When I dropped in on Ernie over the weekend he commented that he hadn’t seen me for a while. I had to confess that my evenings had been very busy recently - first at the local Community College doing a course on creative writing, and then at home writing my first novel.
“It is a spy story – dramatic and gripping but imbued with a sense of gritty realism!” I expounded proudly. I went on to explain how the protagonists were three agents of the Kzijekistanian Secret Service (KSS) who had discovered a sinister plot hatched in the neighboring dictatorship of Spoznikia. The Spoznikians were planning to assassinate the co-chairs of the Kzijekistanian Ruling Triumvirate and replace them with puppet leaders. The agents had indisputable evidence of the plot, but they had been betrayed by a Spoznikian double agent and, after a ferocious gun-battle, were now barricaded in a small room at the top of the radio tower of Spoznikia’s Department of Propaganda, with enemy agents in close pursuit.
“Very dramatic “, commented Ernie. “What happens next?”, so I continued with my plot summary. “The agents, known only by their initials A, B, and C, realize they cannot fight their way out as they have less than a handful of bullets left. But they have a single pair of Acme Inc.® Wing-Suit-Underwear™ (one size fits all) – so one of them could glide from the tower into the Spozkzij River, swim through the rapids, ascend the cliffs below, cross the border minefield, and get the evidence to the Kzijekistanian authorities. But, by use of truth serum, the Spoznikians will gather vital information from the other two if they surrender. So they decide on the ultimate sacrifice – two must die before the Secret Police can break down the barricaded penthouse door”.
“Gritty too”, broke in Ernie, “Who should the survivor be?”, so I continued with my summary. “The Spozknikians will almost certainly see the wing-suit and try to guard the escape route – but they are short of soldiers after a recent army purge and won't only be able to guard the river, the cliffs, and the mine-field simultaneously. The Kzijekistanian agents, who coincidentally are all brilliant mathematical logicians, combine that information with their own respective individual expertise at swimming, cliff-scrambling, and sneaking and use game theory to calculate that the most successful strategy is that escaping agent should be chosen randomly, with the relative probabilities p:q:r, for agents A:B:C respectively”.
“Game theory!”, exclaimed Ernie. “So it is a gripping story as well. How do the agents choose the survivor?”. “I’m glad you asked”, I replied. “The agents will choose via a truel”. I went on to explain how that worked – “The three duelists are named A, B, and C and have accuracy of a%, b%, and c% and...”
“Well – that’s a pity”, Ernie broke in, “you just failed on the realism side of things. The KSS is renowned for the marksmanship of its agents – they always shoot with an accuracy of 100%. But even if you ignored that – with only a ‘handful of bullets left’ they could run out of bullets before completing your truel!”
My heart sank as I realized that Ernie’s criticism was completely justified. “But maybe there is a solution”, Ernie continued. How about the three agents compete in a Круговая дуэль”. When I asked what that was Ernie said that he wasn’t sure how it translated into English – but that it was a traditional historic pastime for Kzijekistanians when they got depressed sharing snowed-in huts during long and bleak winters. “But I can explain the rules”, he continued:
Rules of Круговая дуэль for $n$ players
The $n$ players sit in a small circle, so that in clockwise order they are labelled $1,2,3...n$ (note that player $1$ is of course clockwise to player $n$).
Each player $i$ has a revolver with $C_{i}$ chambers, that is initially loaded with $B_{i}$ bullets
Player $1$ has the first turn, after which the turn goes to the first living player clockwise to the player who has just played. This continues until only one player is left alive.
A turn consists of the following three steps:
a. The player spins the chamber of their revolver.
b. The player aims their revolver at the living opponent with the highest chance of firing a bullet on that opponent's turn (on the assumption that the opponent will still be alive at that time). If two or more opponents have the same maximum chance then the player aims at the first such opponent in the circle (counting clockwise from the player's position).
c. The player pulls the trigger once (inevitably killing the opponent if the revolver fires).
That sounded like a perfect solution to me (especially with the added bonus of historical accuracy) so I asked for him to help with the details.
“OK Ernie”, I started, “I want a solution guaranteed to achieve an $A:B:C$ ratio of $1/6:1/3:1/2$ (that matches the game theory results that I had imagined them calculating).” -- “No problem”, Ernie replied.
“…And I want a solution that uses the smallest possible number of bullets, to emphasize how dire their situation is.” -- “No problem”, Ernie replied.
“…And I want the players to shoot in the order A-B-C…, because that matches the dialog I have already written.” -- “Sorry”, said Ernie, “that order of play isn't consistent with the smallest possible number of bullets.”
“OK”, I said – thinking that I could swap the dialog around a bit – “…how about they shoot in order B-A-C…?” -- “Theoretically possible”, said Ernie, ”but highly unrealistic. I doubt the KSS have revolvers with the required number of chambers in their arsenal.”
“Then”, I said – accepting that the dialog might require a more significant rewrite. “…how about they shoot in order B-C-A…?” -- “There is definitely a solution for that.” Ernie replied, and wrote down the numbers of chambers and bullets each of the three agents needed on the back of an envelope.
Unfortunately, the envelope slipped through a hole in my pocket while I was cycling home so I have lost the information I need to complete my novel. Can anyone help me recover it? It would also be nice to get my head around what precisely was wrong with the A-B-C… and B-A-C… scenarios.