Strategic disclosure of random variables

We consider a game Gn played by two players. There are n independent random variables Z1; : : : ; Zn, 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 := (z1; : : : ; zn) of them. Player 1 begins by choosing an order zk1 ; : : : ; zknof the realizations. Player 2, who does not know the realizations, faces a stopping problem. At period 1, player 2 learns zk1 . 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 o¤ering zk2 euros to player 2. Play continues until player 2 accepts an o¤er. If player 2 has rejected n 1 times, player 2 has to accept the last o¤er at period n. This model extends Moser’s (1956) problem, which assumes a non-strategic player 1. We examine di¤erent types of strategies for the players and determine their guaranteelevels. Although we do not …nd the exact value vn of the game Gn in general, we provide an interval In = [an; bn] containing vn such that the length of In is at most 0:07 and converges to 0 as n tends to in…nity. We also point out strategies, with a relatively simple structure, which guarantee that player 1 has to pay at most bn and player 2 receives at least an. In addition, we completely solve the special case G2 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.