Quantum games and quaternionic strategies

It is well-known that the phenomenon of entanglement plays a fundamental role in quantum game theory. Occasionally, games constructed via maximally entangled initial states (MEIS) will have new Nash equilibria yielding to the players higher payoffs than the ones they receive in the classical version of the game. When examining these new games for Nash equilibrium payoffs, a fundamental question arises; does a suitable choice of an MEIS improve the lot of the players? In this paper, we show that the answer to this question is yes for at least the case of a variant of the well-known two player, two strategy game of Chicken. To that end, we generalize Landsburg’s quaternionic representation of the payoff function of two player, two strategy maximally entangled states to games where the initial state is chosen arbitrarily from a circle of maximally entangled initial states and for the corresponding quantized games show the existence of superior Nash equilibrium payoffs when an MEIS is appropriately chosen.     

Nonlocal sets of orthogonal multipartite product states with less members

Repeated quantum game theory addresses long-term relations among players who choose quantum strategies. In the conventional quantum game theory, single-round quantum games or at most finitely repeated games have been widely studied; however, less is known for infinitely repeated quantum games. Investigating infinitely repeated games is crucial since finitely repeated games do not much differ from single-round games. In this work, we establish the concept of general repeated quantum games and show the Quantum Folk Theorem, which claims that by iterating a game one can find an equilibrium strategy of the game and receive reward that is not obtained by a Nash equilibrium of the corresponding single-round quantum game. A significant difference between repeated quantum prisoner’s dilemma and repeated classical prisoner’s dilemma is that the classical Pareto optimal solution is not always an equilibrium of the repeated quantum game when entanglement is sufficiently strong. When entanglement is sufficiently strong and reward is small, mutual cooperation cannot be an equilibrium of the repeated quantum game. In addition, we present several concrete equilibrium strategies of the repeated quantum prisoner’s dilemma.

     

Three-player conflicting interest games and nonlocality

We outline the general construction of three-player games with incomplete information which fulfil the following conditions: (i) symmetry with respect to the permutations of players; (ii) the existence of an upper bound for total payoff resulting from Bell inequalities; (iii) the existence of both fair and unfair Nash equilibria saturating this bound. Conditions (i)–(iii) imply that we are dealing with conflicting interest games. An explicit example of such a game is given. A quantum counterpart of this game is considered. It is obtained by keeping the same utilities but replacing classical advisor by a quantum one. It is shown that the quantum game possesses only fair equilibria with strictly higher payoffs than in the classical case. This implies that quantum nonlocality can be used to resolve the conflict between the players.     

Characterizing the Nash equilibria of a three-player Bayesian quantum game

Quantum games with incomplete information can be studied within a Bayesian framework. We consider a version of prisoner’s dilemma (PD) in this framework with three players and characterize the Nash equilibria. A variation of the standard PD game is set up with two types of the second prisoner and the first prisoner plays with them with probability p and \(1-p\), respectively. The Bayesian nature of the game manifests in the uncertainty that the first prisoner faces about his opponent’s type which is encoded either in a classical probability or in the amplitudes of a wave function. Here, we consider scenarios with asymmetric payoffs between the first and second prisoner for different values of the probability, p, and the entanglement. Our results indicate a class of Nash equilibria (NE) with rich structures, characterized by a phase relationship on the strategies of the players. The rich structure can be exploited by the referee to set up rules of the game to push the players toward a specific class of NE. These results provide a deeper insight into the quantum advantages of Bayesian games over their classical counterpart.     

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist.     

Existence of equilibria in quantum Bertrand–Edgeworth duopoly game

Both classical and quantum version of two models of price competition in duopoly market, the one is realistic and the other is idealized, are investigated. The pure strategy Nash equilibria of the realistic model exists under stricter condition than that of the idealized one in the classical form game. This is the problem known as Edgeworth paradox in economics. In the quantum form game, however, the former converges to the latter as the measure of entanglement goes to infinity.     

Mini-maximizing two qubit quantum computations

Two qubit quantum computations are viewed as two player, strictly competitive games and a game-theoretic measure of optimality of these computations is developed. To this end, the geometry of Hilbert space of quantum computations is used to establish the equivalence of game-theoretic solution concepts of Nash equilibrium and mini-max outcomes in games of this type, and quantum mechanisms are designed for realizing these mini-max outcomes.     

Evolutionarily stable sets in quantum penny flip games

In game theory, an Evolutionarily Stable Set (ES set) is a set of Nash Equilibrium (NE) strategies that give the same payoffs. Similar to an Evolutionarily Stable Strategy (ES strategy), an ES set is also a strict NE. This work investigates the evolutionary stability of classical and quantum strategies in the quantum penny flip games. In particular, we developed an evolutionary game theory model to conduct a series of simulations where a population of mixed classical strategies from the ES set of the game were invaded by quantum strategies. We found that when only one of the two players’ mixed classical strategies were invaded, the results were different. In one case, due to the interference phenomenon of superposition, quantum strategies provided more payoff, hence successfully replaced the mixed classical strategies in the ES set. In the other case, the mixed classical strategies were able to sustain the invasion of quantum strategies and remained in the ES set. Moreover, when both players’ mixed classical strategies were invaded by quantum strategies, a new quantum ES set was emerged. The strategies in the quantum ES set give both players payoff 0, which is the same as the payoff of the strategies in the mixed classical ES set of this game.     

How Well Do People Play a Quantum Prisoner’s Dilemma?

Game theory suggests quantum information processing technologies could provide useful new economic mechanisms. For example, using shared entangled quantum states can alter incentives so as to reduce the free-rider problem inherent in economic contexts such as public goods provisioning. However, game theory assumes players understand fully the consequences of manipulating quantum states and are rational. Its predictions do not always describe human behavior accurately. To evaluate the potential practicality of quantum economic mechanisms, we experimentally tested how people play the quantum version of the prisoner’s dilemma game in a laboratory setting using a simulated version of the underlying quantum physics. Even without formal training in quantum mechanics, people nearly achieve the payoffs theory predicts, but do not use mixed-strategy Nash equilibria predicted by game theory. Moreover, this correspondence with game theory for the quantum game is closer than that of the classical game.     

Strategic Multiway Cut and Multicut Games

We consider cut games where players want to cut themselves off from different parts of a network. These games arise when players want to secure themselves from areas of potential infection. For the game-theoretic version of Multiway Cut, we prove that the price of stability is 1, i.e., there exists a Nash equilibrium as good as the centralized optimum. For the game-theoretic version of Multicut, we show that there exists a 2-approximate equilibrium as good as the centralized optimum. We also give poly-time algorithms for finding exact and approximate equilibria in these games.     

The Nash Equilibrium Point of Dynamic Games Using Evolutionary Algorithms in Linear Dynamics and Quadratic System

In this paper, examining some games, we show that classical techniques are not always effective for games with not many stages and players and it can’t be claimed that these techniques of solution always obtain the optimal and actual Nash equilibrium point. For solving these problems, two evolutionary algorithms are then presented based on the population to solve general dynamic games. The first algorithm is based on the genetic algorithm and we use genetic algorithms to model the players' learning process in several models and evaluate them in terms of their convergence to the Nash Equilibrium. in the second algorithm, a Particle Swarm Intelligence Optimization (PSO) technique is presented to accelerate solutions’ convergence. It is claimed that both techniques can find the actual Nash equilibrium point of the game keeping the problem’s generality and without imposing any limitation on it and without being caught by the local Nash equilibrium point. The results clearly show the benefits of the proposed approach in terms of both the quality of solutions and efficiency.     

Discrete time dynamic multi-leader-follower games with feedback perfect information

Both Stackelberg games and Nash games play extremely important roles in such fields as economics, management, politics and behavioral sciences. Stackelberg game can be modelled as a bilevel optimization problem. Static multi-leader-follower optimization problems are initially proposed by Pang and Fukushima. In this article, a discrete time dynamic version of multi-leader-follower games with feedback information is given and analyzed. There are two major contributions in this article. On one hand, based on the multi-leader-follower games, discrete time dynamic multi-leader-follower games are proposed. On the other hand, dynamic programming algorithms are presented to attack discrete time dynamic multi-leader-follower games with multi-players under feedback information structure for dependent followers.     

Establishing Nash equilibria of strategic games: a multistart Fuzzy Adaptive Simulated Annealing approach

This paper's proposal is to show some significant results obtained by the application of the optimization algorithm known as Fuzzy Adaptive Simulated Annealing (Fuzzy ASA) to the task of finding all Nash equilibria of normal form games. To that end, a special version of Fuzzy ASA, that utilizes space-filling curves to find good seeds, is applied to several well-known strategic games, showing its effectiveness in obtaining all Nash equilibria in all cases. The results are compared to previous work that also used computational intelligence techniques in order to solve the same problem but could not find all equilibria in all tests. Game theory is a very important subject, modeling interactions between generic agents, and Nash equilibrium represents a powerful concept portraying situations in which joint strategies are optimal in the sense that no player can benefit from changing her/his strategy while the other players do not change their strategies as well. So, new techniques are always welcome, mainly those that can find the whole set of solutions for a given strategic game.     

A game theoretical perspective on the quantum probabilities associated with a GHZ state

In the standard approach to quantum games, players’ strategic moves are local unitary transformations on an entangled state that is subsequently measured. Players’ payoffs are then obtained as expected values of the entries in the payoff matrix of the classical game on a set of quantum probabilities obtained from the quantum measurement. In this paper, we approach quantum games from a diametrically opposite perspective. We consider a classical three-player symmetric game along with a known expression for a set of quantum probabilities relevant to a tripartite Einstein–Podolsky–Rosen (EPR) experiment that depends on three players’ directional choices in the experiment. We define the players’ strategic moves as their directional choices in an EPR setting and then express their payoff relations in the resulting quantum game in terms of their directional choices, the entries of the payoff matrix, and the quantum probability distribution relevant to the tripartite EPR experiment.     

A novel approximate method of computing extended Nash equilibria

In this paper, we propose a new method of computing an approximate Nash equilibrium with additional features. Existing algorithms often fail to produce an exact solution for games involving more than 3 players. Similarly, existing algorithms do not permit additional constraints on the problem. The principle idea of this paper involves proposing a methodology for computing approximate solutions through evolutionary computation. To do so, we first provide formal definitions of these problems and their approximate versions. Following which, we present the details of our solution. One of the most important advantages of the proposed solution is flexibility, which provides solutions to problems related to Nash equilibrium extensions. The proposed idea is tested on several types of games that vary with difficulty and size. All test sets are generated based on the well-known Gamut program. Additional comparisons with classical algorithms are also performed. Results indicate that Differential Evolution is capable of obtaining satisfactory solutions to large random and covariant games. The results also demonstrate that there is a high probability that even large games, in which a set of strategies with a non-zero probability of being chosen are very small, have a solution. The computation time depends mainly on the problem size, and the original Nash equilibrium problem is unaffected by additional modifications.     

Bounded budget betweenness centrality game for strategic network formations

In computer networks and social networks, the betweenness centrality of a node measures the amount of information passing through the node when all pairs are conducting shortest path exchanges. In this paper, we introduce a strategic network formation game in which nodes build connections subject to a budget constraint in order to maximize their betweenness in the network. To reflect real world scenarios where short paths are more important in information exchange in the network, we generalize the betweenness definition to only count shortest paths with a length limit ? in betweenness calculation. We refer to this game as the bounded budget betweenness centrality game and denote it as ?- B³C game, where ? is the path length constraint parameter.We present both complexity and constructive existence results about Nash equilibria of the game. For the nonuniform version of the game where node budgets, link costs, and pairwise communication weights may vary, we show that Nash equilibria may not exist and it is NP-hard to decide whether Nash equilibria exist in a game instance. For the uniform version of the game where link costs and pairwise communication weights are one and each node can build k links, we construct two families of Nash equilibria based on shift graphs, and study the properties of Nash equilibria. Moreover, we study the complexity of computing best responses and show that the task is polynomial for uniform 2- B³C games and NP-hard for other games (i.e. uniform ?- B³C games with ?≥3 and nonuniform ?- B³C games with ?≥2).     

Equilibria,fixed points,and complexity classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are, respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in two-player normal form games, and (mixed) Nash equilibria in normal form games with three (or more) players. This paper reviews the underlying computational principles and the corresponding classes.     

Abstracting Nash equilibria of supermodular games

Supermodular games are a well known class of noncooperative games which find significant applications in a variety of models, especially in operations research and economic applications. Supermodular games always have Nash equilibria which are characterized as fixed points of multivalued functions on complete lattices. Abstract interpretation is here applied to set up an approximation framework for Nash equilibria of supermodular games. This is achieved by extending the theory of abstract interpretation in order to cope with approximations of multivalued functions and by providing some methods for abstracting supermodular games, thus obtaining approximate Nash equilibria which are shown to be correct within the abstract interpretation framework.     

Nash equilibria in risk-sensitive dynamic games

Dynamic games in which each player has an exponential cost criterion are referred to as risk-sensitive dynamic games. In this note, Nash equilibria are considered for such games. Feedback risk-sensitive Nash equilibrium solutions are derived for two-person discrete time linear-quadratic nonzero-sum games, both under complete state observation and shared partial observation     

Discovering theorems in game theory: Two-person games with unique pure Nash equilibrium payoffs

In this paper we provide a logical framework for two-person finite games in strategic form, and use it to design a computer program for discovering some classes of games that have unique pure Nash equilibrium payoffs. The classes of games that we consider are those that can be expressed by a conjunction of two binary clauses, and our program re-discovered Kats and Thisse?s class of weakly unilaterally competitive two-person games, and came up with several other classes of games that have unique pure Nash equilibrium payoffs. It also came up with new classes of strict games that have unique pure Nash equilibria, where a game is strict if for both player different profiles have different payoffs.