To clarify, what I meant was not that they need a source of shared randomness, but that they need a shared probability distribution; ie, having dice isn’t enough, they also need to coordinate on a way of interpreting the dice, which is similar to the original problem of coordinating on an ordering over points.
We already assumed a source of mutual randomness in order to guarantee that the feasible set is convex (caption to Figure 1).
To clarify, what I meant was not that they need a source of shared randomness, but that they need a shared probability distribution; ie, having dice isn’t enough, they also need to coordinate on a way of interpreting the dice, which is similar to the original problem of coordinating on an ordering over points.