Why do acids favor the oxidizer? And bases the reducer?
The effect that can be observed by adding acids to a solution to increase conductivity is known as the Grotthuss mechanism. Protons have a much higher mobility in solution ($3.62\cdot10^{-3}~\mathrm{cm^2~V^{-1}~s^{-1}}$) than any other cation, basically allowing it to tunnel through a solution. (A similar effect can be noted for hydroxyl anions.)
Basically this allows a much faster charge exchange through hydrogen bond networks, also resulting in a higher cell voltage.

(But then again I am not really sure I understood your question correctly.)