The Three Way Duel

If you were to have a gun battle with two other people, would you want to be the most skilled? As it turns out, this is a problem of strategy and probability. And here are the rules:

1 – Each of the three participants get to fire in turn as determined by flipping a coin at the beginning of the match.
2 – Each participant gets to choose who they shoot.
3 – Each participant gets only one shot per turn.

Now here is the tricky part. Let’s assume that the participants have the following skill levels:

Person A hits the target 100% of the time.
Person B hits the target 75% of the time.
Person C hits the target 25% of the time.

Assuming all three choose their best strategy for survival, who has the best chance of being the last person standing?

I won’t take up your time going through the math. The answer is that Person C has the best chance of survival, roughly twice the chance of survival of either of the other two. The math logic that leads to that conclusion is a bit tangled, but the underlying idea is that the person most likely not to be shot at is Person C.

So, what about my first question. Which person would you like to be at the start? The issue is now more than just a math problem. It’s your life. But wait! There’s more! Let’s make the situation even worse. Let’s allow you to choose your skill level in the following way. We will provide three guns, one that fires 100% of the time, one that fires 75% of the time, and one that fires 50% of the time. You then draw straws to see who gets to choose the first gun, who gets to chose the second gun, and who gets the gun that is left. BUT WAIT! THERE’S MORE! Your mother is watching you make your selection. What was a dry problem of mathematics is now a suspenseful novel!

NOTE: This type of duel is indeed called a truel. Various forms and rules have been studied. If you need to look into the matter, I would suggest that you start with the entry about it in Wikipedia.com. For myself, I would rather worry about what my mother would have told me to do.

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