### The Good, the Bad and the Weird: Duels, Truels and Utility Functions

In the excellent (and highly recommended) book "Fifty Challenging Problems in Probability with Solution", Frederick Mosteller poses "The Three-Cornered Duel":
A, B and C are to fight a three-cornered pistol duel. All know that A's chance of hitting his target is 0.3, C's is 0.5, and B never misses. They are to fire at their choice of target in succession in the order A, B, C, cyclically (but a hit man loses further turns and is no longer shot at) until only one man is left unit. What should A's strategy be?
This is problem 20 in Mosteller's book, and it also appears (with an almost identical solution) in Larsen & Marx "An Introduction to Probability and Its Applications".

Mosteller's solution:
A is naturally not feeling cheery about this enterprise. Having the first shot he sees that, if he hits C, B will then surely hit him, and so he is not going to shoot at C. If he shoots at B and misses him, then B clearly shoots the more dangerous C first, and A gets one shot at B with probability 0.3 of succeeding. If he misses this time, the less said the better. On the other hand, suppose A hits B.  Then C and A shoot alternately until one hits. A's chance of winning is $$(.5)(.3) + (.5)^2(.7)(.3) + (.5)^3(.7)^2(.3) + \ldots$$ . Each term corresponds to a sequence of misses by both C and A ending with a final hit by A. Summing the geometric series we get ... $$3/13 < 3/10$$. Thus hitting B and finishing off with C has less probability of winning for A than just missing the first shot. So A fires his first shot into the ground and then tries to hit B with his next shot. C is out of luck.
Is this right? What if B were to follow A's example and fire into the ground? What if all three were to keep firing into the ground? That this type of an outcome isn't unreasonable for certain sets of shot accuracy probabilities can be illustrated by considering the case where A's accuracy is 0.98, B's accuracy is 1.0 and C's accuracy is 0.99. Mosteller's argument is equally applicable in this case, but if B shoots C after A deliberately misses he'll be shot by A with probability 0.98. Is that reasonable?

Under the assumption that deliberately missing is allowed, there are $$2^3-1=7$$ possible outcomes - each participant can be shot or not shot, and there must be at least one participant not shot. The lack of clarity for what the ideal strategies are for A, B and C in the general case arises from the utility of 2- or 3-way ties to each of the participants being undefined.

In the next article I'll analyze 2-way duels where deliberate missing is allowed by using such fully-defined utility functions. These results will be used in a third article on 3-way duels (truels); in particular, I'll re-examine Mosteller's solution.

### A Bayes' Solution to Monty Hall

For any problem involving conditional probabilities one of your greatest allies is Bayes' Theorem. Bayes' Theorem says that for two events A and B, the probability of A given B is related to the probability of B given A in a specific way.

Standard notation:

probability of A given B is written $$\Pr(A \mid B)$$
probability of B is written $$\Pr(B)$$

Bayes' Theorem:

Using the notation above, Bayes' Theorem can be written: $\Pr(A \mid B) = \frac{\Pr(B \mid A)\times \Pr(A)}{\Pr(B)}$Let's apply Bayes' Theorem to the Monty Hall problem. If you recall, we're told that behind three doors there are two goats and one car, all randomly placed. We initially choose a door, and then Monty, who knows what's behind the doors, always shows us a goat behind one of the remaining doors. He can always do this as there are two goats; if we chose the car initially, Monty picks one of the two doors with a goat behind it at random.

Assume we pick Door 1 and then Monty sho…

### Mixed Models in R - Bigger, Faster, Stronger

When you start doing more advanced sports analytics you'll eventually starting working with what are known as hierarchical, nested or mixed effects models. These are models that contain both fixed and random effects. There are multiple ways of defining fixed vs random random effects, but one way I find particularly useful is that random effects are being "predicted" rather than "estimated", and this in turn involves some "shrinkage" towards the mean.

Here's some R code for NCAA ice hockey power rankings using a nested Poisson model (which can be found in my hockey GitHub repository):
model <- gs ~ year+field+d_div+o_div+game_length+(1|offense)+(1|defense)+(1|game_id) fit <- glmer(model, data=g, verbose=TRUE, family=poisson(link=log) ) The fixed effects are year, field (home/away/neutral), d_div (NCAA division of the defense), o_div (NCAA division of the offense) and game_length (number of overtime periods); off…

### Notes on Setting up a Titan V under Ubuntu 17.04

I recently purchased a Titan V GPU to use for machine and deep learning, and in the process of installing the latest Nvidia driver's hosed my Ubuntu 16.04 install. I was overdue for a fresh install of Linux, anyway, so I decided to upgrade some of my drives at the same time. Here are some of my notes for the process I went through to get the Titan V working perfectly with TensorFlow 1.5 under Ubuntu 17.04.

Old install:
Ubuntu 16.04
EVGA GeForce GTX Titan SuperClocked 6GB
2TB Seagate NAS HDD