tengyu maxiaoming sunhuacheng yu institute for interdisciplinary information sciences tsinghua...
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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
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
-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
solve the case from given -RP of :
Hard cases: idea and example(1)
𝑉 1
𝑉 2
000100
010 110
111011
001 101
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