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 Okay, here we go; an analysis of the cowboy's dilemma.
You're a cowboy, and get involved in a three way pistol duel with two other cowboys. You are a poor shot, with an accuracy of only 33%. The other two cowboys shoot with accuracies of 50% and 100%, respectively. The rules of the duel are one shot per cowboy per round. The shooting order is from worst shooter to best shooter, so you get to shoot first, the 50% guy goes second, and the 100% guy goes third, then repeat. If a cowboy is shot he's out for good, and his turn is skipped. Where or who should you shoot first?
Yesterday we noted there are three possibilities:
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Shoot at the 100% cowboy.
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Shoot at the 50% cowboy.
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Shoot anywhere else.
So we want to determine which of these possibilities has the best probability for our survival. To do this, we take each case and determine all the outcomes, along with the probability of each outcome. The overall probability for the case is the sum of all the possible outcomes. Here we go:
n Shoot at 100% cowboy
o 33%: hit him
§ 50% cowboy shoots at you (you’re the only one left)
· 50%: he hits you (he wins – 33*50)
· 50%: he misses, your turn
o You shoot at 50% cowboy
§ 33%: hit him (you win – 33*50*33)
§ 66%: you miss, his turn
· 50%: he hits you (he wins – 33*50*66*50)
· 50%: he misses, your turn
o 33%: hit him (you win – 33*50*66*50*33)
o 66% you miss, his turn
§ 50%: he hits you (he wins – 33*50*66*50*66*50)
§ 50%: he misses, your turn (33*50*66*50*66*50)
§ Probability of winning approx .06%, ignore further…
o 66%: you miss
§ 50% cowboy shoots at 100% cowboy (best chance for him)
· 50%: he hits him (100% cowboy dead)
o You shoot at 50% cowboy
§ 33%: hit him (you win – 66*50*33)
§ 66% you miss, his turn
· 50%: he hits you (he wins – 66*50*66*50)
· 50%: he misses, your turn
o 33%: hit him (you win – 66*50*66*50*33)
o 66% you miss, his turn
§ 50%: he hits you (he wins – 66*50*66*50*66*50)
§ 50%: he misses, your turn (66*50*66*50*66*50)
§ Probability of winning approx 1.2%, ignore further…
· 50%: he misses, 100% cowboy's turn
o 100% cowboy shoots at 50% cowboy (best chance for him)
§ 100%: kills him, your turn
§ You shoot at 100% cowboy
· 33%: hit him (you win – 66*50*33)
· 66%: you miss, his turn
o 100%: he hits you (he wins – 66*50*66)
The total probability of survival is the sum of each outcome where you win, highlighted above:
(33*50*33 + 33*50*66*50*33 + .06 + 66*50*33 + 66*50*66*50*33 + 1.2 + 66*50*33 = 34.4
If we shoot at the 100% cowboy, our probability of winning is 34%.
n Shoot at 50% cowboy
o 33%: hit him
§ 100% cowboy shoots at you (you’re the only one left)
· 100%: he hits you (he wins – 33)
o 66%: you miss
§ 50% cowboy shoots at 100% cowboy (best chance for him)
· 50%: he hits him (100% cowboy dead)
o You shoot at 50% cowboy
§ 33%: hit him (you win – 66*50*33)
§ 66% you miss, his turn
· 50%: he hits you (he wins – 66*50*66*50)
· 50%: he misses, your turn
o 33%: hit him (you win – 66*50*66*50*33)
o 66% you miss, his turn
§ 50%: he hits you (he wins – 66*50*66*50*66*50)
§ 50%: he misses, your turn (66*50*66*50*66*50)
§ Probability of winning approx 1.2%, ignore further…
· 50%: he misses, 100% shooter’s turn
o 100% cowboy shoots at 50% cowboy (best chance for him)
§ 100%: kills him, your turn
§ You shoot at 100% cowboy
· 33%: hit him (you win – 66*50*33)
· 66%: you miss, his turn
o 100%: he hits you (he wins – 66*50*66)
The total probability of survival is the sum of each outcome where you win, highlighted above:
(66*50*33 + 66*50*66*50*33 + 1.2 + 66*50*33 = 26.6
If we shoot at the 50% cowboy, our probability of winning is 27%.
n Shoot at the sky
o 50% shooter’s turn, he shoots at 100% shooter (best chance for him)
§ 50%: he hits him (100% shooter dead)
· You shoot at 50% shooter
o 33%: hit him (you win – 50*33)
o 66% you miss, his turn
§ 50%: he hits you (he wins – 50*66*50)
§ 50%: he misses, your turn
· 33%: hit him (you win – 50*66*50*33)
· 66% you miss, his turn
o 50%: he hits you (he wins – 50*66*50*66*50)
o 50%: he misses, your turn (50*66*50*66*50)
o Probability of winning approx 1.8%, ignore further…
§ 50%: he misses, 100% shooter’s turn
· 100% shooter shoots at 50% shooter (best chance for him)
o 100%: kills him, your turn
o You shoot at 100% shooter
§ 33%: hit him (you win – 50*33)
§ 66%: you miss, his turn
· 100%: he hits you (he wins – 50*66)
The total probability of survival is the sum of each outcome where you win, highlighted above:
50*33 + 50*66*50*33 + 1.8 + 50*33 = 40.2
If we shoot at the sky, our probability of winning is 40%.
There you have it. The best option is door #3, shoot at the sky. We actually have a 40% chance of winning if we do so, despite the fact that both of the other cowboys are better shots than we are.
This is a pretty interesting result. By not shooting at either of the "better" cowboys, we let them shoot at each other first, doing our dirty work for us. Then later we engage and shoot at whoever's left. This same effect is responsible for the complexity of reality TV shows. Many times a "weaker" competitor can survive by being unaggressive in the early going, enabling the "stronger" competitors to take each other out. Then later the weaker competitor can become more aggresive and [possibly] win. Pretty cool!
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