Capacities and Games on Lattices: A Survey of Result

We provide a survey of recent developments about capacities (or fuzzy measures) and ccoperative games in characteristic form, when they are defined on more general structures than the usual power set of the universal set, namely lattices. In a first part, we give various possible interpretations and applications of these general concepts, and then we elaborate about the possible definitions of usual tools in these theories, such as the Choquet integral, the Möbius transform, and the Shapley value.capacity, fuzzy measure, game, lattice, Choquet integral,Shapley value

Cooperative Games with Lattice Structure

A general model for cooperative games with possibly restricted and hierarchically ordered coalitions is introduced and shown to have lattice structure under quite general assumptions. Moreover, the core of games with lattice structure is investigated. Within a general framework that includes the model of classical cooperative games as a special case, it is proved algorithmically that monotone convex games have a non-empty core. Finally, the solution concept of the Shapley value is extended to the general class of cooperative games with restricted cooperation. It is shown that several generalizations of the Shapley value that have been proposed in the literature are subsumed in this model

Generalizations of Boxworld

Boxworld is a toy theory that can generate extremal nonlocal correlations known as PR boxes. These have been well established as an important tool to examine general nonlocal correlations, even beyond the correlations that are possible in quantum theory. We modify boxworld to include new features. The first modification affects the construction of joint systems such that the new theory allows entangled measurements as well as entangled states in contrast to the standard version of boxworld. The extension to multipartite systems and the consequences for entanglement swapping are analysed. Another modification provides continuous transitions between classical probability theory and boxworld, including the algebraic expression for the maximal CHSH violation as a function of the transition parameters.Comment: In Proceedings QPL 2011, arXiv:1210.029

A system-theoretic approach to multi-agent models

A system-theoretic model for cooperative settings is presented that unifies and ex- tends the models of classical cooperative games and coalition formation processes and their generalizations. The model is based on the notions of system, state and transi- tion graph. The latter describes changes of a system over time in terms of actions governed by individuals or groups of individuals. Contrary to classic models, the pre- sented model is not restricted to acyclic settings and allows the transition graph to have cycles. Time-dependent solutions to allocation problems are proposed and discussed. In par- ticular, Weber’s theory of randomized values is generalized as well as the notion of semi-values. Convergence assertions are made in some cases, and the concept of the Cesàro value of an allocation mechanism is introduced in order to achieve convergence for a wide range of allocation mechanisms. Quantum allocation mechanisms are de- fined, which are induced by quantum random walks on the transition graph and it is shown that they satisfy certain fairness criteria. A concept for Weber sets and two dif- ferent concepts of cores are proposed in the acyclic case, and it is shown under some mild assumptions that both cores are subsets of the Weber set. Moreover, the model of non-cooperative games in extensive form is generalized such that the presented model achieves a mutual framework for cooperative and non-co- operative games. A coherency to welfare economics is made and to each allocation mechanism a social welfare function is proposed

Causation does not explain contextuality

Realist interpretations of quantum mechanics presuppose the existence of elements of reality that are independent of the actions used to reveal them. Such a view is challenged by several no-go theorems that show quantum correlations cannot be explained by non-contextual ontological models, where physical properties are assumed to exist prior to and independently of the act of measurement. However, all such contextuality proofs assume a traditional notion of causal structure, where causal influence flows from past to future according to ordinary dynamical laws. This leaves open the question of whether the apparent contextuality of quantum mechanics is simply the signature of some exotic causal structure, where the future might affect the past or distant systems might get correlated due to non-local constraints. Here we show that quantum predictions require a deeper form of contextuality: even allowing for arbitrary causal structure, no model can explain quantum correlations from non-contextual ontological properties of the world, be they initial states, dynamical laws, or global constraints.Comment: 18+8 pages, 3 figure

Extensions of the t-value to NTU-games

Game Theory;econometrics

Optimal dynamic profit taxation: The derivation of feedback Stackelberg equilibria

Game Theory;Corporate Tax;operations research

Unconditional convergence for discretizations of dynamical optimal transport

The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamic formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several disretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space. In this article, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional one for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien and Solomon fit in this framework