There are many good arguments against democracy. The most standard of these arguments is primarily an ethical one: that it is unjust for a majority to be able to vote away the rights of a minority. For if democracy is defined in terms of majoritarianism, it must be dismissed as being inherently incompatible with a universal application of rights to all human beings, since it implies that any larger amount of people can legitimately force their will on any smaller amount of people. This makes democracy nothing but might makes right, cloaked in egalitarian rhetoric. The ethically and logically consistent position would be that if it is wrong for an individual to do X (such as murder), then it is wrong for a group to do X. However, I intend to take a bit of a different approach to arguing against democracy here.
The ideal of political democracy is that of a government controlled by the people as a whole. The idea is that by expanding access to the governmental apparatus to everyone, wether that be through voting or through eligibility for holding political office, we will get rid of exploitation of men by men. This is supposed get rid of the special privileges in society, converting everyone into more or less a state of "equality under the law". But this idea is ridiculous. The government is not actually directly controlled by the people. Ownership by the government, in practice, amounts to ownership by an oligarchy, for the people do not in fact directly control the government. The people who actually constitute our government, in practise, are the politicians, bureaucrats, policemen and soldiers. Another related class of people are a small band of private interests who ally with the government for special privileges in exchange for political support. Combined, those are the real, albeit unjust, owners of "public property", which is stolen from "the people" in reality (including workers and the poor in general).
The only remotely good thing that democracy does as a system is get rid of the monarchal king. But this move in itself becomes meaningless and negated by the proceeding steps in the transformation towards democracy. Democracy may indeed get rid of the king, but it replaces the king with a plurality of rulers (which for all intents and purposes can now function as multiple kings). But think about what this actually does. We have gotten rid of the special privilege of the king, and replaced it with a special privilege to an even larger band of men.
Democracy does not get rid of privileged rulers. It replaces a system in which one person is at the top of the oligarchy with one in which multiple people are at the top of the oligarchy. Indeed, democracy does not get rid of oligarchy. A government (no matter what form), by its very nature, is oligarchical, and a government inherently creates a class division between itself and the populace. The political class is those who constitute the government and the individuals who ally with them for privileges (the tax-consumers), and the subject class is those who are ruled by the political class (the tax-payers). What democracy does is allow more people to become part of the political class, and hence it actually expands special privileges.
The theory of political democracy is wrong. The existence of "representatives" in itself drives a wedge between "the people" and "the government", for the control is not direct. This is what distinguishes "participatory" or "direct" democracy from what most people refer to as "democracy" (representative democracy, which is a sham). The theory of control by the people would only be true if we had participatory democracy, and if participatory democracy were actually put into practise, it would be an anarchy because it would have to be based on unanimous consent and direct control. Participatory democracy could not in practise continue being a government. There is no such thing as a government that is directly controlled by all of the people: they would all have to literally be members of it. Such a notion is absurd.
To add some criticism of the effectiveness of voting and political representation: it is physically and logically impossible for a few men in the government to realistically represent all of the people who voted for them or all of the people in the district or state in general. They are individual human beings, they can only directly represent themselves. It is impossible for one man to represent the diverse desires of an entire society, for all of the people within a given society vary widely and conflict in their desires in the first place. It would be impossible for one man or small band of man to even accurately ascertain or predict those desires. This is essentially the calculation or information problem applied to political participation and representation. As a consequence of the information problem, political representatives inherently must impose either their own will or the will of a special interest group onto the masses, even if every single person voted. In short, they must centrally plan in opposition to the desires of "the people".
The introduction of the institution of voting into society does not magically make the government more controlled by the people, nor does it necessarily make it any more voluntary. It first must be established that the casting of a vote in itself need not be one of enthusiasm, but can be one of resignation or as a reactionary mechanism against encroachment by interests that the individual dislikes or opposes. It also must be established that the mere act of voting is not truly binding on any politician. For all intents and purposes, a politician can run on a platform of X, and then do Y once in office (for example, see George W. Bush preaching about the virtues of non-intervention with foreign nations in the 90s and compare it to his actions in the present). One may counter that in four years or so they can get voted out of office, but:
The demographic reality tells us that large chunks of many country's populations simply don't vote. It would follow that these people cannot be said to be "responsible" for things that came about as a result of voting that they did not take part in. One cannot reasonably argue that someone who has never cast a vote can truly be adequately represented. This rather large group of people, the non-voters, are technically pure subjects. But so are most of the people who do vote, because their votes do nothing but bind them. Both the voter and non-voter are bound all the same, for they are still subject to whatever the government decides. The typical voter has simply been given the illusion that they are in a better position than the non-voter. This is not the case — both are subject to the government's decrees against their will all the same.
Even if we do assume that the government can represent voters in any realistic way, we are almost never dealing with true majorities in a democracy. We are dealing with "numerical majorities". The idea that, say, Texas is a "red state" is absurd in the sense that the actual statistics show us something very close to the following: say, thirty-five percent of the eligible Texas population voted, and out of that thirty-five of the eligible Texas population two percent voted for third parties, fifteen percent voted for Democrats and eighteen percent voted for Republicans. Thus, by literally considering Texas to be a "red state", we are projecting a rather small statistic (eighteen percent of eligible voters, which is probably ten percent or less of the Texas population) into a generalisation encompassing the entire population of the state. "The majority" or "the state of Texas" is not being "represented". We are dealing with a rather small fraction of its population.
This is why I consider the very idea of political democracy to ultimately be a sham, an impossibility.