Which half-cell reaction would have the greatest tendency to occur at the cathode?

In electrochemistry, cathode is the electrode where the reduction reaction occurs. In a redox reaction, the reduction involves decreasing the atom's oxidation number, losing electrons in the process. The opposite is true for oxidation.

I think you forgot to include the choices. Nevertheless, by concept, the greatest tendency for reduction reaction will be based on empirical values. The standard electrical potential, denoted as E0, is gathered from experiments. Every known reduction reaction there is, has a known value of E0. Therefore, the greatest tendency to occur in the cathode is the reaction with the highest standard reduction potential starting from the upper left to the lower right reaction in the picture.