TY - JOUR

T1 - The Big Match with a Clock and a Bit of Memory

AU - Hansen, Kristoffer Arnsfelt

AU - Ibsen-Jensen, Rasmus

AU - Neyman, Abraham

N1 - Publisher Copyright:
Copyright: © 2022 INFORMS.

PY - 2023/2

Y1 - 2023/2

N2 - 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.

AB - 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.

KW - bounded memory

KW - Markov strategies

KW - stochastic games

UR - http://www.scopus.com/inward/record.url?scp=85152196777&partnerID=8YFLogxK

U2 - 10.1287/moor.2022.1267

DO - 10.1287/moor.2022.1267

M3 - Journal article

AN - SCOPUS:85152196777

SN - 0364-765X

VL - 48

SP - 419

EP - 432

JO - Mathematics of Operations Research

JF - Mathematics of Operations Research

IS - 1

ER -