One part of electoral systems that is not really discussed much is how the leader and the government is formed from the parliament. I am Canadian and am most familiar with our systems so I will use the Westminster system as an example. Lets assume the parliament has been elected and the task now is to come up with which members of the parliament form the government and who is the Prime Minister.
In the current system the leader of the party who has the most seats is typically selected as the prime minister and tries to form the government. This method is mostly there for historical reasons and should be evaluated under any process of electoral reform. If the leader of the largest party does not have enough seats for their party to have a majority, a coalition of other parties may form. Coalition governments are formed by two or more parties combining to govern together and blocking the confidence in the leader of the largest party. It would be unexpected for the leader of any party to be the member with the highest confidence of the house. Because of which, this process does not adhere particularly well to the concept of a representative government. This leads to one of the major motivations for Proportional Representation. It stems from wanting the party which forms the government to also be the party with the greatest proportion of the popular vote but this is only because of the parties role in government formation.
This seems exactly like a perfect situation for asset voting but I am not really an expert in that so Ill leave that to others and describe how I would do it.
When designing a selection method, the goal would be to select a broadly appealing member as the Prime Minister and another which opposes their general stance to lead the opposition. The most polarizing systems are Single Plurality Voting and Rank Voting so one might think that these would produce a good Prime Minister and opposition leader. However, in both these systems it is highly likely that the winner and runner-up would be leaders of parties. If the idea is to find a candidate who can get the highest confidence of the house, a different system must be used.
Unlike in general elections where multiple rounds would be logistically and economically unfeasible this is not the case for elections within the members of the assembly. There could be one vote to elect the Prime Minister, and then from the remaining members a second to elect the leader of the opposition. It is suggested that a form of Score Voting would be optimal since it has been established as the best system for single winner elections. It would be desirable that the leader of the opposition is chosen by those members who did not vote for the leader of the government as it would optimize polarization/opposition. In a Score Voting system with more than two gradations, it becomes unclear who those members should be. As such, the version of score voting with a binary choice is best, otherwise known as Approval Voting. It is also suggested that the vote be public and open so the electorate can see who their representative supported.
In summary, there would be one Approval Vote for the Prime Minister. Those who vote for the winning candidate form the government. The leader of the opposition would then be chosen by a second round of Approval Voting from those who are not already in the government. The Prime Minister and the opposition leader would then be free to choose their cabinets as in the current system.
The consequences of strategic voting for Approval Voting would be for all members to vote for any member they could work with. This would be all members of their own party and for all but the largest party, several select candidates from other parties. Typical results would be the same as the current system but in a minority of cases a more unifying leader would be chosen. This would solve the problem of minority governments and coalitions. It can be thought of as a method to find the best coalition government to rule.