Episode 409 - "Graphic

Charlie works to find an extremely rare comic book that was stolen during a deadly robbery at a comic book convention.

Mathematical Topics	Activity Concepts	Appropriate Course
Game Theory	Prisoner's Dilemma activity and references to other NUMB3RS based game theory	Algebra 1, Algebra 2, Precalculus
Fractal Number	Chaos game	All Courses
Jump Bidding and Auction Theory	This topic may be a suitable basis for a class discussion	All Courses

Game Theory

In the beginning of the episode, Charlie says to his class "People wonder how game theory can be applied to casual relationships. With friends, what’s the strategic goal? There’s no commodity being sought, right? But, actually even the most causal relationship is built around a very important commodity: information. When a friend shares information, they place a value on our intention not to use it against them. And we are expected to share our information with them. We call this symmetrical information, and it’s the foundation of partnerships between countries, corporations, lovers, and…friends."

The following NUMB3RS episodes and activities relate to game theory:

• Tic-Tac-Toe from episode #302 "Spree, part II" students explore variations on the game of tic-tac-toe, looking for winning strategies. Winning strategies are a set of moves that guarantee a win, regardless of what moves the opponent makes.
• The Escape Game from episode #201 "Assassin" students explore an example from behavioral game theory where the motives of two players are contradictory and the behavior of one player affects the behavior of the other. The extensions to this activity feature the Prisoner's Dilemma and the well-known game Rock, Paper, Scissors. The concept of a zero-sum game is introduced.
• To C or Not to C from episode #321 "The Art of Reckoning" also features the Prisoner's Dilemma. In this activity, students test different strategies for the Prisoner's Dilemma.
• The NUMB3RS blog for episode #401 "Trust Metric" also focused on game theory. The blog included a discussion of Nash Equilibria and gave an example of a Trust Metric. This example illustrates how game theory can be applied to casual relationships as Charlie mentions in the episode.

Prisoner's Dilemma activity: Two students, Tori and Malcolm, are charged with violating a school's honor code. They are questioned separately by the principal and have no contact with each other. The principal has evidence linking either just one or both students to the violation. The principal tells each student that if they both deny doing anything wrong they will each be suspended for 3 days. If one of them confesses and testifies against the other student, then the student who confesses will not be suspended and the other student will be suspended for 10 days. But if both confess to the violation, they will each be suspended for 6 days.

a) If you were Tori in this situation and you were innocent of the charge, what would you do?
b) If you were Malcolm in this situation and were guilty of the charge, what would you do?
c) Do you think the principal acted ethically?

Fractal Number

A high-value comic was stolen, and then found hours later on the black-market- along with several other copies of the same comic. Obviously, all but one is a forgery, but which one? Charlie offers to determine a fractal number estimate to determine the real from the fake. As Charlie says,

"Fractal Number Estimate. It’s based on Mandelbrot’s use of fractal dimension to measure the jaggedness of a coastline. It’s been used to detect forged handwriting, but we’re applying it because hand-drawn art can be evaluated with the same process. We use the fractal dimension analysis to evaluate the "wrinkliness" of the lines. An authentic piece of handwriting or drawing will have a smoother ink edge than one that is forged. Movements of the hand are fast and fluid, minimizing contact between paper and pen. The faster the hand movement, the smoother the edges of the ink. When someone copies the signature they imitate the movements. It’s not natural for the forger, thus slower. The slower the pen, the longer the contact with the paper, allowing more ink to be absorbed. Creating a more irregular or wrinkly edge. Fractal Dimension allows us to compare the wrinkliness and detect which is the fake."

Fractal classroom activity: Using a permanent marker, place three dots on an overhead transparency (one for each group of students in your class) at the vertices of a large triangle. Make sure that the dots are in exactly the same places on each of the transparencies. Give each group one of the transparencies, a die, a ruler, a dry erase marker (ideally with a small tip), and 6 punched paper holes from a hole puncher.

Instructions for each group: Label each three dots on the transparency with two of the numbers from 1 through 6. For example: The top dot could be {1, 2}, the left one {3, 4}, the right one {5, 6}. Drop a punched paper hole onto the transparency. Roll the die. Place another punched paper hole half way between the first one and the dot corresponding to the number on the die. Repeat this process for the remaining four punched paper holes – each time moving half way from the previous paper hole to the vertex corresponding to the next roll of the die. Now start marking a dot on the transparency using the dry erase marker for each outcome. Continue this process until each group has 100 dots.

Ask the groups to discuss this somewhat chaotic process and its outcome. Do they see any patterns on their transparency? Then collect all of the transparencies. Align them so that the original three dots are in the same position on each transparency. Paperclip the packet together and project the image on a screen using an overhead projector. The result of this process, called the Chaos Game, is a Sierpinski triangle. A Sierpinski triangle is a fractal.

Fractal's are self similar figures. Cynthia Lanius's website gives an excellent explanation of fractal dimension. In summary if you double the length of a one unit line segment, you get two copies of the original segment. If you double the length of a one unit square, you get four copies of the original square. If you double the length of a one unit cube, you get eight copies of the original cube.

Figure	Dimension of the Object	# of Copies
Line Segment	1	2=21
Square	2	4=22
Cube	3	8=23

It appears as if the dimension is the same as the power of two which gives the number of copies. This result can be expanded to get a value for the dimension of fractals as shown below.

The Sierpinski triangle is shown below. If you double the length of the side you get three copies of the original. We can determine its fractal dimension by determining the power of two which yields a value of 3:

The above TI-83/84 screen shot comes from a Key Curriculum Chaos Game activity.

The above TI-Nspire screen shots were made by John Hanna for this blog.

In the episode, Charlie uses fractals to determine the authenticity of an original comic book. This technique has been used by others to identify counterfeits. The physicist Richard Taylor has suggested that newly discovered Jackson Pollack paintings may be fake because they display a different fractal character from his previous works. An article about his work is titled "Splattery will get you somewhere: fractal forgery."

The NCTM Illuminations site features a fractal tool which allows students to explore the iterative process in creating fractals.

Jump Bidding and Auction Theory

The "fake" copies of the stolen comic are being auctioned off for charity – but the FBI suspects that one of those fakes is actually genuine. The auction is a way for the thief to "launder" the real comic. Charlie suspects they can determine which copy is the real deal by monitoring the bidding strategies used at the auction. As Charlie says,

"In Auction Theory we talk about equilibria or symmetry. All bidders should have the same exact information. But in this case, the criminal knows something the other bidders don’t know. This is asymmetrical information. And this can create asymmetrical bidding. This asymmetry will almost certainly lead to something called "jump bidding." That’s when a person enters a bid far larger than what’s necessary to be the current winning bidder. Think of it like a bike race. Typically, there’s no advantage to jumping out ahead of the pack. The racers tend to stay clustered together and switch off the leads. But there is a strategy of taking the lead early to exhaust the competition. Jump bidding is like quickening the speed of the race. It encourages the early exit of bidders, discourages late entry of others. So during the auction, when someone makes a big bid, that person should be the criminal."

As Charlie explains, the purpose of jump bidding is to try to frighten off competing bidders. For example, if an item is being auctioned and the bid is at $70 going up in increments of $10 at a time, then the jump bidder may attempt to frighten off competing bidders by offering a much higher bid, of say $140, which can frighten off the other bidders and grind the auction to a halt. It can of course have the opposite effect too and encourage other bidders to jump in quickly, almost with a sense of panic, assuming that the item must be worth far in excess of this amount.

Auction theory is an applied branch of game theory which deals with how people act in auctions and researches their theoretical properties.

At the end of the episode, Charlie’s father tells a magazine writer "Charlie thinks mathematics is beautiful and he wants everyone to love it the way he does."

Cheers,

John F. Mahoney, Benjamin Banneker Academic HS, Washington, DC