Quantum games of opinion formation based on the Marinatto-Weber quantum game scheme

Quantization becomes a new way to study classical game theory since quantum strategies and quantum games have been proposed. In previous studies, many typical game models, such as prisoner’s dilemma, battle of the sexes, Hawk-Dove game, have been investigated by using quantization approaches. In this paper, several game models of opinion formations have been quantized based on the Marinatto-Weber quantum game scheme, a frequently used scheme to convert classical games to quantum versions. Our results show that the quantization can change fascinatingly the properties of some classical opinion formation game models so as to generate win-win outcomes.

💡 Research Summary

The paper investigates how the Marinatto‑Weber (MW) quantum‑game scheme can be used to quantize three classical opinion‑formation games originally proposed by Di Mare and Latora. The three games are: (i) GM I, a 2 × 2 zero‑sum game with strategies “Change” (C) and “Keep” (K); (ii) GM II, a 3 × 3 zero‑sum game that adds a third strategy “Agree” (A); and (iii) GM III, a 3 × 3 game that further incorporates a distance parameter d between the agents’ opinions, making the game non‑zero‑sum only when d ≤ 1/(b + c).

The authors first recall the MW scheme, which was originally designed for 2 × 2 matrix games. In this scheme each player applies either the identity operator I or a unitary, Hermitian swap operator C with certain probabilities; the initial joint state is a possibly entangled two‑qubit state |ψ_in⟩=∑u_{ij}|ij⟩. Payoffs are obtained as the expectation values of payoff operators built from the classical payoff matrix.

For GM I the authors directly apply the original MW scheme. They derive the expected payoffs ⟨$ₐ⟩ and ⟨$_B⟩, showing that ⟨$ₐ⟩+⟨$_B⟩=0 for any choice of the initial state and probabilities, i.e., the zero‑sum nature is preserved. The Nash equilibrium (NE) remains (Keep, Keep), and the quantum version does not improve the joint payoff.

GM II and GM III require a 3 × 3 extension of the MW scheme. The authors adopt the generalized version proposed by Iqbal and Toor, where each player now has three unitary operators: I (do nothing), C (permutes strategy 1↔3), and D (permutes 1↔2). Players choose these operators with probabilities (p, p₁, 1‑p‑p₁) and (q, q₁, 1‑q‑q₁) respectively. The initial state is a superposition of the nine basis states |ij⟩ with coefficients u_{ij}.

Applying this to GM II, the authors find again that the sum of expected payoffs is zero, so the quantum version remains a zero‑sum game. However, the quantum formulation produces a richer set of Nash equilibria because the strategy space is enlarged by the probabilistic use of the three operators.

The most striking results appear for GM III. The authors compute the joint payoff ⟨$ₐ⟩+⟨$B⟩ (equation 17) and formulate an optimization problem to maximize it under the normalization constraint on the u{ij}. The maximal value is 4/d, achieved when the initial state concentrates probability on off‑diagonal entries (|u_{i* j*}|²=1 for i≠i* and j≠j*). Importantly, this maximum can be reached without the classical restriction d ≤ 1/(b + c).

A concrete example is given with the initial state |ψ_in⟩=√0.5|11⟩+√0.5|33⟩. By setting the probabilities so that both players always apply the “Agree” operator (p₁=q₁=1), the authors obtain a Nash equilibrium where each player’s expected payoff is 1/d and the joint payoff is 2/d, a clear win‑win outcome. This equilibrium exists for any d, showing that quantization removes the distance‑based condition required in the classical model.

In conclusion, the paper demonstrates that quantizing opinion‑formation games via the MW scheme preserves zero‑sum characteristics for games that are inherently competitive (GM I and GM II) but can fundamentally alter the strategic landscape of games that admit cooperative outcomes (GM III). The presence of quantum entanglement enables agents to achieve mutually beneficial outcomes without relying on favorable parameter regimes (such as small opinion distance). The authors suggest that future work should explore more realistic multi‑agent opinion dynamics, dynamic quantum strategies, and empirical validation with real social data.