$\textbf{Color the Grid}$

Two geniuses are playing a game of coloring a $2\times n$ grid where $n$ is an odd integer. Each of them in turn picks a uncolored cell and colors it in either green or red until all the cells are filled. At the end of the game, if the number of adjacent pairs with the same color is greater than the number of adjacent pairs with different colors, then the person who picks and colors first wins the game. (An adjacent pair consists of two cells next to each other.) Otherwise, if there are more adjacent pairs with different colors than those with same color, the person who starts later wins. If these two numbers are the same, the result is a tie. Who will win if both players make no mistake?