Multiple buying or selling with vector offers

We consider a generalization of the house-selling problem to selling k houses. Let the offers, X1, X2, ..., be independent, identically distributed k-dimensional random vectors having a known distribution with finite second moments. The decision maker is to choose simultaneously k stopping rules, N1,...,Nk, one for each component. The payoff is the sum over j of the jth component of XNj minus a constant cost per observation until all stopping rules have stopped. Simple descriptions of the optimal rules are found. Extension is made to problems with recall of past offers and to problems with a discount.