
Re: Beating the Odds?
Posted:
Feb 3, 2013 3:53 PM


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. > > 2player 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, . . ., (2n1)/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 (k1)/(n1).
The value of the game for player 2 is $1/(n1): player 1 wins a dollar if x = y, otherwise the chances are even.

