Sultan's Dowry

The Sultan's Dowry is a sticky problem in probability that first came to light in Martin Gardner's 'Mathematical Recreations' column in the February 1960 issue of Scientific American. Gardner's original version has become known as the Secretary Problem. In the exactly equivalent form called the Sultan's Dowry Problem, a sultan has granted a commoner the chance to marry one of his hundred daughters. The commoner will be shown the daughters one at a time and will be told each daughter's dowry. The commoner has only one chance to accept or reject each daughter; he can't go back and choose one that he has previously rejected. The sultan's catch is that the commoner may only marry the daughter with the highest dowry. What is the commoner's best strategy, assuming that he knows nothing in advance about the way the dowries are distributed?


Many mathematicians have tackled this question and numerous papers have been written on the subject. It has even spawned its own area of study within the field of management science. The consensus among those who have worked on the problem is that the commoner's best strategy is to let a certain fraction of the daughters pass and then choose the next one who has a dowry higher than any of the ones seen up to that point. The exact number to skip is determined by the condition that the odds that the highest dowry has already been seen is just greater than the odds that it remains to be seen and that if it is seen it will be picked. This amounts to finding the smallest x such that:


    x/n > x/n × [1/(x+1) + ... + 1/(n-1)].


Substituting n = 100 leads to the conclusion that the commoner should wait until he has seen 37 of the daughters, then pick the first daughter with a dowry that is bigger than any that have already been revealed. With this strategy, his odds of choosing the daughter with the highest dowry are surprisingly high: about 37%.



1. Mosteller, F. Fifty Challenging Problems in Probability with Solutions, Addison- Wesley, 1965, #47; "Mathematical Plums", edited by Ross Honsberger, pp. 104-110.