Show that this algorithm does not work for determining convex polygons Context

Consider this algorithm. If the set $\{\angle p_ip_{i+1}p_{i+2} : i=0,...,n-1\}$  does not contain left and right turns, output "yes the polygon is convex"; otherwise, "no".

My answer  

Consider this nonsimple polygon having 4 vertices; the algorithm above will output "yes" as the set of points does not contain both left and right turns, yet the polygon is not convex. Is this a good counterexample rendering the above algorithm incorrect? 



Thanks in advance.
A simple counterexample is a pentagram (http://en.wikipedia.org/wiki/Pentagram). It contains right turns (if you go in clockwise direction) only, but it is not convex. The above algorithm will incorrectly say pentagram is a convex polygon. It's actually a complex polygon. 

In other words, algorithm might fail if the input polygon is complex.