Special on-line Bahncard problem and its competitive analysis.

While designing the issue of the discount card, stores need to strictly analyze the effect of factors such as the issue price and discount rate on the consumer's behavior. Using the theory of on-line algorithms and competitive analysis, this paper studied the on-line problem how consumers make decisions between two discount cards which are issued at the same time. On the one hand, the optimal deterministic strategy and its competitive ratio were obtained; on the other hand, a randomized strategy was constructed and an upper bound for the optimal randomized competitive ratio was obtained, and moreover, a lower bound for the optimal randomized competitive ratio was proved by Yao's Lemma. Thanks to numerical examples, the effect of the factors on the on-line strategies and their competitive ratios was analyzed. The results may instruct the stores in determining the issue price and discount rate of the discount card. [ABSTRACT FROM AUTHOR]