We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations R, S, a componentwise reducibility is defined byHere, f is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and f must be computable. We show that there is a Π 0 1 -complete equivalence relation, but no Π 0 k -complete for k ≥ 2. We show that Σ 0 k preorders arising naturally in the above-mentioned areas are Σ 0 k -complete. This includes polynomial time m-reducibility on exponential time sets, which is Σ 0 2 , almost inclusion on r.e. sets, which is Σ 0 3 , and Turing reducibility on r.e. sets, which is Σ 0 4 . 859 860 EGOR IANOVSKI, RUSSELL MILLER, KENG MENG NG, AND ANDRÉ NIES so that we can actually learn something from the reduction. In our example, one should ask how hard it is to compute the dimension of a rational vector space. It is natural to restrict the question to computable vector spaces over Q (i.e., those where the vector addition is given as a Turing-computable function on the domain of the space). Yet even when its domain D E such vector spaces, computing the function which maps each one to its dimension requires a 0 -oracle, hence is not as simple as one might have hoped. (The reasons why 0 is required can be gleaned from [7] or [8].) 1.1. Effective reductions. Reductions are normally ranked by the ease of computing them. In the context of Borel theory, for instance, a large body of research is devoted to the study of Borel reductions (the standard book reference is [17]). Here, the domains D E and D F are the set 2 or some other standard Borel space, and a Borel reduction f is a reduction (from E to F , these being equivalence relations on 2 ) which, viewed as a function from 2 to 2 , is Borel. If such a reduction exists, one says that E is Borel reducible to F , and writes E ≤ B F . A stronger possible requirement is that f be continuous, in which case we have (of course) a continuous reduction. In case the reduction is given by a Turing functional from reals to reals, it is a (type-2) computable reduction.A further body of research is devoted to the study of the same question for equivalence relations E and F on , and reductions f : → between them which are computable. If such a reduction from E to F exists, we say that E is computably reducible to F , and write E ≤ c F , or often just E ≤ F . These reductions will be the focus of this paper. Computable reducibility on equivalence relations was perhaps first studied by Ershov [12] in a category theoretic setting.The main purpose of this paper is to investigate the complexity of equivalence relations under these reducibilities. In certain cases we will generalize from equivalence relations to preorders on . We restrict most of our discussion to relatively low levels of the hierarchy, usually to Π 0 n and Σ 0 n with n ≤ 4. One can focus more closely on very low levels: Such articles as [3,5,18], for instance, have dealt exclusively with Σ 0 1 equivalen...
Individual rankings are often aggregated using scoring rules: each position in each ranking brings a certain score; the total sum of scores determines the aggregate ranking. We study whether scoring rules can be robust to adding or deleting particular candidates, as occurs with spoilers in political elections and with athletes in sports due to doping allegations. In general the result is negative, but weaker robustness criteria pin down a one-parameter family of geometric scoring rules with the scores 0, 1, 1 + p, 1 + p + p 2 , . . .. These weaker criteria are independence from deleting unanimous winner (e.g., doping allegations) and independence from deleting unanimous loser (e.g., spoiler candidates). This family generalises three central rules: the Borda rule, the plurality rule and the antiplurality rule. For illustration we use recent events in biathlon; our results give simple instruments to design scoring rules for a wide range of applications.
We consider equivalence relations and preorders complete for various levels of the arithmetical hierarchy under computable, component-wise reducibility. We show that implication in first order logic is a complete preorder for Σ 0 1 , the ≤ P m relation on EXPTIME sets for Σ 0 2 and the embeddability of computable subgroups of (Q, +) for Σ 0 3 . In all cases, the symmetric fragment of the preorder is complete for equivalence relations on the same level. We present a characterisation of Π 0 1 equivalence relations which allows us to establish that equality of polynomial time functions and inclusion of polynomial time sets are complete for Π 0 1 equivalence relations and preorders respectively. We also show that this is the limit of the enquiry: for n ≥ 2 there are no Π 0 n nor ∆ 0 n -complete equivalence relations.
We examine the history of cake cutting mechanisms and discuss the efficiency of their allocations. In the case of piecewise uniform preferences, we define a game that in the presence of strategic agents has equilibria that are not dominated by the allocations of any mechanism. We identify that the equilibria of this game coincide with the allocations of an existing cake cutting mechanism.
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