The Big Match with a Clock and a Bit of Memory

Kristoffer Arnsfelt Hansen, Rasmus Ibsen-Jensen () and Abraham Neyman ()
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Rasmus Ibsen-Jensen: University of Liverpool, Liverpool L69 3BX, United Kingdom
Abraham Neyman: Hebrew University of Jerusalem, Jerusalem 9190401, Israel

Mathematics of Operations Research, 2023, vol. 48, issue 1, 419-432

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.

Keywords: Primary: 91A15; secondary: 91A05; stochastic games; Markov strategies; bounded memory (search for similar items in EconPapers)
Date: 2023
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http://dx.doi.org/10.1287/moor.2022.1267 (application/pdf)

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Working Paper: The Big Match with a Clock and a Bit of Memory (2018)
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