tengyu maxiaoming sunhuacheng yu institute for interdisciplinary information sciences tsinghua...

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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

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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

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

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

Thank You!

Reference