Decision-Making under Negativity Bias: Double Hysteresis in the Opinion-Dependent $q$-Voter Model

TL;DR

The paper addresses how negativity bias shapes collective opinion dynamics by introducing an opinion-dependent q-voter model with two group sizes, q_+ and q_-, governing the effort to switch from negative to positive and vice versa. Using mean-field analysis and simulations, the authors show that even minimal asymmetry can produce discontinuous transitions and hysteresis, including a novel double hysteresis with both reversible and irreversible components, and that bias terms β_+ and β_- further enrich the phase diagram. They derive explicit fixed-point relations and, in a key case, analytical expressions for critical points, highlighting conditions under which symmetric coexistence arises and when irreversibility dominates. The findings illuminate how loss-averse social influence can generate path dependence and persistent sentiment shifts in markets or brands, with practical implications for reputation dynamics and crisis recovery; the accompanying open-source Julia code enables replication and further exploration.

Abstract

Negative information often exerts a disproportionately strong impact on human decision-making, a phenomenon known as the negativity bias. In behavioral economics, this effect is formally captured by Prospect Theory, which posits that losses loom larger than equivalent gains. For example, a single negative product review can outweigh numerous positive ones, reflecting this principle of loss aversion in consumer behavior. While this psychological effect has been widely documented, its implications for collective opinion dynamics, critical for understanding market stability and reputation dynamics, remain poorly understood. Here, we generalize the $q$-voter model with independence by introducing opinion-dependent influence group sizes, $q_+$ and $q_-$, which represent the social reinforcement needed to change an opinion from negative to positive and from positive to negative, respectively. We study two versions of this asymmetric model: a baseline model that reduces to the standard $q$-voter model when $q_+ = q_- = q$, and an extended model that incorporates an additional asymmetry expressed as a preference for one opinion. In its reduced version, this represents a minimal model in terms of non-linearity within the $q$-voter framework that allows for discontinuous phase transitions and hysteresis. Using mean-field analysis and computer simulations, we show that these modifications lead to rich collective behaviors, including double hysteresis, one form of which is irreversible, providing a mechanism for path-dependence and the sustained, irrecoverable damage to collective sentiment, brand equity, or market confidence.

Figures (5)

• Figure 1: Illustration of the extended model, where red denotes negative opinions, green positive opinions, and target agents are shown in circles: (a) A negative agent may, with probability $p$, act independently and switch to positive, or, with probability $1-p$, be influenced by the $q$-panel, but only if it is unanimous; otherwise, it retains its state. When exposed to a unanimous positive $q$-panel, a negative agent conforms with probability $\beta_+$ and resists with the complementary probability. (b) The process is analogous for a positive agent, but the influence group sizes in (a) and (b) differ: $q_+$ is the group size required for a shift from negative to positive, and $q_-$ the size required for the reverse. For the baseline model $\beta_+=\beta_-=1$.
• Figure 2: Baseline model. The stationary concentration of $+1$ agents $c^*$ as a function of the probability of independence $p$, obtained from Eq. (\ref{['eq:p_stacionary']}) for (a) $q_{+}=2$ and (b) $q_{+}=6$, and for the values of $q_-$ indicated in the legends. Solid lines denote stable steady states, while dashed lines correspond to unstable steady states. (a) Three different behaviors are observed: a smooth change of $c^*$ with $p$ for $(q_+,q_-)=(2,1)$ (no transition), a symmetric continuous transition for $(q_+,q_-)=(2,2)$, and an irreversible discontinuous transition for $(q_+,q_-)=(2,3)$ and $(2,6)$ (lower branch). (b) There is a symmetric discontinuous transition for $(q_+,q_-)=(6,6)$ and an irreversible discontinuous transition for $(q_+,q_-)=(6,5)$ (upper branch). The $(q_+,q_-)=(6,8)$ curves show both, a reversible and an irreversible discontinuous transitions at the upper and lower branches, respectively. The reversible transition is shown in more detail in Fig. \ref{['fig:double_histeresis']}.
• Figure 3: Baseline model: double hysteresis. Stationary concentration $c^*$ vs $p$ for $q_{+}=6$ and $q_{-}=8$. Solid and dashed lines correspond to the stable and unstable fixed points from Eq. \ref{['eq:p_stacionary']}, respectively, while symbols are the results from Monte Carlo simulations of the dynamics. The numerical simulation value of $c^*$ for each $p$ corresponds to the time average of $c(t)$ at the stationary state of a single realization of the dynamics for a system of $N=10^6$ agents, starting from two different initial conditions, $c(0)=0.0$ (circles) and $c(0)=1.0$ (diamonds). (a) Double hysteresis: a reversible hysteresis takes place in the bistable region $0.0805 \lesssim p \lesssim 0.0817$ (upper branch), while an irreversible hysteresis is observed in the bistable region $0 \le p \lesssim 0.055=p_c$ (lower branch). The former corresponds to a classical hysteresis loop [zoomed in panel (b)], while the later consists of a irreversible hysteresis loop that leads to a cusp catastrophe: once the system jumps from the lower to the upper branch as $p$ overcomes $p_c$, it remains in the upper branch as $p$ is varied. (b) Zoom in on the region of the reversible hysteresis curve.
• Figure 4: Extended model. $c^*$ vs $p$ from Eq. \ref{['eq:p_stacionary-1']} for (a) $q_{+}=q_{-}=2$ and (b) $q_{+}=q_{-}=6$. Each curve corresponds to $\beta_+=1$ and the value of $\beta_{-}$ indicated in the legend. (a) The continuous transition observed for $\beta_+=\beta_-=1$ becomes discontinuous for $0<\beta_-<\beta_+$, showing an irreversible hysteresis loop. (b) Besides the discontinuous transition observed for $\beta_+=\beta_-=1$ with an associated reversible loop, an irreversible hysteresis appears for $0<\beta_-<\beta_+$ (double hysteresis). In both panels, the transitions disappear for $\beta_-=0$.
• Figure 5: Extended model. $c^*$ vs $p$ from Eq. (\ref{['eq:p_stacionary-1']}) for (a) $(q_+,q_-)=(2,1)$ and (b) $(q_+,q_-)=(6,1)$. Each curve corresponds to $\beta_+=1$ and a different value of $\beta_-$, indicated in the legend. (a) The set $(q_+,q_-)=(2,1)$ corresponds to the lowest order in the $q$-panel that leads to an irreversible discontinuous transition at a threshold value $p_c$. The symmetric coexistence solution $c^*=1/2$, stable and unstable, is obtained for $\beta_-=0.5$ from Eq. (\ref{['eq:q_vs_beta']}). The stable solution for $\beta_-=0.5$ (solid line) is the same as that found in the symmetric case $q_+=q_-=2$ and $\beta_+=\beta_-=0.5$ (see main text), and shows a continuous transition. For $\beta_-=0.45$ and $0.55$ the transition becomes discontinuous (lower and upper branches, respectively), characterized by an irreversible hysteresis loop. Inset: Threshold $p_c$ vs $\beta_-$ calculated numerically from Eqs. \ref{['eq:p_stacionary-2']} and \ref{['eq:P-c']} (continuous lines). Dashed lines correspond to the analytical approximations Eqs. \ref{['eq:pc-approx']}. The value $p_c=1/9$ of the continuous transition for $\beta_-=0.5$ is indicated by gray lines. (b) The set $(q_+,q_-)=(6,1)$ corresponds to the lowest order in the $q$-panel that leads to a reversible discontinuous transition. A double hysteresis is observed for $\beta_-=0.01$, composed by an irreversible hysteresis loop in the lower branch and a reversible hysteresis loop in the upper branch, while the curves for $\beta_-=0.1$ and $0.03125$ show only an irreversible transition (upper branches).