[Siegmund]

In the dice game, it's true that equal expectation doesn't guarantee an equal chance of winning. (Because it the dice game, player A can only win a 5-0 victory, while Player B might achieve anything from a 5-0 to a 5-4 win.)

However, returning to the OP's game:

In his first post it wasn't clear that his goal was to win all 40 chips.

If we make the further assumption that that the two players agree to continue playing until one or the other of them has 40 chips, then the only two possible outcomes of the game are 40-0 and 0-40.

Since the EV of every bet is 0, the EV of the game as a whole is zero; since the EV of the game is zero AND the only two possible outcomes are 'player A wins all the chips' and 'player B wins all the chips', these two outcomes must also be equally likely.

In other words, it is completely irrelevant what strategy either player uses: the probability of winning remains 50-50 no matter what size the bets are.