[AaronBrown]

It depends crucially on the blind/limit structure. In no-limit, AA always goes all-in preflop, and wins on average.

You never have much more than a 20% chance against AA. Say the other player plays 20% of his hands, with an average preflop winning chance of 20%. That means he wins 4% of the hands. With $1/$2 blinds, and $2/$4 limit bets; out of 25 hands, he'll pay $30 for the 20 hands he folds plus $20 to see 5 flops. Even if he knew for sure whether he would win or lose after the flop, it only costs A's $10 to call to the river. That means the random hand player makes $18 on a win, not enough to make up for the $50 he spent.

Those are just made-up numbers, and the random hand player can improve slightly with game theory, but it will still be a losing proposition. If he could bet $100 on the river, it's a different story.