[marv]

With optimal play on both sides, player 2 will have a positional advantage, but under optimal play each player will be using a best response to the other players strategy.

In my example which wasn't quite optimal for player 1, if P1 checks with [0.1,0.8] and folds to a bet too often (as he does in my example), player 2's best response to a check must be to bet everything in [0,1]. Your proposal for player 2 wasn't betting enough so player 1 was getting more EV than he should.

Note that player 1's distribution after check is not that much weaker than after he bets. This is typical when player 1 may have further decisions after checking, and is why we wouldn't expect a strategy of the form 'always fold after checking' to be optimal - it means P1's initial action gives away too much information.

Marv