Minimal Stable Sets in Tournaments

We propose a systematic methodology for defining tournament solutions as extensions of maximality. The central concepts of this methodology are maximal qualified subsets and minimal stable sets. We thus obtain an infinite hierarchy of tournament solutions, which encompasses the top cycle, the uncovered set, the Banks set, the minimal covering set, the tournament equilibrium set, the Copeland set, and the bipartisan set. Moreover, the hierarchy includes a new tournament solution, the minimal extending set, which is conjectured to refine both the minimal covering set and the Banks set.Comment: 29 pages, 4 figures, changed conten

Dominating the Erdos-Moser theorem in reverse mathematics

The Erdos-Moser theorem (EM) states that every infinite tournament has an infinite transitive subtournament. This principle plays an important role in the understanding of the computational strength of Ramsey's theorem for pairs (RT^2_2) by providing an alternate proof of RT^2_2 in terms of EM and the ascending descending sequence principle (ADS). In this paper, we study the computational weakness of EM and construct a standard model (omega-model) of simultaneously EM, weak K\"onig's lemma and the cohesiveness principle, which is not a model of the atomic model theorem. This separation answers a question of Hirschfeldt, Shore and Slaman, and shows that the weakness of the Erdos-Moser theorem goes beyond the separation of EM from ADS proven by Lerman, Solomon and Towsner.Comment: 36 page

Sub-structural Niching in Estimation of Distribution Algorithms

We propose a sub-structural niching method that fully exploits the problem decomposition capability of linkage-learning methods such as the estimation of distribution algorithms and concentrate on maintaining diversity at the sub-structural level. The proposed method consists of three key components: (1) Problem decomposition and sub-structure identification, (2) sub-structure fitness estimation, and (3) sub-structural niche preservation. The sub-structural niching method is compared to restricted tournament selection (RTS)--a niching method used in hierarchical Bayesian optimization algorithm--with special emphasis on sustained preservation of multiple global solutions of a class of boundedly-difficult, additively-separable multimodal problems. The results show that sub-structural niching successfully maintains multiple global optima over large number of generations and does so with significantly less population than RTS. Additionally, the market share of each of the niche is much closer to the expected level in sub-structural niching when compared to RTS

Approximately Strategyproof Tournament Rules: On Large Manipulating Sets and Cover-Consistence

We consider the manipulability of tournament rules, in which n teams play a round robin tournament and a winner is (possibly randomly) selected based on the outcome of all binom{n}{2} matches. Prior work defines a tournament rule to be k-SNM-? if no set of ? k teams can fix the ? binom{k}{2} matches among them to increase their probability of winning by >? and asks: for each k, what is the minimum ?(k) such that a Condorcet-consistent (i.e. always selects a Condorcet winner when one exists) k-SNM-?(k) tournament rule exists? A simple example witnesses that ?(k) ? (k-1)/(2k-1) for all k, and [Jon Schneider et al., 2017] conjectures that this is tight (and prove it is tight for k=2). Our first result refutes this conjecture: there exists a sufficiently large k such that no Condorcet-consistent tournament rule is k-SNM-1/2. Our second result leverages similar machinery to design a new tournament rule which is k-SNM-2/3 for all k (and this is the first tournament rule which is k-SNM-(<1) for all k). Our final result extends prior work, which proves that single-elimination bracket with random seeding is 2-SNM-1/3 [Jon Schneider et al., 2017], in a different direction by seeking a stronger notion of fairness than Condorcet-consistence. We design a new tournament rule, which we call Randomized-King-of-the-Hill, which is 2-SNM-1/3 and cover-consistent (the winner is an uncovered team with probability 1)

Dynamic stable set

We study a dynamic vNM stable set in a compact metric space under the assumption of complete and continuous dominance relation. Internal and external stability are defined with respect to farsighted dominance. Stability of an outcome is conditioned on the history via which it is reached. A dynamic stable set always exists. Any covering set by Dutta (1988) coincides with the set of outcomes that are implementable via a dynamic stable set. The maximal implementable outcome set is a version of the ultimate uncovered set.vNM stable set, dynamic, history

Computing Tournament Solutions using Relation Algebra and REL VIEW

We describe a simple computing technique for the tournament choice problem. It rests upon a relational modeling and uses the BDD-based computer system RelView for the evaluation of the relation-algebraic expressions that specify the solutions and for the visualization of the computed results. The Copeland set can immediately be identified using RelView's labeling feature. Relation-algebraic specifications of the Condorcet non-losers, the Schwartz set, the top cycle, the uncovered set, the minimal covering set, the Banks set, and the tournament equilibrium set are delivered. We present an example of a tournament on a small set of alternatives, for which the above choice sets are computed and visualized via RelView. The technique described in this paper is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other problems of social choice and game theory.Tournament, relational algebra, RelView, Copeland set, Condorcet non-losers, Schwartz set, top cycle, uncovered set, minimal covering set, Banks set, tournament equilibrium set.