Reservoir computing (RC) is a computer architecture allowing us to utilize any nonlinear dynamical system as a computing device called a reservoir. Exploiting elementary cellular automata (ECAs) as reservoirs is no exception. Recent studies have shown that the RC implemented by asynchronously tuned ECAs (AT_ECAs) has an advantage of enhancing the learning ability to identify multiple patterns, suggesting effects of critical spacetime patterns that the AT_ECAs create in a wide range of transition rules. However, the generalization capability of the AT_ECA-based RC remains unclear. This study evaluated the generalization performance of the AT_ECA-based RC using the temporal parity task in comparison with the ECA-based RC. We found that the AT_ECA-based RC demonstrated higher performance than ECA-based RC in most rules. This might have been a result of critical behaviors that the AT_ECAs universally generate with different mechanisms from the ECAs.

This article studies the location–price Hotelling game. Numerous studies have been conducted on the Hotelling game with simultaneous decisions; however, in real-life scenarios, decisions are frequently sequential. Unfortunately, studies on the sequential Hotelling (SHOT) game are quite scarce. This article contributes to the study of the SHOT game by considering the case in which the location of one of the players, either the leader or the follower, is externally fixed. The game is studied analytically and by numerical simulation to address scenarios where mathematical analysis is cumbersome due to the discontinuous nature of the game. Simulation is found to be particularly useful in evaluating the subgame perfect equilibrium (SPE) solution of these SHOT games, where the follower outperforms the leader as a very general rule, with very few exceptions. This article complements a previous study of the SHOT game where the two locations are parameterized and paves the way to address the analysis of more sophisticated formulations of the SHOT game, such as those with reservation cost and with elastic demand.

This work studies the Hotelling game with sequential choice of prices, that is, the Stackelberg–Hotelling (SHOT) game. The game is studied through numerical simulation, which provides the subgame perfect equilibrium solution not only in the unrestricted game but also in the game with reservation cost and with elastic demand. The simulation technique is tested first in the unconstrained game, where the analytical subgame perfect equilibrium solution was already known. Then, the numerical procedure is generalized to cope with the SHOT game with reservation cost and with elastic demand. These enriched formulations of the SHOT game have not been studied so far, so this article provides an exploratory study of them.

A sequential common-pool resource game with variable elastic marginal profit is studied in this work both analytically and through numerical simulation. The game is studied in both classic and quantum approaches considering symmetric and asymmetric costs. In the classic approach, it is shown how the increase in the level of inelasticity in the model boosts the leader advantage in the perfect equilibrium solution as well as contributes to the depletion of the resource. The quantum approach enables the emergence of the symmetric Pareto optimal solution when the entanglement increases. Furthermore, for high values of the factor of entanglement, the Pareto solution is reached regardless of the level of elasticity of the game. These results are applicable to the model with symmetric and asymmetric costs.

The principle of minimum differentiation stated by Hotelling in his pioneer paper was proved to be invalid 50 years later, since there is a region where the players are so close that Nash equilibrium does not exist. Some authors have contributed with amendments to the original game in the aim of ensuring the existence of Nash equilibrium independently of the location of the players. However, this work analyzes the less frequently studied case of the Hotelling game with close players under different implementations of the game, considering elastic and inelastic demand (conventional Hotelling game, the Hotelling game with reservation cost and Hotelling-Smithies game). The study is supported by means of numerical simulation of the game.

A continuous public goods game with variable elastic marginal profit is studied in this work both analytically and through numerical simulation a la cellular automata The implemented simulation suffers some difficulties in properly finding the Nash equilibrium in games with low elasticity, but proves to be very effective in finding the Pareto optimal solution in games with entangled players.

In the Bertrand–Edgeworth duopoly game, two players compete in price to capture the market demand of a uniform product. The game is studied from a general perspective, so that players with different production costs and capacity constraints as well as the two more important rules dealing with unsatisfied demand (proportional and efficient) are taken into consideration. A quantization scheme is applied to the game with the aim of improving the results compared to the classic game. The quantum Bertrand–Edgeworth duopoly game is studied in this work via spatial numerical simulation, supporting the results analytically when it is possible. In this context, it is found that high entanglement induces a Pareto optimal solution ruled by the lower capacity of the players. The way in which the players’ entanglement acts in the game is examined through simulation, paying special attention to the critical value of entanglement from which the Pareto optimal solution emerges.

In this work, we analyse a common-pool resource game with homogeneous players (both have boundedly rational expectations) and entanglement between players' strategies. The quantum model with homogeneous expectations is a differential approach to the game since, to the best of our knowledge, it has hardly been considered in previous works. The game is represented using a Cournot type payoff functions, limited to the maximum capacity of the resource. The behaviour of the dynamics is studied considering how the fixed points (particularly the Nash equilibrium) and the stability of the system vary depending on the different values of the parameters involved in the model. In the analysis of this game, it is especially relevant to consider the extent to which the resource is exploited, since the output of the players is highly affected by this issue. It is studied in which cases the resource can be overexploited, adjusting the parameters of the model to avoid this scenario when it is possible. The results are obtained from an analytical point of view and also graphically using bifurcation diagrams to show the behaviour of the dynamics.

This work studies the quantum Hotelling game with elastic demand by means of an ad hoc simulation technique that allows to scrutinize how the entanglement of the players induces the emergence of the Pareto optimal solution in Nash equilibrium (NE), even when NE does not exist in the classic game due to the proximity of the players.

The study of Game Theory is widely developed currently and has many applications in several fields such as economics, psychology, biology, etc. The use of quantum theory is a demonstrated way to improve the results comparing to the classic games due to the entanglement between the players. As well, it is known the positive effect of implementing long term memory in the stability of a dynamical system. In this context, memory is applied to the model of a quantum dynamical Cournot duopoly game considering two different cases, depending on the players' expectations and the mechanisms they used to maximize their profits. In this work we analyse the game with homogeneous players, considering two boundedly rational players, and the game with heterogeneous expectations, where one of the players is boundedly rational and the other one is a naive player, comparing the results obtained in both cases. Firstly, we come to the conclusion that neither memory nor the type of players (heterogeneous or homogeneous) produces variations in the stable fixed points comparing to the quantum, or even, classic Cournot duopoly game and, therefore, Nash equilibrium is preserved. Secondly, it is observed that the game with homogeneous players, which is not deeply studied previously in quantum games, can improve the local stability of the system versus the game with heterogeneous players, under certain conditions. Finally, it is shown the role of memory as an effective mechanism of chaos control. This achievement is a remarkable economic advantage, since enables the system to reach the stability faster. Throughout this article, these statements are proved analytically and supported widely with several numerical simulations, using different values of the memory factor.

This work studies the Hotelling game with elastic demand, that is, the Hotelling–Smithies game, through numerical simulation. The implemented simulation technique allows to monitor the correctness of the analytical solution of the game when it is available, for example, the monopolistic model, and to explore scenarios where the analytical solution is not available or cumbersome to find. This is even the case for Nash equilibrium in games with fixed locations. In games with variable location and price, and in games with variable location and fixed price, the locations reached through the simulation drift towards the center, as stated by H. Hotelling in his seminal article.

Previous authors tend to consider a certain range of values of the parameters involved in a game, not taking into account other possible values. In this article, a quantum dynamical Cournot duopoly game with memory and heterogeneous players (one of them is boundedly rational and the other one, a naive player) is studied, where the quantum entanglement can be greater than one and the speed of adjustment can be negative. In this context, we analyzed the behavior of the local stability and the profit in those values. Considering the local stability, it is observed that the stability region increases in the model with memory regardless of whether the quantum entanglement is greater than one or whether the speed of adjustment is negative. However, it is also shown that the stability is greater in the negative than in the positive zone of the speed of adjustment and, therefore, it improves the results obtained in previous experiments. This increase of stability enables higher values of speed of adjustment and, as a result of that, the system reaches the stability faster, resulting in a remarkable economic advantage. Regarding the behavior of the profit with these parameters, the principal effect shown is that the application of memory causes a certain delay in the dynamics. Through this article, all these statements are analytically proved and widely supported with several numerical simulations, using different values of the memory factor, the quantum entanglement, and the speed of adjustment of the boundedly rational player.

This work studies the quantum Hotelling game via spatial numerical simulation. It is concluded that entanglement enables the emergence of the Pareto optimal solution in Nash equilibrium in simulations of the game with variable prices and fixed location of the players. In the rather complicated scenario of variable prices and locations, the implemented simulation technique shows that the players share the market, both locating close to (L/4, 3L/4). The Hotelling game is studied with both linear and quadratic transportation cost, with no relevant discrepancies found in both scenarios.

The quantum Bertrand duopoly game is studied in this work via spatial numerical simulation. It is found that the implemented simulation converges to the Pareto optimal solution if the prices are strongly entangled, even if such a solution is not in Nash equilibrium.

This article studies the quantum Stackelberg duopoly game where the leader player moves as a free-rider. It is found in this scenario that the quantities and payoffs in equilibrium are not far from those achieved when the leader moves according to the backwards induction principle, converging to the Pareto optimal solution as the entanglement factor increases.

The quantum war of attrition game is studied in this work via spatial numerical simulation. It is found that the implemented simulation converges to the Pareto optimal solution, i.e. no fighting at all, when the resign times of the players are entangled with higher factor, whereas larger resign times would be got with weak entanglement. This finding is shown to apply also in a fiercer war game, the war of extermination, in which game the non-entangled (or classical) simulation leads to very high resign times and consequently to very high negative payoffs.

This chapter deals with collective games in which the players are arranged in a spatially structured two-dimensional lattice as explained in Sect. 3.1. Sections 3.2 and 3.3 deal with two-parameter and three-parameter strategy simulations respectively. A variant of the canonical EWL model is considered in Sect. 3.4.

This article studies the quantum Cournot duopoly game via the iterated confronting of a large number of players laying in a two-dimensional lattice. In every iteration, every player plays with his nearest partners and adopts the strategy of his best paid nearest mate. Variable degree of quantum entanglement is taken into consideration in the study.

This chapter focuses on simulations where players are connected at random, instead of in the spatially structured manner as considered so far. Section 5.1 deals with simulations were both player-types update his strategies, whereas in Sect. 5.2 only one of the player-types updates his strategies. The simulations in networks are compared with those in spatial lattices in previous Chaps. 3 and 4 . Both shared and distinctive features of the simulations in both types of layouts are scrutinized.