A New Variation of Hat Guessing Games

1. A New Variation of Hat Guessing Games Tengyu Ma Xiaoming Sun Huacheng Yu Institute for Interdisciplinary Information Sciences Tsinghua University Institute for Advanced Study, Tsinghua University Institute for Interdisciplinary Information Sciences Tsinghua University

2. Hat guessing puzzle • 3 cooperative players • each is assigned a hat of color red or blue • each can only see others’ hat • guess own color or pass • players win if: at least one correct and no wrong guess • goal : to maximize winning probability

3. Hat guessing puzzle • strategy1: only a pre-specified player guesses randomly winning prob. = • strategy2: if other two have same color, guess the opposite, otherwise pass. winning prob. = • is optimal pass pass

4. General hat guessing game • cooperative players: • coordinate a strategy initially • assigned a blue or red hat • uniformly and independently • guess a color or pass • winning condition: • at least correct guesses and no wrong guess • goal: to maximize winning prob.

5. Previous Study • case is well studiedby [?], [?].. • Observation 1: randomized strategy does not help • Observation 2: related to the minimum -dominating set of Definition: A -dominating set for a graph is a subset of , such that every vertex not in has at least neighbors in

6. reduce -DS to strategy design pass pass win! lose pass pass

7. reduce -DS to strategy design(2) pass pass win! lose pass pass winning point losing point

8. reduce -DS to strategy design(3) pass pass win! lose pass pass winning point has at least losing points as neighbors

9. Simple Facts • all losing points • is -dominating set of • winning prob. = • reduction can be done vice versa • by counting argument: • winning prob.

10. Main Theorem • Theorem: • There exists a -dominating set of size , as long as is an integer, for large enough (. • It follows that there exists a strategy of the hat guessing games with winning prob. • theorem is not true for small • example:

11. Perfect -dominating set each has neighbors in each has neighbors in

12. -regular partition of each has neighbors in each has neighbors in

13. Easy and hard cases • -DS of -RP of • possible -RP of : • the parameters are of the following form • possible -DS corresponds to the case • easy case • hard case ,

14. From easy to hard • from the cases to -- nontrivial, [?] • from to

15. Hard cases: idea and example(1) • solve the case from • given -RP of : 111 011 101 001 110 010 000 100

16. Hard cases: idea and example (2) • now construct -partition for • for each • sys. of equations over • , the collection of solutions of • is an independent set

17. Hard cases: idea and example(3)

18. find a perfect matching in • cut each black set by an additional eqn. • for and use eqn.: 6 = 2 * the index of the different bit

19. find a perfect matching in • cut each black set by an additional eqn. • for and use eqn.: 2 = 2 * the index of the different bit

20. all the grey points ,. • is a -RP of • this idea is extendable to general cases

21. Recap • Main contribution: • foy any odd , and , when , there exists a -regular partition of • particularly, it follows that for large enough , there exists -dominating set of size , as long as is integer.

22. Thank You!

23. Reference