Friday, January 26, 2007
Dueling problem (see above post from sunday)
Mr. White has to fire at Mr. Black.
This seems trivially easy, but then I scored a 99% on my LSAT, but needed help last night from two mechanics changing lightbulbs in my car.
or maybe it's harder than I first thought - how does it play out?
If Black is killed, White will then try to kill Brown.
Brown also must try to kill Black.
Black has a 2/3 chance of surviving white times a 1/3 chance of surviving brown = 2/9.
7 of nine tries, Black dies.
If Black lives, he will kill Brown.
Second turn, White gets another shot at Black. 2/9 times 2/3 chance of surving white's second shot. = black kills brown and white = 4/27ths. About 15%. If he misses, black kills him, game over.
If white kills black in the second round, he must then face brown.
If white shoots at brown in the first round and hits, black kills white, game over.
But maybe white shoots at brown and hopes to miss.
Nah, couldn't be.
If white kills black, brown shoots at white and has 2/3 chance of killing and winning, first round.
2/9 chance of this scenario.
If brown misses, it's white's turn. 1/3 chance of winning on second round, 1/3 chance of surviving to round 3.
Now ,what we are dealing with here is a three person version of a re-iterated prisoner's dilemma, with specified probablilites.
This is an example of game theory, the work for which John Nash won the Nobel prize and was played by Russell Crowe in the movie A Beautiful Mind. I've read the book, not seen the movie.
There is probably a short elegant way to work out the answer.
I don't know it.
It may be possible to look up the answer on google.
But what search terms? three-player prisoner's dilemma?
Using brute force, I've sorted out that Black's chances don't look good, that Brown has a pretty good chance, that white has some chance, but I havent worked out the numbers,and I don't want to spend all day on this. I have work I'm avoiding.
If you have the answers, leave a comment. No one ever coments at this blog - maybe I don't have comments enabled?
gtbear at gmail.
I'll ponder this a bit, try a google search, update later.
Thanks for sending in your puzzler answer. Imagine how excited you'll be if we choose your response as the correct, winning answer! In case you were wondering, here's the process we go through to select each week's winner: (pun filled process omitted)
Cordially,
Tom and Ray Magliozzi
Click and Clack the Tappet Brothers
Will Baude took a course on game theory at the university of chicago. This seems to be the notes from that class, discussing the three person prisoner's dilemma. It doesn't translate precisely to this problem, but it's listed for people who want a place to start. Not that I have much reason to think anyone reads this blog.
maybe I'll make this a blog contest, and solicit answers from the five people I know who would at least recognize the problem - 4 ex roommates, if i count my brother as an ex roommate,and Mr. Baude. Grand prize a random used T-shirt - i cleaned my closet today-, or perhaps a whisk, or some hawaii sea salt. Maybe I'll even mail volokh if I get ambitious.
The question isn't "who should white fire at?" It's more, submit a formal proof in English of why white should fire at black, unless I'm wrong, but better, since I wouldn't be able to understand a proof, what are the odds for white brown and black.
Bonus: If they know the odds, do they still play the game? They were already commited to a roughly 2/3 chance of death. A sort of assisted suicide.
If you read this blog, you get a day's head start - I won't send out any emails till tomorrow at least.
Mr. White has to fire at Mr. Black.
This seems trivially easy, but then I scored a 99% on my LSAT, but needed help last night from two mechanics changing lightbulbs in my car.
or maybe it's harder than I first thought - how does it play out?
If Black is killed, White will then try to kill Brown.
Brown also must try to kill Black.
Black has a 2/3 chance of surviving white times a 1/3 chance of surviving brown = 2/9.
7 of nine tries, Black dies.
If Black lives, he will kill Brown.
Second turn, White gets another shot at Black. 2/9 times 2/3 chance of surving white's second shot. = black kills brown and white = 4/27ths. About 15%. If he misses, black kills him, game over.
If white kills black in the second round, he must then face brown.
If white shoots at brown in the first round and hits, black kills white, game over.
But maybe white shoots at brown and hopes to miss.
Nah, couldn't be.
If white kills black, brown shoots at white and has 2/3 chance of killing and winning, first round.
2/9 chance of this scenario.
If brown misses, it's white's turn. 1/3 chance of winning on second round, 1/3 chance of surviving to round 3.
Now ,what we are dealing with here is a three person version of a re-iterated prisoner's dilemma, with specified probablilites.
This is an example of game theory, the work for which John Nash won the Nobel prize and was played by Russell Crowe in the movie A Beautiful Mind. I've read the book, not seen the movie.
There is probably a short elegant way to work out the answer.
I don't know it.
It may be possible to look up the answer on google.
But what search terms? three-player prisoner's dilemma?
Using brute force, I've sorted out that Black's chances don't look good, that Brown has a pretty good chance, that white has some chance, but I havent worked out the numbers,and I don't want to spend all day on this. I have work I'm avoiding.
If you have the answers, leave a comment. No one ever coments at this blog - maybe I don't have comments enabled?
gtbear at gmail.
I'll ponder this a bit, try a google search, update later.
Thanks for sending in your puzzler answer. Imagine how excited you'll be if we choose your response as the correct, winning answer! In case you were wondering, here's the process we go through to select each week's winner: (pun filled process omitted)
Cordially,
Tom and Ray Magliozzi
Click and Clack the Tappet Brothers
Will Baude took a course on game theory at the university of chicago. This seems to be the notes from that class, discussing the three person prisoner's dilemma. It doesn't translate precisely to this problem, but it's listed for people who want a place to start. Not that I have much reason to think anyone reads this blog.
maybe I'll make this a blog contest, and solicit answers from the five people I know who would at least recognize the problem - 4 ex roommates, if i count my brother as an ex roommate,and Mr. Baude. Grand prize a random used T-shirt - i cleaned my closet today-, or perhaps a whisk, or some hawaii sea salt. Maybe I'll even mail volokh if I get ambitious.
The question isn't "who should white fire at?" It's more, submit a formal proof in English of why white should fire at black, unless I'm wrong, but better, since I wouldn't be able to understand a proof, what are the odds for white brown and black.
Bonus: If they know the odds, do they still play the game? They were already commited to a roughly 2/3 chance of death. A sort of assisted suicide.
If you read this blog, you get a day's head start - I won't send out any emails till tomorrow at least.
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