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.

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.

Alon Nir

OK, so I guess these are my 15 min of online fame. I am truly amazed at how things turned out, and incredibly happy they did the way they did. So thank you Stephen Dubner for making it happen, thank you to show winner David Olsen for your participation in the discussion and a very special thank you Peter Norvig for taking it further than I could ever imagine.

A few replies to the comments:

#11: Dan M. – First off, thank you for your kind words.

Second, You asked for critique of your thought process so i'll just say a couple of things. Your way of attacking the problem is very interesting and thought provoking. You start off right, since you assigned numerical values to the couples, which allowed you to give numerical values to the different outcomes. This is a must for a game theory solution, since the best solution is the one which provides the highest utility to the couple. Utility is a pretty elusive concept, so your suggestion is as good as any other. Now, here comes to perplexing part to people without economic training: to find the (optimal game theory) solution, you need to take what you suggested to the next level and build a utility function, which will take under consideration the probabilities of each scenario, and the probability of vengeance. Once you do that, the problem becomes too cumbersome.

Good thought process though for someone never taken economics! Maybe you're a natural.. :)

#15: Jan - I indeed assumed the strength of a competitor is fixed throughout the game, and not challenge specific. Since the players are given (identical) study materials before each challenge, it can be argued that the competency that matters most is the learning ability, which is of course not challenge dependant. We can argue about it a bit, but at the bottom line it's just another assumption that has to be made if you want to formulate this problem.

#16: Mark – Your take is interesting. I'm not sure that throwing the initial challenges is necessarily the best way to go about it. I can see its advantages (and there are quite a few), and David Olsen sides with you, but since there's a chance you'll end up in an elimination round facing a stronger couple, I can't say full heartedly that it's best practice.

Thank you for your thought provoking comment.

#17: Jonathan – As many commented here, including one of the show's winners, revenge plays a big role in the game. And yes, I limited the problem to two time periods only (the show in which you have to choose who to send to elimination and the show after). Later I waived this assumption. After three years of studying economics, I believe this construction of a problem was the best given the complexity. Thank you.

#20 & #24: Kevin H. – Your take in comment #24 is very interesting, to include a subjective 'threat rating' from the POV of each couple. I feel it should be included in the larger problem of multiple time periods. This way, the threat any given couple poses to you is affected by events during the game play, e.g. – how long has it been since you sent a couple to the elimination room, had there been other couples to send that given couple to elimination in another episode, etc. (share with me your thought about this). Great insight! Thank you!

#28 : 'Assumo' : Thank you for your insight. You mentioned several factors affecting the game and the problem I presented. However, it is pretty much impossible to re-write the problem to include such variables. Also, please view my response to Jan (#15).

#35: Adina – Thank you for mentioning the rules regarding the final. It makes the problem a whole lot more difficult, and on the other hand it supports the importance I give for vengeance in the game.

I will address winner David Olsen's interesting comment and Peter Norvig's amazing analysis a bit later. Obviously I have a lot to say to both.

Ben K

How did a medium team end up winning both initial challenges? Doesn't that seem an unlikely set up?

Isn't it more likely that a strong team would be facing this senario trying to decide wether to take out other strong teams or average teams since weak teams should be left for last? I have only watched a little but it seems the chalenges are based on the percieved weekness of geek or beuties learning the opposites culture so is it better to be a generalist: sf buff, model(normals don't get on the show) or specialist chemist, Ms. South Dakota?

David Olsen

Peter (#36), I must tip my hat to you for doing the legwork to generate those results. If only I had that matrix a year ago.

I find your third observation to be the most interesting, however, for it matches my own intuition in doing the "field research," as it were. The game is random enough that strategy takes a backseat to merit, especially since only two (sometimes only one) team gets to make any interesting decisions during each round (from a game standpoint).

Nevertheless, it appears that the strong teams are favored to win. I think the data sample is far too small to see if those results are true, but I don't think it can hurt to look at it. In my opinion, these are the results of the American seasons (I'm not familiar enough with the international versions, but, if someone is, that would increase our data set, though probably by not enough):

Season 1: Strong team wins

Season 2: Strong team wins

Season 3: Medium/strong team wins

Season 4: I'm way too biased to make a call here.

Season 5: Strong team wins.

Hmm, it appears that reality actually seems to match laboratory results. What a pleasant surprise.

Brian

Why do you assume people would seek revenge? If your team is being so careful as to choose teams based on game-theory and probability of winning, who is to say the other teams won't do the same? It's pretty ridiculous and arrogant to just assume teams make choices based on revenge and make this an absolute criteria for your question.

Peter Norvig

I wrote a simulator to play out the game with different strategies. (You can see it at norvig.com/geek.html if URLs are allowed in comments; otherwise search for my name.) I conclude the following:

1. There are no dominant strategies. That is, there is no strategy that works best regardless of what the other players do.

2. There are a few dominated strategies: strategies that do not work as well as another strategy on any combination of other players strategies. For example, for the medium player (the focus of Nir's question), the strategy of choosing weak is dominated, and thus should not be played.

3. The strategies do not seem to matter all that much. There are three strong players; if they were infinitely stronger than the weak and medium players they would each win 33.3% of the time. Instead, they win 28 to 32%. The weak players are stuck in the 0 to 2% range, and the medium player varies from 3 to 10%.

4. The medium player can score 9 or 10% by playing any one of the eight strategies, as long as the strong players play randomly; for some combinations of what the weak players do, this can result in only 7 or 8%.

5. If the strong player does not play randomly, then the best medium can do is 3 to 5%.

6. Overall, it looks like medium should play randomly, or perhaps strong.

7. Curiously, strong's best plays are split among all six non-random strategies.

8. The weak players really need to hope that strong plays randomly (which strong should not), and then it looks like weak should play randomly. But if strong plays rationally (non-randomly in this case), then weak is doomed to a less than 1% chance of winning. To maximize this slim chance, weak should play randomly.

X

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...

Kevin H

@17, revenge is perfectly logical if you assume a team who picked you once will pick you again, which is a reasonable if not perfect assumption.

Judith

Unless this is the first elimination round, shouldn't you take into consideration who has sent you previously to elimination?

I'm not sure you can assume revenge even with 50% accuracy. If the players are rational, they may make similar calculations to yours, and choose on that basis, rather than the basis of revenge.

The show complicates the problem even further: The winning couple is determined, not on the basis of beating the last couple at trivia, but based on votes from all the eliminated teams. Thus, you have to consider how your choice of who to send face-off affects the opinions of the couples whom you spared.

jordan

I think more thought should be given to whether you desire to be one of the top three teams, or the bottom three teams after elimination.

If you are one of the top three teams, you will most likely be set to elimination round the next week if you eliminate one of the top teams, putting yourself into top three.

Choosing weak vs weak moves you to the bottom three, most likely giving you a pass till the following weak where you are again in the middle.

assumo

Before any calculation, one would have to develop a definition of "strength" for the game. This is not as easy as assigning a number or coming up with a neat stratification. Nobody knows what the next challenge is going to be. Furthermore, much of the game is based on social interaction, and half of the contestants are dumbfounded at the sight of their partners at the beginning, the other half being selected more for their looks than their ablitity to appreciate game theory. Therefore, learning and adapting would have to factor into a calculation of strength. The nature of the game does not allow for a sure test of this, so the challenges themselves make for the only reliable proof of strength. Under this assumption, wouldn't you be considered the strongest if you won both challenges in the first round? I would imagine that no matter who you picked to go to elimination, you would be a prime target for the next round.

X

Reduce and conquer:

If 3 teams are left... You have no choice - you must send the other two teams.

If 4 teams are left... Knowing the team that selects next week will not have a choice, send the two strongest teams - one of them will be eliminated and there will be no opportunity for spite for the other.

If 5 teams are left... Send the two strongest. Though the winner may have an opportunity for spite, it will have no incentive (see example above where 4 teams remain).

At some larger number of teams, revenge and cooperation will become factors - how many does the show start off with?

Bill Mill

You make it very public that you're going to pick two random teams, and you do just that. I don't believe that you can be accurate enough with your prediction of team strength in the future, so I argue that the best you can do is to minimize the revenge factor by picking randomly.

X

Calgary (#16) makes a good point - revenge is only feasible early in the game, when your reputation for seeking revenge might matter. Later in the game, you will have less incentive and fewer opportunities to punish teams that try to eliminate you.

Early in the game, you want to form a coalition, a tacit agreement between two or more teams not to send the other to elimination: two is stronger than one. The only way to enforce such a coalition is to also agree to send cheating coalition members (who try to eliminate fellow members) to the elimination round - this is the same concept as revenge. In later rounds, the ability to enforce the coalition vanishes and so the coalition disbands. I believe small coalitions are feasible when there are many rounds left.

Leland Witter

To X (#12) and others wondering why the revenge factor is so clear, note that it is very prevalent on this show due to the "Beauties". Now before I get slammed for that statement, what I really mean is the personalities of the Beauties that are selected for the show. The producers seem to select "drama queens" and self-centered individuals that are successful socially, but not intellectually.

So the revenge factor seems very prevalent among the women, but so much with the Geeks. Since the Beauties are better able to manipulate socially, they many times get the Geek to go along with their revenge "reasoning".

Kevin H

oh geez, I forgot that there were two immunity/choice challenges. that changes all my math above. wasted computation.

Hmm, in thinking about it, the best way to model revenge is probably a generalized 'threat rating' each team would judge (presumably) accurately the overall skill of each team, and then this value could be modified by in-game behavior so that the treat ratings kept by each team diverged over time. So for example, maybe if you sent someone to elimination your threat rating (from the pov of that team only) would go up 2, or maybe multiplied by 2, or 1.878 or whatever.

even better, you probably want the threat ratings to start even between all teams, and then have them change according to who wins a challenge, maybe winning both challenges or two in a row is greater than the sum of its parts, etc.

Alon Nir

Marijn Vervoorn (#47) - The idea of alliances is interesting indeed, and some already suggested doing so in the comments above. however, incorporating alliances in the game theory problem would DEFINITELY make things a lot more complicated. maybe it should be a subject of another problem which will solely investigate if and how to form alliances.

Calgary Grit

Strong vs. Strong...with the caveat that you pick strong teams that you think are less likely to exact revenge. Or teams that have already been sent to elimination because they would have other teams they might more want to exact revenge against.

Revenge isn't a given. If they're a strong team, odds are another team has tried to eliminate them before. They might not win. Or they might also go for their own self-interest rather than trying to go for revenge.

I suspect that it might make more sense to target the strong teams later in the game rather than earlier (later there's less time for revenge and they're more likely to already have more "enemy" teams/other factors contributing to their decisions).

BukaHobbit

Hmmm, interesting take on the issue...but shouldn't you be on the show?

Kevin H

I think we need to know how strongly are teams effected by revenge. Do they always pick the team that picked them, or is there only a slight increase in the % that they will pick you? That % would seem to strongly effect the optimal strategy. Sounds like a couple of late nights with the the Beauty and the Geek DVD to me. =)

@5 however, revenge by itself is not important, what's important is your % of facing elemination next round or in the future. Therefore you have forgotten to factor in the chance that the team will be able to act on their revenge.

lets assume revenge and revenge decay is 100%, that if you pick them and they get to pick, they pick you, but that only lasts for one round.

so you pick strong, other picks strong. 50% for incuring revenge, then there are s s y w w w left, so they would get to act on revenge 4/13 (4 + 4 + 2 + 1 + 1 +1) or ~30% of the time. if you win, you don't face elimination, that would happen 2/13 or ~15%. if someone else wins, we'll just assume they pick at random, so 30% you face 100% elimination challenge, 15% you'll face 0% and 55% of the time you face 20% chance elimination challenge, or a total of ~41% for facing elimination if the team you pick wins. Then if the team you picks looses, its much simpler, and you only get a random choice from the next winner. or 20%. So the final % of facing elimination next time if you pick strong and so does the other is 30%

following the same logic, you pick strong, they pick weak is 36% of facing elimination. This works out if a team is really 4x as strong, the chances are 80/20

you pick weak, vs strong: 19%

you pick weak, vs weak: 20%

so, the conclusion is still the same, even more so that you should pick a weak team to face elimination if your only concern is not facing elimination in the very next round.

but for example, if revenge is only 50% instead of 100%, s vs s = 21% chance of facing elmination in the next round, so that changes things around considerably.

I don't want to work out the math for mult-round, but it certainly seems feasible.