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In this note, we present a compatibility test based on John Nashs game-theoretic notion of equilibrium strategy. The test must be taken separately by both partners, making it difficult for either partner alone to control the outcome. The mathematics behind the test including Nashs celebrated theorem and an example from the film, A Beautiful Mind, are discussed as well as how to customize the test for more accurate results and how to modify the test to evaluate interpersonal relationships in other settings, not only romantic. To investigate the long-term dynamics of give and take in a relationship we introduce the iterated dating dilemma and apply the notion of zero-determinant payoff strategy introduced by Dyson and Press in 2012 for the iterated prisoners dilemma.
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We introduce and analyze several variations of Penneys game aimed to find a more equitable game.
We detail the rules and mathematical structure of Al-Jabar, a game invented by the authors based on intuitive concepts of color-mixing and ideas from abstract algebra. Game-play consists of manipulating colored game pieces; we discuss how these colored pieces form a group structure and how this structure, along with an operation used to combine the pieces, is used to create a game of strategy. We also consider extensions of the game rules to other group structures. Note: While this is an article for general readership originally published online by Gathering for Gardner in honor of Martin Gardners birthday (Oct. 2011), Al-Jabar has been played in university abstract algebra courses as a teaching tool, as well as by game enthusiasts, since its release. Moreover, the algebraic game structure described has sparked further work by other mathematicians and game designers. Thus, we submit this article to the ArXiV as a resource for educators as well as those interested in mathematical games.
Causality is a seminal concept in science: Any research discipline, from sociology and medicine to physics and chemistry, aims at understanding the causes that could explain the correlations observed among some measured variables. While several methods exist to characterize classical causal models, no general construction is known for the quantum case. In this work, we present quantum inflation, a systematic technique to falsify if a given quantum causal model is compatible with some observed correlations. We demonstrate the power of the technique by reproducing known results and solving open problems for some paradigmatic examples of causal networks. Our results may find applications in many fields: from the characterization of correlations in quantum networks to the study of quantum effects in thermodynamic and biological processes.
This article is an updated version of an entry of the Encyclopedia of Mathematics Education (2018). In the same time, it is the seed of the HAL collection DAD-MULTILINGUAL, constituted by the translation of this entry in various languages.The documentational approach to didactics is a theory in mathematics education. Its first aim is to understand teachers professional development by studying their interactions with the resources they use and design in/for their teaching. In this text we briefly describe the emergence of the approach, its theoretical sources, its main concepts and the associated methodology. We illustrate these aspects with examples from different research projects. This synthetic presentation is written for researchers, but also for non-specialists (e.g. master students) interested in a first discovery of the documentational approach.
This paper considers a game-theoretic formulation of the covert communications problem with finite blocklength, where the transmitter (Alice) can randomly vary her transmit power in different blocks, while the warden (Willie) can randomly vary his detection threshold in different blocks. In this two player game, the payoff for Alice is a combination of the coding rate to the receiver (Bob) and the detection error probability at Willie, while the payoff for Willie is the negative of his detection error probability. Nash equilibrium solutions to the game are obtained, and shown to be efficiently computable using linear programming. For less covert requirements, our game-theoretic approach can achieve significantly higher coding rates than uniformly distributed transmit powers. We then consider the situation with an additional jammer, where Alice and the jammer can both vary their powers. We pose a two player game where Alice and the jammer jointly comprise one player, with Willie the other player. The use of a jammer is shown in numerical simulations to lead to further significant performance improvements.
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