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(k1), ..., z(kn) of the realizations. Player 2, who does not know the realizations, faces a stopping problem. At period 1, player 2 learns z(k1). If player 2 accepts, then player 1 pays z(k1) euros to player 2 and play ends. Otherwise, if player 2 rejects, play continues similarly at period 2 with player 1 offering z(k2) 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 max-min and min-max values of the game G(n) in general, we provide an interval I-n = [a(n), b(n)] containing these 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.
- Secretary problem
- Moser's problem
- Incomplete information
- Lack of information on one side
- Optimal strategies