Who Should Tuco Shoot?

A few days ago, I posted one of my favorite movie scenes from The Good, the Bad, and the Ugly. This scene is followed by another great movie moment, the 3-way "duel" between Tuco (Eli Wallach), Blondie (Clint Eastwood) and "Angel Eyes" (Lee Van Cleef). Strictly speaking, it is not a "Dual", but rather, a "Truel".

A few years ago, I wondered who would have the advantage in a 3-way gunfight and created some simple computer simulations to test the outcomes based on variables about the skill of the gunmen. Surprisingly, it appeared that the least skilled marksman might have the highest chance of survival. In the scenarios, I assumed that Blondie was nearly perfect, Angel Eyes was somewhat less perfect and Tuco was the least accurate shooter (we never really find out the relative ability of Tuco vs. Angel Eyes).

It turns out that I was not the only one interested in this scenario. Mathematician Martin Gardner had proposed it as a puzzle:

"Three men are in a pistol duel. Each man will shoot in turn. The three men are identified as A, B, and C. A is a poor shot, and hits only 50% of the time. B is an expert marksman, and always hits. C is a moderate shot, and hits 80% of the time. (In some variants of the problem, A's probability is 25% and C's is 50%; in practice this makes little difference as long as A's is lower than C's.)"

"The most common solution is that A should deliberately fire into the ground until one of the other two men is dead, then shoot at him.

The reasoning for this is as follows: B and C are greater threats to each other than A is to either of them, and thus rationally should target each other first. If B fires first, he will certainly kill C; if C fires first, he may kill B, and if he does not, B will certainly kill him. In the first "round" of the duel one of these two interactions will occur between B and C and it is not in A's interest to disrupt it. If A kills one of the men, the other man will target him, probably killing him; if A fires but does not kill either man, he makes no difference. After either of these interactions completes, it will become A's turn with A never having been targeted and having a chance to kill the one remaining man and win the duel, and this is the best possible position for him."

William Chen's paper provides a more detailed look at the possible outcomes.

The rules are a bit different for this analysis than the movie, since they have not drawn lots to determine shooting order, but the results and outcomes might work out the same way. Tuco knows that Blondie and Angel Eyes will probably shoot at each other, so his best strategy would be to hesitate and then shoot the survivor.

"Gentleman H is a crack shot, with a 100% record of fatal shots on target. Gentleman E is a moderate shot, with an 80% record of fatal shots on target. Gentleman F is the weakest shot, with only a 50% record of fatal shots on target."

"Fifty-fifty Jones (F), against his own best interests, will blaze away when able at the opponent he imagines to be most dangerous. Even so, he still has the best chance of survival, 44.722 per cent. Brown (E) and Smith (H) find their chances reversed. Eighty-twenty Brown’s chances are 31.111 per cent and sure-shot Smith comes in last with 24.167 per cent."

Getting back to the movie, [SPOILER ALERT].

The game was actually rigged since Blondie knew that Tuco's gun was unloaded. It appears that Tuco chose to shoot at Angel Eyes and probably assumed that Blondie would not kill him once Angel Eyes was eliminated. He may have also realized that turning his gun towards Blondie would have been futile because he was fast and accurate enough to kill both of them. By aiming at Angel Eyes, he hoped to re-establish a partnership.

For a Cold-War era scenario, see "Who Should China Nuke?".

A separate but relevant bit of research also suggests that in gun duels, the first person to draw might actually be at a disadvantage.