There is no official rule covering a tie between a batter/runner and a thrown ball to first. Baseball rules do not recognize a tie between a base runner and a force out at any base. The umpire must make the judgment call on which arrived first. However, an umpire is taught to watch the runner and see when his foot touches the bag, and he is supposed to listen for the ball to pop the glove of whoever is playing on or covering first. So if it is a tie then this is where the runner will get the benefit of the doubt.
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There is no such rule as "tie goes to the runner" in baseball or softball. That is a common misconception. The umpire must judge whether the runner beat the ball or vise versa.
In sports, generally a "tie" goes to the offensive player, since you have to have some ruling on what happens in the case of a tie