Download A note on the high-low search problem

A note on the high-low search problem
Kensaku Kikuta
University of Hyogo
School of Business Administration
Gakuentoshi 8-2-1, Kobe 651-2197,
Japan
2012年4月23日月曜日
Robbert Fokkink
University of Delft
Department of Applied Mathematics
Delft
Netherlands
Model
A point h is known to lie on the closed interval [0, 1].
An information on a position of the point : a probability density function f on [0, 1].
With this information, a searcher wishes to locate h
by successively choosing g1 , g2 , . . . , gn ,
where at each choice gi , i ≤ n − 1 he is told whether h = gi , h < gi and h > gi ,
and then he chooses gi+1 , i ≤ n − 1 with informations on g1 , . . . , gi .
If there is 1 ≤ i < n such that h = gi
then the searcher stops at that time and we let gi+1 = · · · = gn = h for simplicity.
The choice gi+1 , i ≤ n − 1 is dependent on informations on the previous choices g1 , . . . , gi .
0
2012年4月23日月曜日
h
g
1
Strategy and Cost, Median
A pure strategy for the searcher : γ = {γ1 , . . . , γn }
where γi is a 2i−1 -th vector
γi =
1
2i−1
(gi , . . . , gi ), 0
≤
gi1
≤ ··· ≤
2i−1
gi
≤ 1.
(1)
For the pair of h and γ, we define the cost to the searcher, written as c(h, γ),
by the gap between the final choice and the position of the point, i.e.,
c(h, γ) = |gnα − h|,
(2)
where 1 ≤ α ≤ 2n−1 . For simplicity we assume that the searcher uses a pure strategy. The
median for (f, γ) is M e(f, γ) such that
!
1
f (h)dh = .
(3)
2
c(h,γ)≥M e(f,γ)
Given the probability density function f , the purpose of the searcher is to choose a pure strategy
γ so that it minimizes the median M e(f, γ).
2012年4月23日月曜日
mini-Median approach
Definition. A strategy γ ∗ is a mini-median strategy if
M e(f ) ≡ M e(f, γ ∗ ) = inf M e(f, γ).
γ
(4)
Assumption. The probability density function f on [0, 1] is continuous and has no mass on
[0, 1].
In [Alpern 1985] three approaches are stated
that a minimizing searcher may take to the problem, that is,
(1) the Bayesian approach,
(2) the minimax approach
(3) the consideration as a game.
In the first approach a distribution for the point is assumed and seeks a pure search strategy
minimizing the expected cost.
In this note perhaps we treat a variation of this approach where a distribution for the point
is assumed and seeks a pure search strategy minimizing the median cost.
2012年4月23日月曜日
References
[1] S. Alpern: Search for a point in interval, with high-low feedback. Math.Proc.Camb.Phil.Soc.
(1985), Vol.98, 569-578.
[2] S.Alpern and S.Gal: The theory of search games and rendezvous. Kluwer (2003). esp. 84-91.
[3] V.Baston and F.Bostock: A high-low search game on the unit interval. Math.Proc.Camb.Phil.Soc.
(1985), Vol.97, 345-348.
[4] T.S.Ferguson: A problem of minimax estimation with directional information. Statistics
and Probability Letters. (1996), Vol.26, 205-211.
[5] S.Gal: A discrete search game. SIAM J. Appl.Math. (1974), Vol. 27, 641-648.
[6] S.Gal: A stochastic search game. SIAM J. Appl.Math. (1978), Vol. 34, 205-210.
[7] S.M.Johnson: A search game. Advances in Game Theory(Princeton University Press,1964),
39-48.
[8] S.Murakami: A dichotomous search with travel cost. J. Oper.Res.Soc.Japan(1976), Vol. 19,
245-254.
2012年4月23日月曜日
Example 1. Problem for n = 1: c(h, γ) = |g11 − h|.
c(h,γ)
f(h)
Me
g1
a
0
1
b
1
Figure 1: Median for n = 1
Suppose
b 1 − a1 =
Let g11∗ =
a1 +b1
2
inf {b − a :
0≤a<b≤1
and γ ∗ = {g11∗ }. Then
∗
M e(f ) = M e(f, γ ) = b1 −
g11∗
!
b
a
=
1
f (h)dh = }.
2
g11∗
(5)
b 1 − a1
− a1 =
.
2
Let γ = {g11 } be any strategy. Define a and b by solving c(h, γ) = M e(f, γ) with respect to h.
If M e(f, γ) > c(1, γ) then let b = 1. If M e(f, γ) > c(0, γ) then let a = 0. Then 0 ≤ a < b ≤ 1
"b
b1 −a1
∗
and a f (h) = 12 . By (5) we have b − a ≥ b1 − a1 . So M e(f, γ) = b−a
≥
=
M
e(f,
γ
).
2
2
Hence the mini-median value is M e(f, γ ∗ ).
2012年4月23日月曜日