logical deduction - The Mexican Standoff

There are three cowboys in a Mexican standoff against each other, named Juan, José, and Jorge.

• Juan is the straightest shooter in the West, and can hit his target 100% of the time.



• José, who has a cataract and can't see clearly, can hit his target 70% of the time.


• Jorge, who has no aim but is very clever, can hit his target only 30% of the time.

They proceed to shoot starting from Jorge and going to José and finally Juan, continuing in the same order until only one of them is left standing. Who has the highest chance of surviving, and what strategies do each of them use to maximize their chances?

Answer

• 
  

  Juan's strategy is simple: if he has to choose between opponents, then his shot will turn it into a two-man contest, and he will have to survive a shot before he can kill the second man. So of course he will shoot Jose first.

  
  


• 
  
  

  José's strategy, similarly, is to shoot at Juan. Leaving Juan alive and Jorge dead cannot possibly benefit him.

  
  


• 
  

  Jorge's strategy is the interesting case.

  
  

Aiming at José is clearly wrong, since hitting him would leave Juan with no other target.

So let's say Jorge aims at Juan and kills him. His chance of surviving is the sum of the chances that he will both survive a round and kill José in the next. Since both are 30% chances, he has a 30%×30% chance of winning on his next shot, a 30%×30%×30%×30% chance of winning on his shot after that, and so on. We can calculate his chances at $\sum_1^\infty 0.3^\left(2n\right)$ = 9.89%.

Can he do better? Yes!

Jorge's best strategy (assuming he's a good enough shot to implement it) is to intentionally miss.

Now if José misses Juan (30% chance), then Juan will kill José. Then Jorge's chances are exactly 30%: he gets one shot and must make it count.

But if José hits Juan (70% chance), then it's Jorge's turn and he has a 30% chance of winning immediately, plus the chances of surviving a duel with Jorge as calculated above. Total chance: 39.89%

This means Jorge's total chances are 70%×39.89% + 30%×30% = 36.9%. Not bad for a guy who can't shoot!

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Now that we have all three strategies, we can calculate the chances for Juan and José.

José will get the first meaningful shot. He must first kill Juan (70% chance) and then survive his battle with Jorge (60.11% chance: 100% minus the 39.89% chance calculated above). Total chance: 70%×60.11% = 42.1%

Juan has a 30% chance of surviving José's shot and then killing him, then a 70% chance of beating Jorge. Total chance: 30%×70% = 21%

Putting it all together:

• Juan: 21%



• José: 42.1%


• Jorge: 36.9%

The moral of the story: never be the best gunfighter around.