Who wins The Hunger Games?

I’ve just been watching The Hunger Games and noticed that Katniss managed to win by killing a small proportion of the participants and I wondered whether this is a typical outcome.  Given the “Careers” who are trained for the event I had imagined something more like Kill Bill where the victor would have had to fight through many people to win.  Let’s consider a statistical model for The Hunger Games to consider what outcomes are typical.

We model participants as meeting randomly for one-on-one fights to the death.  This isn’t a perfect model but hopefully it’s a useful model.  The random meeting assumption seems OK as it is hard for an individual to find a particular individual in the Hunger Games arena.  We imagine that group activity can be broken down to steps of one person administering the coup-de-grace to one victim.   This makes the Hunger Games look like a random binary tree with 24 leaf nodes.

We consider two ways of determining who wins a one-on-one fight to the death: “random” and “deterministic”.  In “random” we flip a coin to determine the victor – this randomness seems realistic and could include the risk of death by infection or any of the other uncertainties that are in the arena.  “Deterministic” is the opposite extreme and we know in advance a strength order for individuals which determines the winner of any fight; this model may more closely model the “Careers” approach.

I couldn’t find a closed form solution for the outcome of these models so I simulated it in R.    I show below the probability mass function and the cumulative distribution function for the number of fights that the victor took part in (and won).

Probability mass function

Cumulative distribution

You can see that in the “random” case the modal number of fights is pretty low at 3.  95% of outcomes lie in the range [1,6].  The “deterministic” case typically gives more fights with a model number of fights of 5 and 95% of outcomes lie in the range [3,9].  In terms of a random binary tree these two models can be thought as the number of branches from the root to take at random until you reach a leaf node and the depth of a random node respectively.

Anyway Suzanne Collins seems to have got things looking statistically typical. You could count Katniss’s encounters in various ways but to me they all lie in the 95% confidence intervals of both models.  Did Suzanne Collins have a statistician on board when writing the books?

Q: Can you advise on how to solve this problem analytically rather than by simulation?