In Axelrod's tournaments of the Prisoner's Dilemma, carried out in the 1980s, a strategy called Tit for Tat was declared the winner, and it has since then been thought of as the strategy to use to do as well as possible in different situations. In this thesis, we investigate whether Tit for Tat will still do as well if we change the game to the Hawk-Dove Game. This is done by comparing Tit for Tat to other strategies -- All C, All D, Joss and Random -- one at a time. First we analyse under which conditions each strategy will be an Evolutionary Stable Strategy, then if it is possible for a population of these two strategies to end up in a stable polymorphism, and finally, if we have a finite population instead of an infinite one, under which conditions selection will favour the fixation of each of the strategies. This leads to the conclusion that how well Tit for Tat will do depends a lot on the different conditions on the game, but in general, the more times that a pair of individuals will meet, and the higher the value of the resource is compared to the cost of fighting, the better Tit for Tat will do.