Related papers: The Four Number Game
Any system that is used for naming or representing numbers is a number system, also known as numeral system. The modern civilization is familiar with decimal number system using ten digits. However digital devices and computers use binary…
Graph games of infinite length are a natural model for open reactive processes: one player represents the controller, trying to ensure a given specification, and the other represents a hostile environment. The evolution of the system…
We define the family of {\it locally path-bounded} digraphs, which is a class of infinite digraphs, and show that on this class it is relatively easy to compute an optimal strategy (winning or nonlosing); and realize a win, when possible,…
We consider a zero-sum continuous time stopping game in which the pay-off is revealed in the maximum of the two stopping times instead of the minimum, which is the case in Dynkin games.
In a guessing game, players guess the value of a random real number selected using some probability density function. The winner may be determined in various ways; for example, a winner can be a player whose guess is closest in magnitude to…
We cosider random dynamical systems with randomly chosen jumps. The choice of deterministic dynamical system and jumps depends on a position. We proove the existence of an exponentially attractive invariant measure and the strong law of…
The voting process is formalized as a multistage voting model with successive alternative elimination. A finite number of agents vote for one of the alternatives each round subject to their preferences. If the number of votes given to the…
A simple weakly frequency dependent model for the dynamics of a population with a finite number of types is proposed, based upon an advantage of being rare. In the infinite population limit, this model gives rise to a non-smooth dynamical…
The numbers game is a one-player game played on a finite simple graph with certain ``amplitudes'' assigned to its edges and with an initial assignment of real numbers to its nodes. The moves of the game successively transform the numbers at…
Probabilistic properties of tennis scoring systems are examined and compared with best-of-K systems. A model, where each player has his/her own probability of winning his/her service point and which remains invariant for the duration of the…
In set theory without the axiom of regularity, we consider a game in which two players choose in turn an element of a given set, an element of this element, etc.; a player wins if its adversary cannot make any next move. Sets that are…
The well known binary and decimal representations of the integers, and other similar number systems, admit many generalisations. Here, we investigate whether still every integer could have a finite expansion on a given integer base b, when…
Counters that hold natural numbers are ubiquitous in modeling and verifying software systems; for example, they model dynamic creation and use of resources in concurrent programs. Unfortunately, such discrete counters often lead to…
Despite their apparent simplicity, random Boolean networks display a rich variety of dynamical behaviors. Much work has been focused on the properties and abundance of attractors. We here derive an expression for the number of attractors in…
We obtain sharp bounds for the number of n-cycles in a finite graph as a function of the number of edges, and prove that the complete graph is optimal in more ways than could be imagined. En route, we prove some sharp estimates on power…
Minority game is a model of heterogeneous players who think inductively. In this game, each player chooses one out of two alternatives every turn and those who end up in the minority side wins. It is instructive to extend the minority game…
In this short note we give an example of a four-person finite positional game with perfect information that has no positions of chance and no Nash equilibria in pure stationary strategies. The corresponding directed graph has only one…
In this work, we begin the study of a new class of dynamical systems determined by interval maps generated by the symbolic action of erasing substitution rules. We do this by discussing in some detail the geometric, analytical, dynamical…
Given entropy's central role in multiple areas of physics and science, one important task is to develop a systematic and unifying approach to defining entropy. Games of chance become a natural candidate for characterising the uncertainty of…
We study zero-sum games, a variant of the classical combinatorial Subtraction games (studied for example in the monumental work "Winning Ways", by Berlekamp, Conway and Guy), called Cumulative Subtraction (CS). Two players alternate in…