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The progression I envisioned would be that each volume magnitude proposed by player 1 and acceded to as a lower bound by player 2 would be greater than that on the previous turn.There is no progression. The question is which player has a winning strategy.

As to the winning strategy question, the second player always gets to make use of the first player's efforts, and gets to use the 'greater than or equal to' property to ensure that he is never wrong unless the first player has already erred.

Citing the 'high confidence' conjecture from your post ...

... you appear to be postulating a universe whose size is at least countably infinite, wherein there would be no winning strategy for such a game, but in the case of a finite universe, the advantage would be to the second player, who would always come out with at least a draw.... we would have high confidence that there are subsets which have more than any specified finite volume. [We would have to nail down with some precision what sort of subsets we had in mind and what volume measure we'd use]