Our Daily Bleg: Beauty and the Geek Game Theory
We recently received this bleg request from Alon Nir, a regular reader who has contributed to this blog before. He is a young Israeli with a bachelor’s degree in economics and business management who is getting started in the Israeli start-up industry.
About a week or so ago I caught a rerun of a Beauty and the Geek [episode] in which Dubner appeared. There was something in Mr. Freakonomics’s appearance on the show that sparked the economist in me. I started pondering the best strategy to play the game and wondered if there’s a game-theory solution for this problem. Soon it dawned on me that this is a thesis- or perhaps a Ph.D.-level question.
For those of you who aren’t familiar with the show, I’ll present the concept and the rules briefly. Eight nerdy guys and 8 beautiful gals are paired together. In each episode the couples compete in two challenges. The couple that wins a challenge is safe from elimination and also has to choose which of the other couples should go to the elimination room. Two couples compete head-to-head in the elimination room; one couple prevails and returns to the game, whilst the other goes home.
[Those are] the rules, now here’s the problem: if you’re one of the couples and you win a challenge, which couple should you send to the elimination room?
I’ve spent some time over the weekend thinking about what is the least complicated way to depict this problem in game theory terms. Believe it or not, the following is actually the best I could do:
For starters, let us assume that the couples’ competences aren’t distributed evenly. Three of the couples are considered “weak” contenders, and they are the least likely to win a challenge/elimination round. Next, there are two couples of medium strengths which are twice more likely to win a challenge than the weak couples; and finally there are three “strong” couples which are twice more able than the medium couples, and hence 4 times more able than the weak couples.
Furthermore, let’s say you’re one of the medium strength couples and the other couple of medium strength was eliminated during the previous episode. (So other than you there are three strong couples and three weak couples still in the game.) Next, let’s assume you have won both challenges during the show and you must decide which two couples to send to the elimination room. (This assumption makes the problem a whole lot simpler; I’ll get back to it later.) Keep in mind that the couple that returns from the elimination room will seek revenge, and if they win a challenge the next week, you will definitely be sent to the elimination room (unless you win the other challenge and get immunity).
Now you have to choose which couples to send to the elimination room. There are three possible combinations:
1) One strong, one weak couple: If the weak couple eliminated the strong one, than you’re twice better off, since one strong couple (which is a bigger threat to you) is out of the game, and the couple that returns and seeks revenge on you is weak and not very likely to be in a position to punish you for sending them to elimination. However, this is the less likely scenario.
It is more likely that the strong couple will prevail; then not only will there be one more strong couple in the competition for the prize money, there’s going to be one angry strong couple that is likely to win a challenge and send you to elimination in the next episode.
2) Two weak couples: On the one hand, the couple that will return from the elimination room will seek revenge, but will be less likely to be able to achieve it. On the other hand, you’re left either way with more strong couples in the game, which diminishes your chances of winning.
3) Two strong couples: One strong couple will be out of the game, which is a good thing of course. Then again, one strong couple will want to get back at you and is very likely to win a challenge in the next episode. (That couple is even more likely to win the challenge now that there’s one less strong couple in the game.)
So what would you choose?
In case this problem is too easy for you, it’s fairly easy to make it a lot more complex. Let’s waive the assumption we made earlier and say that you won only one challenge and another couple won the other. Now you have to pick one couple to send to the elimination room, while trying to predict what the other couple will do. Now this is an interesting interaction in game theory regards.
Still not complicated enough for you? Well I also used an implicit assumption that the problem spreads only across two periods; so if a couple that seeks revenge on you doesn’t get it in the next episode, you’re safe. Well, you can waive that assumption as well and say that when you send a couple to the elimination room, they will hold a grudge against you until the end of the game. There. Complicated enough?
As you can see, in game theory terms this is a very complex problem. Of course I don’t expect anyone to find a Nash equilibrium, but I would still love to read your thoughts on this problem.
It’s been a pleasure writing a guest post for the Freakonomics blog.