arXiv subjects
Publications and source records attributed to Cesco Reale.
We determine the optimal bound for the maximum number of moves required to reach a periodic configuration of open mancala (also called open owari), inspired by a popular African game. A mancala move can be interpreted as a map from the set of compositions of a given integer in itself, thus relating our result to the study of the corresponding finite dynamical system.
Observing the existing relationships between the elementary operations of addition, multiplication (iteration of additions) and exponentiation (iteration of multiplications), a new operation (named incrementation) is defined, consistently with these laws and such that addition turns out to be an iteration of incrementations. Incrementation turns out to be consistent with Ackermann's function. After defining the inverse operation of incrementation (named decrementation), we observe that R is not closed under it. So a new set of numbers is defined (named E, Escherian numbers), such that decrementation is closed on it. After defining the concept of pseudoorder (analogous to the order, but not transitive), it is shown that Escherian numbers are not transitive. Then addition and multiplication on E are analysed, and a correspondence between E and C is found. Finally, incrementation is extended to C, in such a way that decrementation is closed on C too. English keywords: hyper-operations, incrementation, zeration, Ackermann function, intransitive order, not transitive order, intransitive numbers, non transitive numbers, not transitive numbers, new number sets.
In many cases the Nash equilibria are not predictive of the experimental players' behaviour. For some games of Game Theory it is proposed here a method to estimate the probabilities with which the different options will be actually chosen by balanced players, i.e. players that are neither too competitive, nor too cooperative. This will allow to measure the intrinsec cooperativeness degree of a game, only in function of its payoffs. The method is shaped on the Prisoner's Dilemma, then generalized for asymmetric tables, N players and N options. It is adapted to other conditions like Chicken Game, Battle of the Sexes, Stag Hunt and Translators (a new name proposed for a particular condition). Then the method is applied to other games like Diner's Dilemma, Public Goods Game, Traveler's Dilemma and War of Attrition. These games are so analyzed in a probabilistic way that is consistent to what we could expect intuitively, overcoming some known paradoxes of the Game Theory.