The Truel

demonseed
Posers and Puzzles 22 Jun '07 11:26
1. 22 Jun '07 11:26
This is an old one so excuse me if you have encountered it before.

A truel is similar to a duel only there are three participants instead of two: in this case there are Mssrs Black, Grey and White. Mr Black is the worst shot hitting the target 1/3, Mr Grey is slightly better hitting the target 2/3 and Mr White never misses. To make the truel fairer Mr Black is allowed to shoot first followed by Mr Grey (if he is still alive) followed by Mr White (if he hasn't snuffed it too) and this continues, with each taking turns, until one is left standing.

The question is: where should Mr Black aim his first shot?
2. 22 Jun '07 12:50
Mr. White seems the best bet, because then Mr. Grey will probably follow likewise, and Mr. White will most likely die, or if he doesn't, will probably aim at the bigger threat and kill Grey. Then just aim at the remainding one and hope.

If you aim at Mr. Grey, he would probably shoot back, and then Mr. White is free to kill you.

This is probably very wrong.
3. Daemon Sin
I'm A Mighty Pirate™
22 Jun '07 13:071 edit
Originally posted by demonseed
This is an old one so excuse me if you have encountered it before.

A truel is similar to a duel only there are three participants instead of two: in this case there are Mssrs Black, Grey and White. Mr Black is the worst shot hitting the target 1/3, Mr Grey is slightly better hitting the target 2/3 and Mr White never misses. To make the truel fairer Mr B urns, until one is left standing.

The question is: where should Mr Black aim his first shot?
Mr. Black should aim his shot ANYWHERE but at one of the other treulers.

If he did fire at one and hit, it leaves the other with a shot at him at at least 66% chance of hitting. Therefore Mr. Black's only has a 0 - 33% chance of surviving the first round and then a 33% chance of shooting his opponent in the second round. Not exactly great odds.

If he fired and missed, it leaves Mr. Grey a one shot chance to take out his biggest threat (Mr. White). Either way, Mr. Grey will make his 66% chance of hitting Mr. White or he will miss and Mr. White will shoot him as he's Mr. White's biggest threat. Then which ever one of the 2 is left standing one has to duel with Mr. Black. Which is the best possible outcome for Mr. Black as he avoids the 0 - 33% survival chance from the other option and skips straight to dueling with the remaining man.

Therefore, Mr. Black should force the miss on his first shot by not aiming at anyone in order to get down to the one-on-one duel situation with him shooting first.
4. 22 Jun '07 13:27
Originally posted by Daemon Sin
Mr. Black should aim his shot ANYWHERE but at one of the other treulers.

If he did fire at one and hit, it leaves the other with a shot at him at at least 66% chance of hitting. Therefore Mr. Black's only has a 0 - 33% chance of surviving the first round and then a 33% chance of shooting his opponent in the second round. Not exactly great odds.

If h ...[text shortened]... ming at anyone in order to get down to the one-on-one duel situation with him shooting first.
Too easy for you huh?

Well done.
5. Palynka
Upward Spiral
22 Jun '07 13:34
Originally posted by demonseed
Too easy for you huh?

Well done.
I love this new concept of 0 - 33% chance of surviving. 😵
6. Daemon Sin
I'm A Mighty Pirate™
22 Jun '07 13:36
Originally posted by Palynka
I love this new concept of 0 - 33% chance of surviving. 😵
I'm sure we could arrange it for you if you liked! 😉
7. Palynka
Upward Spiral
22 Jun '07 13:41
Originally posted by Daemon Sin
I'm sure we could arrange it for you if you liked! 😉
Mr. Pink?
8. Daemon Sin
I'm A Mighty Pirate™
22 Jun '07 13:58
Originally posted by Palynka
Mr. Pink?
Only on Wednesdays.
9. ark13
Enola Straight
23 Jun '07 18:59
I come to the same conclusion at Daemon Sin. I just have some calculated probabilities.

If Black is left against Gray and is shooting first, he has a 1/3 + 1/3*(2/3*1/3) + 1/3*(2/3*1/3)^2 + 1/3*(2/3*1/3)^3 ... = 3/7 chance of winning. If Gray shoots first, Gray has a 2/3*(2/3*1/3) + 2/3*(2/3*1/3)^2 + 2/3*(2/3*1/3)^3 ... = 6/7 chance of winning, making Black's chance in that case 1/7.

It is clear that Black shouldn't shoot at Gray because his chances against Gray are better than those against White. If Black misses his first shot, intentionally or not his probability of victory (2/3)*(3/7) (gray shoots next and kills white*probability of victory in that situation) + (1/3)(1/3) (gray shoots next and misses white*probability of shooting first against white, who has killed gray) = 25/63. If Black shoots and hits White, his chances (shooting second) are 1/7. Since his chances are clearly better if he misses, he should maximize the likelihood that he misses by aiming elsewhere.

It's baffling that the worst shot should have a 25/63 > 1/3 chance of winning. Although shooting first is an advantage, his best strategy is to pass, or not use this advantage. Another good question is what are the chances of each of the other's victories assuming best play? It seems to me that White has a 2/9 chance winning; 1/3 (surviving gray's first shot) * 2/3 (winning against Black shooting second). Therefore, Gray has a 1- 2/9 - 25/63 = 8/21 chance of winning. So, the best chance is held by Black, followed by Gray and then White.