In the first, there are n i.i.d. fair coin flips and k noncommunicating players (working as a team) each of whom are sent these n coin flips. However, independently for each player and for each coin, the incorrect outcome is sent with probability epsilon. Each player is required to `simulate a fair coin' based on what they receive, using no other randomness. (For example, they might (1) just use the outcome of the first coin they receive or (2) if n is odd, use a majority rule meaning saying heads if and only if there are more heads than tails.) The goal for the team is to maximize the probability that the outcomes of the coins `tossed' by the k players are all equal. The question is what is the maximal probability that can be achieved and what is the optimal protocol that achieves this.
In the second model, player 1 receives n i.i.d. coin flips which she sends to player 2, which in turn are sent to player 3, etc until the kth player. Again, the coins are randomly corrupted as above and again the players want to find a protocol (as above) maximizing the probability that the outcomes of the coins `tossed' by the k players are all equal. In this model, in turns out that the unique optimal protocol is that everyone uses the outcome of the first coin they receive.
There is also a more general model encompassing both of the above situations.
This is joint work with Elchanan Mossel, Ryan O'Donnell, Oded Regev and Benny Sudakov.