M.Sc Student | Dagan Yuval |
---|---|

Subject | Twenty Questions Game Using Restricted Sets of Questions |

Department | Department of Computer Science |

Supervisor | Dr. Yuval Filmus |

A basic combinatorial
interpretation of Shannon's entropy function is via the "20
questions" game. This cooperative game is played by two players, Alice and
Bob: Alice picks a distribution \pi over
the numbers \{1,\ldots,n\}, and
announces it to Bob. She then chooses a number x according to \pi, and Bob attempts to identify x using as few Yes/No queries as
possible, on average.

An optimal strategy for the "20
questions" game is given by a Huffman code for \pi: Bob's questions reveal the codeword
for x bit
by bit. This strategy finds x using
fewer than H(\pi) questions
on average. However, the questions asked by Bob could be arbitrary. In this
paper, we investigate the following question: Are there restricted sets of
questions that match the performance of Huffman codes, either exactly or
approximately?

Our first main result shows that for every
distribution \pi, Bob
has a strategy that uses only questions of the form "x < c?" and "x = c?", and uncovers x using at most H(\pi) questions on average,
matching the performance of Huffman codes in this sense. We also give a natural
set of O(rn^{1/r})questions
that achieve a performance of at most H(\pi), and
show that \Omega(rn^{1/r}) questions
are required to achieve such a guarantee.

Our second main result gives a set \mathcal{Q} of 1.25^{n(n)} questions such that for
every distribution \pi, Bob
can implement an optimal strategy for \pi using
only questions from \mathcal{Q}. We
also show that 1.25^{n-o(n)} questions
are needed, for infinitely many n. If we
allow a small slack of r over
the optimal strategy, then roughly (rn)^{\Theta(1/r)} questions
are necessary and sufficient.