
\(\theta _{j}\), they are
inflexible in running into the run. This incentivizes the Stochastic followers to change their minds and run out of the bank run in their revision if they find out that the Stochastic leaders have not run initially. If these conditions are satisfied leaving the money in the bank becomes a strictly dominant strategy for the Stochastic leader(s). The Stochastic followers know this and select the same course of action. No depositor withdraws and the game ends up in the No-run region. Under Stackelberg leadership, the multiplicity of equilibria disappears and self-fulfilling bank runs are avoided.
An advantage of our framework is that it offers a novel way to assess the changes in the likelihood of bank runs based on strategic interactions and depositor/bank characteristics. It uses the combined size of the No-run regions. The larger that is, relative to the Possible-run region, the less likely the \(\mathcal {W}\) equilibrium is - implying a reduced threat of a bank run. We report below how the size of the No-run region(s) and the implied danger of a bank run depend on the bank’s reserve ratio, its investment return and depositors’ rigidity.
Our analysis emphasizes that in order for withdrawal rigidity to enhance implicit coordination between flexible and inflexible depositors, the proportions (implying aggregate revision probabilities) must be known to all depositors in advance. An essential attribute of our framework is therefore observability of the depositors’ heterogeneous characteristics and their past behavior. This is consistent with the above-cited studies on the importance of social networks and awareness of the decisions of others, and implies a key role for the banking regulator to play in disseminating information.
Section
2 presents our bank-run model, which is first examined within the conventional game-theoretic frameworks (Section
3) and then within our Stochastic Leadership framework (Sections.
4 and
5). For robustness purposes, we consider below two extensions, namely the effects of government deposit insurance (Section
6) and costs of withdrawing funds with a delay at the revision stage (Section
7). A brief discussion of larger numbers of depositors is also provided (Section
8). While some new insights emerge the analysis shows that our baseline findings are qualitatively unchanged. The final section of the paper spells out policy options that may reduce the occurrence of costly bank runs. As such, they may enhance social welfare by ensuring that running remains a healthy physical exercise rather than a threat to the banking system and people’s prosperity.
9 Summary and conclusions
Nine-time Olympic champion swimmer Mark Spitz argued that ‘If you fail to prepare, you’re prepared to fail.’ This statement is arguably true not only in the sports context, but also in relation to financial regulation. There is an ever-present danger of a banking panic. In recent years it was intensified by the 2020-2022 boom-bust financial cycle, the problems induced by the Covid-19 pandemic and excessively high inflation. The March 2023 collapses of several banking institutions in the U.S. provide a further demonstration that the required policy intervention can be costly and magnify the moral hazard problem and systemic risk down the road. As such, financial institutions and public policymakers need to better understand the mechanics of bank runs as well as the solutions available. This paper contributes on both fronts.
The existing bank-run literature in the tradition of Diamond and Dybvig (
1983) generally assumes that withdrawal is a one-off decision that depositors cannot alter. This may be a plausible simplification in some contexts, but it may be too restrictive in others. Furthermore, the two polar cases commonly examined in the literature have very different predictions. In most papers depositors decide simultaneously, i.e. they are unable to observe what others have done. This leads to multiple equilibria and the possibility of bank runs. On the other hand, Kinateder and Kiss (
2014) and others examine sequential timing in which each depositor can observe moves that occurred previously. In such Stackelberg settings the likelihood of bank runs in equilibrium is greatly diminished or fully eliminated.
Our Stochastic Leadership framework nests the simultaneous and Stackelberg frameworks as special cases. It allows depositors to change their minds about withdrawing money from the bank. Each depositor has some ex-ante probability of being able to reconsider her initial decision based on observing what others have done.
19
Our analysis demonstrates how the Stochastic leaders (rigid depositors) and Stochastic followers (flexible depositors) are more likely to achieve an outcome with higher welfare if their actions are implicitly coordinated. Their heterogeneity may help them to avoid self-fulfilling bank runs. This reduces the need for deposit insurance, and can thus help to alleviate its various undesirable incentive effects (see Demirgüç-Kunt and Detragiache (
2002); Hoggarth et al (
2005); Wang (
2013)).
In regards to policy recommendations, our analysis implies that availability of a wide range of banking products (including a good mix of current and term deposits) may be beneficial by ensuring a wide range of depositor rigidities. For example, in the case of the recent failure of Silicon Valley Bank, more diversity among depositors, rather than the high concentration of large tech-firm deposits, may have slowed or eliminated the very fast run that led to its failure. However, heterogeneity must be combined with collection and dissemination of information by the government or regulator to facilitate implicit coordination of flexible and rigid depositors.
Our results suggest that the existence of a large depositor/investor with a credible long-term commitment, acting as a Stackelberg or Stochastic leader, may in principle help to prevent bank runs. There are some related historical examples, e.g. the leadership role of J.P. Morgan during the financial panic of 2007.
More research is required to assess how observable heterogeneous characteristics such as withdrawal rigidity may affect the depositors’ behavior in various institutional settings, and to fully understand the policy implications for optimal regulation of banking services. Nonetheless, let us emphasize that our ‘change-of-mind’ framework may be potentially useful in modeling strategic behavior in many different areas. Examples include macroeconomic policy (e.g. interactions between the central bank, government and the prudential authority), management/business areas (e.g. the interactions between oligopolists), as well as political science (e.g. interactions between political parties or climate-deal negotiating governments).