, 2022

At the start of the game, there are $2021$ pairs of two adjacent cells with different names and the number of such pairs is reduced by two per one operation. If there are three or more pairs of two adjacent cells with different names then a player can do the operation hence the number of such pairs is one at the end of the game. Therefore the total number of operations done by two players is $1010$.

When Alex does the operation, there are three or more pairs of two adjacent cells with different names. Let $(X, Y)$ be the leftmost one among all such pairs and let $(Z, W)$ be the second one from the left. Assume $X$ lies to the left of $Y$ and $Z$ lies to the left of $W$ then $X$ and $W$ have the name of Alex and all cells between $X$ and $W$ have the name of Betty. Therefore by doing the operation to $X$ and $W$, Alex can increase the number of consecutive cells with Alex's name beginning from the leftmost cell by two. When there are consecutive cells with Alex's name beginning from the leftmost cell, they are not replaced by Betty's operation hence the number of such cells is not reduced by Betty's operation. Alex does the operation $505$ times thus $m = 1 + 2 \cdot 505 = 1011$ satisfies the condition. On the other hand, Betty can similarly make $1011$ or more consecutive cells with Betty's name beginning from the rightmost cell hence we have $m \le 2022 - 1011 = 1011$. Therefore the answer is $1011$.