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Quantum games: a review of the history, current state, and interpretation

Faisal Shah Khan, Neal Solmeyer, Radhakrishnan Balu, Travis Humble

arXiv:1803.07919v2quant-ph

TL;DR

Quantum games have been interpreted through different theoretical and experimental approaches, alongside early controversy over how games relate to quantum physics. This review surveys their history and current state, including formal theory, applications, and experiments. Small-scale human experiments report a greater prevalence of altruism than in the classical game.

  • Problem

    Quantum-game research spans distinct interpretations and approaches, including work framed as cooperative games despite its difference from non-cooperative game theory.

    Source: §1 · p. 2, §4.1 · pp. 12–13

  • Method

    The paper reviews theoretical and experimental developments in quantum games, tracing their history and discussing their controversy, applications, and experimental realization.

    Source: §1 · p. 2, §13 · p. 33

  • Results

    Small-scale human experiments produce a greater prevalence of altruism than in the classical game, while full-scale human implementation has not been achieved.

    Source: §13 · p. 33

  • Takeaways & Limitations

    Quantum networks and quantum Internet-of-Things are identified as immediate practical interests for technological implementations of quantum games.

    Source: §13 · p. 33

    High-fidelity, addressable qubits remain expensive to fabricate, store, operate, and measure, although their relative cost may decline over time.

    Source: §12 · p. 32

Abstract

from arXiv · show

We review both theoretical and experimental developments in the area of quantum games since the inception of the subject circa 1999. We will also offer a narrative on the controversy that surrounded the subject in its early days, and how this controversy has affected the development of the subject.

1 Introduction

Quantum uncertainty links information theory with quantum physics and motivates studying simultaneous optimization through non-cooperative games. The introduction situates quantum information and computing, then outlines the review’s coverage of theory, controversy, experiments, and applications.

1 Introduction

  • Quantum uncertainty provides a natural link between physics and information theory, where measurement produces probability distributions and supports a corresponding notion of quantum entropy.The review frames simultaneous optimization of multiple objectives as central to non-cooperative game theory in quantum systems.Source: §1 · pp. 1–2
  • Quantum information theory developed into quantum computation and communication, with quantum computers and commercial quantum communication devices described as emerging technologies.The introduction cites quantum computing work by Feynman and Deutsch and secure public-key distribution proposed by Bennett and Brassard.Source: §1 · p. 2
  • The review covers formal game-theoretic foundations, historical controversy, quantum algorithms and protocols, stochastic and dynamic games, experiments, and future applications.It also contrasts cooperative and non-cooperative games and introduces entanglement as a proposed mechanism for social equilibrium.Source: §1 · p. 2

2 Non-cooperative games

Non-cooperative games model competitive choices through players’ strategies, preferences, and payoffs. Nash equilibrium captures resistance to unilateral deviation, and fixed-point results establish when such equilibria exist.

2 Non-cooperative games

  • A Nash equilibrium is a play where no player benefits from unilateral deviation, whereas Pareto optimality requires that no alternative makes every player better off.A Pareto-optimal outcome is also a Nash equilibrium when the deviating player is among those made worse off.Source: §2 · pp. 3–4
  • A normal-form game specifies each player’s strategy set and payoff, with payoffs typically representing preferences over possible outcomes.Players are assumed to know one another’s preferences over outcomes, which informs their strategic choices.Source: §2 · pp. 3–4
  • In mixed games with finite strategy sets, Kakutani’s fixed-point theorem guarantees at least one Nash equilibrium.Allowing probability distributions over strategies yields convex strategy spaces and expected payoffs; the countering-set map then supplies the fixed-point argument.Source: §2 · pp. 4–5
  • For infinite or potentially uncountable strategy sets, Glicksberg’s fixed-point theorem can establish Nash equilibrium existence under its stated conditions.The theorem applies to compact, convex subsets of convex Hausdorff topological vector spaces with an upper semi-continuous set-valued map whose values are nonempty and convex.Source: §2 · p. 5

3 Non-cooperative quantum games

Quantum games apply quantum strategies and measurements to game-theoretic settings, from Meyer’s Penny Flip to the EWL and MW protocols. The review describes equilibrium results and the conditions under which guarantees do or do not hold.

3 Non-cooperative quantum games

  • Player 2 can always win Meyer’s Penny Flip game by choosing the Hadamard strategy when the input is |0⟩ or |1⟩.Player 1 uses strategies P1 and P2 from a restricted set of quantum operations, while Player 2 has access to a proper quantum strategy.Source: Figure 1 · p. 6, §3 · p. 6
  • The EWL protocol extends Prisoner’s Dilemma with superposition and entanglement, while preserving the original game and permitting a Nash equilibrium that is Pareto-optimal.Its circuit maximally entangles the qubits before the players’ strategies and disentangles them before computational-basis measurement.Source: §3 · p. 7, Figure 2 · p. 7
  • A pure-strategy quantum game is formulated as a unitary function over players’ pure quantum strategies and outcomes with preferences defined over those outcomes.The review presents EWL and MW as dominant quantization protocols, while describing broader circuit-synthesis implementations as largely unexplored.Source: §3 · p. 7
  • The MW protocol differs from EWL by omitting the disentangling gate, so it does not restrict to the original classical game.The original game is encoded in the computational-basis elements of the Hilbert space.Source: §3 · p. 7, Figure 3 · p. 8
  • Pure quantum games can have guaranteed Nash equilibria when payoff functions are linear, but the EWL expected-payoff-after-measurement formulation does not guarantee one.The review also reports that state preparation with state overlap as payoff guarantees Nash equilibrium.Source: §3 · pp. 8, 10

4 Criticism of quantum games - a discussion

The discussion reviews criticism of the EWL and MW protocols, distinguishes formal cooperative and non-cooperative game theory, and proposes interpreting entanglement through social equilibrium. It also notes that alternative quantization protocols and their applications remain largely unexplored.

4 Criticism of quantum games - a discussion

  • The authors answer the claim that measured EWL outcomes are entirely non-quantum by treating EWL as a prescriptive mechanism for designing expected outcomes at Nash equilibrium.They argue that the same objection could be made about probability-distribution extensions of classical games.Source: §4 · p. 10
  • EWL’s entanglement-generated correlations can be viewed as mediated communication, which blurs—but does not eliminate—the distinction between cooperative and non-cooperative games.The authors relate this to classical mediated games, where Nash equilibrium remains suitable for the extended game.Source: §4 · pp. 10–11
  • A more general quantum-physical implementation of EWL may lack a Nash equilibrium, while randomization restores a near-optimal equilibrium that still pays better than the classical game.The authors note that the strategy set has a classical discrete analogue and that Eisert and Wilkens addressed this criticism in a follow-up.Source: §4 · p. 11
  • EWL is one among many circuit-synthesized quantization protocols, but the consequences of alternatives in economics, evolutionary biology, and other fields remain largely unexplored.The authors also conjecture that MW has received less criticism partly because of its formal mathematical foundation, while noting its limited game-theoretic meaning in its original form.Source: §4 · pp. 10–11

4.1 Quantum games: cooperative versus non-cooperative

  • EWL became physicists’ working definition of non-cooperative quantum games, in part because quantum-computational playability matters for practical economic and financial decisions.The authors contrast this practical focus with the expectation that physicists might emphasize equilibrium or optimal behavior of quantum systems.Source: §4.1 · p. 12
  • The authors argue that much work claiming to cooperatively game quantum processes lacks formal cooperative-game mappings, despite cooperative games’ coalition-based agreements and distinct solution concepts.They distinguish these from non-cooperative games, which use Nash equilibrium, and say unclear analogies leave the work’s game-theoretic or quantum-physical merit uncertain.Source: §4.1 · pp. 12–13

4.2 Quantum entanglement: Nash versus social equilibrium

  • The authors interpret entanglement as a restriction on feasible strategies in an abstract economy, allowing a social-equilibrium reading of the dilemma-breaking outcome.They contrast this with mediated communication yielding near-optimal Nash equilibrium in mixed quantum strategies; the existence of social equilibrium in pure-strategy quantum games is conjectured, not established.Source: §4.2 · pp. 13–14

5 Quantum games, quantum algorithms, and protocols

The authors revisit quantum games as tools for reasoning about quantum computation and communication, including a reinterpretation of MW as a constrained optimization protocol. They describe related applications while noting that Meyer-style searches for quantum advantage at Nash equilibrium remain largely unexplored.

5 Quantum games, quantum algorithms, and protocols

  • The original MW protocol lacks game-theoretic substance, but the authors reinterpret it as the first non-cooperative game-theoretic construction of a constrained-optimal quantum computational protocol.This “gaming the quantum” reading applies players’ Prisoner’s Dilemma preferences to computational-basis elements while setting aside proper game extension.Source: §5 · p. 14
  • The review describes applications spanning quantum gambling, communication efficiency, BB84 security, quantum automata, error correction, and voting.Examples include a fair remote two-party game, a model of BB84 as a zero-sum game, state restoration relevant to error correction, and tactical voting using entanglement, interference, and superpositions.Source: §5 · p. 15
  • Meyer-style searches for quantum advantage at Nash equilibrium remain largely unexplored, despite equilibrium guarantees for mixed quantum states and recent guarantees for pure states.The authors note that a pure-state Nash-equilibrium guarantee appeared only recently [25].Source: §5 · p. 15
  • The authors note that Meyer’s work was followed by sparse early research on using quantum games to develop quantum algorithms and communication protocols.They also mention work connecting network-creation games with quantum algorithms for the quantum Internet and adiabatic-program compilation.Source: §5 · p. 16

6 Bell’s inequalities and quantum games

The section reframes quantum-game advantages through the joint probability distributions available to players, relating them to Bell inequalities and classical advice. It argues that Bell-based games can show advantages unavailable to classical correlated equilibria, while clarifying that entanglement is not the only relevant feature of quantum correlations.

6 Bell’s inequalities and quantum games

  • Bayesian games connect player types and actions to Bell-experiment measurements and outcomes, with non-communication corresponding to locality.Players know their own type, but not their opponent’s; the joint-action distribution is factorizable when one player’s action is not influenced by the opponent’s type.Source: §6 · p. 16
  • A probability-based formulation can give classical and quantum players identical strategy choices, with Bell-inequality violation yielding an advantage in some games but not the Prisoner’s Dilemma.The review cites matching pennies as an example of a game where violation can lead to quantum advantage.Source: §6 · p. 17
  • Quantum players can receive non-factorizable outcome distributions through entanglement or mediated quantum communication, extending the distributions available from mixed strategies.Classical shared advice also produces correlated equilibria, while mixed-strategy Nash equilibrium is a product distribution.Source: §6 · p. 17
  • Entanglement can generate joint outcome distributions that violate Bell inequalities and, with appropriate Bayesian-game payoffs, be more favorable than classical distributions.The comparison is against distributions arising from classical shared advice or hidden variables.Source: §6 · p. 18
  • Bell-inequality games can outperform classical games with correlated equilibria without the strategy restrictions that weaken earlier EWL implementations.The review identifies these games as showing a benefit from quantum correlations without the shortcomings of earlier quantization schemes.Source: §6 · p. 18
  • The section treats outcome distributions as more fundamental than entanglement alone, noting advantages from quantum discord and super-quantum correlations as well.Games based on no-signaling super-quantum correlations can outperform quantum versions.Source: §6 · p. 18

7 Stochastic games

The section reviews Markovian stochastic games, their equilibrium and value results, and the discounted MDP special case. It then identifies a route toward quantum stochastic games using quantum probability and Markov semigroups, while limiting the described backward-induction method to perfect state observation.

7 Stochastic games

  • The review proposes extending stochastic games through quantum stochastic calculus and quantum stochastics, noting little existing work on qMDPs.It outlines a progression from stochastic games to MDPs, partially observed quantized MDPs, quantum probability, and quantum Markov semigroups.Source: §7 · pp. 18–19
  • A classical stochastic game is specified by states, players’ action sets and payoffs, transition probabilities, a discount factor, and an initial state.The review relates the discount factor to patience and to uncertainty about how long the game will continue.Source: §7 · p. 19
  • For finite stochastic games, a Nash equilibrium of the constructed game with N*S players is also a Markov perfect equilibrium of the original game.Each constructed player corresponds to an original player and state; the resulting strategy depends only on the current state.Source: §7 · p. 20
  • The unique fixed point α* of the game operator gives the expected discounted payoff at every history in a subgame perfect equilibrium.The theorem states that a strategy pair is subgame perfect if and only if this payoff equals α*(x) after any history leading to state x.Source: §7 · p. 21
  • Optimal strategies for the zero-sum matrix game defined by Rx(α*) are optimal in the stochastic game and form a Markov perfect equilibrium.The proposition applies to optimal, possibly mixed strategies for both players.Source: §7 · p. 21
  • The reviewed backward-induction algorithm solves the Bellman equation for perfect-state-observation games; asymmetric-information games are left to cited work.The authors confine their quantization starting point to games with symmetric information.Source: §7 · p. 22
  • For finite-state, finite-action MDPs with discount factor λ < 1, a pure stationary strategy exists with the proposition’s stated optimality guarantee.MDPs are presented as games in which one player maximizes payoff against nature, which introduces randomness.Source: §7 · p. 22

8 Quantum probability

The section develops quantum probability as a framework in which classical random variables and measures have operator and trace analogues. It introduces quantum probability spaces, quantum conditional expectation, and quantum Markov semigroups as ingredients for quantum stochastic games.

8 Quantum probability

  • A finite-dimensional quantum probability space is a Hilbert space, an operator *-algebra, and a density-matrix state.The state is represented by a trace-class operator.Source: §8 · p. 23
  • Quantum probability replaces classical random variables and measures with self-adjoint operators and trace mappings, while allowing non-commuting observables in one setting.When the observables of interest commute, the classical spaces can be composed as a product probability space and CP = QP.Source: §8 · pp. 22, 24
  • Classical probability spaces can be converted to quantum probability spaces using square-integrable functions, operator algebras, quantum states, and unitary evolutions.The review describes classical probability measures becoming quantum states through Gleason’s theorem [99].Source: §8 · p. 24
  • Quantum conditional expectation is presented in a form adequate for the paper’s purposes, followed by quantum Markov semigroups defined as one-parameter families of completely positive maps.These constructions supply ingredients for defining quantum stochastic games.Source: §8 · pp. 24–25
  • The proposed quantum stochastic games generalize the probability transition mapping to a quantum Markov semigroup and use partial observation to evaluate payoff.The review distinguishes observation-free quantum random walks from non-demolition measurements of compatible Hermitian operators; coherent evolution may produce dynamics such as Bloch-sphere geodesics.Source: §8 · p. 25
  • The game framework considered has a long but finite horizon, with payoffs calculated and quantum measurements made at the final stage.Source: §8 · p. 25

9 Quantum Markov decision processes

The section defines a qMDP using a finite-dimensional quantum state space, actions, a payoff based on partial observation, a completely positive transition mapping, a discount factor, and an initial state. It also describes continuous non-demolition measurement as an alternative basis for payoff that can allow coherent evolution until a selected time.

9 Quantum Markov decision processes

  • A qMDP is specified by a quantum state space, finite actions, a partially observed payoff, a completely positive transition map, a discount factor, and an initial state.The state space is C^2⊗n, and the action set consists of unitary operators on the Hilbert space.Source: §9 · p. 25
  • In the information-theoretic formulation, the game state is represented by n qubits and each player applies a unitary operator from a fixed finite set as a strategy.Source: §9 · p. 26
  • Payoffs can instead be based on estimates from continuous non-demolition measurements, allowing coherent evolution until a desired time.The cited approach estimates the qubit state through a quantum filtering equation for a system coupled to an environment.Source: §9 · p. 26

10 Quantum stochastic games

The section formulates quantum stochastic games as dynamic games over quantum states, with player actions, payoffs, and state transitions. It describes Markov-perfect equilibrium and a route for applying stochastic-game methods to quantum systems.

10 Quantum stochastic games

  • A quantum stochastic game specifies a Hilbert-space state, player actions and payoffs, a state-transition map, a discount factor, and an initial state.The state-transition map may induce a quantum Markov semigroup when time-dependent.Source: §10 · p. 26
  • Players apply strategies to an initial entangled or product state, while a referee advances the game coherently and payoffs and next states follow from Q and T.State updates may use partial observation or non-demolition measurements on a shared bath.Source: §10 · p. 26
  • A quantum Markov perfect equilibrium requires each player's strategy to depend only on the current state, and is a subset of Nash equilibria.Against other players' Markov strategies, a player's best response can also be Markov.Source: §10 · p. 26
  • The proposed construction starts with a classical stochastic game, builds its canonical Hilbert space, and generates a quantum Markov semigroup for further analysis.The framework can be used to investigate games with different entanglement configurations and cost functions, including conditions for Nash and Markov-perfect equilibria.Source: §10 · p. 27

11 Experimental realizations

Experimental quantum games have been implemented on several platforms, usually by running theoretically selected equilibrium circuits rather than having independent agents play in real time. The review describes hardware demonstrations, their reported outcomes and constraints, and separate experiments involving human players.

11 Experimental realizations

  • Demonstrations typically benchmark hardware by running circuits for theoretically determined Nash strategies, rather than having independent agents play in real time.Quantum-game circuits require single- and two-qubit gates, but their small qubit counts make them attractive for early hardware demonstrations; realistic use may also require networking and quantum memory.Source: §11 · p. 27
  • Despite noise or decoherence generally lowering payoffs, reported quantum-game equilibrium payoffs still outperform the classical game with some amount of decoherence.The implemented games were selected to have a larger payoff at quantum equilibrium than in the classical case.Source: §11 · p. 28

11.1 Nuclear magnetic resonance

  • An NMR two-qubit game reported 8 percent error against theory, while a three-qubit game's experimental populations differed from theory by 15 to 20 percent.The two-qubit study computed 19 data points; the three-qubit study varied initial-state noise and showed payoff decreasing as noise increased.Source: §11.1 · p. 28
  • NMR offers relatively large signal sizes, but scaling and communication links are challenging, and approximate mixed-state initialization leaves uncertainty about whether states are entangled.NMR systems cannot initialize pure states, and promising routes to larger qubit counts or communication interfaces are lacking.Source: §11.1 · p. 28

11.2 Linear optics

  • A single-photon interferometer game reported payoff errors of 1 to 2 percent against theory; another optical experiment reported 95 percent fidelity and a quantum advantage without entanglement.The latter used discord, while linear optics can support remote agents but requires physical circuit rearrangement and lacks memory.Source: §11.2 · p. 29
  • Photon-pair approaches reported fidelities around seventy to eighty percent; a four-player implementation reached near 75 percent fidelity with about 10 percent equilibrium-payoff errors.The four-player game used two entangled-pair inputs and could accept a continuous set of initial states.Source: §11.2 · p. 29

11.3 Other proposals

  • Ion-trapped and cavity-QED systems are proposed platforms with potential for computation, memory, and photonic communication; a five-qubit Bayesian game was demonstrated on a quantum computer.The demonstration analyzed payoff and phase-change-like behavior as entanglement varied.Source: §11.3 · p. 30

11.4 Human implementations

  • In human experiments, quantum-rule players chose the altruistic option more often and matched game-theoretic predictions more closely; authors suggest classical preconceptions may contribute to larger classical deviations.These experiments used classical simulations because quantum hardware was limited; a full implementation with real players on quantum hardware had not been performed.Source: §11.4 · pp. 30–31

12 Future applications for quantum game theory

The section considers quantum game theory as a possible model for applications in communications, publishing, and quantum technology. It also highlights implementation costs, the possibility of classical alternatives, and infrastructure needs for future field tests.

12 Future applications for quantum game theory

  • In a wireless spectrum-sharing minority game, distributing an n-partite entangled state reduced transmission-collision probability by a factor of n while retaining fairness.Transmitters apply local strategies and use correlated measurement outcomes to decide whether to transmit or wait.Source: §12 · p. 31
  • A simulated quantum routing game found minimum network-flow cost with partially entangled states, while demonstrating Braess' paradox only at maximal and vanishing entanglement.The model distributes a quantum state among network nodes, which apply local unitary rotations before measurement.Source: §12 · pp. 31–32
  • For open-access publishing choices, quantized versions of three games with distributed quantum resources yielded Nash equilibria favoring open-access publication.The classical game equilibria either discouraged open access or included it as a minority in a mixed strategy.Source: §12 · p. 32
  • Quantum game theory may model competition between users and the environment in quantum control, error correction, and fault-tolerant operations.The section identifies these as application areas where solutions require models of quantum dynamics and interactions.Source: §12 · p. 32
  • High-fidelity addressable qubits are costly, and classical advice can introduce correlations similar to those supplied by distributed quantum states, raising whether quantum formulations are necessary.Future field testing may require quantum networks that distribute resources among players, along with quantum memories, entangling operations, and synchronized classical control.Source: §12 · pp. 32–33

13 Conclusion

The conclusion reviews quantum-game theory’s theoretical and experimental development and applications, while noting limited growth associated with misconceptions and presenting equilibrium-existence results using geometry and topology.

13 Conclusion

  • Small-scale quantum-game experiments have produced counter-intuitive outcomes, including greater prevalence of altruism than in classical games.The review notes that full-scale human implementation has not been achieved.Source: §13 · p. 33
  • Quantum networks and quantum Internet-of-Things are practical-interest settings where distributing a quantum resource among players creates competition.The review also discusses quantum games’ roles in quantum algorithms and communication protocols, but notes that work in this area is limited.Source: §13 · p. 33
  • The review covers classical stochastic games in resource-competing networks, quantum probability in single-player dynamical processes, and quantum games involving Bell inequalities.Source: §13 · p. 33
  • The authors describe quantum game theory as having stagnated with limited success, and say misconceptions and confusions appear to have discouraged physicists, particularly quantum-information scientists.Source: §13 · p. 33
  • Differential geometry and topology can guarantee Nash-equilibrium existence in pure-strategy quantum games, using classic fixed-point ideas also used in classical game theory and mixed-strategy quantum games.Source: §13 · p. 33
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