In many respects, Hobbes' concern has nothing to do with irrationality or even justice. The first and most urgent problem that Hobbes confronts is, very simply, disagreement. This disagreement may arise from any number of causes -- irrationality, selfishness, bias -- but its source doesn't matter. What matters, Hobbes argues, is that in a state of nature, disagreement will inevitably lead to violence.
A War of All Against All
Famously, he describes the state of nature as a "war of all against all":
In such condition there is no place for industry, because the fruit thereof is uncertain, and consequently, no culture of the earth, no navigation, nor the use of commodities that may be imported by sea, no commodious building, no instruments of moving and removing such things as require much force, no knowledge of the face of the earth, no account of time, no arts, no letters, no society, and which is worst of all, continual fear and danger of violent death, and the life of man, solitary, poor, nasty, brutish, and short. (Leviathan, Chapter 13)
Why does this happen? It does not happen because people are irrational; it happens because there is no higher authority to arbitrate disagreements. Without some higher authority to make and enforce judgments about disagreements, those disagreements will persist, eventually escalating into violence.
It doesn't matter whether the higher authority makes just or unjust judgments. All that matters is that the parties to the disagreement abide by those judgments. If they do so, then the disagreement will not escalate. There will be peace, and that peace will allow people to develop art, knowledge, and industry in ways that will improve everyone's standard of living. From this point of view, any government -- even the most unjust government -- is better than no government at all.
The Best Kind of Government
Now, this does not mean that there aren't better or worse governments, according to Hobbes. It only means that some government is always better than no government. However, he does have some things to say about which kinds of government are better. And in fact, he does take into consideration the "tainted" judgement of individuals. Paradoxically, from a modern point of view, that's why he prefers a monarchy!
Among the three types of government that Hobbes thinks exist -- monarchy, aristocracy, and democracy -- he believes that monarchies are most likely to be just. This is because the other two types of government divide power between individuals, who are then apt to be selfish and rule in their own favor. But somewhat paradoxically, by putting absolute power in one ruler, monarchy actually ensures that the selfish interests of the sovereign are inevitably aligned with those of the society as a whole.
In monarchy the private interest is the same with the public. The riches, power, and honour of a monarch arise only from the riches, strength, and reputation of his subjects. For no king can be rich, nor glorious, nor secure, whose subjects are either poor, or contemptible, or too weak through want, or dissension, to maintain a war against their enemies; whereas in a democracy, or aristocracy, the public prosperity confers not so much to the private fortune of one that is corrupt, or ambitious, as doth many times a perfidious advice, a treacherous action, or a civil war. (Leviathan, Chapter 19)
Now, there are a number of ways to argue against this claim. A modern thinker might point out that although the sovereign's interests are theoretically aligned with the society as a whole, the sovereign may not easily recognize those interests, or easily separate them from the barrage of requests for special favors he or she is likely to receive. But remember that Hobbes was working during one of the greatest upheavals in English history, the English civil war (1642–1651); he published Leviathan in 1651. And the civil war resulted from a dispute -- between the monarchy and parliament. He may well have felt that even an imperfect absolute monarchy would have been preferable to a civil war.