The availability of antibiotics presents medical practitioners with a prescribing dilemma. On the one hand, antibiotics provide a safe and effective treatment option for patients with bacterial infections, but at a population level, over-prescription reduces their effectiveness by facilitating the evolution of bacteria that are resistant to antibiotic medication. A game-theoretic investigation, including analysis of equilibrium strategies, evolutionarily stability, and replicator dynamics, reveals that rational doctors, motivated to attain the best outcomes for their own patients, will prescribe antibiotics irrespective of the level of antibiotic resistance in the population and the behavior of other doctors, although they would achieve better long-term outcomes if their prescribing were more restrained. Ever-increasing antibiotic resistance may therefore be inevitable unless some means are found of modifying the payoffs of this potentially catastrophic social dilemma.