This is a guest post by Michael A. Lewis (PDF), a friend of mine who is a professor at the Silberman School of Social Work at Hunter College.

One of the things I found most interesting and surprising about the movie The Hunger Games (HG) is how mathematical it is.

The basic premise of the story is that there is a society in what used to be North America made up of a centralized capital and 12 outer districts. Seventy-four years ago the districts staged an uprising against the capital which was violently put down. As punishment for this transgression, every year each of the districts must send one boy and one girl (it’s not clear what would happen to transgendered persons in this world) to take part in the Hunger Games. This is a televised "contest" in which 24 children between the ages of 12 and 18 (inclusive) fight to the death until there is a sole survivor who is declared the winner. The story centers on Katniss, a smart, brave, and compassionate participant in the Hunger Games who is from District 12. HG is a gripping and suspenseful tale that is masterful at depicting a decadent and oppressive regime in contrast to a desperate, hopeless, and oppressed people.

Let’s focus on two mathematical aspects of the movie: the lottery probabilities, and the game theory of sleeping.

The way districts choose which boy and girl to send to the capital for the Hunger Games is by lot. The movie doesn’t provide a lot of details about how the lottery works. There are lines from a couple of characters which make it clear that the more times one’s name appears in the lottery the more likely one is to be chosen for the game. Luckily the details of the lottery can be found in Suzanne Collins’ book The Hunger Games, on which the movie is based.

Once a child in a district turns 12 years old his or her name goes into the drawing for the Hunger Games. If the kid’s name is drawn, her or his name doesn’t appear in any future drawings either because the kid ends up dying in the Hunger Game or wins the game. That is, the names of dead kids and winners do not re-appear in future drawings. Ignoring, for the moment, certain complications, each previous year that a kid’s name is not drawn her or his name appears one more time the next year. A 12-year-old whose name is not drawn will have her name appear two times when she's 13 (given that her name was not drawn at age 12), three times when she's 14 (given that her name was not drawn at age 13), etc. In other words, the equation that represents how the number of times kid’s names appear in the lottery changes over time is an arithmetic progression.

Suppose the parents in a given district gave birth to only 10 children, five boys and five girls, and that all of these kids were born at the same time. This would mean that they would all turn 12 at the same time and that all their names would go into the lottery at the same time. Since the boys' and girls' drawings are done separately, each boy and each girl would have a 1-in-5 or 20 percent chance of being chosen for the game. Now in any given year, one girl and one boy will be chosen for the game and either because of victory or death, their names won’t appear the next year. Thus, in the next year all the kids that are eligible for the drawing would be 13 years old and all of their names would appear in the drawing two times. There would now be 8 boys’ names in the pool for boys (2*4 = 8 names), 8 girls’ names in the pool for girls, and each boy and girl would have a 2-in-8 or 25 percent chance of being chosen for the game. That is, the number of times that each person’s name appears in the lottery will have increased and the chance of being chosen will have as well. It shouldn’t be too difficult to see that each boy and girl will have a 3-in-9 or 33 percent chance of being chosen when they are 14, a 4-in-8 or 50 percent chance when they are 15, and at age 16 each would have a 5-in-5 or 100 percent chance of being chosen for the game. The figure below shows how the chance of being chosen increases with age:

It shouldn’t be too difficult to tell from the graph that the chance of being chosen not only increases with time but does so at an increasing rate. This can also be shown through the use of difference quotients.

Now let’s consider some of the complications. The simple arithmetic progression discussed earlier is not a good model of how the number of times children’s names appear in the lottery would change as they aged. This is because HG makes it clear that there is another way for children’s names to appear more often in given drawings than merely getting older. The world of HG is one of near starvation for many of those residing in the districts. One way to get more food is for a family to volunteer to have a child’s name entered into the lottery a higher number of times. That is, a family with a 13-year-old whose name would ordinarily appear in the drawing twice could enter the child’s name more than two times in return for a higher portion of food. Also, presumably, parents in the HG world would not all have their kids at the same time and then have no more children. They’d continue to have kids at different times. So some kids would be aging out of the Hunger Game drawings and others would be aging in. The math gets more complicated as these contingencies do.

Alas, changes in the numbers of times names appear in drawings and in probabilities of being selected couldn’t really be figured out unless a lot of detail were known about demographics and the "choices" people made regarding risking a greater chance of peril for their children in the Hunger Games in return for eating a little better. But until we can combine demographics with the mathematics of decision theory — how people make decisions in the face of uncertainty — we won’t be able to know how families decide whether to enter their children’s names more times in exchange for food.

Now to game theory. Game theory is a branch of mathematics that represents interdependent decision making. By "interdependent decision" making I mean situations (arguably most, if not all, of those we face in life) where the outcomes of one’s decision depends on decisions made by others. One of the most frequently discussed models in game theory is the well-known Prisoner’s Dilemma (PD).

Here is the story of the PD. Two people, who are suspected to have been involved in a serious crime, are being interrogated separately by the police. The police inform each man that they know that they were involved in a serious crime but don’t have enough evidence to convict them. They also inform the suspects that they know that they were involved in a more minor crime and that they could easily convict them of this one. They offer each suspect the following deal. If one of them confesses but the other one doesn’t, the one who confessed will be freed and the one who did not will do 15 years in prison. If neither of them confesses, they will be easily convicted of the minor crime and both will do 1 year in prison. If both of them confess to the more serious crime, they will each do 5 years, instead of the full 15 years, as a reward for their cooperation. Assuming the suspects would rather do less time in prison than more time, they would both be better off if they both kept quiet. But some simple tools of game theory can show that each prisoner is under compelling pressure to confess.

In both the movie and the book we see a coalition of some of the players develop where they attack other players as a group. As I considered this, and being aware of game theory, I wondered how such an alliance could be stable, given the powerful incentive all members of the coalition have to kill each other in order to better position themselves to win the game. In fact, I wondered how members of the coalition would even get any sleep, especially given that they slept near each other. This may seem like a strange question but the PD game can show that it isn’t so strange.

Consider the following table:

Don’t Sleep all Sleep all Don’t Sleep 1 Tired, TiredKill, KilledSleep 1 Killed, KillRested, RestedHere subscript "1" refers to any member of the coalition and subscript "all" refers to all other members of the coalition. Let’s consider matters from any given member’s perspective (the subscript 1 player). What if the other players don’t sleep? If you don’t either, then you will be tired and, perhaps, more vulnerable to better-rested contestants. But if you sleep while others are awake, any one of them can kill you in your sleep. Presumably, it’s better to be tired than dead so you are under tremendous pressure to stay awake.

If all participants choose not to sleep and make this choice evening after evening, then all of them will end up being tired and more vulnerable to better-rested contestants. So why do members of Cato’s coalition in HG get any sleep at all?

There is an extensive literature in mathematics and economics addressing the issue of why what would seem to be the most compelling outcomes in PD like situations don’t necessarily occur. Relating this to HG, this literature addresses the question of why coalition members would actually sleep when it would seem that there is a powerful incentive for them not to. Of course, the answer to whether to sleep or not isn't easy, but it's pretty exciting that there actually is a way to mathematically address this question.

As I write these lines HG is a blockbuster movie and top-selling book. This is due, I suspect, to the fact that it is a very interesting political thriller. But it is also a fertile source of mathematical inspiration.

If you're interested in more mathematics in movies, check out this article.

Top image: Moyan Brenn/Flickr/CC-licensed