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wu :: forums    riddles    easy (Moderators: Grimbal, SMQ, Eigenray, towr, Icarus, william wu, ThudnBlunder)    a number picking game « Previous topic | Next topic » Pages: 1  Reply Notify of replies Send Topic Print    Author  Topic: a number picking game  (Read 518 times) Slayer Newbie     Posts: 5 a number picking game   « on: Feb 23rd, 2009, 8:42am » Quote Modify the game to be repeatedly played: Player 1 and Player 2 each pick any natural number (zero is not natural). player Lowest number wins $1 EXCEPT when Lowest number = HighestNumber - 1, then player with Highest number wins $2.   Problem: Develop a long-term strategy for Player 1 such that Player 1 does not lose even if Player 2 knows the strategy Player 1 is using. IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: a number picking game   « Reply #1 on: Feb 23rd, 2009, 8:59am » Quote Modify Each round randomly pick 1, 2 or 3 IP Logged Wikipedia, Google, Mathworld, Integer sequence DB Slayer Newbie     Posts: 5 Re: a number picking game   « Reply #2 on: Feb 23rd, 2009, 9:22am » Quote Modify on Feb 23rd, 2009, 8:59am, towr wrote: Each round randomly pick 1, 2 or 3   Then player B can always choose 3, his expected gain is 1/3*(-1)+1/3*2=1/3 which means player A loses 1/3 on average. IP Logged Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: a number picking game   « Reply #3 on: Feb 23rd, 2009, 9:58am » Quote Modify Just play whatever you like. Nowhere does it say how you can loose money.   If the collaborate, one plays 1,2,1,2,... the other 2,1,2,1,..., they both win $1 per round on average. IP Logged towr wu::riddles Moderator Uberpuzzler Some people are average, some are just mean.     Gender: Posts: 13730 Re: a number picking game   « Reply #4 on: Feb 23rd, 2009, 10:12am » Quote Modify on Feb 23rd, 2009, 9:22am, Slayer wrote: Then player B can always choose 3, his expected gain is 1/3*(-1)+1/3*2=1/3 which means player A loses 1/3 on average. Hmm.   Well, some other strategies I can think of are Don't play Threaten player 2 « Last Edit: Feb 23rd, 2009, 10:13am by towr » IP Logged Wikipedia, Google, Mathworld, Integer sequence DB howard roark Full Member     Posts: 241 Re: a number picking game   « Reply #5 on: Feb 23rd, 2009, 11:48am » Quote Modify I think equilibrium exists only when actions available to players is finite....which doesnt seem to be true in this case IP Logged howard roark Full Member     Posts: 241 Re: a number picking game   « Reply #6 on: Feb 23rd, 2009, 11:49am » Quote Modify What happens when both chose the same number? IP Logged rmsgrey Uberpuzzler     Gender: Posts: 2874 Re: a number picking game   « Reply #7 on: Feb 23rd, 2009, 2:17pm » Quote Modify There's a venerable variant on this, which again has no upper limit on what can be played, in which playing higher numbers scores more points (though, again, only if your opponent plays at least 2 higher or 1 below)   In that variant, as I recall, the unbeatable strategy involves playing nothing over 6...   Despite the fact that playing one more than your opponent's number is best possible, if your opponent never plays higher than, say, 5, and only rarely plays that, then it's trivially not worth playing 7 or higher, and probably not worth playing 6... IP Logged howard roark Full Member     Posts: 241 Re: a number picking game   « Reply #8 on: Feb 23rd, 2009, 5:21pm » Quote Modify on Feb 23rd, 2009, 2:17pm, rmsgrey wrote: There's a venerable variant on this, which again has no upper limit on what can be played, in which playing higher numbers scores more points (though, again, only if your opponent plays at least 2 higher or 1 below)   In that variant, as I recall, the unbeatable strategy involves playing nothing over 6... Is there a formal way to prove that? IP Logged rmsgrey Uberpuzzler     Gender: Posts: 2874 Re: a number picking game   « Reply #9 on: Feb 24th, 2009, 3:55am » Quote Modify on Feb 23rd, 2009, 5:21pm, howard roark wrote: Is there a formal way to prove that? Yes.   I just don't know what it is   (Okay, strictly, I have been told there exists a formal proof by someone who claimed to have seen it...) IP Logged greenfin1 Newbie     Posts: 1 Re: a number picking game   « Reply #10 on: Feb 24th, 2009, 4:22am » Quote Modify its really confusing for me, tell the detail to play this game: IP Logged Grimbal wu::riddles Moderator Uberpuzzler     Gender: Posts: 7527 Re: a number picking game   « Reply #11 on: Feb 24th, 2009, 7:24am » Quote Modify on Feb 24th, 2009, 3:55am, rmsgrey wrote: (Okay, strictly, I have been told there exists a formal proof by someone who claimed to have seen it...) OK.  That's something.  If I tell someone about the problem I can tell that I read on a forum about someone who has been told there exists a formal proof by someone who claimed to have seen it.   IP Logged rmsgrey Uberpuzzler     Gender: Posts: 2874 Re: a number picking game   « Reply #12 on: Feb 25th, 2009, 2:06pm » Quote Modify on Feb 24th, 2009, 7:24am, Grimbal wrote: OK.  That's something.  If I tell someone about the problem I can tell that I read on a forum about someone who has been told there exists a formal proof by someone who claimed to have seen it.   Proof by urban myth? 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