Minimax search:
Always notice there that the process above was in order for player one to choose his and her first move for that. The whole entire process would require to frequenting for player two's first move, so then player one's second move, etc. Most probably in that we see, agents playing games using a mini-max searching just have to calculate the best move at each stage using a new minimax search.
Well don't forget is just means that an agent thinks their opponent will act rationally, does not sense they will, well doing then they cannot assume a player will make a particular move until they have in actual fact done it.