Game Theore*c Hurricane Evacua*on Interac*ons Ms. Elizabeth A. Newell, Undergraduate Student, Industrial and Systems Engineering, University at Buffalo, [email protected]
Dr. Jun Zhuang, Assistant Professor, Industrial and Systems Engineering, University at Buffalo, [email protected]
is low, both choose<(52#:"#=&>2%1?21+@5#$&+21A(*#
not to evacuate. As the probability of the storm occurring
increases past the halfway poin
0$(%)12)3#'4$"%)0'56%78()
private citizens’ evacuate
and the government advises
evacuation.
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Abstract
Preliminary Results
The debate about emergency prepara2on and mi2ga2on has become •  Players: For the two player game, the players are the Government a cri2cal issue for emergency managers, especially in rela2on to and Private Ci2zens each with a decision probability, Y and X hurricane disasters. This research develops a model where the respec2vely. First, we calculate the expected payoffs,
weighted by the probability of the storm occurring, S, and the probability that
government and private ci2zens strategically interact in ita won’t
hurricane occur, 1-S, for both • the Objec*ves: government and the private citizens. We get four pairs of equations:
evacua2on scenario based on evacua2on advisement, and players' payoff
Government: Their objec2ve is tand
o m
damage. (1) Equation ?? is the expected
given the government
chooses Y=1
theinimize private citizens’
choose X=1:
preferences. In this model, the government moves first by advising its Private Ci/zens: Their objec2ve is to minimize cost. ci2zens to evacuate based on the probabilis2c uncertainty of disaster •  Decisions: ) = S(D1 + P1 ) + (1 S)(P2 )
UG (X,Y
occurrence, and then the ci2zens choose to react to the government's Government: They choose whether and to advise evacua2on or not with advisement based on their preferences for social guilt and evacua2on a pUrobability Y. (1)
P (X,Y ) = S(D1 + G1 ) + (1 S)(G2 ) + E1
costs. We analyze these interac2ons in order to find both the Private Ci/zens: They choose whether to evacuate or not with a (2)given Equationthe ?? is the expected payoff given the government chooses Y=1 and the private citizens’ choose X=0:
government’s and the private ci2zen’s op2mal strategy probability of X. /01#1&'2)
likelihood of a disaster occurring and each players’ preferences and !"'3&)
4,"'237'3& )A0BC'3)
UG (X,Y ) = S(D1 + D2 + P3 ) + (1 S)(P4 )
costs. !+%(+24)25#
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(a) Case;(52#<"#=&+>#$&+21?(*#.)4>
1: Governments’ Damage is High and the
Citi- (b) Case
2: Citizens’ Guilt is High and the Government’s #
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#
zens’ Guilt is Moderate
Damage is Moderate
/-probability
85-9))))))))))))/that
First, we calculate the expected payoffs, weighted by the probability of the stormand
occurring, S, and the
-8!-84-)
it won’t occur, 1-S, for both theUgovernment
and the
private
)*)get four
(S)(G
(X,Y ) = S(D
+D
+ Gcitizens.
) + (1 We
) + Epairs of equations:
(2)
2
3
and
4)
UP (X,Y ) = S(D1 + G1 ) + (1 S)(G2 ) + Eand
1
67879#
!+%(+24)25#
1
5<9)))))))))))))))))!:84<)
(1)
UP (X,Y
) =theS(D
(1 S)(G
+ Ethe
(3)
1 + G5 ) +chooses
1 private citizens’ choose X=0:
(2) Equation ?? is the expected payoff
given
government
Y=16 )and
()6)
/ 85 9))))))))))))/ 8! 84 )
- =
- &,27
6
(4) Equation ?? is the expected payoff given the government chooses Y=0 and the private citizens’ choose X=0:
UG (X,Y ) = S(D1 + D2 + P3 ) + (1 S)(P4 )
and 8 )
UG (X,Y ) = S(D15)+ D2 + P7 ) + (1 S)(P
()6)
+
7
,
2
,
&
&
)!"#$S)(G4 ) + E2and 6
UP (X,Y ) = S(D1 + D2 + G3 ) + (1
%#&'
()-.*
)
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"
"
!"
!"#$%&'(')*)+,#&-#(#./%%)0(12#300/%%)14#
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(c) Case 3: Citizens’ Guilt is High and the Government’s (d) Case 4: Citizens’ Guilt is Moderate and the GovernDamage is High
ment’s Damage is Moderate
/-8/:85?9)))))/-(2)
8/:8!:84?)
(4)
The Amermath of Hurricane Next,
we calculate the best response function to find the optimal strategy for the private citizens’.5@9))))))))))))!:84@)
UG (X,Y ) = S(D1 + P5 ) + (1 S)(P6 )
Sandy [1] If the government chooses to not advise evacuation (Y=0), the private citizens’ expected payoff can also be represented
by two cases given the probability of the hurricane:
6787:#
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#"
5>9))))))))))))))))))!-84>)
(3) Equation ?? is the expectedUpayoff
the 1government
theE2private citizens’ choose X=1:
) = S(D
+ D2 + G7chooses
) + (1 Y=0
S)(Gand
P (X,Ygiven
8) +
Path of Hurricane Irene [3] =
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"
"
!+%(+24)25#
4
2
#&'
%
$
#
"
!
5:9))))))))))))))))))!
(1) Equation ?? is the expected payoff given the government chooses
Y=1 and the private citizens’ choose
X=1: -84:)
(3) Equation ?? is the expected payoff given the government
chooses Y=0 and the private citizens’ choose X=1:
5)
/-8/:85;9)))))/-8/:8!:84;)
+,&)
!"#$
UG (X,Y ) = S(D1 + P1 ) + (1%#&S)(P
'()-.2 )
*) 6 )
UG (X,Y ) = S(D1 + P5 ) + (1 S)(P
P
Figure 3: Four Cases for Optimal Player Strategy Given Different Players’ Preferences
Future Work
and
Use real disaster data from Hurricane Irene to model the strategic interac2ons. Modeling
(
4. Conclusion
and Future
Research
Extensions
• 
Develop a 3-­‐player game with 2 types of private ci2zens that interact using social (4) Equation ?? is the expected payoff given the government chooses Y=0 and the private citizens’ choose X=0:
UP (X = 0,Y
= 0) = S(D1 + D2 + G7 ) + (1 S)(G8 ) + E2 , if X = 0
Objec2ve F
unc2ons: media. This research looks
into recent hurricane disasters and studied the various interactions that occur between the go
UP (X,Y = 0) =
(5)
S(D + G ) + (1
S)(G ) + E1 , if X =
1
P (X = 1,Y = 0) = Minimize ment and private
citizens
information
and social
media
interactions.
We tdescribed
the intera
• 
The model given
could probabilistic
also be generalized to other scenarios, such as terrorism, hreats, and •  UGovernment: Damage property, and publicity) UG (X,Y ) = S(D1 + D12 + P75) + (1
S)(P8 )6 (lives, in depth using other game ttheoretic
and modeling,
and
noted the
variousbdecisions
ypes of nprinciples
atural disasters, and also consider irra2onal ehavior oand
f the outcomes
players. of each de
given t
he c
hoices o
f t
he p
rivate c
i2zens a
nd t
he p
robability a
s
torm and
From there we calculate the best response function for the private citizens’ given the government chooses to not advise
for an evacuation scenario. We presented results of this research looking into four distinct cases: (1) where the go
will occur. U
) = S(D1 + D2 + G7 ) + (1 S)(G8 ) + E2
(4)
evacuation:
P (X,Y
UP (X,Y ) = S(D1 + G5 ) + (1 S)(G
6 ) + E1
Game
(3)
• 
ment has a high potential for publicity damage and the private citizens’ have a moderate potential for social gui
G
1
1
2
3
2 +(1-­‐X)P4)]+(1-­‐Y) where the private citizens’ have a high potential for social guilt and the government has a moderate potential for
0, 5if+(1-­‐X)(D
UP(Y=0),
(X = 0,Y
=7))+ 0) <citizens’
UP (X =expected
1,Y
=
0) 8can
If the government chooses to not[S(D
advise1+XP
evacuation
the
private
payoff
also be represented
+P
(
1-­‐S)(XP
+
(1-­‐X)P
)] References
2
6
X̂(Y
=
0)
=
(6)
licity
damage;
(3)
where
the
private
citizens’
potential
social guilt is high and the government’s potential for pub
by two cases given the probability of the hurricane:
1,otherwise
[1] Bloomberg Businessweek. urricane potential
Sandy, Osocial
ctober guilt
30, 2and
012. hnp://
•  Private Ci2zens: Minimize Cost (lives, property damage, evacua2on damage is high;
and (4) where
the privateHcitizens’
the
government’s potential for pub
www.businessweek.com/ar2cles/2012-­‐10-­‐30/new-­‐york-­‐airports-­‐shunered, Accessed in (
damage
are
both
moderate.
By
studying
these
four
cases,
we
are
better
able
to
provide
insight
into
the
optimal
de
and n
on-­‐evacua2on c
osts, a
nd g
uilt). Surfers heading out during Hurricane Similarly, If the government chooses
to =advise
(Y=1),
private
citizens’
expected
payoff
is:
UP (X
0,Y =evacuation
0) = S(D1 +
D2 + Gthe
)
+
(1
S)(G
)
+
E
,
if
X
=
0
7
8
2
March 2013. U
(X,Y
=
0)
=
(5)
P
making
strategies
for
the
government
and
the
private
citizens’
given
incomplete
and
imperfect
information
and
Satellite image of Hurricane Sandy [1] Irene [2] • 
Min UP(X,Y)= Y[S(D1+XG1+(1-­‐X)(D2+G3))+(1-­‐S)(XG2+(1-­‐X) U
(X
=
1,Y
=
0)
=
S(D
+
G
)
+
(1
S)(G
)
+
E
,
if
X
=
1
P
1
1
[2] ABC Ac2on News. Deaths caused by fallen trees, surf and acts of nature: Hurricane 5
6
(
perceived loses
intakes socialat guilt
and
publicity
damage.
Thishresearch
could have a significant impact for the govern
8 1) = S(D1 + D2 + G3 ) +[S(D1+XG5+(1-­‐X)(D2+G7))+(1-­‐S)(XG6+(1-­‐
Irene l
east 3
8 l
ives, A
ugust 2
8,2011. np://www.abcac2onnews.com/dpp/
G4)+XE1+(1-­‐X)E2]+(1-­‐Y) UP (X
= 0,Y =
(1
S)(G
)
+
E
,
if
X
=
0
4 ), if
2 Y = 0 and U (X = 0,Y = 0) < U (X = 1,Y = 0)
U
(0,0)
=
S(D
+
D
+
P
)
+
(1
S)(P
>
G
1
2
7
8
From there
we calculate
for
the
private
citizens’
given
the
government
chooses Pto not advise (7) P
UP (X,Y
= 1) =the best response function
in achievingweather/weather more optimal usenof
evacuation planning and mitigation
in the face of strategic citizens’ with social m
>
>
ews/Death-­‐by-­‐fallen-­‐trees,-­‐surf-­‐ a
nd-­‐acts-­‐of-­‐nature%3B-­‐Irene-­‐takes-­‐
X)G8)+XE1+(1-­‐X)E2] <
UP (X = 1,Y =U1) (1,0)
= UP (X,Y
+ (1), ifS)(G
1 = 0) U (X = 1,Y = 0)
1+
evacuation:
= S(D)1=+S(D
P5 ) +
(1G1 )S)(P
Y =20) +
andE1U,Pif(XX==0,Y
G
P
6
effects, uncertain
information
and players’
preferences.
fourteen-­‐lives, Accessed in March 2013. UG (X̂(Y ),Y ) =
Notation Next, we calculate the best response
function
to find the optimal
strategy
for the private
citizens’.
•  Min U (X,Y)=Y[S(D
+XP
+(1-­‐X)(D
+P ))+ (1-­‐S)(XP
(
The F=unc2ons: Best Response >
U
(0,1)
S(D1 + D2 + P3 ) + (1 S)(P4 ), if Y = 1 and UP (X = 0,Y = 1) < UP (X = 1,Y = 1) [3] The Weather Channel. Storm Coverage. hnp://www.weather.com/weather G
(
>
>
there we calculate the best response:function for the private citizens’ given the government chooses to advise
Although we only focus on social guilt in relation to social media, there are many other implications of social m
•  X=Probability of Ci2zen’s Choice to Evacuate=1-­‐Probability of the From
Ci2zen’s 0,Uif U
(X
=
0,Y
=
0)
<
U
(X
=
1,Y
=
0)
P
(1,1)
= S(D + P ) + P(1 S)(P ), if Y = 1 and U (X = 0,Y = 1) U (X = 1,Y = 1)
*+,-.,/0!
12/!*+,-.,/0!
• 
• 
• 
• 
• 
• 
• 
• 
(2+038908/!
• 
• 
G
1
1
2
P
hurricanecentral/ar2cle/tropical-­‐depression-­‐nine-­‐storm-­‐hurricane-­‐irene_2011-­‐08-­‐20, evacuation:
X̂(Y = 0) =
(6) P
in
disaster
scenarios that may impact the decisions of various players including social pressure and advice. T
Choice to not Evacuate #34+,/0!546708!
1,otherwise
(9)Accessed in March 2013. Y=Probability of Government to Advise one possible future extension of this work in expanding the current model to include different types of social m
(
12/!*+,-.,/0!
*+,-.,/0!
Using the best responses
for =
the0,Y
private
citizens’,
we
can =
use1)backward induction to solve for the government’s best Y=Probability of Government to G
Aovernment dvise Evacua2on=1-­‐Probability of tSimilarly,
he Evacua2on=1-­‐Probability of the to 0, ifevacuation
UP (X
= 1)
UP (X citizens’
= 1,Y
If the government chooses to advise
(Y=1),
the<private
expected payoff is:
interactions.
response:
X̂(Y
=
1)
=
(8)
Government to not Advise Evacua2on !
!
not Advise Evacua2on 1,otherwise
(
"#$%&%'!
"#$)&%'!
Some other possible future research extensions of this work include, but are not limited to:
UP(X,Y)=U2lity of Private Ci2zens UP (X = 0,Y8= 1) = S(D1 + D2 + G3 ) + (1 S)(G4 ) + E2 , if X = 0
!
!
Acknowledgment
UP (X,Y = 1) =
0, if {UG (X = 0,Y = 0) < UG (X = 0,Y = 1) and UP (X = 0,Y = 0) <(7)
UP (X = 1,Y = 0)
>
>
The expected"payoff
given
response
theG1private
UG(X,Y)=U2lity of Government "($%&%'!
>
UP (X = 1,Y
= 1)the
= best
UP (X,Y
) = S(Dof1 +
) + (1 citizens’
S)(G2 ) +is:
E1 , if X = 1
($)&%'! of the government
>
>
and UP (X = 0,Y = 1) < UP (X = 1,Y = 1)}
This research was supported by the United States Department of Homeland Security >
>
S=Probability of a Disaster Occurring >
!
!
>
>
through the Na2onal Center for Risk and Economic Analysis of Terrorism Events (CREATE) or
{UG (Xfor
= the
0,Yprivate
= 0) <citizens’
UG (X =given
1,Y =the1)government
and UP (X =chooses
0,Y = to
0) advise
< UP (X = 1,Y = 0)
From
there
we
calculate
the
best
response
function
>
"
$%&)'!
"
$)&)'!
>
#
#
D1=Property Damage >
>
evacuation:
under award number 2010-­‐ST-­‐061-­‐RE0001 and through the University at Buffalo Center !
!
>
< and UP (X = 0,Y = 1) UP (X = 1,Y = 1)}
D2=Loss of Human Life "($%&)'!
"($)&)'!
Y ⇤ = ( or {UG (X = 1,Y = 0) < UG (X = 0,Y = 1) and UP (X = 0,Y = 0) UP (X = 1,Y = 0)
(10)for Undergraduate Research and Crea2ve Ac2vi2es (CURCA). However, any opinions, >
findings, and conclusions or recommenda2ons in this document are those of the authors >
E1=Cost to Evacua2on > E2=Cost to not Evacua2on 0,
if U
UP(X
(X =
= 0,Y
=
1)
<
UUP (X
==1,Y
==1)1)}
>
and
0,Y
=
1)
<
(X
1,Y
>
P
P
X̂(Y = 1) = >
(8)
>
and do not necessarily reflect views of the United States Department of Homeland 1,otherwise
>
Pn=Government Publicity Damage: P1=P8=0<P6<P2<P4<P5<P3<P7 >
or {UG (X = 1,Y = 0) < UG (X = 1,Y = 1) and UP (X = 0,Y = 0) UP (X = 1,Y = 0)
>
>
Security, CREATE, or CURCA. This research was also supported by the Na2onal Science >
>
Gn=Ci2zens’ Guilt: G5<G4<G2<G1=G8=0<G6<G3<G7 >
andthe
UPbest
(X =
0,Y = 1)
1,Y = 1)}
P (X = citizens’
>
The expected payoff of the government given
response
of theUprivate
is:
Founda2on (NSF) under award #1200899. We also thank John Coles (University at Buffalo) >
:
1,otherwise
for his helpful comments.