🔬 physics

Dynamics of voting strategies and public good funding

This paper employs an evolutionary game theory framework to model how local social interactions and migration toward higher utility areas drive the dynamics of voting strategies and public good funding, revealing that spillover effects create free-rider dependencies between poorly and well-funded regions while leading populations to converge into either politically gridlocked or mixed consensus-zealot states.

Original authors: Jonathan Engle, Bryce Morsky

Published 2026-03-27

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jonathan Engle, Bryce Morsky

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a giant, living map of a country made up of thousands of tiny neighborhoods (we'll call them "nodes"). In each neighborhood, people are deciding how to vote on a very important question: Should we spend our money to build a big, shared park (a "public good")?

There are two political parties:

Party 1: Wants to build the park.
Party 2: Doesn't want to build it (or wants to spend the money elsewhere).

But here's the twist: The park isn't just for the people who vote for it. If your neighborhood builds a park, the people in the next neighborhood over get to enjoy it too, even if they didn't pay for it. This is called a "spillover effect."

The authors of this paper created a computer simulation to see how people's voting habits and their desire to move to better neighborhoods change the outcome. They found that the world is full of "free riders" and that geography plays a huge role in whether we get a park or not.

Here is the breakdown of the four types of people in this simulation, using simple analogies:

1. The Four Types of Voters

Think of these as four different personality types at a town hall meeting:

The Zealots (The Die-Hards):
- Party 1 Zealot: "I love Party 1! I will vote for them no matter what."
- Party 2 Zealot: "I love Party 2! I will vote for them no matter what."
- Analogy: They are the people wearing the team jerseys to every game, cheering regardless of the score. They never change their minds.
The Consensus-Makers (The Peacemakers):
- They just want everyone to agree. They don't care which party wins, as long as one side wins by a landslide. They hate it when the vote is a 50/50 tie.
- Analogy: They are the people who say, "Let's just pick a side and move on so we can get lunch." They are happy if 90% of the room agrees, but miserable if it's a deadlock.
The Gridlockers (The Stalemate Lovers):
- They love a tie. They want the vote to be exactly 50/50 so that nothing gets done and no one party dominates.
- Analogy: They are the people who say, "Let's debate forever. If we can't agree perfectly, let's just sit here and do nothing." They thrive on chaos and inaction.

2. The Two Ways People Move

In the real world, people don't just stay put; they move. The paper looked at two ways people might move:

The "Blind Wanderer" (Undirected Movement):
- People move randomly to a neighbor's house just to see what's there. They don't care about the politics or the money.
- Result: This usually leads to a messy mix. Some neighborhoods become "Zealot zones," but most end up stuck in a Gridlock (a tie) because the random movement keeps the numbers balanced.
The "Smart Mover" (Directed Movement):
- People move to where they are happiest. Happiness comes from two things:
  1. Social Comfort: Being around people who think like you.
  2. Economic Gain: Living where the public park (or good) is funded and available.
- Result: This is where it gets interesting. People flock to neighborhoods where the park is built. But because of the spillover effect, they can also move to a neighborhood next to a park and enjoy it for free.

3. The Big Discoveries (The "Aha!" Moments)

The "Free Rider" Trap

The most important finding is about spillovers.
Imagine Neighborhood A builds a beautiful park. Neighborhood B is right next door.

Without Spillovers: People in Neighborhood B must vote to build their own park to enjoy one.
With Spillovers: People in Neighborhood B realize, "Hey, I can just move next to Neighborhood A, enjoy their park for free, and not pay for it!"
The Consequence: This creates a Free Rider Effect. Poorly funded neighborhoods become dependent on well-funded ones. The rich neighborhoods (with the park) get crowded, but the poor neighborhoods stop trying to build parks because they can just "steal" the benefits from their neighbors.

The "Bistability" (The Two Worlds)

The simulation showed that the system tends to get stuck in one of two states:

The Gridlock Zone: Everyone is stuck in a tie. No park gets built. Everyone is unhappy, but the system is stable.
The Consensus Zone: One party wins big, the park gets built, and everyone (mostly) agrees.

The paper found that if you have a few "Zealots" (die-hards), they can push the whole system toward a Consensus. But if you have too many "Gridlockers," the system gets stuck in a permanent stalemate.

The "Rich Get Richer" (But with a Twist)

When people move based on money (wanting the park), they cluster in the neighborhoods that already have the park.

Without Spillovers: The park neighborhoods get huge and crowded. The non-park neighborhoods become ghost towns.
With Spillovers: The "Free Riders" move to the edges of the park neighborhoods. This actually breaks up the huge crowds. The population becomes more spread out because people are happy to live near the park without living in the expensive, crowded park neighborhood.

The Takeaway

This paper uses math to tell a story about how our voting habits and our desire to move shape our society.

If we only care about our own party (Zealots) or we love arguing without solving anything (Gridlockers), we might never build the things we need.
If we let "spillovers" happen (where one town's good deed helps its neighbors), it creates a weird dependency. Some towns stop trying to improve themselves because they can just rely on their neighbors.
The Lesson: To get a good public park (or any public good), we need a mix of people who are willing to compromise (Consensus-makers) and a system where people can't just "free ride" on their neighbors' hard work forever. Otherwise, the system gets stuck in a gridlock, or the rich towns get so crowded that the whole system breaks down.

1. Problem Statement

The paper addresses the collective action problem regarding the production and maintenance of public goods, specifically within the context of a two-party electoral system. While public goods are essential, they are prone to "free-riding," where individuals benefit without contributing. The authors investigate how voting strategies, spatial dynamics, and inter-regional spillovers interact to determine the level of public good funding.

Key challenges identified include:

Free-riding via Spillovers: Local public goods often benefit neighboring regions, creating incentives for regions to under-fund goods and rely on neighbors.
Political Polarization vs. Gridlock: How different voter psychologies (zealots vs. consensus-seekers) influence election outcomes and subsequent funding levels.
Migration Feedback Loops: How individuals migrate based on social preferences (voting alignment) and economic utility (public good availability), potentially reinforcing or disrupting funding stability.

2. Methodology

The authors employ an Evolutionary Game Theory framework coupled with Spatial Dynamics modeled by a system of differential equations.

A. Model Components

Voting Strategies: The electorate is divided into four types:
1. Consensus-makers ( $c$ ): Prefer a clear majority for either party; dislike gridlock.
2. Gridlockers ( $g$ ): Prefer an even split (gridlock) over a majority for either party.
3. Party 1 Zealots ( $z_1$ ): Always vote for Party 1 (the funding party).
4. Party 2 Zealots ( $z_2$ ): Always vote for Party 2 (the non-funding party).
Dynamics:
- Fast Voting Dynamics: Modeled by differential equations (Eqs. 1a, 1b) describing how Consensus-makers and Gridlockers adjust their votes based on the current proportion of votes ( $v$ ) for Party 1.
- Slow Strategy Imitation: Modeled by Replicator Equations (Eqs. 3a–3c). Voters change strategies based on "fitness" ( $f_i$ ), which represents satisfaction with the current outcome. Fitness functions are cosine-based to reflect specific preferences (e.g., Gridlockers peak at $v=0.5$ , Zealots at $v=0$ or $1$).
Spatial Structure:
- The population is distributed on an $L \times L$ grid (torus topology) representing neighborhoods.
- Movement: Two types of migration are simulated:
  1. Undirected: Random diffusion to neighboring nodes.
  2. Directed: Movement toward nodes with higher Utility ( $u$ ).
- Utility Function: $u = \lambda(\text{Economic Utility}) + (1-\lambda)(\text{Social Utility})$ $u = λ (Economic Utility) + (1 - λ) (Social Utility)$ .
  - Economic Utility: Determined by local public good funding ( $v$ ) and spillover effects ( $s$ ) from neighbors.
  - Social Utility: Based on the fitness of the voter's current strategy.

B. Simulation Parameters

Grid Size: $10 \times 10$ nodes.
Movement Rate ( $M$ ): Low ( $10^{-2}$ ) to distinguish spatial effects from well-mixed ODE results.
Spillover ( $s$ ): Varied from 0 (no spillover) to 1 (neighbors benefit fully, potentially creating a free-rider scenario).
Economic Weight ( $\lambda$ ): Varied to test migration driven purely by public goods ( $\lambda=1$ ) vs. social preferences ( $\lambda=0$ ).

3. Key Contributions

Integration of Spatial and Evolutionary Dynamics: The paper uniquely combines local strategy evolution (imitation) with spatial migration driven by both social and economic factors.
Quantification of Spillover Effects: It mathematically demonstrates how public good spillovers create a "free-rider" effect that spatially ties poorly funded regions to well-funded ones.
Bistability Analysis: The model reveals that the system exhibits bistability, where outcomes depend heavily on initial conditions, leading to either consensus or gridlock.
Migration-Strategy Feedback: It highlights how migration patterns can either reinforce polarization (by clustering like-minded voters) or destabilize funding centers (by draining resources from high-funding areas due to free-riding).

4. Results

A. ODE Model (Well-Mixed Population)

Bistability: The system converges to either a Gridlock state (mixed strategies, $v \approx 0.5$ ) or a Consensus state (dominance of one party).
Role of Zealots: Introducing even a small proportion of Zealots reduces the basin of attraction for gridlock, pushing the system toward consensus for their respective party.
Thresholds: A critical mass of Consensus-makers ( $>3/4$ ) is required to overcome gridlock in the absence of Zealots.

B. Spatial Dynamics (Undirected Movement)

Heterogeneity: Unlike the ODE model, spatial diffusion creates "pockets" of Consensus-makers and Zealots surrounded by a sea of Gridlockers.
Outcome: Gridlock remains the dominant global state, but local enclaves achieve consensus based on initial random fluctuations.

C. Directed Movement (No Spillovers)

Economic Drivers ( $\lambda=1$ ): When migration is driven solely by public good availability, individuals flock to high- $v$ (Party 1) regions.
Paradoxical Outcome: High migration to well-funded areas leads to overcrowding and strategy adaptation (Gridlockers becoming Consensus-makers), but if the influx is too high, it can undermine the majority, causing local gridlock.
Population Disparity: Significant spatial variation in population density emerges; low-population areas tend to be gridlocked and underfunded.

D. Directed Movement (With Spillovers)

Free-Rider Effect: When spillovers ( $s > 0$ ) are introduced, the incentive to migrate to high-funding nodes diminishes because individuals can access the public good from neighbors.
Destabilization of Clusters: Spillovers disrupt the formation of large, high-density population centers. Poorly funded regions become spatially dependent on well-funded neighbors.
Moderating Effect: When social utility ( $\lambda < 1$ ) drives migration, it moderates the extreme population disparities seen when only economic utility drives movement.

5. Significance and Implications

Policy Design: The findings suggest that public good funding mechanisms must account for spatial interdependence. Policies that ignore spillovers may lead to under-provision in border regions or create "free-rider" dependencies.
Political Polarization: The model explains how "voting with your feet" (migration) can reinforce political polarization. If voters migrate to align with their preferred party, they create homogeneous enclaves that may be more prone to extreme outcomes or gridlock depending on the mix of zealots and consensus-seekers.
Real-World Application: The model offers a theoretical framework for understanding phenomena like the "Sunbelt migration" in the US or EU fiscal coordination, where spillover effects from one region's policies significantly impact the economic and political landscape of neighbors.
Limitations & Future Work: The authors note that assuming smooth fitness functions and identical nodes limits the model. Future work could incorporate discontinuous thresholds, multiple public goods, and non-political migration drivers (e.g., lifestyle, family).

In conclusion, Engle and Morsky demonstrate that the interplay between voting strategies, spatial migration, and public good spillovers creates complex, non-linear dynamics that can lead to either stable consensus or persistent gridlock, heavily dependent on initial conditions and the degree of economic vs. social motivation in migration.