For Sums of Bivariate Random Variables

FOR SUMS OF BIVARIATE RANDOM VARIABLES Minoru Sakaguchi Osaka University (Received March 24, 1978) This paper explores optimal strategies for the problem of choosing several best from a set of sequentially observed bivariate random variables. For example, a couple of husband and wife to make a plan of recreations during a year, has this problem when deciding which offer (or amount of satisfaction) to accept and which to reject. Each offer on arrival is examined first by husband and, if accepted by him, then secondly by his wife. If she rejects it, the offer is rejected. Therefore the offer is "selected" only when both of husband and wife accept it. We assume that the offers are iid bivariate r.v.'s and at most n can be observed. Each offer on arrival is either selected or rejected; an offer rejected now cannot be selected later on. The objective of husband (wife) is to maximize the expected value of the sum, from his (her) standpoint, of the offers actually selected. For another example, the problem of optimal selection of r secretaries to be employed by two university professors in the same department belongs also to our problem. By using dynamic programming technique we develope the optimal procedure for this non-cooperative, sequential, bilateral game and discuss several simple examples. It is shown that it does not matter to husband ( or wife) whether he ( or she) decides first or second. It is also shown that it does matter to either side whether it decides first or second, if the number of rejections available to each side is given beforehand.