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Re: Simple Card Game Of Ups and Downs
Posted:
Dec 14, 2009 11:26 AM
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On Dec 14, 10:20 am, scattered <still.scatte...@gmail.com> wrote:
> On Dec 14, 9:44 am, Leroy Quet <qqq...@mindspring.com> wrote:
>
> > Is there any pure strategy to playing this game, or is it totally a
> > psychological game -- trying to guess in what order your opponent will
> > play their cards?
>
> > By the way. More of my games (many of which are more interesting than
> > this game) at:http://gamesconceived.blogspot.com/
>
> > Thanks,
> > Leroy Quet
>
> Interesting game. In effect, each deal specifies a two player zero sum
> game which in principle has an optimal strategy (though perhaps
> mixed). Some deals make the game blatantly unfair (with the extreme
> case of one player getting the top half of the deck).
>
> Suppose that player 1 recieves cards 1,3,5, ..., 2n-1 and player 2
> recieves cards 2,4,...,2n. It is obvious that any pure strategy that
> player 1 adopts is defeated by one of player 2's (the one in which
> player 2 happens to always play the successor of the card that player
> 1 plays - this assures player 2 will get at least n of the 2n-1 total
> points. It is irrelevant that player 2 is unlikely to know what this
> strategy actually is - its existence demonstrates that player 1 has no
> optimal pure strategy). Player 1 can almost do something similar to
> player 2 - but not quite since their 1 always loses and player 2's 2n
> always wins when placed down. Thus most pure strategies of player 2
> are defeated by some strategy of player 1. I suspect that all of
> player 2's are defeated by at least 1 of player 1's (at least for n
> sufficiently large) - but I haven't bothered to prove it. I suspect
> that this deal is (slightly) unfair to player 1. Does *any* deal lead
> to a fair game?
>
> The game reminds me of a modification to the card game war called
> Napolean's War that I read about here:http://www.pagat.com/invented/war_vars.html
>
> -scattered
Math before enough coffee doesn't mix. I misread how the game was
scored, thinking that either player got a point if the card they
played was larger than the previous card. Reading the game more
carefully, in the case where player 1 gets 1,...,n and player 2 gets n
+1,...,2n, player 1 wins in all cases by a score of n to n-1, in the
other case I mentioned it seems that it is player1 with the advantage
since they could (given psychic ability) always play the predecessor
of the card player 2 is about to play.
An interesting variation on the game is to start with the same
scenario but determine the winner in a different way. The two players
jointly generate a permutation of 1,2,...,2n. Declare player 1 the
winner if it is an odd permutation and player 2 the winner if it is
even.
I think that this game is fair given any deal with simply shuffling
your deck randomly an optimal mixed stratgey, but I haven't proved it
rigorously.
-scattered
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