At the moment the most spread system in the world is the democracy with different accents, some parliamentary, some representative, other presidential, etc.
But recently I have asked my self if exist any political system currently being used but not being a explicit democracy but having some characteristics of it like:
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The Kingdom of Bhutan is a fully functioning monarchy, and in spite of that ranks very high in what it calls "Gross National Happiness." To be sure, it is a benign dictatorship, one in which the monarch makes quite a show of being concerned for his subjects. Historically, this paternalistic approach to governance has actually been the norm. It wasn't always as good as hoped for, but most rulers desired to see the needs of their people be well met.
Additionally, Isaac Asimov once postulated a United States in which the "best" person was selected by a computer to be President, after neutrally testing any applicants who desired to run. The crux of the story was that in one year's cycle, the computer refused to pick anyone, because no one was fully up to the task. There is nothing that says that people necessarily choose the "best" person for the job. Indeed, being too much of a policy wonk may help the government run better, but still fail to capture the people's love. Mario Monti, Prime Minister of Italy, is often called a "technocrat," and widely heralded by pundits as a result, but is almost certain to be defeated in the next election. A match-up against his predecessor, Silvio Berlusconi, held in any kind of public forum, would have been a massacre. But when bond yields plummeted, Monti made good.
Indeed, it is real work to be an active participant in one's own governance, and so one could argue that if happiness (utility in economic terms) were the only desired outcome, then a despot attuned to the needs and wants of his people is actually a better system.
Unfortunately, as Madison said, "if men were angels, we wouldn't need government in the first place." As such, such benign dictatorships, while definitely the model of Plato, are rare in practice. And thus, pragmatically, representative democracy's advantage of incentivizing the goal of looking after the people by giving it power as a reward tends to eventually win out. In the word of Winston Churchill, "Democracy is the worst form of government, except for all the others."
Typically when we hear about accepted forms of government and being a member of the "Free World" as it is called it is typically encouraged for the government to have more democratic measures. The United States as leader of the "Free World" so to speak of course has vested interests in expanding its allies, promoting its security, and encouraging trade.
Typically Democracy is encouraged as it is the most fair (and most predictable) of three basic types of government.
Autocratic: Ruled by a single leader unanimously. Dictatorships, Monarchies, etc... Good examples are North Korea and Syria. The countries actions are unpredictable and may or may not favor the political, defense, or economic goals of first world nations.
Oligarchies: Ruled by a small but powerful group of individuals that set policy and lead the country. They usually have a figure head leader that changes from time to time. Good examples include modern China. Also hard to influence the direction of the country and may be unpredictable.
Democracy: Ruled by elected representatives in which all citizens have a right to choose their leaders. It is by its very nature the most fair, although minority groups may get trampled easily. Good example includes USA, India or Japan. These tend to favor capitalism and market forces the most and tend to be mostly predictable, though not always easily controllable. Democracies are ideal to the capitalist driven free market model that is typically pushed by such "Free World" institutions such as the World Bank and the IMF.
The important thing to note however is that other types of governments have throughout history been accepted by the Democratic free world.
Egypt Mubarak was supported financially and militarily by the United States for a time because he was predictably against military conflict with Israel.
The Post WWII Greek Monarchy Shortly after WWII a communist uprising occurred in Greece, where the United Kingdom and the United States supported the reinstatement of the old Monarchy. He was a Far-Right monarch that predictably was anti-communist, and was open to foreign investments from wealthy capitalists. The United States supported a government that tortured and executed its own citizens and was ruled by a single man.
There are also exceptions where true democracies are more or less not accepted by members of the free world because of socialist or nationalist policies that its citizens had more or less voted in favor of.
Venezuela Is a democracy with elections and representative leaders, however it is painted with a rather negative light because of the highly nationalistic and socialistic tendencies of its elected leaders. Venezuela is restrictive of foreign capitalist investments which makes it an enemy of the IMF and the World Bank as well as wealthy capitalists that tend to have a lot of influence in first world democracies.
Your question is in fact open-ended, because every social system can be considered rational from at least the point of view of the ruling group.
If the rational is the synonym of planned, than the communistic system was fully rational. It was based on the intelectual speculation, but it wasn't as effective in practice as it promised to be.
Rational as the rule of the reason - this definition makes the democracy one of the least rational systems - it is based mostly on emotions, not the reason. Its emotionality is caricaturized by Plato, who proposed the dictature of the reason - the system that have nothing to do with what is called democracy.
As others have pointed out, rational is relative. There is no absolute answer to this question, but it may be instructive to compare countries with different systems. As a disclaimer: comparisons always fail because countries have different backgrounds. Therefore, any conclusions from comparisons need to be taken with a pinch of salt.
Cuba is a dictatorship that self-identifies as socialist. In many international rankings, it doesn't do so poorly. For example, Cuba has a high human development and is often considered to have one of the best healthcare and education systems in the developing world. By comparison, neighbouring Haiti has been a shaky democracy since some decades, ridden by coup d'etats, and has a low human development with many serious problems. On the other hand, Costa Rica also does pretty well considering its material wealth, but perhaps with more poverty than Cuba (hard to find reliable figures). Costa Rica has been a stable democracy since the 1950s and doesn't seem to do significantly better or worse than Cuba. As an interesting side note, Costa Rica abolished its army in the 1950s and has since then been one of the few countries in the region to have no wars and no coup d'etats, something that might have contributed to its stability.
China is a dictatorship in transition from a planned to a market economy (but see also this question). In recent years its economy has grown significantly. Although many people live in poverty or work in very poor conditions, the standard of living has quite strongly improved in recent decades. It has a higher HDI than India, and the difference in IHDI is even larger. India has been a democracy for quite a while, and is growing like China, but presently seems to have more and more severe poverty than China. Although some political analysts in the west have predicted that economic growth and economic freedoms necessarily lead to democratisation and an increase in political freedoms, I'm not sure if there is much evidence for this in China.
Now, this is not an argument against democracy, and in a correlative study, I'm pretty sure democracies would turn out to do statistically better than dictatorships (but correlation does not imply causation). One reason western democracies are rich may be because of the cheap labour in eastern and southern dictatorships, or at least poorer countries. But the comparisons above may indicate that democracy is not the only system that has the potential to provide a decent and improving quality of life for citizens.
