Competitive risk management for online Bahncard problem

In the Bahncard problem a traveler decides when to buy a Bahncard, i.e., a railway discount card of the German Deutsche Bundesbahn company, in an online setting. This problem is introduced by Fleischer and some optimal deterministic algorithms are presented with a fixed Bahncard price. In practice, however, travelers are trying to manage their risks by using some forms of rewards and their forecasting skills. We extend Fleischer's model to a new one in a risk management framework. For such an extended problem, we provide some flexible results which can be used by a traveler to obtain an optimal risk algorithm based on his risk tolerance and forecast. We further study another extention of the Bahncard problem with a fluctuated Bahncard price. We propose some algorithms and analyze their competitive ratios with and without risk, respectively. It turns out that a traveler can significantly improve his risk management performance by putting reasonable forecasts in conventional competitive analysis.