The formation of the government (Executive branch ) happens after the election and can be done in multiple ways. This is independent of the elections themselves. There are many systems of government, each of which has an electoral system and a system of government formation as components.
This concept is more relevant for parliamentary systems since the executive branch is formed as a subset of the legislative branch and as such requires a method to do so. Typical parliamentary systems use a two-step process, first an election is called where the representatives are elected by citizens through a balloting system, then the government is formed from the representatives through its own process.
In a presidential systems the process of electing the the executive and legislative branch is totally separate. This means the government formation process will have presidential elections distinct from the election of the legislative branch. In many situation the cabinet is appointed by the president to for the executive branch but these appointments requires approval by the legislative branch.
The Westminster System
The Westminster system is the most common system for parliamentary systems. In Westminster System, the ballots are cast by a single vote where the candidate with the most votes represents the regional constituency. This electoral system, known as Single Member Plurality, is use to elect the representatives for the parliament. Many countries have modified the electoral system but kept the process of government formation the same. The executive is then formed by the member of the parliament who has the confidence of the largest number of other elected members. In practice, the process of obtaining “The Confidence of the House” has many traditions, paramount of which is that the leader of the party with the most elected members is entitled to the first attempt to form a government. This member is known as the Prime Minister. The rest of the remainder executive branch is appointed by the Prime Minister from within the parliament. With whole of the parliament being the legislative branch the executive is a subset of the legislative branch.
Much of the representation of the election of the parliament can be lost in the formation of the government if it only represents one political party or faction. For this reason, in the Westminster System the parties that are not involved in the formation of the government form the opposition. In a similar manner to the formation of government, the opposition is often formed from the largest party not in the government. The opposition is intended to be a safeguard on the actions of the government. Despite this, elected members in the opposition have significantly less influence than those who are in the government. Furthermore, members who are not in the government or the opposition have even less influence. Some representation is invariably lost for some regions and consequently some citizens.
There is not really a winning party in a Representative Government but merely a party who forms the government. For the unity of the government, its formation is done largely by the discretion the Prime Minister. The leader of the party who has the most seats is typically selected as the prime minister and tries to form the government. If the leader of the largest party does not have enough members in 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.
Criticism and alternate methods for Prime Minister Selection
(This section has some opinions from various advocacy groups).
It would be unexpected for the leader of any party to be the member with the highest confidence of the house. Because of which, the standard process of government formation can be claimed to not adhere particularly well to the concept of a representative government where all representatives have equal power. Instead, it would be desirable to choose a Prime Minister through election by the members in a similar manner as the members were chosen in the first place. In principle, this could be a member which leads no party, but has broad general support of the parliament. The role of the official opposition in the Westminster System is an important check on governmental power. The leader of the opposition must also be selected to form this opposition. It would be desirable if the government and the opposition were optimally polarized to each other so that opposition could be ensured.
There is much desire to find a leader who would have high confidence of the house but this is difficult to select. Forming coalitions which comprise a majority has been known to take a very long time as well. There is likely no way to force a majority of members to agree on a leader so minority governments are unavoidable. (One possible majority rule-based alternative would be to use Condorcet voting to find a beats-all winner or at least a Smith set of viable candidates to lead the government, possibly by using Smith//Approval to find the candidate with the largest majority approval in the Smith set. Candidates not in the Smith set could be eliminated over repeated ballotings.)
There are notable examples of government formation through methods without reference to any party. An interesting example in Canada is “Consensus Government” in the Northwest Territories and Nunavut. There are no parties and the legislature elects first the speaker, then the premier, and finally the cabinet members from amongst themselves.
One of the major motivations for Proportional Representation stems from wanting the party which forms the government to also be the party with the greatest proportion of the popular vote. This motivation can be eliminated under different government formation systems which are less partisan. The current method makes the level of Proportional Representation crucial to the formation of the government. An alternative method which is more democratic and representative would be preferred and greatly decrease issues with low Proportional Representation.
A common suggestion is a bottom up method where the selection of the Prime Minister is done by a vote from among the members themselves. This then returns us to considering the balloting methods like any other election. When designing a selection method, the goal would be to select a broadly appealing member as the leader and another which opposes their general stance to lead the opposition. Most voting reform advocates would argue that Single Plurality Voting is a very polarizing method (with many cardinal method advocates further arguing that IRV is also polarizing) so one might think that these types of methods 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 approval of the house, a different system must be used.
Unlike in general elections where multiple rounds would be logistically and economically infeasible 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. Cardinal method advocates typically advise that a form of Score Voting would be optimal since they argue it to be the best system for single winner elections (based on metrics such as Bayesian regret i.e. social utility). 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. 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 likely best, otherwise known as Approval Voting. It would also help the public see their representation if the vote was public and open so the electorate can see who their representative supported.
In summary, the most common suggested replacement method of government formation by cardinal method advocates is to have an 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 most common rebuttal to this is change is that there would be Strategic voting in these elections. The consequences of strategic voting for Approval Voting are argued to 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. It also would eliminate the need for Proportional Representation to justify the government formation.
This system does make a large change in that being an independent would not be nearly as detrimental. There could arise situations where the vote for the leader came down to the independents. The independent’s ability to choose any member without being restricted by a party could in fact give an independent more power than they would have when in a party. This is the second most common criticisms of this change to government formation since it would make the independents "King Makers". While potentially true it has never been seen in practice and may be a more stable situation than coalitions since a coalition can threaten to break the government over sub-partisan issues. Coalitions are often formed through agreements but this system would be binding and not down to the bickering of factions. As with the current system, the government could always be dissolved by a vote of non confidence.
While this is a less partisan system than the current system, one should not expect political parties to disappear under this Approval Voting government formation system. If there are ideological rifts inside parties they may split into smaller parties which are more ideologically homogeneous. However, they may still end up forming the government together. The difference is that the leader of the government would not necessarily be the leader of any party in the coalition. The current SMP system tends to force parties to merge prior to the election but this new method of government formation gives no advantage for two small parties to merge. It would even be possible for some members of a party to be in the government while others are in the opposition.
Representation When Voting on Bills
It is common that changes to laws are not decided by the government alone and many countries have statements in the constitution restricting what decisions can be made by the Executive branch. The government is normally not formed from all parties so some decisions are made based on a vote from the whole of the elected representatives, the legislative branch. There are many ways in which this can happen and they tend to depend on the system under which the representative was elected.
By far the most common method of Legislative voting is that each representative gets one vote to approve or disapprove of a bill or motion. In a regional system where each region has the same number of constituents this makes sense. Even in this case with both Balanced Representation and Proportionate Representation, some have argued that this method is lacking. The representative is expected to vote on behalf of all their constituents but it is argued that they can’t do this on divisive issues because not all the constituents agree. Therefore, it is open for debate if a representative should have the voting power of those who did not vote for them and do not want to be represented by them.
In some systems, it would be possible for the member’s vote to have power weighted the number of votes cast for them. Alternatively, voting power could be based on the percent of votes cast for them. These two differ in that the former is affected by voter turnout. Systems like Score or Approval voting would be able to more accurately weight the power because the votes for them would better represent their support. This type of weighted vote is often proposed but rarely adopted because of the logistical complications.