Balancedness of multiple machine sequencing games revisited

We study m-sequencing games, which were introduced by [Hamers, H., Klijn, F., Suijs, J., (1999). On the balancedness of multiple machine sequencing games. European Journal of Operational Research 119, 678–691]. We answer the open question whether all these games are balanced in the negative. We do so, by an example of a 3-sequencing situation with 5 jobs, whose associated 3-sequencing game has an empty core. The counterexample finds its basis in an inconsistency in [Hamers et al., ibid], which was probably overlooked by the authors. This observation demands for a detailed reconsideration of their proofs.¹     

Balancedness conditions for exact games

We provide two new characterizations of exact games. First, a game is exact if and only if it is exactly balanced; and second, a game is exact if and only if it is totally balanced and overbalanced. The condition of exact balancedness is identical to the one of balancedness, except that one of the balancing weights may be negative, while for overbalancedness one of the balancing weights is required to be non-positive and no weight is put on the grand coalition. Exact balancedness and overbalancedness are both easy to formulate conditions with a natural game-theoretic interpretation and are shown to be useful in applications. Using exact balancedness we show that exact games are convex for the grand coalition and we provide an alternative proof that the classes of convex and totally exact games coincide. We provide an example of a game that is totally balanced and convex for the grand coalition, but not exact. Finally we relate classes of balanced, totally balanced, convex for the grand coalition, exact, totally exact, and convex games to one another.     

Weighted allocation rules for standard fixed tree games

In this paper we consider standard fixed tree games, for which each vertex unequal to the root is inhabited by exactly one player. We present two weighted allocation rules, the weighted down-home allocation and the weighted neighbour-home allocation, both inspired by the painting story in Maschler et al. (1995) . We show, in a constructive way, that the core equals both the set of weighted down-home allocations and the set of weighted neighbour allocations. Since every weighted down-home allocation specifies a weighted Shapley value (Kalai and Samet (1988)) in a natural way, and vice versa, our results provide an alternative proof of the fact that the core of a standard fixed tree game equals the set of weighted Shapley values. The class of weighted neighbour allocations is a generalization of the nucleolus, in the sense that the latter is in this class as the special member where players have all equal weights.     

On the balancedness of multiple machine sequencing games

This paper takes a game theoretical approach to sequencing situations with m parallel and identical machines. We show that in a cooperative environment cooperative m-sequencing games, which involve n players, give rise to m-machine games, which involve m players. Here, n corresponds to the number of jobs in an m-sequencing situation, and m corresponds to the number of machines in the same m-sequencing situation. We prove that an m-sequencing game is balanced if and only if the corresponding m-machine game is balanced. Furthermore, it is shown that m-sequencing games are balanced if m∈{1,2}. Finally, if m⩾3, balancedness is established for two special classes of m-sequencing games. Furthermore, we consider a special class of m-sequencing situations in a noncooperative setting and show that a transfer payments scheme exists that is both incentive compatible and budget balanced.     

Aubin cores and bargaining sets for convex cooperative fuzzy games

In this paper, we deal with Aubin cores and bargaining sets in convex cooperative fuzzy games. We first give a simple and direct proof to the well-known result (proved by Branzei et al. (Fuzzy Sets Syst 139:267–281, 2003)) that for a convex cooperative fuzzy game v, its Aubin core C(v) coincides with its crisp core C ^cr (v). We then introduce the concept of bargaining sets for cooperative fuzzy games and prove that for a continuous convex cooperative fuzzy game v, its bargaining set coincides with its Aubin core, which extends a well-known result by Maschler et al. for classical cooperative games to cooperative fuzzy games. We also show that some results proved by Shapley (Int J Game Theory 1:11–26, 1971) for classical decomposable convex cooperative games can be extended to convex cooperative fuzzy games.     

Balanced contributions for TU games with awards and applications

Calleja et al. [Calleja, P., Borm, P., Hendrickx, R., 2005. Multi-issue allocation situations. European Journal of Operational Research 164, 730–747] introduced multi-issue allocation situations with awards. In this paper, we extend the classical model of cooperative games with transferable utility to the cooperative games with transferable utility and awards. We define a run-to-the-bank rule for cooperative games with transferable utility and awards and characterise it in terms of a property of balanced contributions. We apply our main result to bankruptcy problems and multi-issue allocation situations with awards.     

Strongly balanced cooperative games

Kaneko/Wooders (1982) derived a list of necessary and sufficient conditions for a partitioning game to have a nonempty core regardless of the payoff functions of its effective coalitions. The main purpose of our paper is to provide a graph-theoretical characterization of this family of games whose associated hypergraphs we callstrongly balanced: we show that the strong balancedness condition is equivalent to thenormality of the hypergraph, which is a type ofcoloring property (Lovasz (1972)). We also study interesting economic examples ofcommunication andassignment games and provide direct proofs that their associated hypergraphs are strongly balanced.We wish to thank two anonymous referees of this journal for their useful comments and suggestions. The previous version of this paper was written while the authors were visiting Department of Economics, University of Bonn. The financial support of Sonderfor-schungsbereich 303 is gratefully acknowledged.     

Cooperation under interval uncertainty

In this paper, the classical theory of two-person cooperative games is extended to two-person cooperative games with interval uncertainty. The core, balancedness, superadditivity and related topics are studied. Solutions called ψ ^α-values are introduced and characterizations are given.     

行m-NSD随机变量阵列的完全收敛性

本文研究了行m-NSD随机变量阵列的完全收敛性问题.主要利用m-NSD随机变量的Kolmogorov型指数不等式,获得了行m-NSD随机变量阵列的完全收敛性定理,将Hu等（1998）andSung等（2005）的结果从独立情形推广到了m-NSD随机变量阵列.本文的结论同样推广了Chen等（2008）,Hu等（2009）,Qiu等（2011）和Wang等（2014）的结果.     

Some functional forms of Blaschke–Santaló inequality

We establish new functional versions of the Blaschke–Santaló inequality on the volume product of a convex body which generalize to the non-symmetric setting an inequality of Ball [Isometric problems in ℓ _p and sections of convex sets. PhD Dissertation, Cambridge, 1986] and we give a simple proof of the case of equality. As a corollary, we get some inequalities for log-concave functions and Legendre transforms which extend the recent result of Artstein et al. [Mathematika 51:33–48, 2004], with its equality case.     

Laumon spaces and the Calogero-Sutherland integrable system

This paper contains a proof of a conjecture of Braverman concerning Laumon quasiflag spaces. We consider the generating function Z(m), whose coefficients are the integrals of the equivariant Chern polynomial (with variable m) of the tangent bundles of the Laumon spaces. We prove Braverman’s conjecture, which states that Z(m) coincides with the eigenfunction of the Calogero-Sutherland hamiltonian, up to a simple factor which we specify. This conjecture was inspired by the work of Nekrasov in the affine [^( \mathfrak sl)]_n\widehat{ {\mathfrak {sl}}}_{n} setting, where a similar conjecture is still open.     

Une preuve alternative de l'existence d'un équilibre de Nash dans les jeux discontinus

This Note presents a theorem of the existence of the Nash equilibrium for discontinuous games in a topological vector space. We will use an assumption of better reply secure which is stronger then that of Reny. If the payoff function is upper semi-continuous, the two assumptions coincide. Our proof is simple, independent and based on a version of Fan–Browder theorem of existence of maximal element due to Deguire and Lassonde, which is extended to the non-Hausdorf case. To cite this article: J.-M. Bonnisseau et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A Model for Multi-timescaled Sequential Decision-making Processes with Adversary

Extending the multi-timescale model proposed by the author et al. in the context of Markov decision processes, this paper proposes a simple analytical model called M timescale two-person zero-sum Markov Games (MMGs) for hierarchically structured sequential decision-making processes in two players' competitive situations where one player (the minimizer) wishes to minimize their cost that will be paid to the adversary (the maximizer). In this hierarchical model, for each player, decisions in each level in the M-level hierarchy are made in M different discrete timescales and the state space and the control space of each level in the hierarchy are non-overlapping with those of the other levels, respectively, and the hierarchy is structured in a "pyramid" sense such that a decision made at level m (slower timescale) state and/or the state will affect the evolutionary decision making process of the lower-level m+1 (faster timescale) until a new decision is made at the higher level but the lower-level decisions themselves do not affect the transition dynamics of higher levels. The performance produced by the lower-level decisions will affect the higher level decisions for each player. A hierarchical objective function for the minimizer and the maximizer is defined, and from this we define "multi-level equilibrium value function" and derive a "multi-level equilibrium equation". We also discuss how to solve hierarchical games exactly.     

An exotic shuffle relation for multiple zeta values

In this short note we will provide a new proof of the following exotic shuffle relation of multiple zeta values: This was proved by Zagier when n = 0, by Broadhurst when m = 0, and by Borwein, Bradley, and Broadhurst when m = 1. In general this was proved by Bowman and Bradley. Our new idea is to use the method of Borwein et al. to reduce the above general relation to some families of combinatorial identities which can be verified by Zeilberger’s algorithm [9, 10] that is part of the WZ method. Received: 27 November 2007 Revised: 28 June 2008     

Limit behaviour of solutions to boundary value problem with equivalued surface

In this paper, we discuss the limit behaviour of solutions to boundary value problem with equivalued surface with m inner holes and give a different proof from that of Li Ta-tsien et al. (1998).     

A GAME-THEORETIC PROOF OF ANALYTIC RAMSEY THEOREM

We give a simple game-theoretic proof of Silver's theorem that every analytic set is Ramsey. A set P of subsets of ω is called Ramsey if there exists an infinite set H such that either all infinite subsets of H are in P or all out of P. Our proof clarifies a strong connection between the Ramsey property of partitions and the determinacy of infinite games.     

An algorithm producing a standard spine of a 3-manifold presented by surgery along a link

We provide a simple algorithm which produces a (branched) standard spine of a 3-manifold presented by surgery along a framed link inS ³, giving an explicit upper bound on the complexity of the spine in terms of the complexity of a diagram of the link. As a corollary, we get an easy constructive proof of Casler’s result on the existence of a standard spine for a closed 3-manifold. We also describe an o-graph which represents the spine.     

On the convexity of step out–step in sequencing games

The main result of this paper is the convexity of step out–step in (SoSi) sequencing games, a class of relaxed sequencing games first analyzed by Musegaas et al. (Eur J Oper Res 246:894–906, 2015). The proof makes use of a polynomial time algorithm determining the value and an optimal processing order for an arbitrary coalition in a SoSi sequencing game. In particular, we use that in determining an optimal processing order of a coalition, the algorithm can start from the optimal processing order found for any subcoalition of smaller size and thus all information on such an optimal processing order can be used.     

Probabilities of Independent Choices Being Ordered

A simple proof of the following result of Baumert et al. is given. If A and B are chosen independently according to a probability distribution defined on subsets of an n element set, then the probability that A contains B is at least . It's shown to lead to a stronger conclusion, and to suggest definition of concepts of “partition number” and of “orthogonality of partitions into chains” as well as several conjectures.