Deal or no deal and rational choice theory

As my students are aware, I have been under the weather since the beginning of January and am finally feeling somewhat like a human being again. During my down time, I took some rest and had time to do some non-school-related activities, one of which was trying out the Deal or No Deal app on my smartphone. You do remember the TV show hosted by Howie Mandel, right?

Anyway, the basic idea of the show is this:

• There are 26 suitcases on state, each with a card containing a dollar amount between $1 and $1 Million.
• The game begins when the contestant chooses one of the 26 suitcases as “their” suitcase. If the contestant keeps the suitcase until the end of play, they win the dollar amount written on the card inside the suitcase.
• The contestant must begin opening a certain amount of suitcases during each round of play–5 the first round, 4 the next, etc.
• After each round, the game pauses and the contestant receives an offer from the mysterious banker via telephone with Howie as the intermediary.
• The contestant is then asked whether there is a “deal, or no deal.” The contestant may accept the banker’s offer or continue. [There is where the drama gets ramped up to 11!]
• If you have watched the show, you’ll notice that the banker’s offer depends upon which dollar amounts have been revealed. If the contestant reveals many high-value suitcases, it becomes like likely (probable) that the suitcase s/he chose at the beginning is a high-value suitcase.

The smartphone version is slightly different from the TV show in that the suitcases do not have dollar amounts attached but point multiples (that is, you win 1X, 2X, 3x, etc. 1000X the pot).

Take a look at the images above screenshot (is that the past participle?) from my smartphone. What do you notice about the banker’s offer? What’s of importance here is the red boxes in each picture. These are two separate games, btw.

These are two separate games. In the top game, there are only two suitcases left–one of them is the 20X and the 200X, Therefore, I have either the 20X or the 200X. That’s quite a big difference in winnings–ten times. So, what would you do? What would a rational choice theorist say you should do? Are the bankers offers rational in each case? Why or why not?