Fattest Stack

After beating the odds with Monty Hall, you're now getting to try your luck on The Fattest Stack, a new popular game show.

After various rounds of wearing silly costumes, answering ridiculous questions, and other sundry foolishness, you're now trying to win the "Fattest Stack".

Here's how it works:

There are four boxes on the stage, each with a stack of brand new one dollar bills inside. Before the game starts, you don't know how much money is in each, or even what the range of values is. You do know that no two boxes have the same size stack.

The host will let you open the boxes, one at a time, in whatever order you wish and examine the stack inside. You may then choose (above the inane hooting of the audience) to take the visible stack or open another box.

Once passed, you may not return to a previous stack; e.g., if upon seeing the first stack you decide to open the second box, you may not then opt for the first stack. If you pass the first three boxes, the fourth stack will automatically be your choice.

Now here's the catch: You only get to keep the money in the box you chose if it turns out to be, in fact, the Fattest Stack (hence the name of the game).

Can you beat the odds with this game? Or is blind guessing just as good a system as any?

Assume that you can tell whether one stack is bigger than another just from observation.