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Topic: Beating the Odds?
Replies: 35   Last Post: Feb 6, 2013 3:44 PM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Beating the Odds?
Posted: Feb 3, 2013 3:53 PM
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On Feb 3, 11:36 am, quasi <qu...@null.set> wrote:
> Here's a finite version of the problem, cast as a game.
>
> Perhaps the game is well known, I'm not sure, however the
> resolution of the game is nice, and not immediately obvious,
> so I'll pose it here as a challenge.
>
> 2-player play a game, win or lose, for 1 dollar, based on a
> fixed positive integer n > 1, known in advance to both players.
>
> Player 1 chooses two distinct integers from 1 to n inclusive,
> writes them on separate index cards, and places them face down
> on the table.
>
> Player 2 then selects one of the cards and turns it face up,
> exposing the hidden value. Player 2 can then either "stay",
> yielding the value on the chosen card, or "switch", yielding the
> value on the other card instead. Player 2 wins (and player 1
> loses) if player 2's final value is the higher of the 2 values,
> otherwise player 2 loses (and player 1 wins).
>
> In terms of n, find the value of the game for player 2, and
> specify optimal strategies for both players.


I haven't tried to prove anything, but I'll guess that the obvious
strategies are optimal.

Let S = {3/2, 5/2, . . ., (2n-1)/2}.

Player 1's strategy: choose random x in S (uniform distribution) and
play {x - 1/2, x + 1/2}.

Player 2's strategy: choose random y in S (uniform distribution), stay
on values > y, switch on values < y; otherwise put (in the form of a
behavior strategy), if the observed value is k, stay with probability
(k-1)/(n-1).

The value of the game for player 2 is $1/(n-1): player 1 wins a dollar
if x = y, otherwise the chances are even.



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