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#1
05-06-2005, 07:17 PM
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Game theory problem related to poker (academic content)
The situation below (Know/Don't Know Game) is related to poker. Think of it as you are up against a good player, it is capped preflop so both of you know sort of what kind of hands the other player could have. Many people advocate 3-betting and capping the turn and river with good hands in situations like these without thinking about what the other player has, what he thinks you have and what he thinks you think he has. Against a good thinking player a raise and a reraise gives you information. If you can solve this problem (you need a calculator!) or realize how it can be solved, you will benefit. [img]/images/graemlins/smile.gif[/img]
Game theory problem: Two positive integers are chosen. The sum is revealed to logician A, and the sum of the squares is revealed to logician B. Both A and B are given this information and the information contained in this sentence. The conversation between A and B goes as follows: B starts. B: "I can't tell what the two numbers are." A: "I can't tell what the two numbers are." B: "I can't tell what the two numbers are." A: "I can't tell what the two numbers are." B: "I can't tell what the two numbers are." A: "I can't tell what the two numbers are." B: "Now I can tell what the two numbers are." What are the two numbers? Very curious if anyone will bother trying. Please post comments if this is a bad post. 
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#4
05-06-2005, 07:25 PM
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Re: Game theory problem related to poker (academic content)
This would be a good post for the Poker Theory forum or Other Other Topics (if you post in OOT though, do NOT mention the word poker). I am quite perplexed by this (did you type it up right?), and I definetly will check back for the answer, because I cannot solve it given how it is stated.
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#6
05-06-2005, 07:30 PM
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Re: Game theory problem related to poker (academic content)
[ QUOTE ]
This would be a good post for the Poker Theory forum or Other Other Topics (if you post in OOT though, do NOT mention the word poker). I am quite perplexed by this (did you type it up right?), and I definetly will check back for the answer, because I cannot solve it given how it is stated. [/ QUOTE ] Yeah, I only post in this forum though and thought it fitted in with many posts recently telling "raise raise you are ahead" without thinking about what information your bet, your opponents raise, your 3-bet and his cap gives you about his holding and him about your.
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#7
05-06-2005, 07:38 PM
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Re: Game theory problem related to poker (academic content)
A hint:
When B says he doesn't know the numbers, the sum revealed to him has to be a sum that can be attained by two squared numbers in at least two different ways. A knows this, and his sum has to be a sum that can be attained by summing up two numbers in at least two ways (he has just the sum of the numbers so it can be many ways), AND there has to be at least two ways any pair of those numbers can be squared and summed to form the same square sum. When B hears A saying he doesn't know, B knows that... etc
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#10
05-06-2005, 07:57 PM
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Re: Game theory problem related to poker (academic content)
The only way A could know what the numbers are right off the bat is if the sum was 2 or 3, and that assumes 0 can't be one of the numbers. If zero is allowed, the only uniquely identified sums are 0 and 1. I think zero has to be allowed, though, because the sum of of the squares of two integers should be unique if zero isn't, meaning B should know the answer. Now, if zero is allowed, B can be faced with an ambiguity of 3 and 4, or 5 and 0 yielding 25, 5 and 12 or 13 and 0 yielding 169, 8 and 15 or 17 and 0 yielding 269, etc. All three (and other such sets) should be unclear to A, and B knows this. A value of 7 given to A can be obtained 4 ways. If B was told 0, 1, 2, or 5, he should know the answer right away.
Actually, instead of giving up, I'll speculate that the numbers are 3 and 4, since it was after the third iteration of the process that B figured it out. I made up my mind on this because I decided that the process would have iterated more if any of the other pairs of numbers I listed above was used, but mostly I'm pretending this is a good reason because I'm going back to play some poker.
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