Uniformly Optimal Strategies in Search Problems
В. И. Аркин · Theory of Probability and Its Applications · 1964
Let $f(x)$ be the density function of the a priori distribution of a particle in $R^n $. The strategy of search is defined by a function $\alpha = \alpha (x,t) \geqq 0$, $\int_{R^n } {\alpha (x,t)} dx = 1$. The probability of finding the particle at a point x during time t, under the condition that it is there, using the strategy $\alpha $, is given by the functional II $(\int_0^t {\alpha (x,t)dt,x} )$. Let $P_\alpha (T)$ be the probability of finding the particle using the strategy $\alpha $ during the time T. A strategy $\alpha ^ * $ is uniformly optimal if $P_{\alpha ^ * } (T) = \sup _\alpha P_\alpha (T)$ for any $T > 0$. In a very general case we prove the existence of the strategy $\alpha ^ * $ and find its explicit form.