Than in the two previous cases, withdrawal cascades have a higher

Than in the two previous cases, withdrawal cascades have a higher chance to emerge. We can indeed observe on Fig 3 that if either the sample size is low (N = 10) or the share of impatient depositors is large ( = 0.9), the only crossing point is at k = 0, a bank run occurs. On the contrary, we obtain that there is no bank run if either the sample size is large enough or the share of impatient depositors is low. The analysis provided by Eq (13) shows the average long-run share of depositors who keep their money in the bank. However, due to randomness in the initial conditions (that is, the actual share of patient depositors at the beginning of the sequence may vary) and in the sampling (e.g. patient depositors may happen to observe more impatient depositors), the actual dynamic process may not always converge to this average long-run outcome. Therefore, we use simulation methods to verify the previously obtained results. In the simulations, we consider a large population of 107 depositors, because our results are obtained for an infinite sequence so we need many depositors. We compute the decision threshold following Lemma 2. We assume that the first 50000 depositors act according to their type (m = 50000), that is, impatient depositors withdraw, patient depositors keep their money in the bank. After these initial set of depositors, we start the process whereby patient depositors decide about withdrawal based on a random sample of size N and the threshold decision rule. We are interested in the long-run share of waitings (k), we measure it using the last 20000 depositors and record the percentage of depositors who keep their money in the bank. In line with Definition 1, we say that a bank run occurred if this percentage is below 3 . We have experimented with alternative thresholds (1 , 5 ), the results remain unchanged, therefore we omit them from the paper. For each parameter setting, we carry out 100 simulation runs and assess the ZM241385 chemical information probability of bank runs based on how many times out of the 100 RG1662 solubility simulations a bank run happened. We tested whether this number of simulation runs gives robust results by running the wcs.1183 model for several sets of 100 runs. We carried out this test for several parameter settings. The different simulation sets of 100 runs give exactly the same probability of bank runs, thus we concluded that it is sufficient to run 100 simulations per parameter setting.PLOS ONE | DOI:10.1371/journal.pone.0147268 April 1,17 /Correlated Observations, the Law of Small Numbers and Bank RunsFig 2. The long-run theoretical share of depositors who do not withdraw (k) in the case of random sampling and Scenario 2. The black line represents the left-hand side of Eq (13) (i.e. the 45-degree line), the colored lines represent the right-hand side of Eq (13) for different parameter values as shown in the legend. The long-run share of j.jebo.2013.04.005 depositors who do not withdraw is given by the largest (rightmost) crossing point of the 45-degree line and a givenPLOS ONE | DOI:10.1371/journal.pone.0147268 April 1,18 /Correlated Observations, the Law of Small Numbers and Bank Runscolored line. The parameter values are as in Scenario 2 (R = 1.3, = 2.5). And on the first Panel: N = 85, is varied as = 0.1 (blue line), = 0.5 (red line), = 0.9 (green line). On the second Panel: = 0.5, N is varied as N = 10 (blue line), N = 85 (red line), N = 160 (green line). doi:10.1371/journal.pone.0147268.gTable 3 shows the simulation results for the different Scenarios desc.Than in the two previous cases, withdrawal cascades have a higher chance to emerge. We can indeed observe on Fig 3 that if either the sample size is low (N = 10) or the share of impatient depositors is large ( = 0.9), the only crossing point is at k = 0, a bank run occurs. On the contrary, we obtain that there is no bank run if either the sample size is large enough or the share of impatient depositors is low. The analysis provided by Eq (13) shows the average long-run share of depositors who keep their money in the bank. However, due to randomness in the initial conditions (that is, the actual share of patient depositors at the beginning of the sequence may vary) and in the sampling (e.g. patient depositors may happen to observe more impatient depositors), the actual dynamic process may not always converge to this average long-run outcome. Therefore, we use simulation methods to verify the previously obtained results. In the simulations, we consider a large population of 107 depositors, because our results are obtained for an infinite sequence so we need many depositors. We compute the decision threshold following Lemma 2. We assume that the first 50000 depositors act according to their type (m = 50000), that is, impatient depositors withdraw, patient depositors keep their money in the bank. After these initial set of depositors, we start the process whereby patient depositors decide about withdrawal based on a random sample of size N and the threshold decision rule. We are interested in the long-run share of waitings (k), we measure it using the last 20000 depositors and record the percentage of depositors who keep their money in the bank. In line with Definition 1, we say that a bank run occurred if this percentage is below 3 . We have experimented with alternative thresholds (1 , 5 ), the results remain unchanged, therefore we omit them from the paper. For each parameter setting, we carry out 100 simulation runs and assess the probability of bank runs based on how many times out of the 100 simulations a bank run happened. We tested whether this number of simulation runs gives robust results by running the wcs.1183 model for several sets of 100 runs. We carried out this test for several parameter settings. The different simulation sets of 100 runs give exactly the same probability of bank runs, thus we concluded that it is sufficient to run 100 simulations per parameter setting.PLOS ONE | DOI:10.1371/journal.pone.0147268 April 1,17 /Correlated Observations, the Law of Small Numbers and Bank RunsFig 2. The long-run theoretical share of depositors who do not withdraw (k) in the case of random sampling and Scenario 2. The black line represents the left-hand side of Eq (13) (i.e. the 45-degree line), the colored lines represent the right-hand side of Eq (13) for different parameter values as shown in the legend. The long-run share of j.jebo.2013.04.005 depositors who do not withdraw is given by the largest (rightmost) crossing point of the 45-degree line and a givenPLOS ONE | DOI:10.1371/journal.pone.0147268 April 1,18 /Correlated Observations, the Law of Small Numbers and Bank Runscolored line. The parameter values are as in Scenario 2 (R = 1.3, = 2.5). And on the first Panel: N = 85, is varied as = 0.1 (blue line), = 0.5 (red line), = 0.9 (green line). On the second Panel: = 0.5, N is varied as N = 10 (blue line), N = 85 (red line), N = 160 (green line). doi:10.1371/journal.pone.0147268.gTable 3 shows the simulation results for the different Scenarios desc.