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.
