Summary: We consider a game \(G_n\) played by two players. There are \(n\) independent random variables \(Z_{1}, \ldots , Z_n\), 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 \(z := (z_1, \ldots , z_n)\) of them. Player 1 begins by choosing an order \(z_{k_1},\ldots ,z_{k_n}\) 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 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.