Can a game theory strategy go in an endless loop?

Question

Suppose there are two players in a game. Player A arrives at their best strategy by game theory but they also know that Player B knows about game theory, so Player B will also go with their best strategy. Now knowing B's best strategy, A adjusts their strategy to a new strategy. Now A knows that B will arrive at the some new strategy by using the same reasoning. So A again adjusts their strategy and this keeps going on...

Answer 1

Certainly! Let Players 1 and 2 have two strategies each, $A$ and $B$. They play a game in which the winner is determined as follows:
Player 1 wins if they choose the same strategy and Player 2 wins if they choose different strategies.

$p_1$'s strategy depends on $p_2$'s strategy, and clearly $p_1$ will match whatever strategy $p_2$ chooses. After matching, $p_2$ will then choose the other strategy not chosen by $p_1$ and this process repeats indefinitely.

Mathematically, this means that the game just described has no Nash equilibrum. That is, there is no pair of strategies such that any single player cannot deviate and improve their situation in the game.