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Centipedes Leap into the Quantum Realm

Kaytki Chakankar, Xinhui Tang, Yiguo Zhang

TL;DR

This paper applies quantum game theory to the multi-round centipede game by adapting the Eisert–Wilkens–Lewenstein protocol into a CTZ framework, representing each round as an entangled qubit and encoding decisions as local unitaries. Through analytic expansions and Qiskit simulations, it identifies two quantum Nash equilibria at $[0,0,0]$ and $[\pi,\pi,\pi]$ that yield cooperative payoffs of $(2,2)$, suggesting a degeneracy where distinct quantum strategies produce identical outcomes. The results demonstrate that maximal entanglement and quantum interference can suppress last-round defection ($P_{3d1}=0$) and promote sustained cooperation, aligning better with observed human behavior than classical backward induction. The work introduces a general conjecture about quantum enhancements in sequential two-player games and lays groundwork for scaling quantum game-theoretic analyses to more complex strategic interactions and real hardware.

Abstract

The centipede game is a two-player non-zero-sum game. Each turn, a player can choose whether they want to take or pass a growing reward. The classical, rational solution of this game shows defection in the first round, when in reality, players cooperate much more often. Inspired by prior work employing quantum strategies in the prisoners dilemma, we showed that when similar quantum mechanics principles are applied to the centipede game, it leads to two new quantum Nash equilibria that are superior to the classical solution. Furthermore, by implementing our algorithm on Qiskit, we confirmed that leveraging quantum strategies, rather than strategies like backward induction, to solve the centipede game provided better payoffs for both players and more accurately modeled the games real-life outcomes. Ultimately, we propose a generalized conjecture for similarly structured quantum games.

Centipedes Leap into the Quantum Realm

TL;DR

This paper applies quantum game theory to the multi-round centipede game by adapting the Eisert–Wilkens–Lewenstein protocol into a CTZ framework, representing each round as an entangled qubit and encoding decisions as local unitaries. Through analytic expansions and Qiskit simulations, it identifies two quantum Nash equilibria at and that yield cooperative payoffs of , suggesting a degeneracy where distinct quantum strategies produce identical outcomes. The results demonstrate that maximal entanglement and quantum interference can suppress last-round defection () and promote sustained cooperation, aligning better with observed human behavior than classical backward induction. The work introduces a general conjecture about quantum enhancements in sequential two-player games and lays groundwork for scaling quantum game-theoretic analyses to more complex strategic interactions and real hardware.

Abstract

The centipede game is a two-player non-zero-sum game. Each turn, a player can choose whether they want to take or pass a growing reward. The classical, rational solution of this game shows defection in the first round, when in reality, players cooperate much more often. Inspired by prior work employing quantum strategies in the prisoners dilemma, we showed that when similar quantum mechanics principles are applied to the centipede game, it leads to two new quantum Nash equilibria that are superior to the classical solution. Furthermore, by implementing our algorithm on Qiskit, we confirmed that leveraging quantum strategies, rather than strategies like backward induction, to solve the centipede game provided better payoffs for both players and more accurately modeled the games real-life outcomes. Ultimately, we propose a generalized conjecture for similarly structured quantum games.

Paper Structure

This paper contains 10 sections, 27 equations, 2 figures, 1 table.

Table of Contents

  1. INTRODUCTION
  2. METHODOLOGY
  3. 1) Expansion of $U_1\ket{0}\otimes U_2\ket{0}\otimes U_3\ket{0}$.
  4. 2) Expansion of $U_1\ket{1}\otimes U_2\ket{1}\otimes U_3\ket{1}$.
  5. The resulting theta values are:
  6. RESULTS
  7. DISCUSSION
  8. REFERENCES
  9. ACKNOWLEDGEMENTS
  10. AUTHOR CONTRIBUTION STATEMENT

Figures (2)

  • Figure 1: Illustration of the 3 rounds played in the centipede game
  • Figure 2: The quantum circuit above is the strategy circuit of a 3-round quantum centipede game. The $q_0, q_1, q_2$ represent the 3 rounds of the centipede game. The $H$ and CNOT gates create entanglement, the $S$ gate adds the necessary phase difference, $R_y$ boxes apply the $y$ rotation, and the $c$ line stores the results of the quantum measurements. Qiskit was used to visualize the circuit.