Yugoslav Journ al o f Operations Research
3 (1993), Num ber 1, 11 -1 9
SEARCH GAME ON AN ODD NUMBER OF ARCS
WITH IMMOBILE HIDER
Ljiljana PAVLOVIC
Faculty of Natural Sciences and Mathematics, D epartment of M athematics,
34000 Kragujevac, Yugosla via
AbstracL The search game for a n immobile hider .o n an odd num ber o f a rcs o f e q ua l
lengths, connecting two points is so lve d. The result o bta ined here so lve the prob le m which
has been open since 1980.
Key words: search game, graph, zero sum.
AMS (MOS) Subject Classification, 90005.
1. INTRODUCTION
Let Q(k ) be a set o f k no nintersec ting a rcs b( 1),b(2),... ,b(k) of the un it le ngt h
which join two points 0 and A, a nd k i a n odd number gre ater th an I. Th e h ider chooses
any point H on one of th e arcs a nd hid es a t th is point - th at is a p u re stra tegy of the hider.
A pure stra te gy of the sea rche r is a co ntin uo us traje cto ry S with ve loci ty no t exceeding I (/\
is th e starting p oint). Since the hider is immobile it is obvious th at th e searc her w ill a lway
-
use his maximal velo city 1. The payoff c(S, H) to th e hid er i th e time spe nt unt il th e h ide r i
cap tured. /\ m ixed sea rc he r stra tegy s is a ra ndo m cho ice a mo ng his pure strategies, while a
•
hider mixed strat egy It is a p ro bab il ity meas u re in the gra ph Q, th at is to say a mixed stra tegy
s (It) of th e searcher (h ider) is a regul ar Bo rel pro bab ility me asu re on the se t of a ll p ure
stra te gies. If th e pl ayer usc m ixed stra tegic, th en th e cap ture time is a random variable
a nd th e p ayo ff be com es a n expec te d co st C(5, 11 ). T he maximal expec ted cos t v(s) of using a
•
sea rc h stra tegy
S :
I'(s) = sur c ( s ,h ) = su pc(s, II ) will be ca lled "the va lue o f s tra tegy v",
h
- - - --------- -
Co mmunica ted by V.Kovacevic· Vujcic
If
Lj.Pavlovic
12
and
v( h)
minimal
the
v(lt)
cost
expected
of
•
USlDg
hider
a
strategy
It:
= in f c(s, h) = inf c( S, h) will be called "the value of strategy It". If there exists a real
s
S
number v which satisfies v = infv(s) = supv(h), then we say that the game has a value v.
s
h
G al has proved [4] that any search game on a graph described above has a value and an
optimal search strategy.
Gal showed in his book [4] that the value of this game satisfies: k/2 < v < k/2
+ 1/2,
where the hider can guarantee the lowest value by using the completely randomized
stra tegy, while the searcher can guarantee the upper value by going along a random
per mu ta tio n of the arcs. He considered the search game on three arcs and a set of
trajectories SCi, j, o.), where i and j are two distinct integers in the set {l, 2, 3} and 0 < ex. < 1.
The trajectory sCi,
i. ex.) starts from A,
moves along arc b(i) to 0, moves along arc 00) to the
poin t A (ex.) which has a distan ce of ex. from 0, moves back to 0, moves to A along b(m),
whe re
111 E
{ 1, 2, 3 } \ {i, j}, a nd then moves from A to A( «) along bO). The distance ex. is a
random va riable. He also showed that the search strategy s' (3) from the set of strategies
SCi, j, ex.), where the sea rc he r chooses equiprobably i
rando m va riable with th e foll owing distribution
E
{I, 2, 3 }, j
function
E
{1, 2, 3} \ { i } and ex. is a
F:• F • (o.) = 0 for ex. < 0,
F· (ex.) = 1/2 + 1/4 exp(ex.) for 0 <ex. < In 2 a nd F· (ex.) = 1 for ex. > In2, is optimal among search
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stra tegies of the similar type . T he sea rcher ca n gua rantee the value (4
+
In2) / 3 by using
th e stra te gy s'(3). On th e o the r hand , the hider ca n guarantee th e same value by using th e
stra te gy It' (3), nam ely, he makes eq u ip ro bably ch oice of an arc b(i) , i
a t the po int o f b(i ) wh ich has a distance
the probabili ty density 2 exp( - P) fo r 0 <
p from
0 , wh ere
p is a
E
{l, 2, 3}, and hides
random variable which has
P < In2 a nd ze ro othe rwise.
Bostock [3 J consid ered th e th ree a rcs ga me as a limit ing case of a dis crete gam e o n
•
threc a rcs a nd sho we d that s (3) is op tima l a mo ng a wider class of sea rc h stra tegies.
As for the case wh en k is eve n ( the Eu leria n case), Ga l showed in his book [41 th at
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th e va lue o f this gam e is k/2. Th e sea rc her ca n guara ntee th i va lue using th e strate gy s' (k),
whe re the sea rche r choos es e q u ip ro ba bly a ny pe rmutati on be twee n th c a rcs (fo r exa mp le :
bU I ) , b(i 2 ) ,
.. .
,b(i k ) ) a nd goes a lo ng th cm fro m o ne end poi nt to a no the r (thc searcher
starts fro m th e poin t A, mo ves a lo ng the a rc b(i ! ) to th e po int 0 , moves a lo ng th e arc b(i~ )
~
to A an d so o n) . The hid er ca n guaran tee th i. va lue u in g co mplete ly ra ndom stra tegy .
13
Sea rch game with immobile hider
2. OPTIMAL STRATEGIES
We will prove that s"(3) and h' (3) a re th e optimal stra tegies with out a ny
restriction and that the optimal strategies for a game on an odd number of a rcs a re almos t
the same as for the game on three arcs. Namely, the hider optimal stra tegy is It· (k), wh e re
he equiprobably (11k) chooses an arc and hides at the point of this are, wh ich has a distance
p from
0, where
p is a random variable with
the probability density 2 exp ( - P) for 0 < P < In2
and zero otherwise. As for the searcher, he can divide this game on two ga mes: on the ga me
on an even number of arcs (k - 3) and the game on three arcs. First, he uses optimal
,
strategy for a game on an even number of arcs and then strategy s (3). In other words, the
searcher chooses any permutation from th e set {b( 1), b(2), ... ,b(k)}, fo r exa mple b(i I)'
b(i z), b(i3)' ... ,bUk- 1) , b(ik)· Th e sea rcher sta rts from the point A, moves along the a rc bUI)
°
to 0, moves along the arc bUz) to A, then moves along the arc b(i 3) to
a nd so o n unt il the
arc b(ik_l) (the last but one). Then he goes along the arc b(ik_l ) to the poi nt A (a) which has
a distance a from 0, returns to
° a nd goes along the arc bU
k)
to A a nd fin ally moves along
the arc b(i k _ 1) from A to A (a ). The di stan ce a is a random va riab le which has F' (a ) as the
distribution function. Let us denote this stra tegy by s' (k). Th e va lue of thi ga me is:
v(k) = (k - 3)z 12k + 3 [k - 3 + (4 + In 2) 13] I k = (k z - I + 2 In2) 12k ( I)
l)Il~REM 1. The strategies s' (k) and It' (k ) a re o ptima l in th e ga me o n Q(k) without a ny
restriction a nd th e value is (k 2 - I + 2 In 2) / 2k.
PROOF.
In o rde r to prove th at we o ught to show:
c(s ' (k) , H) s v(k),
,
c(S, h (k)) > v(k) ,
VH
E
"IS
E
TH(Q (k ))
(2)
TS (Q (k ))
(3)
where TS (Q(k)) ( TH(Q (k) ) ) is the se t o f all pure sea rc he r (h ider) stra tegies o n Q(k) .
Let th e hid er use his pure stra tegy H , nam ely he choose o ne arc (fo r exa mp le, the
a rc b(2)) a nd hid es a t the poi nt whic h has a d ist a nce x from
° and let the searcher usc his
mixed stra tegy s' (k). Th is a rc (b(2)) ca n occ upy the firs t, or the second , or th e third ,..., or
the last place in an y permutat ion from th e se t {b( I), b(2), ... ,b(k) }. The num be r of pure
sea rche r stra tegies bel ongin g to s , (k), wh ich choose this a rc on ilh place is (k - I ) ! a nd
tak in g in acco u nt th at the probabi li ty mass of every pure sea rc he r stra tegy enterin g
s' (k) is k !, we have :
•
Lj.Pavlovic
14
c(s' (k ), H ) = (1 - x )(k -1)!+(1 + x)(k - l )!/ k !+
(3 - x )(k -l)!jk! + (3 + x)(k -l)!/k! +
(5 - x )(k -l)!jkl + (5 + x)(k -l )!jk !+... +(k - 2 - x)(k -l)!jk! +
•
f[o,x )(k - 2 + 2y + 2 - x )dF" (y) + f[x,'n2j(k - 2 + x )dF" (y) +
j(k-2+2y+x)dF"(y)
(k-l)!jkl
JI O,ln 2
r[
= ((e
- 3)/2 + f[o,x )(2y + 2 - x )dF" (y ) + f[x,ln2]xdF" (y) +
2f[o,ln2]ydF"(y)
k
•
o
for 0 < x < In 2
- 1/ k
for In 2 < x
Thus, it is sufficient to calculate c(s" (k),H) for x =
E,
where
E
is small and
ca lcu latio n re ad ily s hows th at:
c(s' (k) ,ri) s c(S"(k ), E) =
(e- 3)/ 2+ 2(3/4)+2f[o,ln2] ydF' (y)
(k
k=
(2')
2
- 1+2 In2)/2 k
In order to prove c(S, II '(k)) ~ v(k ) "IS, we will use the fact (which we will prove
la te r) th at th e probabi li ty d en ity fun ct ion II' (k+ I) = 2exp( - x) / (k+ 1) , 0 < x < In 2 a nd 0
otherwise a long each arc is op tima l mixed s tra tegy for th e hide r in th e game on k + 1 arcs (k
i an od d num ber) both if th e searche r sta rts fro m 0 o r I\. Let us take a ny pure . ea rc h
s tra tegy S
E
TS(Q(k)) in th e ga me o n k arcs. W e ass oc ia te with th e strategy S on e s tra tegy
S· E TS(Q(k+ I )), wh ere the earc he r s tarts fro m 0 a nd mo ves a lo ng o ne arc ( t he firs t. for
exa mp le) to A an d the n moves fu rt her according to the s tra tegy S. ln
sea rc he r, arrived a t A, plays the game
0 11
uch a way the
k a rcs. Since II ' tk + I) is o p ti ma l stra tegy for the
hid er on Q(k+ 1), we have c(::/ ,II ' (k+ I)) ~ (k+ I ) / 2 a nd :
•
Search game wi th immobile hider
15
(k+ l)/2< c(S' ,h' (k +1 )) =
f[o,ln 2]xh' (k + l )dx + k{ 1 + c(S,h'( k))} j(k + 1) =
{t -ln2+k(1 + c(S,h·(k)))} (k +l )
therefore
\;IS E TS(Q(k))
(3')
3. OPTIMALITY OF h•
It remains to prove the former fact, namely that It' (k+ 1) is optimal strategy for the
hider in the game on Q(k+ 1) (k is an odd number). The graph Q(k+ 1) is the union of
(k + 1) / 2 graphs Q(2) joined at two po ints 0 and A There fo re, to p rove o ptima lity of
Jt·(k+1) we will use Theorem 2 from [5], which we cite in a mod ifi ed for m:
1llEOREM
e
2 [5]. The value of strategy It (k+ l ) o n the grap h Q(k+l) is (k+ l) /2 times
greater or equal than the value C o f strategy It· (2) o n Q(2) un de r co ndi tio n th at: 1. the
searcher ca n start from 0 or A and he ca n jump from o ne e nd poi nt to an othe r, but o nly
afte r returning to the initial point; 2. in f 1";/ P; ~ 2 C, whe re infimum takes p lace over all
~
possible co mbinatio ns o f th e parts
(?j
of the trajecto ries, which begin at 0 (A) and end at 0
(A) o r whic h begin at 0 (A) and end at A (0).
•
L EMMA
1. T he strategy It· (2) guarantees the valu e ~ 1 un der condi tio ns th at the searcher
sta rts from th e point 0 (A) .
(4)
where TS o(Q(2)) ( 1'SA(Q(2))) is the set of all pure searc her trajecto ries on Q(2) where the
searc her sta rts fro m 0 (A).
P RO O F.
At fi rst, we will prove o ptima lity o f h' (2) in the ga me on two arcs und e r condi tion
th at th e sea rc he r sta rts from the po int O. To simp lify ou r no tat io n we will de no te It' (2) by
•
h .
•
Lj.Pavl ovic
16
We will divid e the se t of a ll pure stra tegies TS o(Q(2)), where the searcher sta rts
from the poin t 0 , ( the case where the sea rcher starts from the point A is similar) in the
fo llowing way: TS o (Q(2) ) = TS oO(Q(2)) u TSO l (Q(2)) u ... u TS on(Q(2)) u ... . D enote by
A (e ) a point in Q(2) which has a distance e from O. The strategy S" belongs to the set
TS o(Q(2)) iff the searche r, who starts from the point 0, makes exactly n loops, for example,
the searcher starts fro m 0 , moves along the first arc to the point A( a l )
(
0
s al s
In2),
comes back to 0 (the first loop), then moves along the other arc to the point A(13 l ) , comes
back to 0 (the seco nd loop), moves along the firs t a rc to the point A(~)
(a l
comes back to 0 (the third loop), moves along the second arc to the point
A(13 2) (13 1 < 13 2
::;
a 2 < In2),
::; In2), comes back to 0 and so on the searcher moves up to the point A(a(n +l)n) (if n is
odd number a nd a l
::; ~::; ...
a(n+l)n::; In 2) or A (13 nn ) (if n is even number and
13 1 ::; 13 2 ::; ...
13 nn::; In2), co mes back to 0 and then moves along the other arc to A and continues moving
a long the first arc to the poi nt A (a (n +l )n) o r A (13 nn) . We have neglected unreasonable
trategies where the searcher move twice along the same arc successively, for example, he
moves to the point A(a t) , co mes back to 0 a nd moves along the same arc to the point
A( ~ ). We ca n eas ily
al <
~
cc th at eve ry reaso nable stra tegy must sa tisfy the co nd itio ns:
< ... an < ... < In2 a ncl 13 1 < 13 2 < ... 13 n < ... < In2. We will prove c(S,It °) > 1
VS E TS on(Q(2)), VII by induc tio n.
Let us tart wi th the stra tegy SO from the e t TS o o(Q(2)), wh ere the searcher
moves simply along o ne arc to A and then a long the other arc to O.
c(SO, h' )= J~n 2 t cxpl - t )dt + J~n 2 (2 - t )exp(- t )dt
(5)
Let us rega rd any stra tegy fro m the se t TS O I(Q(2)) , where the sea rche r sta rts from
0 , moves a long o ne a rc to the poin t A( a l ) , co mes bac k to 0 and moves alo ng the o the r arc
to A and along the fir st arc to the poi nt A(a l ) .
(6)
In the sa me way, for a ny S2 - TS 0 2(Q (2) ) , whe re the searcher .turts fro m 0 ,
moves along one arc
10
A (a l ) , co mes back to 0 and the n moves alo ng the fir. I ar
co nti nues moving along the seco nd a rc
10
A (/\I ) ' we have:
10 1\
a nd
•
17
Searc h ga me wit h immo b ile hider
C(S2,
h" )= J;1t expl - t )dt + J:l(2 0.1 + t ) exp( - t )dt +
f
n2
0.
1
1n2
( 2 0.\ + 2B1 +t )exp(-t )dt + Jp
r ( 2 0.1 + 2 B1+2 -t )exp(-t )dl =
l
2exp( -0.1 )(0.1 + P1 )+ 1- 2 B1
ac / a~l
= 2 (expj--cc.) - 1) < 1
and c(S2,ho) attains min at ~1 = In2
c(S2,h·) > 2 exp (-0. 1) (0.1 + In2) + 1 - 2 In2 ~
(because this function is concave at 0.1)
~
(7)
1
Let us suppose that c(S,h·) ~ 1,\1'S E TS o o(Q(2» , TS 0 1(Q(2» , ... , TS0 2n- 1(Q(2» ,
for example S2n-l:= (0.1' ~1' 0.2, ~2' ~, ~3' ... , an_I ' ~n-l ' an)' we have:
c(s2n-l ,ho) =
131
10
5: t exp] - t )dt +
1
( 2 0. 1 + t ) exp( - i )dt + t~(2al
+
2P
+
t
)
exp(
- t )dt +
1
Cl
1
132
rJI3I (2 0. 1 +2P1 +20.2 +t )exp(-t)dt+... +
t · (20.
a"ol
1+2P1
(8)
+2 0. 2 + · · · +2a n_1 +2Pn.1 +t)cxp(- t) dt+
1'n2( 2 0. 1 +2 P1 + 2 0. 2+ · .. +2 a n_1 +2P n-1 +20. n+ t) exp(- t) dt +
13 . ·1
t
2
0..
(2 a l + 2P I +20. 2 +· · ·+2a n_1 + 2P n-1 +20. n +2 - t )cxp(-t) dt
=
= 2 exp( -0.1 )(0.1 + B1) + 2 CXp(- PI )( PI + 0.2) +
•
2eXp( -a2)(0.2 +P2 )+ '" +2 CXP(- P,, _I)(P n.1 +0.,.)+
1 - 2 P I - 2 0.2 - 2P 2 - ... -2P n_1 - 2 0." > 1
o). We have:
n
c(
_
s" ,h") =
c (S 2,, - 1
f
n2
0..
,h")+ r·P.,l (2 0.1 + 2 PI + 20.2 + ... +20.,. + t)expl - I )dt +
( 2 0. 1 + 2P I + 20. 2+ ... +20. ,. + 2 p " +t) exp(- t)dt +
J~~2 (20.\
+ 2PI + 2 0. 2 +
+20.,. + 2 p" + 2
-I) exp( - t )dt -
r (20. \ +2P I + 20.2+ +20.,. +t) exp(- t) dtn2
t ( 2 0. +2P I + 20.2+ ... +20." +2- t)exp(- /) dt =
n2
P•.1
•
1
C( S:'.,,-I , h ")+ 2 ex p(-0.,.)(o.n +pJ- 2P,. > 1
(9)
•
Lj .Pavlovic
I
W e have to prove also that if the result holds "IS
TS o2n(Q(2)), then it holds for S
where S 2n + l : = (0.1'
E
E
TS oo(Q(2)), TS O l (Q(2)), ... ,
TS o2n +l (Q(2)). Let us choose
S2n+l E
TS 0 2n +l (Q(2)),
PI' <X:z, P2, ... , an' Pn , a n + l ) . In the same way, we have:
(10)
Thus,
c(S, h -)~ l .
"IS E TS o(Q(2))
(11)
We ca n prove in the same way, using mathematical induction, that the strategy his op timal in the ga me on two arcs under the condition that the searcher starts from the
point A . The best behavio r for the searcher is to move along one arc to 0 and then along
the other arc to A. All o the r strategies give math ematical expectation greater th an 1. Using
the simi lar notatio n , we have:
The proof is similar to that for the case when the searcher starts from the point O.
Lemma 2. The stra tegy It -(2) guarantees the va lue ~ 1 under co nd itio ns that the searcher
can jump from one end point to ano the r, but o nly after returning to th e initial po int.
,
1 = min
-inf
SeTSo (Q(2 ))
where
c(S, h"(2)), _i nf
TSo{Q(2)) ( TS A{ Q(2)) ) is
SeTS A (Q(2 ))
c(S, h"(2))
( 13)
a se t of all pure searcher traje cto ries o n 0 (2), wh e re
the earcher tarts from 0 (A) and it is allowed to him to jump fro m o ne e nd po int 0 o r A
to ano the r one, but on ly after returning to the initia l point.
We will o mit the proof of Lemma 2. because it i
irni lar to the one of Lem ma I.,
but more po sibi litie have to be co nsidered.
It is easy to se e that the stra tegy It" sa tisfie co ndi tion 2 of T heorem 2.
20./f: exp( - x )dx = 2a/{ 1 - expl - a )) _ 2
( 14)
7j /Pj ~ 2 ~/f::~exp( -x)dx =2B/( - 1/2 e 'p( -1+1~)) ~2
( 15)
7ie / r,
( 16)
7; / I~ ~
>
1/1 /2 = 2
•
Search game with immobi le hid er
19
Thus, th e value of stra tegy It°(k + 1) (k is odd number) is grea ter or eq ua l (k+ 1)/2
o n the base of Theorem 2, which means that the stra tegy It "(k+ I) is optimal in the gam e o n
Q(k + 1).
REFERENCES
[1] Alpern,S., and ASic,M ., "The search value of a netwo rk", Networks, Vol.15, (1985),
229-238.
[2] Alpern,S., and ASic,M ., "Am bush strategies in search game
on graph ", SIAM
I .Control Optim., Vol. 24, ( 1986), 66-75.
[3] Bostock,F.A., "O n a discre te se arch problem on three arcs", SIAM J.Alg.Disc.Math. ,
Vo1.5, (1984), 94-100.
[4] Gal,S., Search game, Academ ic Press, New Yo rk, ( 1980).
[5] Pavlovic.Lj., "Search game on the un ion of n ide ntical graphs joined at one or two
points", YUJOR,VoI.3, No.1, (1993), 3-10.
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•