For an N players coordination games, Tanaka (2000) proved that the notion of N/2 stability defined by Schaffer (1988) is a necessary and sufficient condition for long run equilibrium in an evolutionary process with mutations (in the sense of Kandori, et. al. (1993)). We argue that the critical number in Schaffer's stability is not unique in every application, but can vary with variables determined before the coordination games. In our specific model, these variables are the portfolio choices of the banks. We derived a Z* stability condition for the long run equilibrium for the banking system, in which there is no speculative bank run. This critical number of players is a function of the size for risky investment, and varies with total risky investments when there are more than two banks. We use this framework to analyze the effect of speculative behavior on banks' risk taking and the phenomenon of system risk, calculating the probability when more than one bank fail together (system risk). Our specific results include: first, we propose a Z* stability condition, which is proved to be a necessary and sufficient condition for such a long run equilibrium in the sense of KMR. This critical number of Z* is a function of the total risky investment in the banking system. In the case with two banks, this value could vary across banks. Second, speculative behaviors do not frustrate single bank's risky taking, but rather, encourage the bank to maintain a high enough level of risky investment, to keep the system stay in the equilibrium of no run. This indicates that although the speculative run equilibrium will be eliminated in the long run, the probability of fundamental run will increase with the mere possibility of speculative behavior. It is well known that sufficiently large exogenous shocks can cause a crisis. For example, Allen and Gale (1998) describe a model in which financial crises are caused by exogenous asset-return shocks. Following a large (negative) shock to asset returns, banks are unable to meet their commitments and are forced to default and liquidate assets. Third, the single bank case does not necessarily apply to the case with multiple banks. Symmetric banks can take different level of risks, which induces a different in the probability of bank failures. The probability of joint failures increases, compare to the case without speculation, but the individual probability of bank failures do not necessarily increase.
