Abstract
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 (El-Yaniv et al. 2001). 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.
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This work was partially supported by KAKENHI (16092216, 19700015, and 19740059).
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Fujiwara, H., Iwama, K. & Sekiguchi, Y. Average-case competitive analyses for one-way trading. J Comb Optim 21, 83–107 (2011). https://doi.org/10.1007/s10878-009-9239-4
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DOI: https://doi.org/10.1007/s10878-009-9239-4