We investigate computability theoretic and descriptive set theoretic contents of various kinds of analytic choice principles by performing a detailed analysis of the Medvedev lattice of $\Sigma ^1_1$ -closed sets. Among others, we solve an open problem on the Weihrauch degree of the parallelization of the $\Sigma ^1_1$ -choice principle on the integers. Harrington’s unpublished result on a jump hierarchy along a pseudo-well-ordering plays a key role in solving this problem.

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The problem we are considering is the following. A colorblind player is given a set B = {b1, b2, . . . , bN } of N colored balls. He knows that each ball is colored either red or green, and that there are less green than red balls (this will be called a Red-green coloring), but he cannot distinguish the two colors. For any two balls he can ask whether they are colored the same. His goal is to determine the set of all green balls of B (and hence the set of all red balls). We study here the case where the Redgreen coloring is such that there are at most p green balls, where p is given ; we denote by Q(N, p, ≤) the minimum integer k such that there exists a method that finds for sure, for any Red-green coloring, the color of each ball of B after at most k (color) comparisons. We extend the cases for which the exact value of Q(N, p, ≤) is known and provide lower and upper bounds for Q(N, p, =) (defined similarly as Q(N, p, ≤), but for a Red-green coloring with exactly p green balls).

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