Hats, auctions and derandomization

Ben-Zwi, Oren; Newman, Ilan; Wolfovitz, Guy

doi:10.1002/rsa.20512

Cited by 4 publications

(4 citation statements)

References 33 publications

(68 reference statements)

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“…Hat guessing games have drawn much attention among mathematicians, computer scientists, coding theorists, and even the press due to their relations to graph theory, circuit complexity, network coding, and auctions [27][28][29][30][31][32]. Some versions of hat guessing games are used in derandomizing protocols in circuit complexity and derandomizing auctions in auction mechanism design due to the innate similarities between these games, and the number-on-forehead models in complexity [33,34] and the bid-independent auctions [35]. Moreover, these games are attractive because their optimal solutions have many unexpected connections to coding theory [33,34].…”

Section: Cooperation Via Ordered Designs In Hat Guessing Gamesmentioning

confidence: 99%

Latin Matchings and Ordered Designs OD(n−1, n, 2n−1)

Jin

Taikun

et al. 2022

Mathematics

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This paper revisits a combinatorial structure called the large set of ordered design (LOD). Among others, we introduce a novel structure called Latin matching and prove that a Latin matching of order n leads to an LOD(n−1,n,2n−1); thus, we obtain constructions for LOD(1,2,3), LOD(2,3,5), and LOD(4,5,9). Moreover, we show that constructing a Latin matching of order n is at least as hard as constructing a Steiner system S(n−2,n−1,2n−2); therefore, the order of a Latin matching must be prime. We also show some applications in multiagent systems.

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Section: Cooperation Via Ordered Designs In Hat Guessing Gamesmentioning

confidence: 99%

Latin Matchings and Ordered Designs OD(n−1, n, 2n−1)

Jin

Taikun

et al. 2022

Mathematics

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“…Hat-guessing games have been studied extensively in a broad area due to their relations to graph entropy, circuit complexity, network coding, and auctions [1,4,6,11,16,20]. In this appendix we show applications of the 1-factorization of the bipartite Kneser graphs in the following variant of hat-guessing game:…”

Section: Proofs Omitted In Sectionmentioning

confidence: 99%

On 1-Factorizations of Bipartite Kneser Graphs

Jin

2019

Lecture Notes in Computer Science

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It is a challenging open problem to construct an explicit 1factorization of the bipartite Kneser graph H(v, t), which contains as vertices all t-element and (v − t)-element subsets of [v] := {1, . . . , v} and an edge between any two vertices when one is a subset of the other. In this paper, we propose a new framework for designing such 1-factorizations, by which we solve a nontrivial case where t = 2 and v is an odd prime power. We also revisit two classic constructions for the case v = 2t + 1 -the lexical factorization and modular factorization. We provide their simplified definitions and study their inner structures. As a result, an optimal algorithm is designed for computing the lexical factorizations.(An analogous algorithm for the modular factorization is trivial.)On 1-factorizations of Bipartite Kneser Graphs 3 PreliminariesThe subset lattice is the family of all subsets of [v], partially ordered by inclusion. Let P t denote the t-th layer of this subset lattice, whose members are the telement subsets of [v]. Throughout the paper, denote d = v − 2t. Let the words clockwise and counterclockwise be abbreviated as CW and CCW respectively. A representation of the edges of H(v, t). We identify each edge (A, A ) of H(v, t) by a permutation ρ of t 's, t 's, and d ×'s: the (positions of) t ' 's indicate the t elements in A; the t ' 's indicate the t elements that are not in A (recall that A has v − t elements); and the '×'s indicate those in A − A. We do not distinguish the edges with their corresponding permutations.Denote [t , t , d×] as the multiset of 2t + d characters with t ' 's, t ' 's, and d '×'s. Giving a 1-factorization of H(v, t) is equivalent to giving a labeling function f from the 2t+d t,t,d permutations of [t , t , d×] to 1, . . . , t+d d so that(a) f (ρ) = f (σ) for those pairs ρ, σ who admit the same positions for t 's; and (b) f (ρ) = f (σ) for those pairs ρ, σ who admit the same positions for t 's.

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“…The aim is then to construct a guessing function f which guarantees a score for any possible configuration of hats [48]. The relation between Winkler's hat game and auctions has been revealed in [1] and developed in [7].…”

Section: Introductionmentioning

confidence: 99%

Finite Dynamical Systems, Hat Games, and Coding Theory

Gadouleau¹

2018

SIAM J. Discrete Math.

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The dynamical properties of finite dynamical systems (FDSs) have been investigated in the context of coding theoretic problems, such as network coding and index coding, and in the context of hat games, such as the guessing game and Winkler's hat game. In this paper, we relate the problems mentioned above to properties of FDSs, including the number of fixed points, their stability, and their instability. We first introduce the guessing dimension and the coset dimension of an FDS and their counterparts for digraphs. Based on the coset dimension, we then strengthen the existing equivalences between network coding and index coding. We also introduce the instability of FDSs and we study the stability and the instability of digraphs. We prove that the instability always reaches the size of a minimum feedback vertex set. We also obtain some non-stable bounds independent of the number of vertices of the graph. We then relate the stability and the instability to the guessing number. We also exhibit a class of sparse graphs with large girth that have high stability and high instability; our approach is code-theoretic and uses the guessing dimension. Finally, we prove that the affine instability is always asymptotically greater than or equal to the linear guessing number.

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Hats, auctions and derandomization

Cited by 4 publications

References 33 publications

Latin Matchings and Ordered Designs OD(n−1, n, 2n−1)

Latin Matchings and Ordered Designs OD(n−1, n, 2n−1)

On 1-Factorizations of Bipartite Kneser Graphs

Finite Dynamical Systems, Hat Games, and Coding Theory

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