Related papers: The Measure Game
This paper deals with different concepts for characterizing the size of mathematical objects. A game theoretic investigation and generalization of two size concepts, which can both be formulated in topological terms, is provided: the so…
We introduce two new mathematical games, the Banach-Mazur-Schmidt game and the Banach-Mazur-McMullen game, merging well-known games. We investigate the properties of the games, as well as providing an application to Diophantine…
Game-theoretic probability uses the structure of gambles to define a concept like probability, but which is more flexible and robust. We show that results in game-theoretic probability can be thought of as minimax theorems for specific…
Schmidt games and the Cantor winning property give alternative notions of largeness, similar to the more standard notions of measure and category. Being intuitive, flexible, and applicable to recent research made them an active object of…
This work contains the mathematical exploration of a few prototypical games in which central concepts from statistics and probability theory naturally emerge. The first two kinds of games are termed Fisher and Bayesian games, which are…
Pursuing a new approach to the study of infinite games in combinatorics, we introduce the categories $\mathbf{Game}_{A}$ and $\mathbf{Game}_{B}$ and improve some classical results concerning topological games related to the duality between…
The game of Prisoner Dilemma is analyzed to study the role of measurement basis in quantum games. Four different types of payoffs for quantum games are identified on the basis of different combinations of initial state and measurement…
Matthew Baker investigated, in previous work, an elegant, infinite-length game that may be used to study subsets of real numbers. We present two accessible examples of how an important technique from set theory, or a different technique…
We present a version of the Banach-Mazur game, where open sets are replaced by elements of a fixed partially ordered set. We show how to apply it in the theory of Fraisse limits and beyond, obtaining simple proofs of universality of certain…
The classical Banach-Mazur game characterizes sets of first category in a topological space. In this work, we show that an effectivized version of the game yields a characterization of sets of effective first category. Using this, we give a…
We develop a systematic approach to quantum probability as a theory of rational betting in quantum gambles. In these games of chance the agent is betting in advance on the outcomes of several (finitely many) incompatible measurements. One…
Schmidt's game is a powerful tool for studying properties of certain sets which arise in Diophantine approximation theory, number theory, and dynamics. Recently, many new results have been proven using this game. In this paper we address…
Games are useful tools for introducing new concepts to students. This paper describes a competitive two-player game for sophomore students in a modern physics survey course or junior/senior students in an introductory quantum mechanics…
We present new game semantics of Martin-L\"of type theory (MLTT) equipped with One-, Zero-, N-, Pi-, Sigma- and Id-types. Our game semantics interprets MLTT more accurately than existing ones. Another advantage of our game semantics over…
An asymmetric generalization of classical Cournot's duopoly game was introduced and the simulation scheme of its quantized version was analyzed. In this scheme, the player assigned by a 'classical' measurement scheme always wins the player…
We reason about possible future development of quantum game theory and its impact on information processing and the emerging information society. Two of the authors have recently proposed a quantum description of financial market in terms…
We consider strong law of large numbers (SLLN) in the framework of game-theoretic probability of Shafer and Vovk (2001). We prove several versions of SLLN for the case that Reality's moves are unbounded. Our game-theoretic versions of SLLN…
Schmidt's game is generally used to deduce qualitative information about the Hausdorff dimensions of fractal sets and their intersections. However, one can also ask about quantitative versions of the properties of winning sets. In this…
This paper introduces a measure of uncertainty in the determination of the Shapley value, illustrates it with examples, and studies some of its properties. The introduced measure of uncertainty quantifies random variations in a player's…
We study a basic sequential model for the discovery of winning coalitions in a simple game, well known from its use in defining the Shapley-Shubik power index. We derive in a uniform way a family of measures of collective and individual…