Recently personalized medicine and dynamic treatment regimes have drawn considerable attention. Dynamic treatment regimes are rules that govern the treatment of subjects depending on their intermediate responses or covariates. Two-stage randomization is a useful set-up to gather data for making inference on such regimes. Meanwhile, the number of clinical trials involving competing risk censoring has risen, where subjects in a study are exposed to more than one possible failure and the specific event of interest may not be observed because of competing events. We aim to compare several treatment regimes from a two-stage randomized trial on survival outcomes that are subject to competing risk censoring. The cumulative incidence function (CIF) has been widely used to quantify the cumulative probability of occurrence of the target event over time. However, if we use only the data from those subjects who have followed a specific treatment regime to estimate the CIF, the resulting estimator may be biased. Hence, we propose alternative non-parametric estimators for the CIF by using inverse probability weighting, and we provide inference procedures including procedures to compare the CIFs from two treatment regimes. We show the practicality and advantages of the proposed estimators through numerical studies.