I need to educate you some, John. I hope you don't mind.
When there is a prime-minister, this person is based on the majority of the Representatives. This is actually always the case; there are circumstances that appear to be minority-based (but they are not).
Under specific circumstances, the government may be based on a number of Representatives that constitute a minority, yet they are tolerated by a majority of the Representatives. As such, they are not voted away and as such they have the majority of the Representatives (until voted away).
Any time a vote of No Confidence is answered with a majority of the votes, then the cabinet 'falls'. Either new elections must be held or new negotiations are started up to see if there is another majority of Representatives possible.
What is interesting is that the cabinet does not fall for real until the next cabinet is formed and put in place.
When the vote of no confidence is expressed, the government remains in place until the next government is formed. As long as the majority of the Representatives agree, anything the 'fallen' cabinet proposes does indeed pass. They govern until there is a new cabinet and they do get stuff passed (just not the biggest items because that made the cabinet fall).
So, there are no lame duck situations. That is simply possible because the power is always based on the Representatives in the House. Power is never ever in the hands of anyone else. So the cabinet that is managing the nation is based on the majority of the Representatives giving them the green light to be the government.
When elections are held, the next day all seats are proportionally taken up fully in line with the election results.
The UK, I actually do not even want to talk about them. They have the worst democracy in the world, and the only good thing is that they don't have an empowered president on top of their two chambers.
For Israel, I can only say that I admire Israel trying to be a full-representative democracy. They have been living in a war zone ever since conception. Can you imagine the pressure, day in, day out, for these people. I hope their current troubles of government pass real quickly and that the Israelis don't split in the middle. I can see an exodus quite easily of half of Israel leaving for other places and the other half being chopped into little pieces by their enemies. No matter their voting system, they have bigger fish to fry.
Now, if you want to talk about a lousy proportional voting system, then Italy is the best way forward. Italy's biggest flaw is that both houses are equally empowered to drop the government. It is like building a house and then on top of the house you build another house, and you don't have to take any measures to support that second house other than by the basement preparations for the first house.
Italy is a weird nation that wanted to unite into a larger nation, but the church and the big companies did not want to give too much power to their government. They came up with the weakest form of democracy possible, while shielding the church from any government, simply giving it its own status of a State (smart, yet at the same time making it very obvious what they were doing and how they did not want a strong government at all).
Nations without a strong government can actually have a very strong industry, a very strong economy. The United States is one example, but Italy is the other prime example. Italy is EU's third largest economy. But also the nation with a lot of government debt. You see? The companies own Italy, and when they want the government to pay, then they wiggle and wiggle and wiggle until the government pays. That's how they rake up their enormous government debt; they hand out too much to the companies. Third strongest EU economy.
The power of the voters is vested in the Representatives. In Denmark, Sweden, and Norway, there is just one House. That's it. This makes the voters very powerful.
In the Netherlands and Spain, there are two chambers, and the focus is particularly on one of the chambers where the real power lies. The other chamber has correcting powers (they can say No to a new law, but not write the law themselves); that chamber won't let cabinets 'fall.' The voters are still empowered, but not like the Scandinavian countries.
Germany is the most interesting country for us because they got our system after WW II. Yet they did not like our system in its original version, so they fixed it up a bit. They have our system, but they do not have gerrymandering and they do not have just red and blue in their House of Representatives. They fixed up our system, and they are quite the example. The Founding Fathers were channeled by the American forces in Germany, but they channeled them for the 20th century. They did a pretty good job.
Thank you, John. I hope this helps you understand a bit more about the enormous variety and differences in voting systems in the democratic world.
When admiring basketball, one should be honest enough to say that the game is rigged and that particularly tall people end up playing the big league. Basketball is not a system of fair representation, and neither is winner-take-all.