Dynamics of voting strategies and public good funding

Abstract

We model an electorate voting on the funding of a public good in a two-party system in an evolutionary game theory framework. Voters adopt one of four strategies: Consensus-makers, Gridlockers, Party 1 Zealots, and Party 2 Zealots, which they may change via imitation. The public good benefits both individuals locally and those in neighbouring regions due to spillover effects. A system of differential equations governs the spatial movement of individuals and shifts in their voting strategies. Local social interactions drive strategy evolution, while migration occurs toward areas of higher utility, which is a function of both social and economic factors. Our results reveal bistability and significant spatial variations. Locally, populations converge to a politically gridlocked state or a mix of consensus-makers and zealots, determining public good provisioning. We find that public good spillovers generate a free-rider effect and poorly funded regions become spatially tied to, and dependent upon, well-funded ones.

Figures (8)

• Figure 1: An assortment of bifurcation diagrams in the $cv$-plane for $g=1-v-c$, $z_2=0$, and varying $z_1$. The bifurcation diagrams show bistability in the system. As we increase the number of party 1 Zealots in the system, the number of unstable equilibria decreases, and fewer Consensus-makers are needed to obtain a consensus result.
• Figure 2: Stability analysis of the boundary of the $cgz_1$-simplex. Where • is a stable equilibrium and • is a saddle equilibrium. The arrows indicate the direction of strategy change.
• Figure 3: Representative time series for the undirected movement scenario with and different initial conditions. Panel (a) depicts a representative example of convergence to a high prevalence of Gridlockers (and thus gridlock). Initial conditions were drawn from a uniform distribution. For panel (b), the densities of Consensus-makers and Party $1$ Zealots are initially high, resulting in convergence to a high vote for Party $1$. Note that movement conserves population size and thus $z_2$ is not plotted.
• Figure 4: Heatmaps for the undirected movement scenario at time $t=50$. Pockets of Consensus-makers and Zealots emerge, though the majority of the population are Gridlockers. Note that movement conserves population size and thus $z_2$ is not plotted.
• Figure 5: Representative time series for the directed movement scenario without spillovers. The preferences for the public good are $\lambda=0,0.5,$ and $1$ for the columns left to right.