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
zk1,…,zkn of the realizations. Player 2, who does not know the realizations, faces a stopping problem. At period 1, player 2 learns zkn. If player 2 accepts, then player 1 pays zk1 euros to player 2 and play ends. Otherwise, if player 2 rejects, play continues similarly at period 2 with player 1 offering zk2
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 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.
