Quantum Game Theory: Overview, Example, Issues, and the Future

Background and Motivation

Game theory is the field of mathematics that focuses on mathematical models of “games”, or sets of strategic interactions between rational agents to which some value can be applied. While game theory has been considered since at least 1650, formalized game theory as we know it today was pioneered by John von Neumann with his 1928 paper On the Theory of Games of Strategy. In the 1950s game theory was greatly expanded, and used to model real world situations, as well as to consider a more varied spread of game situations.

Classical game theory is both academically and practically interesting, particularly in the field of computer science, where it has applications everywhere from security to online markets that can execute thousands of transactions per second. However, many problems cannot be quickly solved in a classical manner, or do not have a satisfactory classical solution. One new area to be explored is that of quantum game theory, where additional mathematical tools such as quantum entanglement and superposition of states or strategies can literally change the game.

Early Quantum Game Theory and Example

Quantum game theory was first considered around 1999, with one of the first studies on the topic being David A. Meyer’s Quantum Strategies where he looks at the quantum penny flip game. In a classical penny flip game there are two players, Alice and Bob. Alice always starts by setting the initial state of a penny to heads. Next, Bob gets to choose whether to flip the penny, or leave it as is. However, Bob does not get to know the initial state the penny was set to. Alice then gets the same choice, to flip or not, but without knowing Bob’s previous choice. Finally, Bob gets to choose one last time whether or not to flip, still without knowing any of Alice’s moves. If the penny ends with heads showing, Alice loses a point and Bob gains one, otherwise the point goes to Alice and Bob loses one.

This classical version of the game is known as a zero-sum game, as no points can be gained without being taken away from another player. This game can easily be analyzed by a payoff matrix such as Fig. 1, where Alice starts the penny with heads, with her moves represented on the vertical axis, and Bob’s potential moves represented on the horizontal.

As is clear from Fig. 1, the game should, over time, leave both players with a net change in score of 0. However, when playing the game Bob is winning every round. This is because Bob is cheating by playing a quantum game, not the classical one. In this version of the game, heads can be represented by the quantum state

with tails being represented by

While Alice can still use the two moves introduced earlier, with the flip being represented by

and not flipping represented by I, Bob also gets access to a quantum move,

If Bob plays H for his first turn, the quantum penny is put into superposition where it has equal chance of being heads or tails, represented by

This means that neither of Alice’s moves (ₓ or I) can affect the outcome, as

If on Bob’s next move is another H, the penny is taken out of superposition with a probability of being heads equal to 1, letting Bob win every round, something not possible with classical game theory.

Issue with Quantum Game Theory

If both Alice and Bob are given access to quantum strategies, however, the advantage is taken away, as shown in Meyer’s 1999 paper. This leads to one of the arguments against quantum game theory, that it can be made practically equivalent to classical game theory, simply taking into account a more advanced version of the original games. An example of this would be replacing the penny in the above game with a point on a sphere. If the point lands nearer the top, it is heads, and nearer the bottom is tails. While for this example they are indeed equivalent, it has been repeatedly shown that quantum entanglement cannot be replicated by considering hidden or additional variables, and in fact quantum entangled versions of the prisoners’ dilemma can resolve the dilemma even when a classical correlation would fail.

Another potential issue with quantum game theory is that of practicality. When first considered in the early 2000s, no hardware existed to run practical versions of any quantum games. While today such hardware does exist, and in fact works well with the low overall number of qubits needed for many quantum game theory applications, there are still hardware issues. Specifically, game theory generally relies on input from rational actors, and the low decoherence times of current offerings generally would make it impossible for a human to provide this input, requiring classical computers or other automated systems to take the place of players.

Looking to the Future of Quantum Game Theory

As quantum computing continues to rapidly develop, the level of knowledge, practical use, and available hardware for quantum game theory will also develop. Quantum game theory related concepts are already being implemented and used in real world applications, such as Quantum Key Distribution, a way of securely passing secret keys between two actors with the ability to detect any listeners. This is already real world tested and offered as a service by several companies. As quantum networks advance and allow the passing of qubits more easily, quantum game theory will be needed to consider all possible interactions of these systems, helping to secure, speed up, and advance future quantum systems.

Sources:

Khan, Faisal Shah, Solmeyer, Neal, Balu, Radhakrishnan, and Humble, Travis S. Quantum games: a review of the history, current state, and interpretation. United States: N. p., 2018. Web. doi:10.1007/s11128–018–2082–8.

Ghosh, I. Quantum Game Theory — I. Reson 26, 671–684 (2021). https://doi.org/10.1007/s12045-021-1168-2

Ghosh, I. Quantum Game Theory — II. Reson 26, 791–812 (2021). https://doi.org/10.1007/s12045-021-1180-6

Ghosh, I. Quantum Game Theory — III. Reson 26, 939–951 (2021). https://doi.org/10.1007/s12045-021-1193-1

Meyer, D., 1999. Quantum Strategies. [online] Cs.princeton.edu. Available at: <https://www.cs.princeton.edu/courses/archive/fall06/cos576/papers/meyer99.pdf> [Accessed 29 January 2022].

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