ECCC-Report TR21-096https://eccc.weizmann.ac.il/report/2021/096Comments and Revisions published for TR21-096en-usFri, 09 Jul 2021 00:32:56 +0300
Paper TR21-096
| Keep That Card in Mind: Card Guessing with Limited Memory |
Boaz Menuhin,
Moni Naor
https://eccc.weizmann.ac.il/report/2021/096A card guessing game is played between two players, Guesser and Dealer. At the beginning of the game, the Dealer holds a deck of $n$ cards (labeled $1, ..., n$). ForÂ $n$ turns, the Dealer draws a card from the deck, the Guesser guesses which card was drawn, and then the card is discarded from the deck. The Guesser receives a point for each correctly guessed card.
With perfect memory, a Guesser can keep track of all cards that were played so far and pick at random a card that has not appeared so far, yielding in expectation $\ln n$ correct guesses. With no memory, the best a Guesser can do will result in a single guess in expectation.
We consider the case of a memory bounded Guesser that has $m < n$ memory bits. We show that the performance of such a memory bounded Guesser depends much on the behavior of the Dealer. In more detail, we show that there is a gap between the static case, where the Dealer draws cards from a properly shuffled deck or a prearranged one, and the adaptive case, where the Dealer draws cards thoughtfully, in an adversarial manner. Specifically:
1. We show a Guesser with $O(\log^2 n)$ memory bits that scores a near optimal result against any static Dealer.
2. We show that no Guesser with $m$ bits of memory can score better than $O(\sqrt{m})$ correct guesses, thus, no Guesser can score better than $\min \{\sqrt{m}, \ln n\}$, i.e., the above Guesser is optimal.
3. We show an efficient adaptive Dealer against which no Guesser withÂ $m$ memory bits can make more than $\ln m + 2 \ln \log n + O(1)$ correct guesses in expectation.
These results are (almost) tight, and we prove them using compression arguments that harness the guessing strategy for encoding.
Fri, 09 Jul 2021 00:32:56 +0300https://eccc.weizmann.ac.il/report/2021/096