On the core: complement-reduced game and max-reduced game

This paper presents two characterizations of the core on the domain of all NTU games. One is based on consistency with respect to “complement-reduced game” and converse consistency with respect to “max-reduced game”. The other is based on consistency with respect to “max-reduced game” and weak converse consistency with respect to “complement-reduced game”. Besides, we introduce an alternative definition of individual rationality, we name conditional individual rationality, which is compatible with non-emptiness. We discuss axiomatic characterizations involving conditional individual rationality for the core.     

Probabilizing parking functions

We explore the link between combinatorics and probability generated by the question “What does a random parking function look like?” This gives rise to novel probabilistic interpretations of some elegant, known generating functions. It leads to new combinatorics: how many parking functions begin with i? We classify features (e.g., the full descent pattern) of parking functions that have exactly the same distribution among parking functions as among all functions. Finally, we develop the link between parking functions and Brownian excursion theory to give examples where the two ensembles differ.     

A theorem on the number of Nash equilibria in a bimatrix game

We show that ify is an odd integer between 1 and 2ⁿ ? 1, there is ann × n bimatrix game with exactlyy Nash equilibria (NE). We conjecture that this 2ⁿ ? 1 is a tight upper bound on the number of NEs in a “nondegenerate”n × n game.     

Intersection properties in geometry

In dealing with geometries and diagrams we often need some axioms on the intersections of shadows. Here are the most usual ones: the Intersection Property (see (IP) in [3]), conditions (Int) and (Int′) of [8], and the Linearity Condition (see (GL) in [3]). An example due to Brower shows that the Linearity Condition (GL) is weaker than the Intersection Property (IP). In this paper we point out some conditions which have to be added to (GL) in order to get (IP), and we describe some of the relations between these conditions and each of the four ‘intersection’ properties given above. We summarize most of these connections in the appendix to this paper. The main open question is: ‘Which are the “right” axioms for “good” geometries?’     

Generalizations of bold play in red and black

The strategy of bold play in the game of red and black leads to a number of interesting mathematical properties: the player's fortune follows a deterministic map, before the transition that ends the game; the bold strategy can be “re-scaled” to produce new strategies with the same win probability; the win probability is a continuous function of the initial fortune, and in the fair case, equals the initial fortune. We consider several Markov chains in more general settings and study the extent to which the properties are preserved. In particular, we study two “k-player” models.     

On the optimality of the uniform random strategy

Biased Maker‐Breaker games, introduced by Chvátal and Erd?s, are central to the field of positional games and have deep connections to the theory of random structures. The main questions are to determine the smallest bias needed by Breaker to ensure that Maker ends up with an independent set in a given hypergraph. Here we prove matching general winning criteria for Maker and Breaker when the game hypergraph satisfies certain “container‐type” regularity conditions. This will enable us to answer the main question for hypergraph generalizations of the H‐building games studied by Bednarska and ?uczak as well as a generalization of the van der Waerden games introduced by Beck. We find it remarkable that a purely game‐theoretic deterministic approach provides the right order of magnitude for such a wide variety of hypergraphs, while the analogous questions about sparse random discrete structures are usually quite challenging.     

Affine semigroups acting properly discontinuously on a hyperbolic space

In this paper we study the dynamics of properly discontinuous and crystallographic affine semigroups leaving a hyperbolic form, i.e. a quadratic from of signature (n, 1) invariant. The motivating question here is a question stated by H. Abels, G. Margulis and the author: Is the Zariski closure of a crystallographic affine semigroup leaving a hyperbolic form invariant a virtually solvable group? We proved that that for n = 2 the answer is “yes”.     

Time traps in supply chains: Is optimal still good enough?

Operations Researchers support Supply Chain Management and Supply Chain Planning by developing adequate mathematical optimization models and providing suitable solution procedures. In this paper we discuss what adequate could mean. Therefore, we may ask several questions concerning “optimality” in Supply Chain Planning under causal and temporal uncertainty: What is an optimal solution? When is it optimal? For how long is it optimal? How should the design of a supply chain be changed when conditions and requirements ask for new structures? In particular, we discuss new approaches to Supply Chain Planning in order to give an optimal transformation from an initial solution over multiple periods to a desired one rather than just specifying an optimal snapshot solution. Time and uncertainty are the factors triggering the whole discussion. In particular, several flaws often found when dealing with these factors result in so-called “time traps”. We look at the impact of recent technological developments like the Internet of Things or Industry 4.0 on operational supply chain planning and control, and we show how online optimization can help to cope with real-time challenges. Moreover, we re-coin the concept of risk in the realm of Supply Chain Planning. Here the question is how to measure supply chain specific risks and how to incorporate them “adequately” into mathematical models.     

HOW RAPID SHOULD EMISSION REDUCTION BE? A GAME‐THEORETIC APPROACH

Abstract In this paper, we search for multistage realization of international environmental agreements. To analyze countries' incentives and the results of their interactions, we mathematically represent players' strategic preferences and apply a game‐theoretic approach to make predictions about their outcomes. The initial decision on emissions reduction is determined by the Stackelberg equilibrium concept. We generalize Barrett's static “emission” model to a dynamic framework and answer the question “how rapid should the emission reduction be?” It appears that sharper abatement is desirable in the early term, which is similar to the conclusion of the Stern review. Numerical example demonstrates that abatement dynamics of the coalition and the free‐rider differ when discounting of the future payoffs increases. We show that without incentives from external organizations or governments, such pollution reduction path can actually lead to a decline in the agreement's membership size.     

On the cardinalities of finite topologies

Following the ideas of Sharp [2,3], we will give a partial answer to the question: “Let k be an integer, k ? 2. What is the smallest integer m for which there is a topology on m points with k open sets.” We state several results in the theory of finite topologies by introducing the idea of generating topologies. Using this concept, it is possible to derive existence theorems and get numerical results in an easy manner.     

The shelahP-point independence theorem

In this paper, we present S. Shelah's example of a model of set theory in which there are noP-points in βN/N. This settles the famous open question: “Is ‘ZFC+there are noP-points in βN/N’ consistent?”     

Chip-Firing and Riemann-Roch Theory for Directed Graphs

Baker and Norine developed a graph theoretic analogue of the classical Riemann-Roch theorem. Amini and Manjunath extended their criteria to all full-dimensional lattices orthogonal to the all ones vector. We show that Amini and Manjunath?s criteria holds for all full-dimensional lattices orthogonal to some positive vector and study some combinatorial examples of such lattices. Two distinct generalizations of the chip-firing game of Baker and Norine to directed graphs are provided. We describe how the “row” chip-firing game is related to the sandpile model and the “column” chip-firing game is related to directed G-parking functions. We finish with a discussion of arithmetical graphs, introduced by Lorenzini, viewing them as a class of vertex weighted graphs whose Laplacian is orthogonal to a positive vector and describe how they may be viewed as a special class of unweighted strongly connected directed graphs.     

Mathematics learning and tools from theoretical, historical and practical points of view: the productive notion of mathematics laboratories

In our research work, we have looked at the way in which artefacts become, for teachers as well as for students, instruments of their mathematical activity. The issues related to the use of tools and technologies in mathematical education are now widely considered. A look to history highlights the different ways in which the same questions have been studied at different times and in different places. This suggests that the contribution of artefacts to mathematics learning should be considered in terms of various contexts. Our “visits” to these contexts will be guided by the coordination of two main theoretical frameworks, the instrumental approach and the semiotic mediation approach from the perspective of mathematics laboratory. This journey through history and schooling represents a good occasion to address some questions: Are there “good” contexts in which to develop mathematical instruments? Are there “good” teaching practices which assist students’ instrumental geneses and construct mathematical meanings? How is it possible to promote such teaching practices? Some study cases are discussed.     

The generalized continuum hypothesis revisited

We can reformulate the generalized continuum problem as: for regular κ<λ we have λ to the power κ is λ, We argue that the reasonable reformulation of the generalized continuum hypothesis, considering the known independence results, is “for most pairs κ<λ of regular cardinals, λ to the revised power of κ is equal to λ”. What is the revised power? λ to the revised power of κ is the minimal cardinality of a family of subsets of λ each of cardinality κ such that any other subset of λ of cardinality κ is included in the union of strictly less than κ members of the family. We still have to say what “for most” means. The interpretation we choose is: for every λ, for every large enoughK < ? _w . Under this reinterpretation, we prove the Generalized Continuum Hypothesis.     

Sufficiency and fuzziness in random experiments

In previous papers, the consequences of the “presence of fuzziness” in the experimental information on which statistical inferences are based were discussed. Thus, the intuitive assertion «fuzziness entails a loss of information» was formalized, by comparing the information in the “exact case” with that in the “fuzzy case”. This comparison was carried out through different criteria to compare experiments (in particular, that based on the “pattern” one, Blackwell's sufficiency criterion). Our purpose now is slightly different, in the sense that we try to compare two “fuzzy cases”. More precisely, the question we are interested in is the following: how will different “degrees of fuzziness” in the experimental information affect the sufficiency? In this paper, a study of this question is carried out by constructing an alternative criterion (equivalent to sufficiency under comparability conditions), but whose interpretation is more intuitive in the fuzzy case. The study is first developed for Bernoulli experiments, and the coherence with the axiomatic requirements for measures of fuzziness is also analyzed in such a situation. Then it is generalized to other random experiments and a simple example is examined.     

Une conduite cylindrique n'est pas optimale pour minimiser l'énergie dissipée par un fluide

We consider an incompressible fluid in a three-dimensional cylindrical pipe, following the Navier–Stokes system with classical boundary conditions on the boundary of the cylinder. We are interested in the following question: is the cylinder the optimal shape for the criterion “energy dissipated by the fluid”? We prove that it is not the case. For that purpose, we explicit the first order optimality condition, thanks to adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system. To cite this article: A. Henrot, Y. Privat, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

On the support of complete intersection zero-cycles

On an algebraic varietyY ? ? _N we will call complete intersection a 0-cycle when it is the intersection of Y with a codimension n complete intersection of ? _N . We consider the following problem: Let E?Y be given. Does E contain the support of a complete intersection 0-cycle? The two main theorems shown in this article give the answers in some cases: first, a negative answer for E some “big” subset of a singular irreducible algebraic variety; secondly, a positive answer for some “small” subset, on any algebraic variety.     

The propensity to disrupt and the disruption nucleolus of a characteristic function game

Gately [1974] recently introduced the concept of an individual player's “propensity to disrupt” a payoff vector in a three-person characteristic function game. As a generalisation of this concept we propose the “disruption nucleolus” of ann-person game. The properties and computational possibilities of this concept are analogous to those of the nucleolus itself. Two numerical examples are given.     

A Bridge Between Bilevel Programs and Nash Games

We study connections between optimistic bilevel programming problems and generalized Nash equilibrium problems. We remark that, with respect to bilevel problems, we consider the general case in which the lower level program is not assumed to have a unique solution. Inspired by the optimal value approach, we propose a Nash game that, transforming the so-called implicit value function constraint into an explicitly defined constraint function, incorporates some taste of hierarchy and turns out to be related to the bilevel programming problem. We provide a complete theoretical analysis of the relationship between the vertical bilevel problem and our “uneven” horizontal model: in particular, we define classes of problems for which solutions of the bilevel program can be computed by finding equilibria of our game. Furthermore, by referring to some applications in economics, we show that our “uneven” horizontal model, in some sense, lies between the vertical bilevel model and a “pure” horizontal game.     

Discrete hamstrung squad car games

Isaacs introduced a game of perfect information played on the integer grid in which a pursuer with certain restrictions on his movements seeks to capture a slower moving evader. He showed that in a special case it was possible for the evader to avoid capture, and went on to pose the research question as to whether the same situation arose when the pursuer's “speed” was increased. We show that the answer to his question is “No”.