Few want to do business with a partner who has a bad reputation. Consequently, once a bad reputation is established, it can be difficult to get rid of. This leads on the one hand to the intuitive idea that a good reputation is easy to lose and hard to gain. On the other hand, it can lead to a strong form of history dependence in which a single beneficial or adverse event can cast a shadow over a very long period of time. It gives rise to a reputational trap where an agent rationally chooses not to invest in a good reputation because the chances others will find out is too low. Nevertheless, the same agent with a good reputation will make every effort to maintain it. Here, a simple reputational model is constructed and the conditions for there to be a unique equilibrium that constitutes a reputation trap are characterized.