Cross-border political competition

Abstract

Individuals are increasingly exposed to news and opinion from beyond national borders. This news and opinion are often concentrated in clusters of ideological homophily, such as political parties, factions, or interest groups. But how does exposure to cross-border information affect the diffusion of ideas across national and ideological borders? Here, we develop a non-linear mathematical model for the cross-border spread of two ideologies. First, we describe the standard deterministic model where the populations of each country are assumed to be constant and homogeneously mixed. We solve the system of differential equations numerically by the Runge-Kutta method and show how small changes in the influence of a minority ideology can trigger shifts in the global political equilibrium. Second, we simulate recruitment as a stochastic differential process for each political affiliation and fit model solutions to population growth rates and voting populations in US presidential elections from 1932 to 2020. We also project the dynamics of several possible scenarios from 2020 to the end of the century. We show that cross-border influence plays a fundamental role in determining election outcomes. An increase in foreign support for a national party's ideas could change the election outcome, independent of domestic recruitment capacity. One key finding of our study suggests that voter turnout in the US will grow at a faster rate than non-voters in the coming decades. This trend is attributed to the enhanced recruitment capabilities of both major parties among non-partisans over time, making political disaffection less prominent. This phenomenon holds true across all simulated scenarios.

Fig 1. Ilustrative figure of the dynamical system.

Fig 2. Number of agents supporting each ideological bloc over time.

I: Simulations for same initial population size (A₁ = B = C = A₂ = D = E) and same influence, with parameters β_B = β_C = β_D = β_E = 0.05 and ϕ_B = ϕ_C = ϕ_D = ϕ_E = 0.02. II: Simulations for different initial population size (A₁/2 = B/2 = C/2 = A₂ = D = E) and same influence, with parameters β_B = β_C = β_D = β_E = 0.05 and ϕ_B = ϕ_C = ϕ_D = ϕ_E = 0.02. III: Simulations for same initial population size (A₁ = B = C = A₂ = D = E) and different influence, with β_B = 0.04 and parameters β_C = β_D = β_E = 0.05 and ϕ_B = ϕ_C = ϕ_D = ϕ_E = 0.02. IV: Simulations for different initial population size (A₁/2 = B/2 = C/2 = A₂ = D = E) and different influence, with β_B = 0.04 and parameters β_C = β_D = β_E = 0.05 and ϕ_B = ϕ_C = ϕ_D = ϕ_E = 0.02. V and VI: Simulations for different initial population size (A₁/2 = B/2 = C/2 = A₂ = D = E) and different influence, with β_B = 0.12, β_C = 0.04, β_D = 0.06, β_E = 0.12 and ϕ_B = 0.01, ϕ_C = 0.03, ϕ_E = 0.01. With these parameters, there is an inflection point around ϕ_D = 0.0267 whose values above and below determine the domains of the success function of one or the other competing ideology.

Fig 3. Phase portraits of scenarios V (ϕ_D = 0.02) and VI (ϕ_D = 0.03).

Representation of the trajectories of solutions of the dynamical system in the phase plane for the initial conditions in V and VI. Parameters: A₁/2 = B/2 = C/2 = A₂ = D = E, and k_B = 0.6, k_C = 0.4, k_D = 0.6, k_E = 0.6, p_B = 0.2, p_C = 0.1, p_D = 0.1, p_E = 0.2, ϕ_B = 0.01, ϕ_C = 0.03, ϕ_E = 0.01.

Fig 4. Supporters of each political group over time.

The simulated scenarios correspond to the parameters described in Tables 1 and 2. Small shifts in political influence outside the country translate into changes in the domestic political map without the need for increases in domestic recruitment capacity. Voter turnout will grow at a faster rate than non-voters in the coming decades.