In what ways does Troy resemble the antihero? In what ways does he resemble the tragic hero?

An antihero is a protagonist of a work who does not possess traditional heroic qualities. Troy Maxson, the self-aggrandizing protagonist of August Wilson's play Fences, can be viewed as an antihero due to his many character flaws. At the start of the play, Troy is 53 years old and what one might consider "all washed up." He was a star in the Negro Baseball League but his glory days have long since passed. Troy lives in the past, spinning tall tales of his own former "heroics," and refusing to see any reality other than the one he himself inhabits. He butts heads with both of his sons, one who wants to be a musician and the other who wants to play football, and his quick temper and stubborn refusal to be swayed in their father-child conflicts, combined with his tendency towards hyperbole (if not outright lies) are all excellent examples of traits Troy exhibits that would never be exhibited by a traditional hero. Over the course of the play, Troy cheats on his wife, the extremely sympathetic Rose, alienates both of his sons, and, through his ignorance, confines his brother to an asylum.
All that said, there are many ways in which Troy is sympathetic. Perhaps if the audience had met him in his heyday, during the height of his baseball career, he would have appeared to be a true hero, as opposed to the antihero or tragic hero he is now that he's past his prime. He's the patriarch and the breadwinner in his family, and, though perhaps its old-fashioned, he believes that, as such, he deserves respect. Though this belief sometimes translates into him behaving like a tyrant, it is also understandable in the context of the time. Troy is also consistently portrayed as a good brother to Gabriel before signing the papers to commit him because he is too proud to admit that he doesn't know how to read.
Outside of his family life, in his professional one, many of Troy's actions could actually be seen as heroic. Troy successfully campaigns to become the first black garbage truck driver in the city, calling out a racist system and getting it abolished. His words and actions are often hypocritical, his worst moments balanced out by his brighter ones.
Given Troy's many sympathetic, even heroic, traits, the appellation of tragic hero better applies to the character than antihero does. A tragic hero, in the classical sense, is one who, through a specific personality defect/error in judgement, brings about his own downfall. Troy's fatal flaw is his pride, which, given how deep it runs, may even be called hubris. It is this pride that leads to his brother's commitment to the asylum. It is this pride that prevents Troy from seeing eye-to-eye with his sons, who, in their differences of opinion from their father, represent changing times and voices of a younger generation.
The main thing distinguishing a tragic hero from an antihero is while an antihero can descend into total amorality, a tragic hero always believes himself to be right and acts according to a set of moral principles that just may turn out to not be the right ones. Each of Troy's actions throughout the play is motivated by his belief that he knows what's best. And it's this exact belief that, in true tragic hero style, is his ultimate demise.