Quantum Information, Game Theory, and the Future of Rationality
Spooky Action In Strategic Play
In previous posts, I discussed how quantum superpositions of coins used in gameplay can create a decisive advantage for players with access to quantum technology. Even more dramatically, employing quantum technology as a referee to mediate communication between players in games like the Prisoner’s Dilemma can resolve the dilemma entirely! This remarkable result arises from a specific quantum superposition of two qubits known as quantum entanglement, famously referred to as "spooky action at a distance" by Einstein during the early days of quantum physics. However, experiments have since confirmed that quantum entanglement is a fundamental aspect of the physical world. The challenge now lies in leveraging it effectively to unlock novel value and opportunities.
Faisal Shah Khan, PhD
11/24/20243 min read
In my post titled Beyond Classical Mediation: Enter the Quantum Referee, I explored how quantum computers can act as quantum referees, enabling higher-order correlations between players' strategic actions and facilitating better-paying Nash equilibria. For example, in the case of the Prisoner’s Dilemma, we saw how a quantum referee could resolve the dilemma by eliminating the strong dominance of the Defect strategy over the Cooperate strategy for both players, thus achieving a more favorable equilibrium. In this post, I’ll describe in some more detail the physical mechanisms of both the classical and quantum referees.
When playing a game, a physical mechanism is typically required for implementation. In classical strategic games like Matching Pennies or the Prisoner’s Dilemma, we saw how players can signal their strategic choices using coins. By positioning a coin in either heads (H) or tails (T), they can indicate one of two possible strategic moves. This same mechanism can be adapted for mixed strategies by introducing randomness through a coin toss. Additionally, coins can be correlated—for instance, by connecting them with a cable—to facilitate the coordination of players' strategic moves and achieve a correlated equilibrium. In this post, I will treat the mechanism of the two cable-connected coins as a referee and compare its role and capabilities to those of a quantum referee. Let's consider the game Prisoner's Dilemma again with two strategies Cooperate and Defect per player.
Depending on the physical nature of the referee, both players are aware of the joint probability distribution (p1, p2, p3, p4) that the referee generates over the four possible outcomes of the game: both players choosing Cooperate, one choosing Cooperate while the other chooses Defect, one choosing Defect while the other chooses Cooperate, and both players choosing Defect. When one of the players, say Bob, tosses his coin, the referee instructs him to play either Cooperate or Defect, based on probabilities derived from the marginal distribution of the joint distribution. In this way, the referee mediates, or correlates, the players' choices with respect to the predefined joint probabilities.
Now, Bob must decide whether to follow the referee's advice or disregard it. If both players always agree to follow the referee’s advice, the outcome is a correlated equilibrium—a more refined version of a Nash equilibrium. Suppose the referee advises Bob to Cooperate. As we’ve seen before, the strong dominance of Defect over Cooperate makes it rational for both Bob and Alice to always ignore the referee's advice. Thus, Bob will flip his coin over to state T to indicate that he will Defect. The same reasoning applies to Alice, leading her to Defect as well, leading to a result that is consistent with the sub-optimal Nash equilibrium in Prisoner's Dilemma.
Next, we introduce a quantum mechanism to serve as the referee by replacing the classical coins with qubits (quantum coins). These qubits start in the state H and are "quantum correlated" through quantum operations, analogous to connecting the qubits via a cable that exists only in the quantum realm. This quantum connection maximally entangles the two qubits, enabling them to share intricate correlations beyond classical limits. Suppose Bob interacts with this quantum referee mechanism by tossing his qubit and measuring the result. Based on the quantum referee’s instructions, aligned with a specific probability distribution over the four possible outcomes, Bob is advised to Cooperate. Thinking classically, Bob disagrees and places his qubit in the state T representing Defect. But now, something truly spooky happens.
Bob’s quantum coin remains unchanged, even though he just flipped it! Moreover, Alice’s coin, located elsewhere—possibly millions of miles away—flips to Defect!! This kind of phenomenon is completely alien to our everyday experiences (and if it has ever happened to you, please let me know!). Yet, this is precisely what quantum physics predicts when you flip a qubit in a quantum game. This is the "spooky action at a distance" that Einstein famously criticized, and it forms the foundation of quantum advantage. This counterintuitive behavior enables breakthroughs in scenarios like games mediated by a quantum referee, where it becomes counterproductive for Bob (or Alice) to ignore the referee's advice. Recent experiments on quantum computers, some of which I have had the privilege of contributing to, have confirmed this astonishing prediction in quantum games, revealing how profoundly the quantum world diverges from classical intuition—and hinting at the vast untapped potential of quantum innovation.
Let me leave you with a challenge: can you devise a purely classical physical mechanism that replicates the behavior of the quantum referee described above? If you think you have a solution, I would love to hear from you—get in touch!
