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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