[donkey69]

The unexpoitable strategy is not difficult to find. In fact its staring everyone in the face but they fail to realise it. Case of missing the forrest for the trees. The following strategy WILL DOMINATE ALL OTHER STRATEGIES in the long run. No ifs or buts. It will be EV+ against ANY other strategy. It will be EV=0 against itself.

Basically its a tight aggressive strategy:
If you draw 1: Fold
If you draw 2-999,999 check if you are first mover and if the other person bets call it.
If you draw 2-999,999 and are second mover, call the bet if there is one.
If you draw 1,000,000 raise infinitely until you are called.

Reasoning:
Folding the 1 is obvious since you are the dog with certainty.
Limiting the betting to maximum of 1 round for 2-999,9999: since you know you dont have the nuts but the other person may you try to limit your lossess as much as possible. On average you will beat your opponent half the time and he will beat you the other half. Should break even in the long run.
1,000,000 The Nuts: All bets made and reraises made are EV+ so you want as much action as possible.

I believe the above strategy will beat all the other strategies mentioned above. If both players are expert players betting should not proceed for more than 1 round.