The only therapy for patients with end-stage liver disease is transplantation. In order to be eligible for a cadaveric liver transplantation in the United States, a patient must join a waiting list maintained by the United Network for Organ Sharing System, which allocates livers using a complex priority system. When a liver offer is made, each patient must decide whether to accept the offer. Although several other models considered this decision-making process, they maximize life expectancy rather than expected utility, and hence do not apply to the vast majority of patients, who are risk-averse. We extend previous research by including patient risk preferences in our modeling approach. We formulate a risk-sensitive infinite-horizon Markov decision process in which the state is defined by the patient's health and the quality of the liver available for transplantation. Our model maximizes total expected utility for an exponential utility function, which provides good approximations for many other types of utility functions. We establish structural properties based on liver quality, health state, and patient's risk sensitivity, and we conduct numerical studies based on clinical data.