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The Big Match with a Clock and a Bit of Memory

Author

Listed:
  • Kristoffer Arnsfelt Hansen

    (Aarhus University, Aarhus 8000, Demark)

  • Rasmus Ibsen-Jensen

    (University of Liverpool, Liverpool L69 3BX, United Kingdom)

  • Abraham Neyman

    (Hebrew University of Jerusalem, Jerusalem 9190401, Israel)

Abstract

The Big Match is a multistage two-player game. In each stage, player 1 hides one or two pebbles in his hand, and his opponent has to guess that number. Player 1 loses a point if player 2 is correct; otherwise, he wins a point. As soon as player 1 hides one pebble, the players cannot change their choices in any future stage. The undiscounted Big Match has been much-studied. Blackwell and Ferguson (1968) give an ε -optimal strategy for player 1 that hides, in each stage, one pebble with a probability that depends on the entire past history. Any strategy that depends on just the clock or just a finite memory is worthless (i.e., cannot guarantee strictly more than the least reward). The long-standing natural open problem has been whether every strategy that depends on just the clock and a finite memory is worthless. The present paper proves that there is such a strategy that is ε -optimal. In fact, we show that just two states of memory are sufficient.

Suggested Citation

  • Kristoffer Arnsfelt Hansen & Rasmus Ibsen-Jensen & Abraham Neyman, 2023. "The Big Match with a Clock and a Bit of Memory," Mathematics of Operations Research, INFORMS, vol. 48(1), pages 419-432, February.
    • Handle: RePEc:inm:ormoor:v:48:y:2023:i:1:p:419-432
    DOI: 10.1287/moor.2022.1267
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