Economics of the Bachelor Pad

Note this economic analysis came not of Dr. Buckley's desire to watch the Bachelor Pad. He has a very convincing wife who loves reality tv and strategy gameshows.

Tonight was the finale of The Bachelor Pad, which if you haven't been watching, let me catch you up:
A bunch of the favorite and not so favorite contestants of previous seasons of the Bachelor (the dating game show with 1 guy and 25 women or 1 lady and 25 men) come together in one giant sexy house to compete for $250,000 and possibly a chance at love. Various alliances were made; many relationships formed; competitions were intense. For the final six, the top three guys picked a girl to create a team, or a couple. Everyone has been voted out except one couple Natalie and Dave.

Okay, you're caught up. We have Natalie and Dave. Natalie is a party girl, and she has kissed the most guys, drank the most and played on different sides of the ladies. Dave has backstabbed some of the guys, broken hearts of two ladies and had a good time. But, through all of this, the pair claimed they were developing a slow relationship which might possibly lead to love.

Here they are, so close to winning, and the host Chris Harrison offers them a choice. They can choose to share the $250,000 evenly, or they can choose to keep it all for themselves. Only, it gets more interesting.

If they both choose to share it, they will split the money evenly and each get $125,000.
If one chooses to share it and the other chooses to keep it, the one who chose to keep it gets the whole $250,000 and the other gets nothing.
If they both choose to keep it, the money gets split evenly among the contestants who've been kicked out and are sitting onstage.

Okay, get ready for some economics.

This is a Prisoner's Dilemma. What you need to think about is a Best Response Function. What is your best move in response to whatever they do?

Let's think about it from Dave's point of view. If she's definitely going to keep the money for herself, does he prefer to play keep and give the prize up to the losers of cast or does he prefer to share, so she can win? If Natalie's going to play share, what is his best response? Would he prefer to split the money with her or screw her over? If he preferred giving her the money rather than the chance of having to give the money to the rest of the cast, then share is his best response in either case.

Natalie might know that, and then she has her best response function. Does she want to share the prize money of $250,000 with him or does she want to keep it all? What do you think happened?