Where does the values of alpha-beta search get updated? 

The correct answer is Along the path of search.

Key Points

• Alpha-beta pruning is an algorithm that seeks to minimize the number of nodes that are evaluated by the minimax algorithm in its search tree. The alpha-beta values get updated as the search algorithm delves deeper into the game tree, i.e., along the path of search.
• Alpha is the best value that the algorithm can guarantee at that level or above, while beta is the best value that the algorithm can guarantee at that level or below. As the algorithm searches through the game tree, alpha and beta are continually updated, hence providing bounds on the values that can be obtained by sections of the tree remaining to be analysed. This allows branches which cannot possibly influence the final decision to be pruned, hence making the algorithm more efficient.
• The initial state introduces alpha and beta values, typically initialized to negative and positive infinity respectively. The end state does provide definite alpha-beta values, but these aren't updated at the end - they are updated along the path as the search progresses.

Therefore option 1 is the correct answer.