The Big Match with a Clock and a Bit of Memory

• 成果类型: Article
• 署名作者: Hansen, Kristoffer Arnsfelt; Ibsen-Jensen, Rasmus; Neymanc, Abraham
• 署名单位: Aarhus University; University of Liverpool; Hebrew University of Jerusalem
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.2022.1267
• 发表日期: 2023
• 页码: 419-432
• 关键词: games

摘要

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 epsilon-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 epsilon-optimal. In fact, we show that just two states of memory are sufficient.