Beauty & the Geek & Game Theory

In the reality show “Beauty and the Geek”, the contestants were first paired off in the following fashion:

7 beauties were located in 1 room and 7 geeks are located in another room and there is a curtain seperating the two. First, a geek would be asked to voulenteer to present himself to the group of beuaties and give a brief speil about who he was. Then, 1 of the beauties would choose to pair up with that geek and they would form a pair. Then, one of the beauties would voulenteer to go over to the geek side and give a brief speil and one of the geeks would voulenteer to pair up with the beauty. This process continues until there is only 1 beauty and 1 geek left and they pair up with each other.

Now, using game theory, is it possible to figure out what would be the optimum strategy for a beauty/geek in order to pair up with the most attractive/intelligent of the opposite kind?

Assume, for the sake of this problem, that all people would rank the attractiveness and intelligence in the same manner, so personal preference has nothing to do with it. Also, assume that a person is aware of his own ranking within his group since he can compare himself with everyone else in the group. But beauties can only compare geeks to the geeks that have been previously presented and geeks can only do the same for beauties.

To me, this seems like some variation on the optimal marriage problem. I’m having problems finding a link to it, but, IIRC, I think either Neumann or Nash or one of the biggies pulled this out as a trivial example of game theory where they managed to prove that a woman should reject all suitors before she is 40 and then marry the first person she meets after the age of 40 which is better than the average of all the suitors who proposed to her before 40. It runs under the same set of arbitrary assumptions but I thought it was quite a neat way to go about the problem.

So, any thought about how to tackle this?

My initial intuition is that you should approach the other group instead of having them come to you and that around 1/2 way through the choosings would be the optimal time. People who start off early on are likely to be rejected in the hopes that someone better will come along later. People later have slimmer pickings of beauties/geeks. I guess it would also depend on your percieved rank within the group. If you rank last, then perhaps it would be better going earlier on so you can exploit the other groups ignorance. Since theres a fixed quantity of attractiveness/intelligence, that makes this a zero sum game which, if I understand the theory correctly, must mean theres a stable optimal solution. I doubt that the theory is advanced enough for us to figure out what it is though.

I wouldn’t approach the other group as you suggest because then you have no choice in the decision - you are likely to get the ugliest girl as she would reckon that no handsome guy would choose her if she went over thus she better choose.

Your best bet is probably a variation on Neumann (or whoever) - though seven is too small really to put it into effective practice. Wait until about a quarter* of pretty girls have paraded infront of you and then pick the first that beats them after that.

*There is an optimal fraction but I will have to find it later

I think approaching is better because you can exploit the information differential. When a geek approaches beauties, the beauties need to. When a beauty approaches a geek, the geek gets to pick. Because geeks know about every other geek in the room, theres a lot more transparancy. On second thoughs, this would probably only apply if you were uglier than average. Okay, how should the most attractive member of both groups act? I’m almost certain theres no way for the most attractive people to find each other. So they will end up sub-optimal compared to the ideal pairing no matter what.

Having posed the question this way, I wonder which room you would start out in? :wink:

(I’m sort of transfixed by the show myself, and having seen the introduction where one of the participants walks into the room, says his or her name and maybe a sentence about him- or herself, I don’t think there will be the sort of information transparency you would require.)

The optimal fraction is actually 1/e, or about 37%. Here’s an article that discusses it.

You guys are such naturals for that show! :wink: The first week, when Richard the Übergeek and his lovely partner (GREAT smile!) both won and had to choose two teams to compete to stay on, I imagined him growing an ulcer while he stood at a whiteboard all night trying to determine the best course of action based on too little information.

We assume for the purposes of this hypothetical that we can, otherwise it would turn intractable.

the problem is more complicated than I originally thought because of ties - e.g. if everyone plays like you then there will be 5 or so people bidding on the same lass and you end up with a dud later on. The problem is that optimal way to play the the game depends on how the others (both girls and guys) are playing it. Of course if they are playing optimal strategy then you can modify yours to beat theirs, but they are probably thinking the same so they are changing theirs. It may well be unsolvable?

I don’t think your formulation of the problem is rigorous enough to analyze.

Question: What if zero or more than one person want to perform the current action (going to the other side or pairing up)?