This diversion is only practiced by those persons who are candidates for great employments, and high favour at court. They are trained in this art from their youth, and are not always of noble birth, or liberal education. When a great office is vacant, either by death or disgrace (which often happens,) five or six of those candidates petition the emperor to entertain his majesty and the court with a dance on the rope; and whoever jumps the highest, without falling, succeeds in the office. (1.3.2)
The Emperor of Lilliput uses this rope dance to test "candidates for great employments." The person who can jump the highest wins a government job. So, basically, if you can impress the Emperor by doing something completely unrelated to government, bam! You can get a government job. How might this rope dance work as a larger metaphor for politics at court?
Part 1, Chapter 5
When I had for some time entertained their excellencies, to their infinite satisfaction and surprise, I desired they would do me the honour to present my most humble respects to the emperor their master, the renown of whose virtues had so justly filled the whole world with admiration, and whose royal person I resolved to attend, before I returned to my own country. Accordingly, the next time I had the honour to see our emperor, I desired his general license to wait on the Blefuscudian monarch, which he was pleased to grant me, as I could perceive, in a very cold manner; but could not guess the reason, till I had a whisper from a certain person, "that Flimnap and Bolgolam had represented my intercourse with those ambassadors as a mark of disaffection;" from which I am sure my heart was wholly free. And this was the first time I began to conceive some imperfect idea of courts and ministers. (1.5.7)
Gulliver sucks up to Lilliput's enemy emperor, the Emperor of Blefuscu. Somehow, it doesn't seem to occur to him that this will make him look bad to Lilliput's own emperor. So, he's pretty naive. But what really gets to Gulliver is the discovery that his enemies of court have been secretly trying to turn the Emperor against him. They use Gulliver's friendly words to the Blefuscu Emperor as ammunition to prove to the Lilliput Emperor that Gulliver is not loyal to him. Again, we have to say, Gulliver claims to be shocked by these activities, which suggests that he is the most naive guy in the world. This makes his final, total mistrust of all people at the end of Part 4 all the more stark.
This open bold declaration of mine was so opposite to the schemes and politics of his imperial majesty, that [the Emperor of Lilliput] could never forgive me. He mentioned it in a very artful manner at council, where I was told that some of the wisest appeared, at least by their silence, to be of my opinion; but others, who were my secret enemies, could not forbear some expressions which, by a side-wind, reflected on me. [...] Of so little weight are the greatest services to princes, when put into the balance with a refusal to gratify their passions. (1.5.5)
Even though Gulliver manages single-handedly to stop the Blefuscu naval fleet from invading Lilliput (that's the "greatest service to princes" he's talking about), he still becomes the Emperor's enemy. Why? Because Gulliver refuses to use his great strength to enslave the Blefuscudians, even though the Emperor of Lilliput asks him to. This is the problem with politics in Lilliput: keeping the Emperor happy often means lying, flattery, or hypocrisy. And acting according to a pretty basic moral rule – do not make people slaves – makes the Emperor angry. Politics and morality do no seem to be compatible.
Part 1, Chapter 6
These false informations, which I afterwards came to the knowledge of by an accident not proper to mention, made the treasurer show his lady for some time an ill countenance, and me a worse; and although he was at last undeceived and reconciled to her, yet I lost all credit with him, and found my interest decline very fast with the emperor himself, who was, indeed, too much governed by that favourite. (1.6.22)
Here, Gulliver is telling us about a rumor that he has been sleeping with Flimnap's wife. Flimnap's six inch tall wife. This rumor is obviously not true, but it makes Flimnap jealous enough to decide to ruin Gulliver's reputation with the Emperor. When a government is run by powerful, incompetent, ambitious, petty people, there is no oversight to make sure that the job still gets done in a fair, equitable way. Once politics becomes personal, you may as well kiss justice and fair play goodbye.
In relating these and the following laws, I would only be understood to mean the original institutions, and not the most scandalous corruptions, into which these people are fallen by the degenerate nature of man. For, as to that infamous practice of acquiring great employments by dancing on the ropes, or badges of favour and distinction by leaping over sticks and creeping under them, the reader is to observe, that they were first introduced by the grandfather of the emperor now reigning, and grew to the present height by the gradual increase of party and faction. (1.6.9)
The whole Lilliput section is probably one of the most integrated and subtle attacks on politics that Swift makes in the whole book. Gulliver seems to believe that "the degenerate nature of man" (in other words, the tendency of human society to get worse and worse as time goes on) is necessarily linked to politics and the "gradual increase of [political] party and faction." The more politics there are, the worse a society is, according to Gulliver's logic.
Part 3, Chapter 8
But when some confessed they owed their greatness and wealth to sodomy, or incest; others, to the prostituting of their own wives and daughters; others, to the betraying of their country or their prince; some, to poisoning; more to the perverting of justice, in order to destroy the innocent, I hope I may be pardoned, if these discoveries inclined me a little to abate of that profound veneration, which I am naturally apt to pay to persons of high rank, who ought to be treated with the utmost respect due to their sublime dignity, by us their inferiors. (3.8.7)
When Gulliver goes to Glubbdubdrib and speaks to the ghosts of the recent European dead, he gets really disillusioned. He finds out that all the people with great reputations have gotten their power from lying, cheating, and bribing their way to the top. This is Gulliver's idea of politics in a nutshell. Do we see any models of honest politicking in this novel?
Part 4, Chapter 7
The Master Horse
He said, "he had been very seriously considering my whole story, as far as it related both to myself and my country; that he looked upon us as a sort of animals, to whose share, by what accident he could not conjecture, some small pittance of reason had fallen, whereof we made no other use, than by its assistance, to aggravate our natural corruptions, and to acquire new ones, which nature had not given us; that we disarmed ourselves of the few abilities she had bestowed; had been very successful in multiplying our original wants, and seemed to spend our whole lives in vain endeavours to supply them by our own inventions; that, as to myself, it was manifest I had neither the strength nor agility of a common Yahoo. (4.7.4)
The Master Horse thinks that the "reason" that Gulliver says European Yahoos have is really just a new kind of vice, allowing Yahoos to cheat, lie, and steal on a grander scale. And in order to practice this fake rationality, European Yahoos have given up the few natural gifts (long nails, hair to protect us from the sun) that humans have. So, what's the point of human politics and society? Does Gulliver give any kind of credit to human civilization? Do you think the book wants us to agree with Gulliver's assessment?
For if," said he, "you throw among five Yahoos as much food as would be sufficient for fifty, they will, instead of eating peaceably, fall together by the ears, each single one impatient to have all to itself. [...]
My master further assured me, which I also observed myself, "that in the fields where the shining stones abound, the fiercest and most frequent battles are fought, occasioned by perpetual inroads of the neighbouring Yahoos." (4.7.6, 4.7.8)
Humans are greedy. Yes, we get it, already. But Gulliver's skewing both his and the Master Horse's examples against us by never pointing out that there have been examples of humans who have given freely of themselves. Like, what about Mother Theresa? How would you argue against Gulliver's assessment of humankind?
I durst make no return to this malicious insinuation, which debased human understanding below the sagacity of a common hound, who has judgment enough to distinguish and follow the cry of the ablest dog in the pack, without being ever mistaken. (4.7.14)
The "malicious insinuation" to which Gulliver refers is the Master Horse's observation that all Yahoo groups choose the weakest, ugliest one of them to lead. Gulliver makes one of his few protests against Houyhnhnm logic, saying that it can't be true that humans are less wise than dogs, who always choose the strongest dog of the pack to lead. What kinds of models of leadership do we get in this book? Doesn't it seem kind of odd that a Tory like Swift, who supports the power of kings (as long as they're not George I) seems to satirize leadership all over the place in this novel? In fact, Swift began as a Whig and later became Tory, so many of his ideas remain pretty liberal. Classic Tory beliefs like divine right of the monarchy don't seem to have been so important to him, which may explain part of his frustration with the extremes of partisanship under George I.