While I understand the MiniMax tree and alpha-beta pruning concept, I don't understand why in many (for example Wikipedia) resources about alpha-beta pruning there is a condition like α >= β. Specifically, equals is confusing. As I understand, the alpha-beta returns move that minmax would return, but mostly does it faster. But this example contradicts it:
/ | \
1 3* 2
/ | / \ | \ \
1 1 5 3 4 3 2
Above is the original min-max tree. As we can see it would choose the one move with score 3. Now let's do the alpha-beta:
/ | \
1 3* 3*
/ | / \ | \
1 1 5 3 4 3
It cuts off the right-most move because of 3 >= 3. But then the algorithm can choose between 2 moves because they have the same score, but as we saw in min-max the right choice is slightly worse. That wouldn't happen if algorithm specified simply α > β, so it would need to search 2 as well.
So was that a typo in Wikipedia's pseudocode (and many other resources)? Or I misunderstood something really big here.