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This puzzle is taken from Schelling's Game Theory: How to Make Decisions by R.V. Dodge, in which bargaining power is gained, paradoxically, by burning bridges and removing options for oneself. Here it goes:

Anderson, Barnes, and Cooper are to fight a gun duel. They will stand close to one another, so that each can kill one of the others or deliberately miss. The first to fire will be chosen at random, and they will rotate in the order Anderson, Barnes, and Cooper, each firing one shot at a time.

If there is more than one survivor after a number of rounds, one of the contenders will be chosen at random and required to shoot one of the others, and this will be repeated if there is still more than one alive.

Before the duel starts, Anderson may make any statement, followed by a statement from Barnes, and finally one from Cooper. They will adhere to the following rules:

  1. A contender may not act to contradict his statement.
  2. He will act in his own best interest when it does not conflict with Rule One.
  3. He will act randomly when it does not conflict with Rules One and Two.

There are referees to ensure that the rules are followed. If a contender commits himself to a choice of action on a statistical basis (for example, if Anderson commits himself to miss with a probability of 1/3), the choice will be determined objectively (by tossing dice, etc.).

Q1:What statement will Anderson make? What's his best strategy and his probability of surviving?

Q2:If three contenders are to make their statements in the order of ACB, what would be the best statement for Anderson?


Update:

Warmup example 1: What happens when no one makes a statement?

If no one makes any statement, it would be foolish for a contender to shoot, because he's going to be shot by the other remaining player. So everyone just wait out a number of rounds, let the referees randomly choose one and force him to shoot, hoping to be the survivor under his gun. By Rule Three, this shooter will randomly shoot one of the other two contenders. So if you're not chosen as the shooter, you have 1/2 chance to survive. All in all, when no one speaks, each one has a survival probability of 1/3.

Warmup example 2: What happens when only Anderson is allow to make a statement?

A can guarantee near certain victory by making this statement to B and C:"If you don't kill each other at your first opportunities, I will kill the first of you who fail to do so at my first opportunity; otherwise, I'll shoot the survivor of you with 1% chance of missing." If C fires first, he'll be dead if he doesn't kill B; if he does kill B, he'll have 1% survival chance. So C will kill B. Similarly, if B fires first, his only chance (also 1%) to live is to kill C. If A fires first he'll shoot in the air and let B kill C. All in all, A has 99.33% chance to survive, B and C each have 0.33% percent, ignoring rounding error.


Hint (I decide to give this hint because I'm tired of trying to convince people about the flaws in their answers. Major spoilers inside.)

The originator of this puzzle, Nobel Laureate Thomas Schelling himself, was able to gave what he thought was a best answer he could think of for Q1, in which Anderson achieves a certain surviving probability less than 90%. Although there's no proof that his is the best, You should check really carefully if you think you find a way to give Anderson a surviving chance close to 100%.

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  • $\begingroup$ Is ACB in Q2 a typo or the intended order? $\endgroup$ – John Meacham May 18 at 12:18
  • $\begingroup$ If the judge requires someone to shoot one of the other two are they still bound by their statement or will that be a free choice? Seems like a situation can easily develop where they can't shoot anyone without violating their statement when randomly chosen by the judge. $\endgroup$ – John Meacham May 18 at 12:23
  • $\begingroup$ @JohnMeacham It's intended order. The judge will require someone to shoot the others only when no one is killed after a number of rounds. $\endgroup$ – Eric May 18 at 12:42
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    $\begingroup$ Having looked at the answer, I think it can be fairly trivially shown that A cannot do better than they can by following Schelling's strategy. $\endgroup$ – H Rogers May 18 at 22:49
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Assuming the statement can contain multiple commitments, A can just about guarantee victory with the following statement.

I will shoot according to the following rules (listed in order of precedence):

If it is round 1:

  1. If B does not commit to shooting C with a probability of x*, I will shoot B in round 1.
  2. If C makes any statement at all, I will shoot C.
  3. I will miss in round 1.

If it is round 2:

  1. If C is alive I will shoot B
  2. If C is dead I will shoot B with a probability of x*

*A can choose any probability x strictly less than 100% and have a x^2 chance of being the last survivor. So A can pick x=0.99999999999 and be almost certain to win the duel.

We can walk through the sequence of events as follows:

B knows that if they don't commit to shooting C with probability x they will die for certain because of A's first commitment, however they can still have a (very slim) chance at life if they go along with A. So B commits to shooting C with probability x. Once B has done this, C knows that if they make any statement at all they will die, so they make no statement. In round 1, A misses intentionally, and B shoots C with probability x. In round 2, A shoots B with probability x and (with effective certainty) wins the duel.

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  • $\begingroup$ I think you missed the sentence "The first to fire will be chosen at random" in the 2nd paragraph. $\endgroup$ – Eric May 17 at 15:32
  • $\begingroup$ My mistake, back to the drawing board $\endgroup$ – H Rogers May 17 at 15:33
  • $\begingroup$ Multiple commitments are definitely allowed in a statement, btw. $\endgroup$ – Eric May 17 at 15:34
  • $\begingroup$ By the way, even if A shoots first, there's still a flaw in your answer: what happens if B doesn't commit and C makes a statement? What would A do? He can't shoot two of them at the same time! $\endgroup$ – Eric May 17 at 15:37
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    $\begingroup$ If B doesn't commit and C makes a statement, A will shoot B. A follows the rules in order. $\endgroup$ – H Rogers May 17 at 15:39
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Anderson's best statement is

! If no one but me makes a statement then If there are 2 people left I will shoot with 99% accuracy, if there are 3 people left and I was not chosen first, I will shoot C with 99% accuracy. However if anyone makes a statement I will shoot whomever made the last statement with 100% accuracy and no one else.

! Now when given the chance to make a statement, doing so will guerantee a loss. If B makes one then C can choose to not make one and A will only shoot B and not be able to shoot C. And if C makes one B is free to shoot C. Since neither makes a statement it falls back to them shooting one another to get the 1% survival rate (Anderson shoots C no matter what after the first round to keep people from stalling for the judges to step in)

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  • $\begingroup$ C can't guarantee a win by being silent if B makes this statement:"If C doesn't kill A at his first opportunity, I'll kill him at my first opportunity." Since first shooter is chosen randomly, B has 1/3 chance being the first, in which case C dies. $\endgroup$ – Eric May 18 at 14:29
  • $\begingroup$ Except C shoots after B. So C won't have an opportunity to kill A before B goes (without A picking off B in between) $\endgroup$ – John Meacham May 18 at 14:35
  • $\begingroup$ "The first to fire will be chosen at random" $\endgroup$ – Eric May 18 at 14:48
  • $\begingroup$ Not an issue, it's still in Cs best interest to not make a statement. At most B can threaten C with a 1/3 chance of death but by making any statement C is now As target so has 1/3 plus whatever chance b shoots them chance of death. $\endgroup$ – John Meacham May 18 at 14:54
  • $\begingroup$ Will B make a statement? If B makes no statement. C can make this statement:"If B doesn't kill A at his first opportunity, I'll kill him at my first opportunity; but if he does kill A, I'll shoot him with 98% accuracy." Given this, if B goes first he will kill A, diminishing A's survival chance to less than 2/3. $\endgroup$ – Eric May 18 at 15:25

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