A poisson process of death
Around the end of April every year when the weather begins to reach tender 40 degree highs in Massachusetts, Harvard undergraduates emerge from their winter-time hibernations — noses come out of books, Harvard Hill-flint sweaters are tossed aside, and salmon-colored shorts are donned by those who are the kind to don salmon-colored shorts.
This period marks an increasing proportion of exposed skin to the unexposed. Collectively as an undergraduate body thousands of square-meters of kill-space appears in time for the well-natured but friendship-terminating games of assassins that happen to start about now. Kirkland’s twisted world has zombies flocking after humans, Quincy house has begun blasting cannon-sounds and names of the dead into their court-yard in a true Hunger-Games-styled ritual, and one of the last two senior-class assassins refuses to eat any meals in Harvard dining halls.
Mayhem ensues. Everyone’s on high-alert for something and, for once, it’s not midterms (of course, those who are always on high-alert for midterms wouldn’t even sign up to play).
A few real quotes from kills and deaths in Eliot house about
Trust no one. Not your neighbors, not your friends, not yourself. Don’t even trust your parents anymore. My only solace is that I wasn’t the person killed 2 minutes after Assassins started
Jeff was selling a TV. My friend was “interested” in the TV. TVs for my friend = 0. Shots to Jeff’s chest with a water gun = 4.
and C) Sacrifice
I’ve been carrying water balloons around in my bag for two days now. I made the trek over to the SOCH to set up my booth and guess who was sitting across from me … Two water balloons to the back. Perfect shot. Elimination.
Assassins is a crazy time and yours truly happened to be the game-master of Eliot’s own game in both 2014 and 2015. Seeing as to how I know some stuff about statistics and computer science and now had complete access to the data of the game from the first year I ran it (2014). I wondered what we could say about the kills? Could we predict when kills would happen? Or maybe who’s to make the next kill?
Idk, let’s give it a shot.
*For those who haven’t taken a probability class, please feel free to just check out the conclusion of this article and look at the pretty pictures:
Model 1: Naive Poisson
We can try to model the wait-times between every kill as a simple exponential distribution with a single parameter, lambda. Boom:
W_i ~ Expo(L)
Where W_i is the wait time between the (i-1)th and the (i)th person dying. Trying to fit the data to this model results in the following graph:
Needless to say, this model kind of sucks. If this can pass as a satisfactory exponential distribution, then I should be able to pass as a first-round draft-pick for Miami in 2015. Both are dreams not too far out of reach, but clearly with some gapping flaw. The model looks pretty much uniform after the third bin, and my free-throw percentage isn’t good enough for the NBA. Everything else seems just about fine.
Okay, but here’s the truth about the model, there are two issues:
First, people ain’t dying at night:
What happened to the thieves in the night? Oh right, that’s actually creepy and assassinations while the victim is in their own bed are strictly forbidden by the Eliot Assassin Rules.
Second, less people alive = less people dying:
As time went on and more people died, the kills were slowing down. Whoops…that makes a lot of sense.
Think about it this way if there were 10,000 people playing a game of assassins, you’d expect kills to happen every few minutes, or even every few seconds. If there are only 2 people left, they’d take much longer to kill the other.
Which leads us to…
Model 2: Adjusted Poisson
After counting the hours from 1am-10am as 1 hour (ie time-contracting those hours deep in the night as 1/9 of an hour each), we get a new model statement.
W_i ~ Expo(L*(N-i))
Where W_i is the wait time between each kill, and N is the number of players at the start of the game. This model deflates the parameter of the exponential as the game progresses, which makes sense. Another way to interpret this model is that the “L” now represents the average time that someone stays alive in the game.
Fitting our new model to the data:
Woooooo!!!!. That looks great! Because each day in our new model is now only 16 hours long, this also gives us the easily interpretable solution that each person on average lives
77.936 games hours = 77.936 / (16 game hours / real day) = 4.87 real days
That’s pretty cool. Now let’s go full-Bayesian:
If we give the L an easily dominated prior-distribution of
We get the following posterior of the lambda parameter
Here’s a graph:
And we can easily calculate known posterior-predictive distributions from it! All-in-all this model fits quite well on the 2014 data and gives us a 95% posterior interval of the mean being between 60.15 and 105.00 game hours.
Ultimately, I took the posterior distribution generated from the 2014 data as a prior for the game this year (2015). The predictions worked quite well and, on my website, featured a section predicting when the next kill would be:
Here’s another example of the predictiveness of the model. This is a chart of the expected number of people left in the assassins game (based on the 2014 data) and the actual number of people left in 2015.
Shoot. That’s quite good. This helped me predict how many people would be left by the time reading period was going to end and when we needed to orchestrate a free-for-all shootout to determine the winner. Yeeee.
In the end, I hope this model can provide calm to a few minds playing the game and to intrigue a few others studying it.
This world is a poisson process of death — an inevitable step-function to asphyxiation — and, in the end, only one person can stand victorious. For anyone still playing, remember:
May the odds be ever in your favor.
PS: thanks for reading my post! If you’re interested in running an assassins game or would like the R-code to any of this data or to the website itself, it’s all here. Also, if interested in a proposed solution in which we model each player specifically in both their effectiveness in killing and the stubbornness of their lives, just hit me up! More data-blog stuff to come soon!