Since cardinal method’s Nash Equilibrium in the single-winner case is the Condorcet winner, it’d seem as though a similar result ought to be derivable in the PR case. However, since cardinal PR methods may not pass Droop PR even when Droop Quotas vote maximally strategically, and Condorcet PR methods are expected to pass Droop PR even under honesty, this is in some doubt.
It may help to consider this possibly superior definition of Hare PR, which most cardinal PR methods pass:
since it shows that both cardinal and Condorcet methods pass Droop PR under maximally strategic conditions in the single-winner case, but not necessarily in the multi-winner case.
(I’d argue that the Nash Equilibrium for cardinal methods in the single-winner case is, with certain assumptions, actually the Smith Set in general, since if voters observe, in pre-election polling, a cycle between the same candidates for who should win, then they may well decide to place their approval threshold between everyone in the cycle that they prefer to everyone else, and everyone else.
One example with ranked preferences:
Cycle between A, B, and C, and 60 voters, a majority prefer any of them to D.
Image showing equilibrium and post for the image:
The image was done for Asset, but should be modifiable to the cardinal methods if you consider voters moving their approval thresholds around for each of the pairwise matchups.)
(Incidentally, this makes me wonder whether some voters could actually start Condorcet cycles in the cardinal polling and thus make maximally strategic voters elect some non-CW even when a CW exists, though I’d need to think further on it.)
Edit: Something to ponder: if Score can be thought of as a Condorcet method where voters can give partial votes in runoffs, and under this definition Score always elects the CW and can never have cycles, then is it possible to run a Condorcet PR method on Score ballots, processing them as mentioned above, and always find a Condorcet Winner PR set? Because if so, this method would seem to me to have a strong claim to being the or one of the conceptually best ways to formalize cardinal PR.
@Toby_Pereira it’d seem that by doing the KP transform on Score ballots, and then turning the resulting Approval ballots into ranked ballots by treating all approved candidates as co-favorites and disapproved candidates as co-equal last, and then running a Condorcet PR method on this, that this is one way of making a cardinal PR method derived from Condorcet.
(A question that remains: Consider that in the single-winner case, with scores of A5 B4 C3, we could allow a voter to actually control the scores they give to each pair of candidates in runoffs (and then tabulate that with a Condorcet method) so that they instead give, say, A5>B0 but B4>C3 and A5>C0, or some such. Can some variation of the KP transform be done with Condorcet PR to allow a voter to have exactly this much control in pairwise matchups between sets?)