I can't speak to AlphaGo specifically, but these parameters (depth and width) are typically chosen through trial and error. People joke about "graduate student

I can't speak to AlphaGo specifically, but these parameters (depth and width) are typically chosen through trial and error. People joke about "graduate student descent" but it's true. There are empirical rules of thumb that arise from knowing what has worked in the past. Empirically, more depth is better and nets should not be very wide at all.