Game theory in Numb3rs: hide and seek

I occasionally enjoy watching the TV show Numb3rs. The show is about a team of FBI agents who solve crimes with the help of a fictional mathematics professor, Charlie Eppes. During a crime investigation, Charlie comes on the scene to analyze all of the numbers and look for patterns that can help the FBI agents track down the criminal.

Charlie seems to be a master at mathematics, physics, economics, game theory, computer science, and any related technical subjects. The setup is a bit of a stretch, and Charlie can be annoyingly smug at times, but the show is amusing nonetheless.

I’ve seen only a handful of episodes, but I’ve already found the show discusses many game theory topics. This makes sense given the amount of strategic thought involved in crime-fighting. Today I want to discuss a part of episode 205 “Assassin.” There is one scene where Charlie discusses the game hide and seek and specifically mentions game theory.

Here is the clip on Youtube:

Youtube video Numb3rs hide and seek

My transcript of the scene

Don: We’re thinking Gabriel’s assassination is going to look like an accident.

Charlie: Awesome. That gives us a better chance of winning this game.

Megan: What game?

Charlie: Hide and seek.

Don: What are you talking about—like the kids’ version?

Charlie: A mathematical approach to it, yes. See, the assassin must hide in order to accomplish his goal. We must seek and find the assassin before he achieves that goal.

Megan: Ah, behavioral game theory, yeah. We studied this at Quantico.

Charlie: I doubt you studied it the way that Rubinstein, Tversky, and Heller studied two person constant sum hide and seek with unique mixed strategy equilibria.

Megan: No, not quite that way.

Don: Just bear with him.

Charlie: The concept is simple, it’s almost instinctive. But, when an assassin has many opportunities to hit his target, it gets complex. Imagine the game Battleship. It’s essentially a hide and seek game. But let’s say the ships can be moved. One player tries to get his ship on several squares of opportunity before all of his ships are found. Now suppose that the player hiding has limited options. He can only move one ship at a time, one square at a time. That will give the player seeking him the ability to calculate his likely moves.

Don: So you’re saying if we know Condor’s opportunities we can predict the attacks?

Charlie: Not predict, as much as calculate the probabilities. If Condor has to make this look like an accident, then he can’t use say, a gun or a bomb. He has to avoid witnesses. He’s limited.

Don: He’s limited by locations, methods.

Charlie: That’s right.

Don: How long will it take you to come up with some probabilities?

Charlie: Soon. Of course the accuracy will improve the more I know about Condor’s abilities. So, I think you should talk to the man you arrested again.

Explanation of terms

I love how Numb3rs is entertaining and mentions math, but often it leaves something to be desired in actually explaining concepts. So I want to explain a few of the terms that Charlie mentions for clarity:

“winning the game”
Charlie explains that he is going to use game theory. The name is somewhat of a misnomer. The name game theory has historical origins in the groundbreaking 1944 text “Theory of Games” by polymath John von Neumann and economist Oskar Morgenstern. This book explains how certain games can be put to math, and how strategy can be analyzed.

Modern game theory is is not really about games, but rather about strategic behavior. This is apparent because game theory is most applied to situations in economics, politics, and biology. It is often convenient to think about the situation in terms of a game like football or hide and seek, but this is where the analogy should end. Anything that involves competition or division of resources can also be thought of as a game, and can likely be analyzed using game theory.

“behavioral game theory”
Behavioral economics is a relatively new field of economics. It distinguishes itself from classical economics by modifying the assumption of rationality and further analyzing human behavior. Several examples are discussed in Dan Ariely’s popular book Predictably Irrational.

One of the biggest names in behavioral game theory is Colin F. Camerer at Caltech. In researching this article, I came across one of his papers which describes a bit more on the subject of behavioral game theory (pdf) .

There appear to be two main refinements in behavioral game theory over classical. The first is that rationality is not infinite but bounded. This means we can only think ahead for finitely many terms, levels, or steps. This would change our predictions on situations that mimic p-beauty prizes, a topic I earlier described in the article How my game theory professor gambled $250 teaching a lesson

The second refinement concerns how we learn in games. Learning is a critical component in repeated games. People are expected to learn quickly in classical economics, and the old adage comes to mind, “Fool me once, shame on you; fool me twice, shame on me.” That assumption is relaxed and investigated in behavioral game theory which uses experiments to better model human interaction and thinking.

“constant sum game”
A constant sum game is one in which all of the players’ payoffs add up to a fixed number. For instance, let’s say an experiment involves a game between two subjects. Suppose the winner gets $5 and the loser is compensated $1 for his time. In such a game, the sum of the payoffs will always be $6–hence this is a constant sum game.

The important feature in constant sum games is that the gains of one player come at the expense of another. The winner of the experimental game in a sense is taking away money from the loser.

A special type of constant sum game is one in which the payoffs add up to zero, and they are called zero-sum games. You can think about chess, football, tennis, or most sports as zero-sum games. As you would expect, zero-sum games are often highly competitive and players can get ruthless, bending rules to win if they must.

“mixed strategy equilibria”
Equilibrium is a concept that seems to come naturally for chemistry or physics. But what is an equilibrium of a strategic situation?

This is a somewhat controversial question, but most will agree on one standard as a baseline of the Nash equilibrium. The Nash equilibrium is a situation where no player can deviate profitably. Often the equilibrium is not a simple strategy, but rather a probabilistic one. A standard example is the game rock-paper-scissors where the best strategy is to play each of the three choices randomly.

Similarly, in hide and seek games, the best strategy usually involves mixing amongst choices. This is why Charlie cannot predict what the criminal will do, but rather he says he can calculate the probabilities that the criminal will make certain choices.

Rubinstein, Tversky, and Heller
Charlie’s dialogue is not specific about a paper, so I’m going to go out on a limb and make a guess. Ariel Rubinstein, Amos Tversky, and Dana Heller wrote a paper called “Naive Strategies in Competitive Games” (1996) regarding hide-and-seek games with experiments done on students at Stanford University (pdf full paper at Rubinstein’s website).

The authors set up the following experiment between two people. One person was assigned the role of hider and had to place a “treasure” in one of four locations–written on a piece of paper. The other person was the seeker and had to guess the hider’s choice. If the seeker guessed correctly, the seeker won $10 and the hider won nothing. If the seeker guessed incorrectly, then the seeker won nothing and the hider won $10.

Rubinstein, Tversky, and Heller were investigating the following question: how should this game play out?

Before getting to the results, let’s consider the problem under classical game theory.

Classical game theory
Classical game theory has a precise prediction. Both hider and seeker should be rational and maximize the chance of winning.

The hider can choose any of the four locations, as can the seeker. There is no reason that any location is special. The best strategy for the hider is to hide the treasure randomly among the locations, with a probability of 0.25 at each location. The seeker, in turn, can do no better than guessing each location randomly with probability of 0.25.

Under classical game theory, the hide and seek game is more or less like a game of guessing the flip of a coin.

But this is where behavioral game theory diverges. It supposes that such psychological questions may matter, and it uses experiments to quantify.

Behavioral game theory
There are three psychological considerations Rubinstein, Tversky, and Heller investigated. They are whether spatial positions of the locations mattered, whether the distinctiveness (focal or non-focal) of the locations mattered, and whether the assignment of the role of hider or seeker mattered.

Of these three factors, the experiments showed one overwhelming trend: both players avoided picking either of the two endpoints. As stated in the paper:

Perhaps the most striking feature of these data is the players’ tendency to avoid the endpoints. Overall, the two endpoints were selected by the hiders and seekers on 31% and 28% of the games, respectively, significantly less than the 50% expected under random choice (p<0.001).

Later the authors place the result in the perspective of other experiments:

The main finding of the present study is that both hiders and seekers tended to avoid the endpoints, thereby departing from the classical game theoretical solution…The tendency to avoid endpoints has also been observed in other contexts. When people are faced with the choice among 3, 4, or 5 identical items, people tend to avoid the endpoints and select the middle item. This bias has been observed in picking products from a supermarket shelf, selecting a bathroom stall, or picking an arbitrary symbol (Christenfeld, 1995).

On a personal note, I have noticed this bias in picking urinals (as famously demonstrated by the right and wrong answers to the urinal game).

Discussion questions

1. How might Charlie use this game theory result to predict the assassin’s move?

2. How might supermarkets stack high and low profit items?

3. Do you think this pattern might be useful for placing items on a restaurant menu?

4. Why do you think people avoid endpoints?

5. Hide and seek is a game of dis-coordination, that is, the players have opposing goals. How might the results be different in a game of coordination, such as avoiding traffic collisions or meeting friends in a bar on a crowded night

References and further reading

Rubinstein, Tversky, and Heller; “Naive Strategies in Competitive Games” (full paper)

Crawford, Vincent P., Iriberri, Nagore; “Fatal Attraction: Focality, Naivete, and Sophistication in Experimental ‘Hide-and-Seek’ Games” (paper)

(lecture slides to above paper)