The antonyms of the phrase "draw lots" are many, as the act of drawing lots implies some level of chance or randomness. One could choose from a variety of antonyms, depending on the context, including select, choose, elect, or appoint. These words suggest a deliberate and intentional action, as opposed to the outcome of drawing lots. Other antonyms for "draw lots" might focus on the idea of fairness or equality, such as divide or distribute, as these words suggest a way to apportion resources or opportunities in a more even or equitable way, rather than leave it to chance or fate.