• Be careful to identify if we’re talking about gains or costs (because the search cost is always negative)
• Check carefully if the payoff cost allows you to use advance formulation of the original model

• You have 5 boxes on the table
– Each box can generate a random number from a uniform distribution (0,100)
– In order to activate a box you need to feed it with a
$1
– In addition, in order to approach the table you need to pay $2
– You can approach the table up to 5 times, and activate as many boxes as you want each time
– Your revenue will be the maximum prize revealed

• Show that 81 is an upper bound for your expected revenue

• Searching sequentially, with a search cost of c=1 and an infinite decision horizon will always yield a better expected net revenue thus this is an upper bound:
V ( x )
  c
 y


 x yf ( y ) dy

F ( x ) V ( x )
 x x

1
 y
100

 x
0 .
01 ydy

0 .
01 x * x

85.86
 x
Good, but not good enough… x

114 .
14

Parallel Search is an even tighter bound (if infinite horizon)…
• We know that:
V
F
N f
N
( x )
( x )
( x )



 c ( N )
 y


 x yf
N
( y ) dy

F
N

F ( x )

N 

0 .
01 x

N
N

F ( x )

N

1 f ( x )

0 .
01 *
( x ) V ( x )
 x
N

0 .
01 x

N

1

N

2

100 y

 x y 0 .
01 * N

0 .
01 y

N

1 dy


0 .
01 x

N x
 x
N=1: x=75.5
N=2: x=79.2
N=3: x=80.5
N=4: x=80.95
N=5: x=81

רשאכ , םידמעומ n=5 םע הריכזמה תייעב הנותנ
שופיחה תא ונמייס םא םג , תירוקמה היעבל דוגינב
ירה רתויב בוטה דמעומה תא ונאצמ אלו
היהי ןתינו יונפ היהי ןיידע אוה α תורבתסהבש
ותוא רוכשל
?
α=0.2
רשאכ שופיחל תילמיטפואה היגטרטסאה יהמ –
.
תינש רותפו ךנוצרכ α ו ) 10 מ לודג ( N ךרע רחב –
•

 n
• n
  best applicant?
– For r=1 or r=n, 1/n
– For r>1:
 j n 
 r
P

 j th applicant and you select is it best


 j n 
 r
1 n

 r j

1

1


 r n
1 j n 
 r

 j
1

1

 max of r-1 > max of j-r the probability that the maximum number in a sample of j-1 is one of the first r-1 numbers timeline n

• The revised equation:
A
 n
 n
1 j n 
 r

 j
1

1


 r n
1

1
2 r 1/(j-1) sum(1/(j-
1)) (r-1)/5 A (r-1)*a/n
 n
0.2
1 2.08
0.2
0.42
0.04
0.46
3 0.5
1.08
0.4
0.43
0.08
0.51
4 0.33
0.58
0.6
0.35
0.12
0.47
5 0.25
0.25
0.8
0.2
0.16
0.36

• Just choose α=1 and the probability of ending up with the best equals 1…

• Assume you are a comparison shopping agent and a buyer offers you sqrt(x) dollars for each saving of x dollars below a price of $50 for a product
• You have 3 stores to look at: f(x) f(x) f(x)
0
Store A c
A
=1.3
100 10
Store B c
B
=0.5
90 20
Store C c
C
=0.3
• How will you search and what is the probability of getting to store C?
80

c i
  x z i


50
 x i
 
50
 z i

 f i
  i dx i
Seller
A
B
C fi(x)
1/100
1/80
1/60
And the rule: go to store with minimal zi unless found a price below zi

c i
  x z i


50
 x i
 
50
 z i

 f i
  i dx i
Seller
A
B
C zi
45.23
36.98
37.41
• Optimal strategy: start with B, if price above
37.41 then go to C and if best price is above
45.23 then go to A

P

P ( B

37 .
41 )

0 .
66

• You go to NY for a month
• There are N pubs you might find attractive
• Each pub has a different a-priori distribution of value for you and different exploration cost
• Your performance will be the accumulated value
• What is the optimal strategy?

• All search should be conducted during first day.
• Cost to be used is c/30 c / 30
  z
  y
 z
   dy

• We have two boxes:
– One with payoffs $10 with probability 0.99 and $K with probability 0.01. Cost of opening is $1.
– The other with probability 0.5 has a prize of 100$ and with probability 0.5 has a prize of $200. its cost for opening is c->0.
– What should be the value of K so that the first box will be opened first

• RV of second box is 200
• RV of first box is:
(K-200)*0.01=1
0.01K=3
K=300 c
  z
  y
 z
   dy

דבעמ לכ יונב םידבעמ תבורמ בשחמ תכרעמב
תוביל יתשמ
הביל לכ לע
)
) הנתמה ןמז
למרונמ , 0<x<1
( יעגרה סמועהש עודי
רובע ( f(x)=2x גלפתמ
לש
הדימבו
הינשה
0.05
אוה דבעמב הבילל השיג ןמז
הבילל םג תשגל ןכמ רחאל םיטילחמו
0.02
– דבעמ ותוא
איבהל איה הרטמהו הדימב המיגדה תוינידמ יהמ
םומינימל ללוכה ןמזה תא
•
•
•
•

ינש תלעב תונמדזהכ דבעמ לכ לע לכתסהל ןתינ
.
םיבלש
: ינשה בלשהמ ליחתנ c
 
0 z  z
 y
   dy
0 .
02
 x

0 .
39

0 z  z
 y

2 y dy
םוגדל הצרנ 0.39
מ לודג ךרע ונאצמש ןמז לכ רמולכ
הינשה הבילה תא םג
•
•
•

: הנושארה הבילה לע לכתסנ תעכ • c
  y
0 .
39  z
 y
   dy

0
  y z

0 .
39 z

0 .
52 z
  w y

0 wf ( w ) dw
 y

1

F ( y )


0 .
02 f
  dy
הצרנ 0.52
מ ןטק ךרע ונאצמ אלש ןמז לכ רמולכ
הנושארה הבילה תא םוגדל
•