Due to a Bitcoin deal gone wrong, you have been convicted along with Alex and Bob, two members of rival gangs. The only other way to settle the case is with a triangular shootout, where the last man standing goes free. As you are a much more seasoned crypto-analyst than marksman, you will hit any target you shoot at with probability 0.4. On the contrary, Alex is a sure shot who hits with probability 1, and Bob's rate is 0.7.
The moderator has declared that you will shoot first, Bob will shoot second, and Alex will shoot third. If no one is dead after the first round (raising suspicions of sabotage), or if anyone shoots out of turn, you will all be sentenced to 20 years in prison. You are confident that Alex and Bob will do whatever it takes to a) stay out of jail and b) remain the last man standing. You have 5 minutes to decide on a strategy. What do you do?
It may sound counterintuitive, but you should throw your shot away.
The key from Alex and Bob’s respective POV’s is that they’d rather defeat their strongest opponents (each other) first and deal with the weaker ones (yourself) later. For example, if it was Bob’s turn to shoot and he shot you, Alex would definitely kill him. Instead, if he took out Alex first, he’d have an easier time fending you off because you only hit with p=0.4.
By similar logic, if you had to deal with Alex on your own, you’d survive with p=0.4. If Alex got to shoot first, which he would if you got rid of Bob with your first shot, you’re toast. So there is no incentive for you to shoot at Bob.
If you shot at Alex, you will hit with p=0.4. Your chances in a subsequent duel with Bob are p=0.3*0.4 (P(Bob misses) P(you hit))
+ 0.3*0.6*0.3*0.4 (P(Bob misses) P(you miss) P(Bob misses) P(you hit))
+ …, which is an infinite geometric series with initial value 0.12 and ratio = 0.18. This series sums up to 0.12/(1 – 0.18) = 0.146. Doesn’t sound promising, but it’s better than Alex finishing you off for sure.
Now, what if you shoot at someone and miss (p=0.6)? Then it’s Bob’s turn. He would almost certainly shoot at Alex first. This gives a probability of 0.7 that you get to deal with Bob one-on-one and 0.3 that you have to face Alex instead. But in either case, you get to shoot first, and this turns out to be a significant difference.
If you face Alex, the calculations are simple. There is a 0.4 chance you’ll hit him; if not, game over. For Bob, the probability is p=0.4 (P(you hit))
+ 0.6*0.3*0.4 (P(you miss) P(Bob misses) P(you hit))
+ 0.6*0.3*0.6*0.3*0.4 (P(you miss twice) P(Bob misses twice) P(you hit))
+…, which is the same geometric series as the previous case, except with a starting rate of 0.4, which is the probability of a hit when you shoot first. The sum is 0.4/(1 – 0.18) = 0.488. The weighted probability of survival, given you miss your first shot, is 0.488*0.7 + 0.4*0.3 = 0.461.
And if you knew your chances were that much better if you missed and pitted Alex and Bob against each other, why would you not make certain of it, by firing a blank with your first shot?
At this point, if you’re wondering if Bob would consider shooting a blank as well, the answer is that he would not. If he did, Alex’s theoretical options are to either shoot Bob, or shoot a blank. He would not shoot you because that would decrease his chances of survival to 0.3. But if Alex did not shoot Bob either, everyone goes to prison—a fate Alex knows he can avoid by killing Bob. Bob should be aware of this and take matters into his own hands by shooting Alex.