Randomization, derandomization and antirandomization: three games

Three games are given between two players, Paul and Carole, with a common theme. In each round Paul does a split and Carole chooses. Random play by Carole allows a bound for the game value. Through derandomization this becomes a deterministic strategy for Carole minimizing a weight function. Paul can use that same weight function to give a bound for the game value in the other direction.