[EM] CPW = CW in the 3 by 1 by 1 case.

[Forest Simmons]

Here's an heuristic argument showing why proxy approval is apt to pick the
CW when there is one, regardless of the dimension of the issue space.

When there is a CW, rational approval players with perfect information
will concede the win to the CW, because the CW is the one and only stable
equilibrium winner.

In proxy approval the proxies have near perfect information about
preferences of the other players, and perfect information about the number
of votes that they control.

The proxies may not be entirely rational, but they do have more access to
strategists than the average voter.

Because of the above facts, the CW of the (weighted) proxies is almost
sure to be elected by them.

The preferences of the proxies are approximately the same as those of
their supporters, so the proxy CW is apt to be the same as the CW of the
voters.

[end of heuristic]

This last link is the weakest in the chain, so ways of strengthening it
are all to the good.

We showed that in one dimensional issue space, that the apparent weakness
of this last link was of no consequence.  The proxy CW was always the same
as the voter CW.  Perhaps there is a way of carrying this argument over
into higher dimensional issue spaces as well.

Forest

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