Although backward reasoning is the safe way to solve this problem, we can jump ahead a little by using some forward-looking arguments. We start by examining each of Larry’s options in turn. What happens if Larry shoots at Mo? What happens if Larry shoots at Curly?

If Larry shoots at Mo and hits, then he signs his own death warrant. It becomes Curly’s turn to shoot, and he never misses. Curly will not pass at the chance to shoot Larry, as this leads to his best outcome. Larry shooting at Mo does not seem to be a very attractive option.

If Larry shoots at Curly and hits, then it is Mo’s turn. Mo will shoot at Larry.

[Think about how we know this to be true.] Hence, if Larry hits Curly, his chance of survival is less than 20 percent (the chance that Mo misses).

So far, neither of these options looks to be very attractive. In fact, Larry’s best strategy is to fire up in the air! In this case, Mo will shoot at Curly, and if he misses, Curly will shoot and kill Mo. Then it becomes the second round and it is Larry’s turn to shoot again. Since only one other person remains, he has at least a 30 percent chance of survival, since that is the probability that he kills his one remaining opponent.

The moral here is that small fish may do better by passing on their first chance to become stars. We see this every four years in presidential campaigns.

When there is a large number of contenders, the leader of the pack often gets derailed by the cumulative attacks of all the medium-sized fish. It can be advantageous to wait, and step into the limelight only after the others have knocked each other and themselves out of the running.

Thus, your chances of survival depend on not only your own ability but also whom you threaten. A weak player who threatens no one may end up surviving if the stronger players kill each other off. Curly, although he is the most accurate, has the lowest chance of survival—only 14 percent. So much for survival of the fittest! Mo has a 56 percent chance of winning. Larry’s best strategy turns his 30 percent accuracy into a 41.2 percent chance of winning.

1 Today’s duels are more likely to be fought between takeover specialist T.

Boone Pickens and the target management over who will end up with control of the board of directors. Our next case presents the story of a firm that tried to preempt a takeover duel through the use of a poison pill. But things don’t always come out as planned, especially if you don’t think far enough ahead.