Do transitive preferences always result in indifferent divisions?

The transitivity of preferences is one of the basic assumptions used in the theory of games and decisions. It is often equated with rationality of choice and is considered useful in building rankings. Intransitive preferences are considered paradoxical and undesirable. This problem is discussed by many social and natural sciences. The paper discusses a simple model of sequential game in which two players in each iteration of the game choose one of the two elements. They make their decisions in different contexts defined by the rules of the game. It appears that the optimal strategy of one of the players can only be intransitive! (the so-called \textsl{relevant intransitive strategies}.) On the other hand, the optimal strategy for the second player can be either transitive or intransitive. A quantum model of the game using pure one-qubit strategies is considered. In this model, an increase in importance of intransitive strategies is observed -- there is a certain course of the game where intransitive strategies are the only optimal strategies for both players. The study of decision-making models using quantum information theory tools may shed some new light on the understanding of mechanisms that drive the formation of types of preferences.Comment: 16 pages, 5 figure

Penney's game between many players

We recall a combinatorial derivation of the functions generating probability of winnings for each of many participants of the Penney's game and show a generalization of the Conway's formula to this case.Comment: 6 page

Model-free reinforcement learning for stochastic parity games

This paper investigates the use of model-free reinforcement learning to compute the optimal value in two-player stochastic games with parity objectives. In this setting, two decision makers, player Min and player Max, compete on a finite game arena - a stochastic game graph with unknown but fixed probability distributions - to minimize and maximize, respectively, the probability of satisfying a parity objective. We give a reduction from stochastic parity games to a family of stochastic reachability games with a parameter ε, such that the value of a stochastic parity game equals the limit of the values of the corresponding simple stochastic games as the parameter ε tends to 0. Since this reduction does not require the knowledge of the probabilistic transition structure of the underlying game arena, model-free reinforcement learning algorithms, such as minimax Q-learning, can be used to approximate the value and mutual best-response strategies for both players in the underlying stochastic parity game. We also present a streamlined reduction from 112-player parity games to reachability games that avoids recourse to nondeterminism. Finally, we report on the experimental evaluations of both reductions

The Penney's Game with Group Action

Consider equipping an alphabet $\mathcal{A}$ with a group action that partitions the set of words into equivalence classes which we call patterns. We answer standard questions for the Penney's game on patterns and show non-transitivity for the game on patterns as the length of the pattern tends to infinity. We also analyze bounds on the pattern-based Conway leading number and expected wait time, and further explore the game under the cyclic and symmetric group actions.Comment: 32 pages, 1 figur