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.
With this request, he is following firmly in the footsteps of economists who see treasure in reality/game shows. See if you can help him out, and feel free to send your own bleg requests here.
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.
and 
. You can also watch it on 
Another way to frame the game:
A given contest consists of eight parties competing in rounds composed of two challenges and an elimination. In each challenge all of the parties compete and the winner of the challenge selects a party to participate in the elimination. In each elimination two parties compete such that the losing party is removed from the game. The information given for each party is a relative probability of success in each event P(Male Challenge) P(Female Challenge) and P(elimination) and probability that they would select you for elimination P(hate). Also under consideration is that after attempting to eliminate a team there is an increase for P(Male Challenge) P(Female Challenge) and P(hate). Given these values after any given challenge select a party to send to the elimination round. This assumes that you are not required to act on your P(hate).
Strategy:
For the early game I would eliminate parties which would maximize my chances of winning at least one of the challenges so that I could control the field in the late game. In the late game I would eliminate teams who have a hi probability of winning the elimination as I will very likely be involved in an elimination round in the late game.
Host a lottery.
Put all the couples’ names in a hat (maybe heavily weighted toward strong couples) and make another couple draw the name(s).
Assuming you’re not playing against other game theorists, this has the wonderful consequence of deflecting the revenge toward the drawer, rather than you. Human nature is to blame the one doing the drawing – don’t poker players blame the dealer?
No PHD or any serious economic training here, but I will give it a shot, working with the easier assumptions and a small amount of common sense…
Two rounds, 3 strong couples, 3 weak couples
I will assign numbers to couples as follows:
weak=1
medium=2
strong=4
2 will beat 1, 4 will beat 2, etc.
Optimal outcome is lowest total at the end of the second round, when you add numbers of each couple remaining.
Test 1-
Round 1 (weak v. weak, one weak eliminated)
1+1+4+4+4=14
Round 2 (strong v. strong, one strong eliminated)
My assumption here is that weak looking to take revenge can’t, because they are not likely to win a challenge. Strong that did win would be looking to eliminate other strong couple.
1+1+4+4=10
Test 2-
Round 1 (weak v. strong, one weak eliminated)
1+1+4+4+4=14
Round 2 (previous strong takes revenge, strong v. medium, medium eliminated)
BAD IDEA
Test 3-
Round 1 (strong v. strong, one strong eliminated)
1+1+1+4+4=11
Round 2 (previous strong takes revenge, strong v. medium, medium eliminated)
BAD IDEA
Optimal strategy, weak v. weak.
I would love to hear a critique of my thought process.
Alon, great post and good luck with your start up.
I haven’t really thought about this thoroughly yet (and I just returned from lunch so I might not have especially cogent thoughts even if I did). However, I note here that you probably shouldn’t assume that any team you send to elimination will automatically seek vengeance the following week if it returns triumphant. Rather, assume they will go through the same thought process you did – why didn’t you consider whether any of the teams had sent you to elimination in making your decision?
Having said that, the empirical evidence suggests that people like revenge, in “Beauty and the Geek” and otherwise…
Choosing a couple for potential elimination does not guarantee that they will seek revenge. Perhaps another couple has previously tried to eliminate them and that couple will be targeted. Or perhaps the aggrieved couple will find a superficial reason to choose another couple for elimination, as is often the case on reality shows. The probability of revenge must be calculated before assigning values to the choices.
Dan, two issues with your discussion. First, in the strong v. weak, you need to add the probability (.25) that weak will defeat strong and vice versa. In weak v. weak and strong v. strong, probability is unnecessary in the first round because the two are equal, but in weak v. strong you must include it.
I think your assumption that the strong couple will want to defeat the other strong couple needs more proof. It’s possible the strong couple might just want to remain on the show for another week, thus choosing to face the weak couple. An additional reason for the strong couple to do this is to allow for a strong-strong or medium-strong matchup that doesn’t include them in the weeks to come. This way they’d ensure they remained on the show and take the chance that other couples will do the dirty work of eliminating the stronger teams. Of course, your assumption could be totally valid, but I’m just adding some counters.
It’s a fun puzzle, and this is not a criticism, but one team is never universally stronger than any other team across multiple challenges. Strengths are challenge dependent. Since it is not possible to know the revenge challenge and so not possible to know the relative future team strengths, there should be a discount to presumed strong teams in the revenge iteration when finding the Nash equilibrium.
Roughly, this would shift the balance toward one weak, one strong.
It seems the true strategy in this game, for a medium strength couple, is to throw the initial challenges, and not pick who is up to be eliminated.
In all all of the reality shows that I have seen it seems that the average players have an advantage. They are not seen as a threat nor an easy cull.