Abstract
<p> Consider a model where there are <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n"> <mml:semantics> <mml:mi>n</mml:mi> <mml:annotation encoding="application/x-tex">n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> items, with item <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i"> <mml:semantics> <mml:mi>i</mml:mi> <mml:annotation encoding="application/x-tex">i</mml:annotation> </mml:semantics> </mml:math> </inline-formula> having an unknown value <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v Subscript i Baseline period"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>.</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">v_i.</mml:annotation> </mml:semantics> </mml:math> </inline-formula> At any time, we can can either stop and declare which item has the largest value or else choose a subset of items to compare. If subset <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is chosen, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="i element-of upper S"> <mml:semantics> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">i \in S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> will be preferred with probability <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="v Subscript i Baseline slash sigma-summation Underscript j element-of upper S Endscripts v Subscript j"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:munder> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>j</mml:mi> <mml:mo> ∈ </mml:mo> <mml:mi>S</mml:mi> </mml:mrow> </mml:munder> <mml:msub> <mml:mi>v</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding="application/x-tex">v_i/\sum _{j \in S} v_j</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . Assuming a Bayesian prior on the values, and subject to the proviso that the policy employed will make the correct choice with probability at least <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha"> <mml:semantics> <mml:mi> α </mml:mi> <mml:annotation encoding="application/x-tex">\alpha</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , we are looking for a policy that needs a relatively small mean number of comparisons before making a decision. Some heuristic policies are presented and analyzed. </p>