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Game Theory Question

  1. Nov 19, 2014 #1
    Hey PF!

    Can you help me with something:

    Players alternately choose 0's or 1's. A play of this infinite game is thus a sequence of 0's and 1's. Such a sequence can be considered as the binary expansion of a real number between 0 and 1. Given a set ##E## of real numbers satisfying ##0 < x < 1 \forall x \in E##, say that player 1 wins if the play corresponds to a number in ##E## and player two wins if the way corresponds to a number in ##[0,1] \backslash E##.

    Evidently the Axiom of Choice implies there exists a set ##E## for which the game has no value. Can you help me out with showing this?
     
  2. jcsd
  3. Nov 19, 2014 #2

    Erland

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    Science Advisor

    1. What about ambiguities similar to 0.99999....=1, for example the two sequences 011111...... and 100000, which both correspond to the real number 0.1?

    2. If this ambiguity is resolved, it is certain that either player 1 or player 2 wins, since every real number in [0,1] lies in either E or its complement. But you meant perhaps someting else with the "value" of the game?
     
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