Average-Case Competitive Analyses for One-Way Trading

Consider a trader who exchanges one dollar into yen and assume that the exchange rate fluctuates within the interval [m,M ]. The game ends without advance notice, then the trader is forced to exchange all the remaining dollars at the minimum rate m. El-Yaniv et al. presented the optimal worst-case threat-based strategy for this game [EFKT01]. In this paper, under the assumption that the distribution of the maximum exchange rate is known, we provide average-case analyses using all the reasonable optimization measures and derive different optimal strategies for each of them. Remarkable differences in behavior are as follows: Unlike other strategies, the average-case threat-based strategy that minimizes E[OPT/ALG] exchanges little by little. The maximization of E[ALG/OPT] and the minimization of E[OPT]/E[ALG] lead to similar strategies in that both exchange all at once. However, their timing is different. We also prove minimax theorems with respect to each objective function.