We consider a game G(n) played by two players. There are n independent random variables Z(1),...,Z(n), each of which is uniformly distributed on [0,1]. Both players know n, the independence and the distribution of these random variables, but only player 1 knows the vector of realizations z := (z(1),...,z(n)) of them. Player 1 begins by choosing an order z(k(1)),...,z(k(n)) of the realizations. Player 2, who does not know the realizations, faces a stopping problem. At period 1, player 2 learns z(k(1)). If player 2 accepts, then player 1 pays z(k(1)) euros to player 2 and play ends. Otherwise, if player 2 rejects, play continues similarly at period 2 with player 1 offering z(k(2)) euros to player 2. Play continues until player 2 accepts an offer. If player 2 has rejected n-1 times, player 2 has to accept the last offer at period n. This model extends Moser''s (1956) problem, which assumes a non-strategic player 1.We examine different types of strategies for the players and determine their guarantee levels. Although we do not find the exact value v(n) of the game G(n) in general, we provide an interval I(n) = [a(n),b(n)] containing v(n) such that the length of I(n) is at most 0.07 and converges to 0 as n tends to infinity. We also point out strategies, with a relatively simple structure, which guarantee that player 1 has to pay at most b(n) and player 2 receives at least a(n). In addition, we completely solve the special case G(2) where there are only two random variables. We mention a number of intriguing open questions and conjectures, which may initiate further research on this subject.
