Ehrenfeucht-Fra\"iss\'e Games for Continuous First-Order Logic

We define a version of the Ehrenfeucht-Fra\"iss\'e game in the setting of metric model theory and continuous first-order logic and show that the second player having a winning strategy in a game of length $n$ exactly corresponds to being elementarily equivalent up to quantifier rank $n$. We then demonstrate the usefulness of the game with some examples. Finally, we discus connections between the game of length $\omega$ and infinitary logic.