A duel in a novel
Frederick Marryat's novel "Mr Midshipman Easy" has a three-way duel in which each man aims only at one set opponent. Choosing a target, as in our puzzle, is not allowed there.
Ann hits with one shot in three, Bob with two in three, and Sam never misses. They take turns, Ann, Bob, Sam and round again, until only one is left. Each picks their own target and may also fire into the air. Who has the best chance of being the last one standing?
Ann, if she fires her first shot into the air. Bob and Sam fear each other most and shoot at each other. Whichever of them survives, the next shot is Ann's. Ann is the last one standing with probability 25/63 ≈ 39.7%, Bob 38.1%, and Sam the sharpshooter only 22.2%.
A duel for three is called a truel. Think paintball or water pistols: whoever is hit is out of the game. Here are the rules.
Ann hits her target with probability 1/3, Bob 2/3, Sam 1, that is, always. Each knows how well the others shoot.
They fire one shot each in turn: Ann, Bob, Sam, then Ann again and so on. Whoever is out skips their turns.
On their turn a shooter picks a target: anyone still in the game. Or they fire into the air, deliberately at nobody. That is allowed. A miss harms no one.
The game goes on until one is left. Each wants to be that last one and plays as well as they can. The others also play their best, and everyone knows it.
Which of the three has the best chance of being the last one left?
"Sam, of course: he never misses"
That is exactly why he is the most dangerous, and Bob aims at him. Ann has the best chance: while Bob and Sam are busy with each other, nobody aims at her.
All three play as well as they can. Lock in your answer before reading on.
A breakdown of your answer will appear here.
Hard to decide? Open the hints one at a time: the easy one first, then stronger ones.
Look through Sam's eyes while Ann and Bob are both in. Which of them is it better for him to knock out: which one leaves him the easier duel?
Now through Bob's eyes: what happens to him if Sam lives to take his turn? So whom does Bob shoot at?
Finally through Ann's eyes: what happens if she hits Sam, or Bob? And if she hits nobody, who shoots first in the duel that is left?
On each of your turns choose whom to shoot, or fire into the air. Bob and Sam play the best strategy for themselves, and hits are decided by chance at their hit rates. How many of ten truels can you win?
Ann has three options for her first shot: into the air, at Bob or at Sam. For each one the computer plays 100,000 truels and counts who is left. Next to them are the exact probabilities from the formulas in round 4. Change the hit rates with the sliders: Bob and Sam pick their best targets afresh each time.
In a truel your chances depend less on how well you shoot than on how dangerous you are to the others.
“In plain words” is hidden on all puzzles.
Let's start with a simple, if incomplete, explanation. Look through Sam's eyes: if he knocks Ann out, he is left alone with Bob, and Bob shoots first. If he knocks Bob out, he is left with Ann, who hits only once in three. The second is far better, so Sam shoots at Bob.
Bob knows this. If Sam is still in the game when Sam's turn comes, Bob is out for sure. So Bob shoots at Sam. While both of them are in the game, nobody aims at Ann.
If Ann hits Bob, Sam hits her on the next turn, and that's it. If she hits Sam, she is left with Bob, and he shoots first: two in three that Ann is out at once. But when Ann fires into the air, Bob and Sam settle it between themselves, and whoever survives, she gets the first shot at him.
“Rigorous” is hidden on all puzzles.
Total: 2/7 + 1/9 = 18/63 + 7/63 = 25/63 ≈ 39.7%
Where does 3/7 come from in Ann's duel with Bob when Ann shoots first? Take one round of two shots. Ann hits with probability 1/3, that is 3/9. Bob gets his shot if Ann missed, and hits: 2/3 × 2/3 = 4/9. The other 2/9 both miss, and the round repeats with the same odds. So decisive rounds go to Ann and Bob as 3 to 4, and Ann wins the duel in 3 cases out of 7.
Bob wins if he hits Sam and then wins the duel: 2/3 × 4/7 = 8/21 ≈ 38.1%. Sam wins only if Bob misses and then Ann misses: 1/3 × 2/3 = 2/9 ≈ 22.2%. Together 25/63 + 24/63 + 14/63 = 1.
| Who | First shot | Survival chance |
|---|---|---|
| Ann | into the air | 25/63 |
| at Sam | 59/189 | |
| at Bob | 50/189 | |
| Bob | at Sam | 8/21 |
| at Ann or into the air | 0 | |
| Sam | at Bob | 2/9 |
| at Ann | 1/9 | |
| into the air | 0 |
Each row shows what happens if only this player changes their choice while the other two play as in the highlighted rows. No change helps, so the highlighted trio is a Nash equilibrium. We checked all 27 combinations of targets with exact fractions: there is no other equilibrium here.
The first shooter hits with probability p, the second q. The first wins with the first shot, or after two misses with the third, or after four with the fifth…
Ann first against Bob, and Bob first against Ann:
Ann's three first shots:
Here 1/7 is Ann's chance in a duel where Bob shoots first: 1 − 6/7. A miss at Sam or Bob leads to the same game as a shot into the air, so the second part of each sum is 2/3 · 25/63.
“Deeper” is hidden on all puzzles.
If everyone must aim at someone, Ann shoots at Sam: the lesser evil. There is again just one equilibrium, but now when Ann hits she hands Bob the first shot in the duel. The weak player's whole advantage rested on the right to do nothing.
Our order favours Ann a lot: she shoots first and Sam last. If a lot decides the order, then in half the cases Sam shoots before Bob, knocks him out at once, and Ann is left with one shot in three. Averaging over the six possible orders gives Sam 4/9. So "the weakest survives" is not a law but a consequence of the rules.
That is what the economist Martin Shubik, one of the founders of game theory, called his 1954 paper. In it he also coined the word "truel". His question went beyond shooting: if everyone plays their best, must the strongest win? The truel answers "not necessarily" with the simplest possible example.
In 1975 the Canadian mathematician Marc Kilgour analysed the sequential truel, where the shots come in turn as in our puzzle, in his paper "The Sequential Truel". In 1997 he and Steven Brams summed up the known variants in the review "The Truel" in Mathematics Magazine. We checked our numbers independently: with exact fractions in Python and by simulation.
Mathematically we can see for which hit rates the first shooter keeps her edge. Logically it is a story about a leader everyone aims at. And in practice it is a question worth asking in any three-way contest: whom do my rivals fear most?
Mathematics
Across is Ann's hit rate, up is Bob's. At each point the page tries all 27 combinations of targets, finds the Nash equilibrium and colours the point by who has the best chance. The order is the same: Ann, Bob, Sam. Click the map to see the exact chances and moves.
Above the diagonal Ann shoots worse than Bob. The wider the pink zone there, the more often the weakest wins. Lower Sam's hit rate: he becomes less frightening for Bob, and the picture shifts.
Logic
Here n players hit with probability 1/n, 2/n, …, 1. They shoot in turn from the weakest to the strongest by one simple rule: aim at the most accurate player still in the game. For n > 3 this is not a proven equilibrium but a model of a coalition against the leader, computed exactly.
The logic is the same as in the truel: the strongest is dangerous to everyone, so everyone silently teams up against him without any deal. He shoots last and often does not live to his turn. The chances go to those nobody is looking at. In 2006 the physicists Pau Amengual and Raúl Toral also noted that the best shooter's chance falls as the number of players grows.
Practice
Test yourself on a classic. Martin Gardner gave his readers "the triangular duel": Smith always hits, Brown hits 80% of the time, Jones 50%, and the order is drawn by lot. Press "Gardner": Jones, the worst shot, survives with probability 47/90 ≈ 52.2%, Smith 3/10, Brown only 8/45.
In life nobody knocks anyone out with one shot, and the model gives not a forecast but a question. In an election with three candidates, two front-runners attacking each other may clear the way for the third. In a market with three firms, a price war between the two big ones may suit the smaller one. But whether it happens depends on the rules, as in the truel: who moves first, whether you may sit it out, whether the loser is really out.
Frederick Marryat's novel "Mr Midshipman Easy" has a three-way duel in which each man aims only at one set opponent. Choosing a target, as in our puzzle, is not allowed there.
Truel researchers usually name Clark Kinnaird's Encyclopedia of Puzzles and Pastimes (1946) as the puzzle's first appearance. Later tellings often soften it: the players shoot not at each other but at each other's balloons.
Martin Shubik built the word truel from three and duel in his paper "Does the Fittest Necessarily Survive?" in the collection Readings in Game Theory and Political Behavior.
In Martin Gardner's version Smith always hits, Brown 80%, Jones 50%, and the order is drawn by lot. Jones, the worst shot, survives most often: 47/90. We recompute the number in the Practice section.
The final scene of Sergio Leone's "The Good, the Bad and the Ugly" is often called a truel: three men stand in a circle in a cemetery. But it is not our puzzle: there they draw at once, and the outcome was settled in advance, because Blondie had secretly unloaded Tuco's gun.
Pau Amengual and Raúl Toral titled their review "Truels, or the survival of the weakest". They studied random and sequential truels, truel "leagues" and games played on networks.
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