=Henry Baker What do you do if the interval is (-oo, oo) on the real line? Does this change things in any meaningful way?
Don't think so; don't you just guess 0 first? As Don points out, domain quantization is probably more significant than domain size. In a casino version I think the round-off would constitute a not-inconsequential house "vigorish".
What is the relationship of this game to continued fractions?
Generally, "continued-f" expansions alternate excreting a spoor of integral floor(x), and transforming the residue: x-->f(x-floor(x)). When f is multiplying by B it gives base-B expansions, reciprocating give continued fractions, squaring first gives continued surds, etc. I suspect the no-three-in-a-row constraint is more akin to "base" Zeckendorf expansion than it is to continued fractions.
=Don Reble [...] Nice analysis, Don, thanks!