This is a concept I had in mind which may already have been described, although not all of the logistics are necessarily hashed out and there may be issues with it. The idea is as follows, and forgive my lack of education regarding how political parties work:
There should be a government body that registers political parties and demands the compliance of all political parties to its procedures in order for them to acquire seats for representation;
(Eyebrow raising, but you might see why...) Every voter must register as a member of exactly one political party in order to cast a ballot (?);
Each political party A is initially reserved a number of seats reserved in proportion to the number of voters with membership in A; the fraction of seats reserved for A is P(A). however
For each pair of political parties A and B (where possibly B=A), a fraction of seats totaling P(A|B):=P(A)P(B) will be reserved for candidates nominated by A, and elected by B; these seats will be called ambassador seats from A to B when B is different from A, and otherwise will be called the main platform seats for A;
Let there be a support quota Q(A|B) for the number of votes needed to elect ambassadors from A to B, and call P(A|B) the ambassador quota of party A for B. If E(A|B) is the fraction of filled A-to-B ambassador seats (as a fraction of all seats), I.e. nominees from A who are actually elected by members of B, then A will only be allowed to elect P(A|A)*min{min{E(A|B)/P(A|B), E(B|A)/P(B|A)}: B not equal to A} of its own nominees. That is, the proportion of reserved main-platform seats that A will be allowed to fill is the least fraction of reserved ambassador seats it fills in relation to every other party, including both the ambassadors from A to other parties, and the ambassadors from other parties to A.
This procedure forces parties to nominate candidates that compromise between different party platforms in order to obtain seats for any main-platform representatives. If a party fails to meet its quota for interparty compromises, it will lose representation. On the flip side, this set up will also establish high incentives for other parties to compromise with them in order to secure their own main-platform representation. In total, this system would give parties high incentives to compromise with each other and find candidates in the middle ground, which will serve as intermediaries between their main platforms which they can maintain.
Basically, here the outlines indicate seats open to be filled by candidates who are nominated by the corresponding party, and the fill color indicates seats open for election by the corresponding party:
Cake Cutting PR.png
Seats with outlines and fills of non-matching color are ambassador seats, and seats with matching outline and color are main platform seats. In terms of party A, by failing to nominate sufficiently-many candidates who would meet the support quota Q(A|B) to become elected as ambassadors from A to B, or by failing to elect enough ambassadors from B to A, party A restricts its own main platform representation and that of B simultaneously. By symmetry the reciprocal relationship holds from B to A. Therefore all parties are entangled to nominate compromise candidates and to elect ambassadors in order to secure main-platform representation.
To see that all needed seats are filled in the case of a stalemate, where parties refuse to nominate acceptable candidates to other parties and/or refuse to elect ambassadors, the election can be redone with the proportions being recalculated according to the party seats that were actually filled.
The support quotas collectively serve as a non-compensatory threshold to indicate sufficient levels of inter-party compromise. Ordinary PR is identical to PR with ambassador quotas but with all support quotas set to zero, whereby there is no incentive to nominate compromise candidates.
The purpose of this kind of procedure is twofold: firstly, it should significantly enhance the cognitive diversity of representatives, and secondly, it should significantly strengthen more moderate platforms (namely those of the ambassadors) that can serve as intermediaries for compromises between the main platforms of parties. Every party A has a natural “smooth route” from its main platform to the main platform of every other party: The main platform of A should naturally be in communication with ambassadors from A to B, who should naturally communicate with ambassadors from B to A, who should naturally communicate with the main platform of B.
