Maybe this description of the two games might help clarify the link. Game 1: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say

Maybe this description of the two games might help clarify the link. Game 1: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b) dollars. Game 2: Player 1 chooses a dice, then player 2 chooses a dice. They both roll numbers, say a and b. Then player 2 gives player 1 (a - b > 0 ? 1 : -1) dollars. You can kind of squint and see that game 2 is the same as game 1 just with (a - b) "rounded" to either 1 or -1.