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It seems to me that the fewer sides, the more "perfect" the die can be, and hence the more fair.

Unless of course near sides have almost equal effect in the particular game of interest, in which case imperfections will have not a big impact.

The bigger issue is that once you go past the 5 platonic solids (4,6,8,12,20 sided) you have to play games with the shape and placement of faces.

You can also use the duals of (convex) uniform polyhedra (as here) - uniform polyhedra have equivalent vertices, their duals have equivalent faces.