What does the blindfold symbolize in chapter 1, "Battle Royal," of Invisible Man?

A blindfold, in isolation, is imbued with many different symbolic connotations, but in the context of the "battle royal" in Ralph Ellison's Invisible Man, the blindfold takes on even more representational significance.

The boys taking part in the battle royal are blindfolded by the white leaders of their community. The men doing the blindfolding are taking charge, symbolically and literally, of what the boys are able to perceive while fighting one another. That these perceptions are controlled and managed by blindfolds applied by the white men is significant; even now, many might argue that much of the American experience is controlled and managed by the white male population, no matter who is wearing the metaphorical blindfold.

A blindfold is an item that ensures vulnerability, as the ability to see is compromised. Anyone wearing a blindfold is forced to trust the people who act as guides, or they must grasp helplessly in darkness. The white men who force vulnerability on the boys do so gleefully, enjoying their power—perhaps forgetting that when one of the five senses is compromised sometimes the other four can strengthen. The narrator experiences the blindfold himself, but the blindfold actually opens his eyes, ironically, to the racist state of the world in which he finds himself.

As he lashes out blindly at his equally exploited opponents, the narrator is assailed on all sides by the sounds of a drunken, baying mob of white men, for whose delectation these young African American boys have been forced to degrade themselves. The blindfold that he wears during the fight is symbolic of awareness; while the narrator has metaphorically "blinded" himself to the harsh realities of life in this deeply racist society, the literal blindfold he wears in this scene allows him to finally understand the full extent of his own oppression.

Before he participated in this unseemly spectacle, the young boy never really understood the depths to which prejudice can so often run. In that sense, one could say that he wore a metaphorical blindfold that hid the truth of what society is really like. Yet ironically, the experience of wearing a literal blindfold opens his eyes to the status of African Americans in society and the abominable treatment they receive from their white oppressors.

The blindfold is also a symbol of the narrator's sense of helplessness in a white supremacist system in which he is both used against his will and complicit in being used so that he can secure minor gains. It's significant that the blindfold is described as a white cloth and that the narrator can only see through a narrow slit that he creates when he frowns. He is, thus, literally blinded by whiteness and can only see through it when he registers a gesture of discontent.

The narrator describes how his inability to see renders him to a state similar to that of a drunk or a baby. All he can do is follow voices and push his weight against the bodies of the other fighters in the ring so that he can win this foolhardy battle. Through the slit he creates, he can see gold coins and greenbacks strewn about the floor of the ring. It is not until he wins the Battle Royal and the blindfold is removed that he recognizes the money as worthless—the gold coins are merely tokens. When he was unable to see, the narrator believed that the battle for money and the prestige of winning the fight justified the brutality that the young black men expressed to each other in the ring. However, when his sight is restored, he realizes that his efforts were for naught. The promise of money was merely an illusion; he'll never really possess the capital of those who forced him into the ring. Later, too, he will see that his ambitions will always be stunted by racism.

The blindfold symbolizes how the narrator himself is blind to the world around him and it also symbolizes how others are blind to who he really is.  They stereotype him instead of looking at him as an individual person.  Enotes states:

The story unfolds by narrating a scene in which those who are "blind" are not only the narrator, who literally wears a blindfold, but also those who abuse the narrator, sizing him up as mere stereotype, erasing his individuality and human dimension.

The narrator is blind simply because he has been raised in a world where the African-Americans and Caucasians have been kept separate; therefore, their lives are, in many ways, mysteries to one another.

In "Battle Royal," Chapter 1 of Invisible Man, why are the boys blindfolded and what does it symbolize?

The "Battle Royal" is probably the most memorable and pivotal scene in the novel. As you've guessed, the scene operates as a metaphor for a larger idea that Ellison wants to present about how black men, in particular, operate within a power structure in which white men are supreme.

The narrator introduces his memory of the battle by talking about conduct. The narrator learned from his grandfather, "a quite old man who never made any trouble," how to behave (Ellison 17). The narrator had made a reputation for himself as "an example of desirable conduct," a reputation that unsettles him because he suspects that it is the opposite of what white people actually want (17). He surmises that they would actually prefer for him to be "sulky and mean," but he chooses to be upstanding because he is afraid to act any other way (17).

In this recollection, we see that, as a young man, the protagonist's sense of identity is completely determined by the white gaze. There is already a sense of himself as a spectacle. When the white male spectators force the boys to watch a nude, blonde white woman dance, they make a spectacle of black male sexuality. When they force the confused boys to stay in the ring, while threatening them from their seats in the audience, they make a spectacle of their fear and powerlessness. There is a reminiscence here of instances in which slave masters pitted slaves against each other in crude boxing matches, forcing them, at times, to fight to the death.

On the surface, the blindfold may be a plot device that heightens the tension and desperation of the characters in the ring. Metaphorically, the blindfold symbolizes the way in which black men fight against each other in the interest of surviving (in the case of Tatlock) or of achieving recognition (in the case of the narrator) in a white man's world. This desperation is more vividly recreated in a subsequent scene in which the young men crawl around the floor of the ring, picking up coins and greenbacks thrown in by the white spectators.

It is not clear if the narrator is really "different" from the others. The only other fighter who is presented is Tatlock, the narrator's final opponent in the ring:

"I found myself facing Tatlock, the biggest of the gang....His face was a black blank of a face, only his eyes alive -- with hate of me...I wanted to deliver my speech and he came at me as though he meant to beat it out of me" (Ellison 24).

Tatlock is a man who has internalized racism, who sees the narrator as an enemy because white supremacy has taught him that it is another black man who stands between him and the resources that he needs to get along. While the narrator remains focused on his speech (incidentally, inspired by Booker T. Washington's ideas on self-reliance and social responsibility among blacks), Tatlock is one who has eschewed language in favor of physical violence. In other words, he has become the "sulky and mean" brute the narrator is trying not to be. 

The narrator is not really "different" from the others. His goals may be different but, like the other young black men in that ring, he has no individual identity that is recognized. He must also co-exist with them in a power structure that renders them not only invisible but also desperate and, ultimately, powerless.