Asymmetric Outcomes in Two-Actor Conflict Dynamics: Stability, Bifurcations, and Emergent Behaviors

TL;DR

The paper analyzes a generalized two-actor CODA-style conflict model by introducing threshold-steepness parameters $p_1$ and $p_2$, forming a three-parameter non-dimensional system $(p,q,r)$. The authors derive the nondimensional ODEs $dx/d\tau=-x+p\tanh(y)$ and $dy/d\tau=-q y+r\tanh(x)$, classify feedback regimes, and perform equilibrium and stability analyses using an auxiliary function $h(x)$, complemented by a nonstandard discretization to preserve stability numerically. They show a supercritical pitchfork bifurcation for cooperative and competitive feedback and demonstrate that no limit cycles occur, mapping out basins of attraction under varying initial conditions. The work reveals phenomena such as false support and non-strict consensus in cooperative settings and polarized outcomes in competitive ones, offering enhanced descriptive power for social dynamics and a foundation for extending to larger networks and stochastic settings.

Abstract

In this paper we present an analytical and numerical study of a generalized model of two-actor cooperative-competitive conflict of the Continuous Opinions and Discrete Actions (CODA) type. Theoretically, we note that the in troduction of a new parameter allows generalizing feedback as strong and weak. Furthermore, we show that for positive-positive and negative-negative feedback there exists a supercritical pitchfork bifurcation, and that the model does not admit limit cycles in any case, and we study the effect of different parameter values and initial conditions by using a difference equation approximation of the model. Additionally, our model offers important insight on social phenomena such as false levels of support among cooperators, often observed in agreement negotiations; instances of ``non-strict consensus'' when two people support the same political position, albeit with different intensities; and competitive situations, such as in competitions with disproportionate profit and losses. Thus, this generalized model offers an enhanced descriptive power compared to previously proposed models.

Figures (6)

• Figure 1: Time evolution of the opinions of the two actors in the system with positive-negative feedback. Three types of initial conditions are considered: both positive (red crosses), both negative (blue crosses) and with different signs (grey markers). The observed behavior agrees with Theorems \ref{['0_onlyEquilibrium__mixed_feedback']} and \ref{['Limit_cycles']}. In all cases, $r=-p=4/3$, so the threshold value was given by $4|pr|\approx 7.11$. (a) If $(q-1)^2<4|pr|$, the states oscillate towards equilibrium, (b) while if $(q-1)^2\geq 4|pr|$, the states decay monotonically towards the equilibrium.
• Figure 2: Time evolution of the states, or opinions, of the two actors in the system with positive-positive and negative-negative feedback. Three types of initial conditions are considered: both positive (red crosses), both negative (blue crosses) and with different signs (grey markers). (a)-(c) show positive-positive feedback, (d)-(f) show negative-negative feedback. In all cases, $|p|=|r|=4/3$, so the threshold value is $pr\approx 1.77$, thus if $q<pr$ (strong feedback), then the system evolves to one of the three existing equilibrium points (see (a)-(b) and (d)-(e)). On the other hand if $q\geq pr$ (weak feedback), then the system evolves to the only existing equilibrium point, $\hat{0}$ (see (c) and (f)).
• Figure 3: Individual effect of parameters $m_i$ and $p_i$ on systems with strong positive-positive feedback. Three types of initial conditions are considered: both positive (red crosses), both negative (blue crosses), and with opposite signs (grey markers). In (a) $m_1=1, m_2=0.7$ and $p=r=2$, $x_1$ converges to a final state closer to its initial state than actor $x_2$. In (b) $m_1= m_2 =1$, $c_1=c_2=3/2$, $p_1= 8/9$ and $p_2 = 2$, $x_1$ converges to a value farther away from its initial state than actor $x_2$.
• Figure 4: Functions related to Lemma \ref{['Lemma_h_function']}. (a) A sketch of the graph of the function $f(x)= \operatorname{sech}^2(x) \operatorname{sech}^2(\operatorname{arctanh}(x/p)), x \in (-p, p)$, if $q<pr$, there exists $u\in (0,p)$ such that $f(u)=q/(pr)$. (b) A sketch of the graph of the function $h(x)=(r/q)\tanh(x)-\operatorname{arctanh}(x/p), x \in (-p, p)$, when $q<pr$ (red) and when $q\geq pr$ (blue).
• Figure 5: Bifurcation diagrams of the system (\ref{['Reduced_model']}) with $|p|=|r|=4/3$ (particularly $pr\approx 1.77$). The system has three different equilibrium points when considering strong feedback (i.e. when $q<pr$), see (a) and (b) for positive-positive and negative-negative feedback, respectively. Note that the actors' equilibrium states (configurations) have the same sign with positive-positive feedback and opposite signs otherwise. When the feedback is positive-negative (see (c)) or $q> pr$, then $\hat{0}$ is the only equilibrium point (both values of each possible configuration of the system are equal to $0$). In these diagrams $s$ and $u$ denote stable and unstable entries of equilibrium points, respectively.

Theorems & Definitions (23)

• Lemma 1
• proof
• Proposition 1
• proof
• Proposition 2
• proof
• Theorem 1
• proof
• Theorem 2
• proof