Playing The Dirty Big Number Duel Optimally

In MIT’s Big Number Duel, the rule simply states:

it would be considered unsporting to win by adding one to your opponent’s last entry

But what if the game had no such rule? Will there be a winning strategy, then?

Let us consider P1 and P2 as the two players in the game, and explore possible outcomes.

Outcome 1: Only P1 plays dirty, and wins

P1 | P2 + 1

P2 | X

Outcome 2: Only P2 plays dirty, and wins

P1 | X

P2 | P1 + 1

Outcome 3: Both play dirty

P1 | P2 + 1

P2 | P1 + 1

This would be a cyclic reference, and the game would be a failed exercise.

Therefore, is it optimal to just not play dirty at all? Are there other outcomes?

Feel free to comment or discuss.