Trashy Game Theory

Of the trashy TV that I watch, I think that Paradise Hotel has got to be the trashiest.  The first week there were 6 girls and 5 guys on the show.  In order to continue their stay at the Paradise Hotel, the girls would each pick a guy, one by one, to be her roommate for the week.  One pair would then have a "third wheel" (since the girls outnumber the guys), and the guy would then choose which girl he would keep as his roommate.

Now, since I'm a complete nerd, I stated working out odds and percentages and maximizing how to play the game.  Game theory has to do with making decisions based on the decisions of other people.  It's incredibly useful and valuable, so let's take a second to break it down in a more simple example.  Let's say there are two competing coffee shop companies, Staribou Coffee and Carbucks Coffee.

Both are national chains and are looking to expand their business in either New York City or Oklahoma City.  Since NYC real estate is much more expensive than OKC real estate, each chain could either open one shop in NYC or 2 shops in OKC.  Because these are two competing businesses, if both choose to open shops in the same cities, neither company will turn a profit.  How does each business choose where to open the new shops?

We probably need a little bit more information, but each shop is going to use game theory.  If you're in Staribou's position, the most important thing is to not open shops in the same city as Carbucks.  So you're going to put yourself in Carbucks' position.  And you may ask yourself, "well, how did I get here? what would Carbucks do?  I'm going to open my shops where they won't open shops."  Well, Carbucks is doing exactly the same thing, and putting itself into Staribou's position, quoting Talking Heads....Game Theory tends to get very circular ;)

Let's get back to the Game of Paradise Hotel:

In the Game of Paradise Hotel, you either put out, or you go home.  I told you it was trashy!

The first girl and the last girl are trivial choices.  The first girl has open season, and the last girl will be forced to (we'll see why she's forced to in a second) double up on a man.  Let's work on the second girl.  She has a choice; either she can double up on the first girl's choice, or she can pick a new hunk.  IF she doubles up, then she has a fifty percent chance of going home, depending on who the guy chooses (every other girl will choose an open man because that is the choice that gives a one hundred percent chance of staying).  IF she does not, and picks a new man, then we need to find the odds of another girl choosing the same man.  Recursion, ho!

3RD GIRL: We've assumed that the 2nd girl picked a new man.  There are now 3 open men.  She can either pick a taken man (50%) or pick a new man.  Recursion, ho!

4TH GIRL: Assuming 3rd girl picked a new man.  2 open men.  She can either pick a taken man (50%) or pick a new man.  Recursion, ho!

5TH GIRL: Assuming 4th girl picked a new man.  1 open man.  If she picks a taken man, she has a 50% chance of going home.  Since there's one girl to go after her, she has a .2*.5=10% chance of going home if she picks the open man.  The 5th girl will, naturally, pick the last man standing.  Ho, recursion!

4TH GIRL:  OK, how we know what the 5th girl's choice is.  That means that the 6th girl is the only girl that can challenge you for your man.  10% chance of going home if she picks a new man.  The 4th girl will pick an single man.  Ho, recursion!

3RD GIRL:  Now we know what 4th and 5th girls' choices are!  They're going to pick new men!  10% chance of going home for picking a single man, so the 3rd girl will pick a single man.  Ho, recursion!

2ND GIRL: That brings us to the end of the recursive argument.  Now we know what each girl after us is going to do; they're picking new men.  So, to recap, there's a 50% chance of going home if she doubles up.  Since each girl after us is picking a new man, that means there's a 10% chance that, if you pick a new man, the 6th girl will 1) choose your man AND 2) your man chooses girl #6 over you.  Therefore, the 2nd girl will choose a new man.

Thus, each girl will choose a new man (unless she thinks that a particular man has more than a 90% chance of picking her over the other woman).  The 6th girl must be doubled up, and thus has a trivial decision (any man she picks will result in a 50% chance to go home).

Due to the nature of the game, she has a 50% chance of going home.  Every other girl has a 10% chance of going home.

I pity the girl who chooses last!