Modeling the Occurrence or Non-occurrence of Belief-based “runs”

“Runs” on banks and on similar financial intermediaries have been a recurrent phenomenon. In some cases, a run has occurred when depositors or investors have received information that the intermediary lacks sufficent resources to honor all of its claims. Then there is a race to present claims to the intermediary before those resources run out. But many people—including market participants and economics researchers who have studied runs—believe that not all runs can be explained simply as a response to insufficiency of resources. Motivated by that suspicion, Bryant [1] and Diamond and Dybvig [2] have formulated theories of an expectations-based run. These theories emphasize that a depositor’s decision whether or not to withdraw funds from the bank, if there is not actually an immediate need for money, depends on beliefs about whether or not other depositors without urgent needs will keep their money in the bank. If so, then there will be enough resources for everyone. If not, then the bank’s resources will be exhausted, so each depositor will race to withdraw his/her funds before that happens. Either belief can be self fulfilling. The upshot is that, for no particular reason, a run on a conservatively managed bank can occur. A bank can structure and operate itself as a narrow bank—one that can honor all of its claims, even if a run occurs. Essentially, to do so, it must not pay interest on deposits. But prospective depositors generally are willing to take a small risk of there being a run, in order to receive interest. Diamond and Dybvig, especially, have emphasized that a narrow bank would unanimously be considered worse than an interest-paying bank, if everyone were confident that no one else would make withdrawals that they did not genuinely need to make. To restate Diamond and Dybvig’s point in the language of game theory, each way of strucuring and operating the bank entails a specific game among the depositors. The narrow-bank game has a unique equilibrium. The interest-paying-bank game has multiple equilibria, one of which is unanimously better than, and the other of which (that is, equilibrium in which a run occurs) is unanimously worse than, the narrow-bank equilibrium, when viewed from an ex ante perspective. Thus, Diamond and Dybvig suggest that, acting in accord with the preferences of their depositors, who initially are confident that everyone will “play by the rules of the game,” bankers opt to pay interest. In their terminology, bankers opt to offer a standard deposit contract. A run occurs when, possibly without any good reason, depositors lose confidence that other depositors will exercise their withdrawal options under a standard deposit contract only when they have genuine need to do