In the formal description of Deterministic Pushdown Automata, they allow $\epsilon$ moves, where the machine can pop or push symbols onto the stack without reading a symbol from the input. If these $\epsilon$ moves aren't allowed, and the stack can only be modified once after each symbol read, are the resulting automata equal to power to DPDAs?
There may be something trivial I am missing with regards to using the powerset of $\Gamma$ as your new $\Gamma$, allowing you to "compress" $\epsilon$ moves into the equivalent automaton without them, similar to how you can compress $\epsilon$ moves in a DFA. Just it seems that such a conversion is not as trivial as for DFAs, and I'm not sure it's even possible.
So are the two equivalent in power? I'm just asking because everyone seems to assume that DPDAs have $\epsilon$ moves and I'm wondering why that assumption exists, since it seems like a more complex model.