Competitive risk management for online Bahncard problem

Abstract

In the Bahncard problem a traveler decides when to buy a Bahncard,i.e., a railway discount card of the German Deutsche Bundesbahncompany, in an online setting. This problem is introduced byFleischer and some optimal deterministic algorithms are presentedwith a fixed Bahncard price. In practice, however, travelers aretrying to manage their risks by using some forms of rewards andtheir forecasting skills. We extend Fleischer's model to a new onein a risk management framework. For such an extended problem, weprovide some flexible results which can be used by a traveler toobtain an optimal risk algorithm based on his risk tolerance andforecast. We further study another extention of the Bahncard problemwith a fluctuated Bahncard price. We propose some algorithms andanalyze their competitive ratios with and without risk,respectively. It turns out that a traveler can significantly improvehis risk management performance by putting reasonable forecasts inconventional competitive analysis.