You are being asked to make a choice between two situations. First, you can choose to kill one person. If you select this choice, you will be responsible for killing one person and the lives of nineteen others will be saved. Second, you can choose not to kill anyone. If you select this choice, twenty people are sure to die. The locals know that if you choose to shoot one person, they each have a 1 in 20 chance of being shot. If you do not choose to shoot anyone, they each have a 20 in 20 chance of being shot. If you choose to act, they stand a much better chance of surviving. Consequently, each of them would prefer that you act rather than refrain from acting.
One of the first questions you must ask yourself is will you play this game at all. You did not create this situation and so do you have any obligation to consider the choices placed before you. Jonathan Baron, in his analysis of morality and rational choice, notes that most legal systems do not hold people responsible for failing to aid someone in need. Consequently, you probably have no legal obligation to participate. Thus, you must ask yourself whether you have a moral obligation to do so. Is it better to hurt someone through an action or through failing to act? Generally, we tend to believe that an act is worse than the omission. Thus, if you fail to act you may feel you are not responsible for whatever the outcome of the captain's game. If you feel no moral obligation to participate, then you should walk away now and end it. If you feel morally obligated to participate, you will need to ask yourself certain questions.
Utilitarianism and, more generally, consequentialism maintain that the act-omission distinction is morally irrelevant. Nonetheless, these theories do place a moral obligation upon you to act. However, I believe you will see that beyond that they are very skewed in their treatment of morality. But, if you accept a moral obligation to...