On the theory of exponential integer parts

We axiomatize the first-order theories of exponential integer parts of real-closed exponential fields in a language with $2^x$, in a language with a predicate for powers of $2$, and in the basic language of ordered rings. In particular, the last theory extends $\mathsf{IOpen}$ by sentences expressing the existence of winning strategies in a certain game on integers; we show that it is a proper extension of $\mathsf{IOpen}$, and give upper and lower bounds on the required number of rounds needed to win the game.