I think your proposal is more complicated than, say, mutually randomly choosing an ordering in one step. Does it have any superior properties to just doing that?
I don’t think the mechanics of the problem, as specified, let them mutually specify random things without something like an externally-provided probability distribution. This is aimed at eliminating that requirement. But it may be that this issue isn’t very illuminating and would be better addressed by adjusting the problem formulation to provide that.
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.
I think your proposal is more complicated than, say, mutually randomly choosing an ordering in one step. Does it have any superior properties to just doing that?
I don’t think the mechanics of the problem, as specified, let them mutually specify random things without something like an externally-provided probability distribution. This is aimed at eliminating that requirement. But it may be that this issue isn’t very illuminating and would be better addressed by adjusting the problem formulation to provide that.
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.