The Bahncard Problem with Interest Rate and Risk

Abstract

This paper investigated a new framework for the competitive analysis of the Bahncard problem. In contrast to the earlier approach we introduce the interest rate i and the risk tolerance t into the model, in which the traveller can develop the optimal trading strategies based on his risk preference. Set $\alpha=\frac{1}{1+i}$. We prove that the Bahncard problem with the interest rate is $1+(1-\beta)\alpha^{{m^*}+1}\,$-competitive, where m ^* is the critical point. Then we further design a t-tolerance strategy and present a surprisingly flexible competitive ratio of $1+\frac{(1-\beta)\alpha^{m^*}}{tr^*-(1-\beta\alpha^{m^*})}$ , where r ^* is the optimal competitive ratio for the Bahncard problem with the interest rate and β is the percentage of discount.

This research is supported by NSF of China under Grants 10371094 and 70471035.