a new variation of hat guessing games
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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. - PowerPoint PPT PresentationTRANSCRIPT
A New Variation of Hat Guessing GamesTengyu Ma Xiaoming Sun Huacheng YuInstitute for
Interdisciplinary Information Sciences Tsinghua University
Institute for Advanced Study,
Tsinghua University
Institute for Interdisciplinary
Information SciencesTsinghua University
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
Hat guessing puzzle
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
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.
General hat guessing game
case is well studied by [?], [?].. Observation 1: randomized strategy
does not help Observation 2: related to the minimum
-dominating set of
Previous Study
Definition: A -dominating set for a graph is a subset of , such that every vertex not in has at least neighbors in
win! losepass
pass pass
pass
reduce -DS to strategy design
win! losepass
pass pass
pass
winning point losing point
reduce -DS to strategy design(2)
win! losepass
pass pass
pass
winning point has at least losing points as neighbors
reduce -DS to strategy design(3)
all losing points ◦ is -dominating set of ◦winning prob. =
reduction can be done vice versa by counting argument:
◦ winning prob.
Simple Facts
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:
Main Theorem
Perfect -dominating set
{0,1 }𝑛∖𝐷𝐷
each has neighbors in
each has neighbors in
𝑉 1𝑉 2
each has neighbors in
each has neighbors in
-regular partition of
-DS of -RP of possible -RP of :
◦the parameters are of the following form
possible -DS corresponds to the case
easy case
hard case ,
Easy and hard cases
from the cases to -- nontrivial, [?] from
to
From easy to hard
solve the case from given -RP of :
Hard cases: idea and example(1)
𝑉 1
𝑉 2
000 100
010 110
111011001 101
now construct -partition for for each sys. of equations over , the collection of solutions of
◦ is an independent set
Hard cases: idea and example (2)
{0,1 }6=𝑠𝑜𝑙 (𝐸000)∪𝑠𝑜𝑙 (𝐸001)∪…∪𝑠𝑜𝑙(𝐸¿¿111)¿Hard cases: idea and example(3)
𝑉 1
𝑉 2
𝑠𝑜𝑙(𝐸011)
𝑠𝑜𝑙(𝐸001) 𝑠𝑜𝑙(𝐸101)
𝑠𝑜𝑙(𝐸100 )
𝑠𝑜𝑙(𝐸010)
𝑠𝑜𝑙(𝐸000)
𝑠𝑜𝑙(𝐸000)
𝑠𝑜𝑙(𝐸110)
𝑉 1
𝑉 2
𝑠𝑜𝑙(𝐸011)
𝑠𝑜𝑙(𝐸001) 𝑠𝑜𝑙(𝐸101)
𝑠𝑜𝑙(𝐸100 )
𝑠𝑜𝑙(𝐸010)
𝑠𝑜𝑙(𝐸000)
𝑠𝑜𝑙(𝐸000)
𝑠𝑜𝑙(𝐸110)
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
𝑉 1
𝑉 2
𝑠𝑜𝑙(𝐸011)
𝑠𝑜𝑙(𝐸001) 𝑠𝑜𝑙(𝐸101)
𝑠𝑜𝑙(𝐸100 )
𝑠𝑜𝑙(𝐸010)
𝑠𝑜𝑙(𝐸000)
𝑠𝑜𝑙(𝐸000)
𝑠𝑜𝑙(𝐸110)
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
𝑉 1
𝑉 2
𝑠𝑜𝑙(𝐸011)
𝑠𝑜𝑙(𝐸001) 𝑠𝑜𝑙(𝐸101)
𝑠𝑜𝑙(𝐸100 )
𝑠𝑜𝑙(𝐸010)
𝑠𝑜𝑙(𝐸000)
𝑠𝑜𝑙(𝐸000)
𝑠𝑜𝑙(𝐸110)
all the grey points , . ◦ is a -RP of
this idea is extendable to general cases
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.
Recap
Thank You!
Reference