The game of normal numbers

Research output: Contribution to journalArticlepeer-review


We introduce a two-player game where at each period one player, say, Player 2, chooses a distribution and the other player, Player 1, chooses a realization. Player 1 wins the game if the sequence of realized outcomes is normal with respect to the sequence of distributions. We present a pure winning strategy of Player 1 and thereby provide a universal algorithm that generates a normal sequence for any discrete stochastic process. It turns out that to select the nth digit, the algorithm conducts O(n2) calculations. The proof uses approachability in infinite-dimensional spaces (Lehrer 2002).

Original languageEnglish
Pages (from-to)259-265
Number of pages7
JournalMathematics of Operations Research
Issue number2
StatePublished - May 2004


  • Normal number
  • Two-player game


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