Russian Roulette Again — With 2 Bullets — Spin or Shoot?

Why an empty click makes pulling the trigger safer than spinning the cylinder again

Two bullets are adjacent to each other. The first player got an empty click and hands you the gun. Should you spin first and then shoot, or just shoot?

SPIN OR SHOOT?

I’ve done a version of this puzzle before, here, but no one noticed! So let’s revisit it since Matt Leung recently had a look here. But I want to model this with a Bayesian Network (I use Netica).

Let’s model a simpler version first, where the two bullets are just randomly placed in the chambers. We have 6 variables, one for each chamber, with two states to denote what happens. He either lives or dies.

The network with default uniform probabilities

Let live = 0 and die = 1. The simple constraint is that there are 2 bullets in total, randomly placed anywhere. Hence the plus function below.

There are two bullets randomly placed anywhere

Once compiled, we have the updated network:

Once compiled the network is updated

Let’s see what happens with this version after we activate the constraint. We want to know what the second player should do once the first player either lives or dies on his turn.

We activate the constraint by setting the constraint to True

First, we see that the initial probability of dying is about 33% ( = 1/3, since 2 of 6 chambers have a bullet). If the first player lives, then you have a 40% chance of dying if you don’t spin, so you should spin and reset the probability to 33%. But note also that if the first player dies, should you just shoot, not spin, since you have an 80% chance of surviving? Almost, but not quite. Only 1 bullet remains, so spinning again will give you about an 83% ( = 5/6) chance of surviving.

We reuse the same network, but we add another constraint to ensure that the two bullets are side by side. Since the two states (live and die) are indexed at 0 and 1, two bullets will be adjacent if and only if the product of two adjacent nodes is 1. We add this constraint below:

C1 C2 = 1 if and only both C1 and C2 equal 1, meaning each have a bullet*

The resulting network:

The network before activating the constraint

Let’s watch the network in action: we will activate the constraint and then choose some chamber (it does not matter which) and test both cases, when the first player either lives or dies, and we pay attention to the changing probability distribution of the next chamber which the second player must worry about.

Watch what happens as we activate the network

(We don’t know which chamber is aligned with the barrel and the hammer, but it does not matter).

What this shows is that if the first player lives, you should shoot since 75% > 67%%, but if the first player dies, you should spin again since 83% > 80%.

To summarize, let’s see the two models side by side, comparing 2 adjacent bullets vs 2 random bullets.

If the first player lives, just shoot and don’t spin if 2 adjacent bullets, but spin first then shoot if two random bullets

If the first player dies, do spin in the first version, and also spin in the 2 random bullets version (since just barely better chances)

ANOTHER APPROACH

Here is another approach, modeling this as a decision network.

Network before compiling equations

Two equations needed:

The utility node for the value of empty shot (1) vs a loaded one (0, since you die)

Let’s see the network in action:

We compile and activate the network and then optimize

When we optimize the network we get our optimal decision in both cases

Just as before, if the first shot is empty, then the second player should just continue and shoot, but if it was loaded, then he should first spin and then take his chances.

Note, by the way, that the second player should hope that the first player shoots himself in the head, since he then gets to spin and gets the best chances of surviving this lethal game!

SOURCES & REFERENCES

[embed]Should You Roll Or Not? — A Counterintuitive Maths Puzzle Two bullets are loaded into a gun’s round barrel consecutively.medium.com

[embed]SPIN OR SHOOT? Russian Roulette On Brilliant.org, I encountered a series of puzzles involving variations on Russian Roulette. The challenge is not only…medium.com

[embed]Norsys - Netica Application Netica is a powerful, easy-to-use, complete program for working with belief networks and influence diagrams. It has an…www.norsys.com