arXiv subjects

Azhar Iqbal

Publications and source records attributed to Azhar Iqbal.

At least 19 recordsLinked to original sources

arXiv2026

Entanglement, Evolutionary Stability, and Strategy-Space Dependence in an EWL Quantum Game

We examine how Eisert--Wilkens--Lewenstein (EWL) quantization affects evolutionary stability in a symmetric $2\times2$ game whose interior mixed Nash equilibrium is not evolutionarily stable. In the restricted two-parameter EWL strategy space, the quantum equilibrium $s^*=(\pi/2,\pi/4)$ becomes a strict symmetric Nash equilibrium, and hence an evolutionarily stable strategy, for every $\gamma>0$. Extending the admissible pure strategies to the full three-parameter $SU(2)$ family preserves the Nash equilibrium but introduces a continuum of payoff-neutral mutants, causing strictness and the second ESS condition to fail. Thus, entanglement stabilizes the equilibrium only within the restricted EWL strategy space; the effect does not survive enlargement to full $SU(2)$.

quant-ph↗

arXiv2026

Quenching Speculation in Quantum Markets via Entangled Neural Traders

Speculative trading can drive pronounced market instabilities, yet existing regulatory and macroprudential tools intervene only after such dynamics emerge. Quantum technologies offer a fundamentally new means of shaping economic behavior by introducing non-classical correlations between decision-makers. Here we demonstrate a prototype quantum stock market in which entanglement between traders' valuations mitigates the runaway devaluation characteristic of speculative busts. Using reinforcement-learning agents trading a single commodity, we show that replacing classical valuations with quantum-correlated qubit-encoded valuations stabilizes prices and increases the AI traders' net worth relative to a classical market, where instead agents rapidly converge to liquidation strategies that collapse the asset value. To explain this behavior, we formulate and analyze a quantized version of the $p$-guessing game, a canonical model of speculative dynamics. Quantum entanglement and phase coherence reshape the strategic landscape, eliminating the pathological pure-strategy Nash equilibrium that drives market collapse in the classical game, while mixed-strategy equilibria remain non-degenerate and avoid bust-type outcomes. These results identify quantum correlations as a novel, endogenous mechanism for market stabilization and, more broadly, demonstrate the utility of multi-agent reinforcement learning algorithms for uncovering optimal strategies in complex decision-making frameworks with quantum degrees of freedom.

quant-ph↗

arXiv2025

Analytical Stackelberg Resource Allocation in Sequential Attacker--Defender Games

We develop an analytical Stackelberg game framework for optimal resource allocation in a sequential attacker--defender setting with a finite set of assets and probabilistic attacks. The defender commits to a mixed protection strategy, after which the attacker best-responds via backward induction. Closed-form expressions for equilibrium protection and attack strategies are derived for general numbers of assets and defensive resources. Necessary constraints on rewards and costs are established to ensure feasibility of the probability distributions. Three distinct payoff regimes for the defender are identified and analysed. An eight-asset numerical example illustrates the equilibrium structure and reveals a unique Pareto-dominant attack configuration.

cs.GT↗

arXiv2025

Vulnerability and Defence: A Case for Stackelberg Game Dynamics

This paper examines the tactical interaction between drones and tanks in modern warfare through game theory, particularly focusing on Stackelberg equilibrium and backward induction. It describes a high-stakes conflict between two teams: one using advanced drones for attack, and the other defending using tanks. The paper conceptualizes this as a sequential game, illustrating the complex strategic dynamics similar to Stackelberg competition, where moves and countermoves are carefully analyzed and predicted.

cs.GT↗

arXiv2023

Resolving game theoretical dilemmas with quantum states

We present a new framework for creating a quantum version of a classical game, based on Fine's theorem. This theorem shows that for a given set of marginals, a system of Bell's inequalities constitutes both necessary and sufficient conditions for the existence of the corresponding joint probability distribution. Using Fine's theorem, we re-express both the player payoffs and their strategies in terms of a set of marginals, thus paving the way for the consideration of sets of marginals -- corresponding to entangled quantum states -- for which no corresponding joint probability distribution may exist. By harnessing quantum states and employing Positive Operator-Valued Measures (POVMs), we then consider particular quantum states that can potentially resolve dilemmas inherent in classical games.

quant-ph↗

arXiv2021

Equating the area sums of alternative sectors in a circle

We determine the conditions resulting from equating the area sums of alternative sectors in a circle generated by four, two, and three straight lines, respectively, that connect opposite points on its circumference while passing through a point that is arbitrarily placed within the circle.

math.GM↗

arXiv2021

Two-player quantum games: When player strategies are via directional choices

We propose a scheme for a quantum game based on performing an EPR type experiment and in which each player's spatial directional choices are considered as their strategies. A classical mixed-strategy game is recovered by restricting the players' choices to specific spatial trajectories. We show that for players' directional choices for which the Bell-CHSH inequality is violated, the players' payoffs in the quantum game have no mapping within the classical mixed-strategy game. The scheme provides a more direct link between classical and quantum games.

quant-ph↗

arXiv2019

Game theoretical modelling of network/cybersecurity

Game theory is an established branch of mathematics that offers a rich set of mathematical tools for multi-person strategic decision making that can be used to model the interactions of decision makers in security problems who compete for limited and shared resources. This article presents a review of the literature in the area of game theoretical modelling of network/cybersecurity.

cs.GT↗

arXiv2019

Quantum strategies and evolutionary stability (in Urdu)

This article in Urdu presents an introduction to extension of an established branch of mathematics called game theory towards the quantum domain. We describe concepts of quantum games and evolutionary stability and go through some of the unanswered questions in this research area.

quant-ph↗

arXiv2018

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

In the standard approach to quantum games, players' 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' 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.

quant-ph↗

arXiv2018

The equivalence of Bell's inequality and the Nash inequality in a quantum game-theoretic setting

The interaction of competing agents is described by classical game theory. It is now well known that this can be extended to the quantum domain, where agents obey the rules of quantum mechanics. This is of emerging interest for exploring quantum foundations, quantum protocols, quantum auctions, quantum cryptography, and the dynamics of quantum cryptocurrency, for example. In this paper, we investigate two-player games in which a strategy pair can exist as a Nash equilibrium when the games obey the rules of quantum mechanics. Using a generalized Einstein-Podolsky-Rosen (EPR) setting for two-player quantum games, and considering a particular strategy pair, we identify sets of games for which the pair can exist as a Nash equilibrium only when Bell's inequality is violated. We thus determine specific games for which the Nash inequality becomes equivalent to Bell's inequality for the considered strategy pair.

quant-ph↗

arXiv2017

Kidnapping Model: An Extension of Selten's Game

Selten's game is a kidnapping model where the probability of capturing the kidnapper is independent of whether the hostage has been released or executed. Most often, in view of the elevated sensitivities involved, authorities put greater effort and resources into capturing the kidnapper if the hostage has been executed, in contrast to the case when a ransom is paid to secure the hostage's release. In this paper, we study the asymmetric game when the probability of capturing the kidnapper depends on whether the hostage has been executed or not and find a new uniquely determined perfect equilibrium point in Selten's game.

cs.GT↗

arXiv2016

Generalizing the Lorentz transformations

In this paper we develop a framework allowing a natural extension of the Lorentz transformations. To begin, we show that by expanding conventional four-dimensional spacetime to eight-dimensions that a natural generalization is indeed obtained. We then find with these generalized coordinate transformations acting on Maxwell's equations that the electromagnetic field transformations are nevertheless unchanged. We find further, that if we assume the absence of magnetic monopoles, in accordance with Maxwell's theory, our generalized transformations are then restricted to be the conventional ones. While the conventional Lorentz transformations are indeed recovered from our framework, we nevertheless provide a new perspective into why the Lorentz transformations are constrained to be the conventional ones. Also, this generalized framework may assist in explaining several unresolved questions in electromagnetism as well as to be able to describe quasi magnetic monopoles found in spin-ice systems.

physics.gen-ph↗

arXiv2016

The vector algebra war: a historical perspective

There are a wide variety of different vector formalisms currently utilized in engineering and physics. For example, Gibbs' three-vectors, Minkowski four-vectors, complex spinors in quantum mechanics, quaternions used to describe rigid body rotations and vectors defined in Clifford geometric algebra. With such a range of vector formalisms in use, it thus appears that there is as yet no general agreement on a vector formalism suitable for science as a whole. This is surprising, in that, one of the primary goals of nineteenth century science was to suitably describe vectors in three-dimensional space. This situation has also had the unfortunate consequence of fragmenting knowledge across many disciplines, and requiring a significant amount of time and effort in learning the various formalisms. We thus historically review the development of our various vector systems and conclude that Clifford's multivectors best fulfills the goal of describing vectorial quantities in three dimensions and providing a unified vector system for science.

physics.hist-ph↗

arXiv2016

Geometric Algebra: A natural representation of three-space

Historically, there have been many attempts to produce an appropriate mathematical formalism for modeling the nature of physical space, such as Euclid's geometry, Descartes' system of Cartesian coordinates, the Argand plane, Hamilton's quaternions and Gibbs' vector system using the dot and cross products. We illustrate however, that Clifford's geometric algebra (GA) provides the most elegant description of physical space. Supporting this conclusion, we firstly show how geometric algebra subsumes the key elements of the competing formalisms and secondly how it provides an intuitive representation of the basic concepts of points, lines, areas and volumes. We also provide two examples where GA has been found to provide an improved description of two key physical phenomena, electromagnetism and quantum theory, without using tensors or complex vector spaces. This paper also provides pedagogical tutorial-style coverage of the various basic applications of geometric algebra in physics.

physics.hist-ph↗

arXiv2015

A brief study of time

Understanding the nature of time remains a key unsolved problem in science. Newton in the Principia asserted an absolute universal time that {\it `flows equably'}. Hamilton then proposed a mathematical unification of space and time within the framework of the quaternions that ultimately lead to the famous Minkowski formulation in 1908 using four-vectors. The Minkowski framework is found to provide a versatile formalism for describing the relationship between space and time in accordance with relativistic principles, but nevertheless fails to provide deeper insights into the physical origin of time and its properties. In this paper we begin with a recognition of the fundamental role played by three-dimensional space in physics that we model using the Clifford algebra multivector. From this geometrical foundation we are then able to identify a plausible origin for our concept of time. This geometrical perspective also allows us to make a key topological distinction between time and space, with time being a point-like quantity. The multivector then allows a generalized unification of time and space within a Minkowski-like description.

physics.gen-ph↗

arXiv2015

On the equivalence between non-factorizable mixed-strategy classical games and quantum games

A game-theoretic setting provides a mathematical basis for analysis of strategic interaction among competing agents and provides insights into both classical and quantum decision theory and questions of strategic choice. An outstanding mathematical question, is to understand the conditions under which a classical game-theoretic setting can be transformed to a quantum game, and under which conditions there is an equivalence. In this paper, we consider quantum games as those that allow non-factorizable probabilities. We discuss two approaches for obtaining a non-factorizable game and study the outcome of such games. We demonstrate how the standard version of a quantum game can be analyzed as a non-factorizable game and determine the limitations of our approach.

quant-ph↗

arXiv2015

Social optimality in quantum Bayesian games

A significant aspect of the study of quantum strategies is the exploration of the game-theoretic solution concept of the Nash equilibrium in relation to the quantization of a game. Pareto optimality is a refinement on the set of Nash equilibria. A refinement on the set of Pareto optimal outcomes is known as social optimality in which the sum of players' payoffs are maximized. This paper analyzes social optimality in a Bayesian game that uses the setting of generalized Einstein-Podolsky-Rosen experiments for its physical implementation. We show that for the quantum Bayesian game a direct connection appears between the violation of Bell's inequality and the social optimal outcome of the game and that it attains a superior socially optimal outcome.

quant-ph↗