Additionally, UNGA has a procedure that allows a simple majority to call a vote that subjects a resolution (or amendment) to a 2/3 majority threshold for adoption. This surely because of their broader focus on consensus.
If 192 Member States agreed on the text but there is just one Member State that requests a vote, then consensus is not reached.
Today [...] roughly 80% of the General Assembly resolutions are adopted by consensus, that is, without taking a vote.
The 2/3 vote is from rules 83-85:
[83:] Decisions of the General Assembly on important questions shall be made by a two-thirds majority of the members present and voting. [...]
[85:] Decisions of the General Assembly on questions other than those provided for in rule 83, including the determination of additional categories of questions to be decided by a two-thirds majority, shall be made by a majority of the members present and voting.
So, the latter essentially allows a blocking minority (above 1/3) provided that simple majority considers the matter important enough. But this isn't too uncommon in other places too, like e.g. the US Senate, considering filibustering, and the votes needed to overcome that, although the threshold there has been lowered to 3/5, but it's absolute number, whereas at the UN is relative (to those actually voting).
Prior to 1917 the Senate rules did not provide for a way to end debate and force a vote on a measure. That year, the Senate adopted a rule to allow a two-thirds majority to end a filibuster, a procedure known as "cloture." In 1975 the Senate reduced the number of votes required for cloture from two-thirds of senators voting to three-fifths of all senators duly chosen and sworn, or 60 of the 100-member Senate.
And yeah, the famous amendments/resolutions at UNGA to condemn Hamas (specifically) failed like that, e.g. in 2018:
The Assembly had voted to apply the two-thirds majority requirement for the adoption of draft by a vote of 75 in favour to 72 against, with 26 abstentions.
(Mike Scott pointed out in a comment that US Senate rules could be changed by simple majority to not allow filibuster [anymore]. But the UNGA also has some such flexibility, as you can see in the quote above, the vote whether to apply the 2/3 rule to some issue only required simple majority; this procedure is codified article 18 of the UN Charter. True though, unlike in the US Senate case, UNGA is mandated by the Charter to apply the 2/3 rules in a few specific cases like the election of UNSC non-permanent members, admission/expulsion of UN members, "budgetary questions", etc. -- the full list of topics is in art 18.)
The number of countries which voted down the actual measure was lower though.
But the vote on the resolution to condemn Hamas was 87 in favor against 57 opposed, with 33 abstentions — a plurality but below the two-thirds requirement to adopt it.
And as deftly noted in a comment of Obie 2.0: both the UN and the US senate apportion seats to states/countries, not based on population. So that makes a blocking minority (of the population) even more possible.
And to mention the EU here too, their rules are more complicated, but do essentially also allow for a blocking minority in their most commonly used procedure (called "qualified majority", which is used for about 80% of the votes):
When the Council votes on a proposal by the Commission or the High Representative [of the Union for Foreign Affairs and Security Policy], a decision is adopted if 55% of the participating Council members, representing at least 65% of the population of the participating member states, vote in favour. In this case, a blocking minority must include at least the minimum number of member states representing more than 35% of the EU population, plus one member, failing which the qualified majority is deemed attained.
When the Council is not acting on a proposal by the Commission or the high representative, a decision is adopted if 72% of the participating Council members, representing at least 65% of the population of the participating member states, vote in favour.
Also
Prior to November 2014 in the EU 28 a blocking minority required 93 votes of the 352 votes in the Council. [...]
That was about 26.4%.
And although you inquired about UNGA, the UNSC is where most binding security decisions are made. As the P5 all have a blocking vote (veto) there, the smallest countries of them, population-wise are France and the UK, both with about 68 million out of around 8 billion on the planet--so around 8.5% each. True, these two countries seldom block UNSC decisions on their own, although in theory they could do it much more often. (Apparently, the last time France did that was in 1976 regarding Mayotte and the UK in 1972 regarding Southern Rhodesia.) A similar napkin calculation is much harder to do for the UNGA.
There is however one 2013 book that mentions that:
The African group currently constitutes about 28.2 per cent of total UNGA membership. Similarly,
The Asian group holds about 27.6 per cent. Accordingly, both of these regional groupings are almost able to forma a blocking minority by themselves within the UNGA. The other regional groupings [...] have a combined share of about 44 per cent of UNGA membership.
I strongly suspect they don't mean by population, but number of countries though.
The same book notes (p. 57) that during the Cold War, at UNGA:
The Soviet Bloc was isolated in the early days and almost always lost on Cold War
issues [...]
Colonialism issues and South Africa occasionally caused rifts between
the US and its Western allies. Consequentially, the colonial powers rather than the US occupy
the most extreme Western position during this period.
During the first five years there was an important orthogonal dimension of contestation
concerning the issue of Palestine. Arab and most Asian countries opposed partition while the
West and East stood united in favor of partition, although this unity proved to be short-lived.
After Joseph Stalin’s death, the Soviet Union started courting non-aligned states and generally voted with the Arab states. The Middle East issue thus became part of the first dimension.
[...] The Soviet Union started to aggressively pursue the allegiance of the former
colonies. This challenged the US to choose between these newer states and their most important allies, the “Old Europeans,” against whom the many anti-colonialism resolutions were
targeted.
[...] This essentially created a
‘third party’, commonly referred to as the Group of 77 (G77) or the Non-Aligned Movement
(see Iida 1988). Indeed, Figure 4.2(b) shows three fairly cohesive voting blocs with a largely
empty space between them. In the literature on the US Congress such empty channels are
seen as indicative of polarization, in the sense that they are indicative of voting behavior
where few cross party lines. The same can be said for this period in the UNGA, which was
heavily dominated by “partisan” attempts to set the agenda and maintain cohesion.
That analysis stops in 2011 (and fails to consider China's rising influence in Africa and the former Soviet Central Asian space as a trend of its own), but it's still somewhat interesting:
In the UNGA, the disintegration of the Soviet voting bloc started in 1989 when
some Eastern European states, led by Czechoslovakia, shifted toward the West. The Soviet
Union followed in 1991 as Russia. Yet, subtle changes occurred before then [...]. By 1987 the Soviet Bloc was no longer furthest removed from the US.
Instead, this distinction belonged to Algeria, Angola, Cuba, Iraq, Libya, Syria, and Yemen.
Figure 4.3 shows the post-Cold War ideological structure in the UNGA. A model that
assumes that states hold static ideal points along a single dimension explains 94 percent of vote
choices on non-consensus UNGA resolutions in both the 1991–2000 and 2001–11 periods.
This ideological structure contains new elements but also resembles patterns of the Cold War
structure. The main exceptions to the stability are the Eastern European states, which
switched sides. Russia and the newly independent former Soviet republics now take a position
in between the West and the rest of the world. Their position resembles that of states such as
Turkey and South Korea. These are all states with strong ties to both the West and the nonWestern part of the world (but for different reasons). Yet, the position of developing countries before the end of the Cold War is a very good
predictor of their position since the end of the Cold War (Voeten 2000).
So, while we established that there are some voting patterns (and there's even a possibly for a blocking minority at the UNGA--but that's pre-approved only on a select list of topics--otherwise you need a simple majority to enable the rule), does that make it a tyranny [of the majority]? I would suggest (in concurrence with some other answers) the answer is no, because typically more than polarization is required to call something a tyranny. E.g.
Majority tyranny arises when winning majorities are fixed, and when there are no checks on the majority’s ability to dominate the minority (Guinier 1994, 4). Hence, the tyranny of the majority can be defined as the permanent exploitation of the minority by the majority in democratic decisions over time. [Footnote:] In accordance with Guinier (1994, 6), I argue that majority tyranny can only be avoided when minority voters have a chance to influence outcomes. Hence, conceding formal participatory rights to the minority is insufficient as long as the minority’s interests are not at least partially reflected in collective decisions.
As was noted in some other answers, the UNGA doesn't make very consequential decisions thus "permanent exploitation" is difficult to argue as being a result of [most] UNGA votes, despite polarization on some issues. Even if, say, the UNGA condemned China on Xinjiang, they'd probably not care too much to change their policies substantially. Russia is probably a good/recent example of ignoring the UNGA like that.
143 countries supported a resolution condemning the recent referendums in Ukraine and demanding Russia reverse its annexation declaration. [...]
All resolutions and decisions of the General Assembly are ‘recommendations’ under Articles 10-14 of the UN Charter. This means that, unlike some Security Council decisions, General Assembly resolutions are not legally-binding and Member States are not bound by them.
And yeah, the UNGA rules allow the minority interests to be reflected in resolutions to some degree, although the degree to which this is guaranteed [beforehand] depends on the topic.
Addendum: as for the 1966 article by Bailey (that the OP mentioned in a comment), the controversy then was that the UNGA was trying expunge some South African and South Rhodesian speeches from the UN record altogether. E.g.
In 1961, a similar attempt had been made to
remove from the records a South African speech, but after a disagreeable
debate the Assembly decided by an overwhelming majority to leave the
speech in the official records but to 'censure' the South African representative for making a statement which was 'offensive, fictitious and
erroneous'.
Likewise he complains that some UNGA resolutions of then, like one Cyprus passed "with substantial African backing" but without any UNSC member voting in favor. He doesn't exactly explain how that was/is tyranny, but he quotes the rep of Saudi Arabia, who objected that that was
'illegal, unconstitutional, unparliamentary and a rejection of fair play and the sense of equity'.
And that Cameroon then proposed that that Saudi speech be expunged from the record, but then backed down.
So, Bailey appears to claim that that was the kind of tyranny that was within the remit of the UNGA. Pretty weak claim, IMHO.
OTOH he also notes (more substantively) that there had been variation in what matters were deemed important enough to be subject to the 2/3 majority requirement (art. 86 of the rules).
During the period 1956-60, when the United States opposed any U.N.
discussion of Chinese representation, the Assembly did in fact decide by
only a simple majority 'not to consider' proposals to exclude representatives of Nationalist China or to seat representatives of the People's Republic of China.
In 1961 and 1965, however, when the question of Chinese representation had been inscribed on the agenda, the Assembly decided, also by a simple majority, that any proposal to change the representation of China was an 'important' question and would therefore require a two-thirds majority.
The United Kingdom was the only U.N. member in 1965 to vote both that the question was 'important' and in
favour of the representation of the People's Republic of China. [...]
The 'simple majority' decisions regarding Chinese representation in
the period 1956-60 were, however, exceptional. Prior to the twentieth
session of the General Assembly, over 2,000 resolutions had been adopted
by the Assembly, of which only twenty were by a simple majority.
And apparently, the African countries tried to do a bit of the same once they had acquired a slim majority:
a group of Afro-Asian States plus Yugoslavia had introduced an
anti-colonial resolution which included the following operative paragraph: 'Requests the colonial Powers to dismantle the military bases installed in colonial Territories and to refrain from establishing new ones'.
The representative of Mali, on behalf of the sponsors, proposed that the
resolution be adopted by a simple majority.
And despite Western objections that that fell under the pre-dermined issues (peace and security) that were covered by article 83...
After a lengthy debate, the
proposal of Mali was put to the vote and approved by 59 votes to 45
(4 abstentions, 9 absent or not participating in the vote).
The main resolution was then put to the vote and, although the paragraph referring to
military bases did not receive a two-thirds majority, the resolution as a whole was adopted.
So, Bailey charges that that's when the UNGA clearly violated its own procedural rules. The resolution cited is UNGA 2105(XX) from 1965.
