Agent Based Models

Computational Analysis of Social Complexity

Prerequisites

None 😏

Outcomes

Understand what a model is
Know the difference between what we call equation based models and agent based models
Understand the key building blocks of agent based models
Learn the key components of the Schelling segregation model

References

Why Models?¶

Many topics of interest for social scientists are either unethical or unreasonable to study in a laboratory
- Impact on communities of upsurge in illicit drug usage (can’t give people drugs to see impact)
- Flow of traffic given a new infrastructure updates (too costly to experiment with)
- Impact of new tariffs in international trading relationships (too costly to coordinate legislation and implement)
For this reason, we as social scientists turn to models to study our problems
A model is a probability distribution over outcomes
I’ll repeat for emphasis: a model is a probability distribution over outcomes
- Example: A model of housing prices doesn’t predict your house will sell for exactly $350,000
- Instead, it might say there’s a 60% chance it sells between $340,000-$ 360,000, 30% chance between $320,000-$ 340,000, etc.
- The model describes the likelihood of different outcomes, not a single deterministic answer

Types of Models¶

A model is a mathematical object: equations, rules, distributional assumptions.
At its heart, a model is a simplification of some real-world system or phenomenon
- Much complexity is abstracted away (or not included directly in model)
- Key aspects relevant for study are modeled explicitly (e.g. trading response to tariffs)
For our purposes, we will think of models as belonging to one of two families
- Equation based models
- Agent based models

This is a simplification and not a perfect classification (because equation based models have agents and agent based models have equations), but we will be able to draw useful distinctions with this classification.

Equation Based Models¶

An equation based model describes the decision making setting for each agent using mathematical equations
Typically, these are posed as (constrained) optimization problems
A set of equations is also developed that describe interaction between agents
These equations can feature random variables and will require specification of model parameters
Most models I study and develop in my economics research are equation based
Pros:
- Allow precise specification of assumptions, incentives, and outcomes
- Have wide toolbox of numerical optimization, and statistical fitting to “solve” model
Cons:
- Optimization and calibration of parameters can be very difficult
- Often subject to the “curse of dimensionality”, which limits size and complexity of model

Agent Based Models (ABMs)¶

An agent based model describes rules for how individual agents respond to their environment
There are usually many agents, each with a set of properties
One common property is the type of the agent: usually drawn from a small/finite set (buyer-seller, parent-child-teacher, sheep-wolf)
All agents of the same type have the same set of additional properties
Each agent has a state at each time step $t$
The rules are equations that specify how the state of an agent is updated between periods $t$ and $t+1$
- Rules are common for all agents of a type, but vary based on that agent’s state and property values
- Rules will often have random variables as well as parameters
- Rules often include notion of “neighboring” agents
Pros:
- Focus on how an individual should respond in a given state without requiring optimization
- Because rules are typically mathematically simple, can have many many agents
Cons:
- Often lacks notion of equilibrium (could be a feature)
- Not very “reusable” -- to study specific topic you usually have to create whole new model
- Sometimes too many parameters: need for careful calibration

Example: Farmer’s Market Pricing¶

Consider modeling prices at a local farmer’s market for tomatoes:

Equation-based approach:

Define supply and demand curves
Solve for equilibrium price where supply equals demand
Result: Market clears at $3.50/lb with 500 lbs sold
Best for: Understanding average market price, total quantity sold, analyzing policy impacts (e.g., effect of a $0.50/lb subsidy)

Agent-based approach:

Individual vendors: each has costs, quality, inventory, and pricing strategy
Individual buyers: each has budget, quality preferences, and willingness to pay
Rules: Buyers visit stalls, compare prices/quality, purchase or move on
Vendors adjust prices based on remaining inventory and time of day
Emergent patterns: Price dispersion, quality segments, end-of-day discounts
Best for: Understanding why some vendors charge more, how relationships form, impact of vendor reputation

The equation model tells us the average outcome; the ABM shows us the rich variety of individual transactions that create that average.

ABMs¶

For the next few lectures we’ll focus on agent base models
We’ll start by outlining the main components of an ABM
Then we’ll talk about how we could represent them in Julia using the Agents.jl library
- This will require a step up in our Julia skills, so we’ll spend some time covering these concepts in greater detail
Finally we’ll see a few examples of ABMs in practice

NOTE: Most of the study of the Julia skills and ABM examples are not in this notebook

ABM components¶

ABMs are made up of 3 distinct components:
1. Agents
2. Environment
3. Rules

Agents¶

Have state at discrete time steps $t$ (state is value of properties, some properties might be fixed)
Always aware of its own state
Autonomous: can make a decisions independent of other agents
Reactive: can respond to changes in environment or state of other agents
Proactive: can behave in a way to achieve a goal
Communicate: can make some attributes visible to other agents

Environments¶

One of two types
1. Natural Environments: biophysical landscapes and settings
2. Artifical environments: classrooms, economic markets, parks, transportation streets, buildings, etc.
Agents reside within an environment
Properties of environment can be fixed (size, dimensions) or varying (weather, congestion, unused capacity)
Agents can observe and potentially respond to properties of the environment

Rules¶

Rules are the key feature that makes ABMs dynamic
Types of rules:
- Inter-agent: how agents communicate and respond to one another (e.g. information spread)
- Agent-environment rules: How an agent responds to an environment (e.g. avoid park if raining), or how an agent’s decisions and behaviors impact environment (e.g. more cars => more pollution)
- Intra-environmental rules: cause and effect mechanisms within the environment (e.g. more rain => more vegetation)

ABMs are particularly powerful for social science because:

No equilibrium required: Social systems rarely reach stable equilibria
- Fashion trends, social media virality, political movements constantly evolve
Heterogeneity matters: Individual differences drive social outcomes
- Not everyone responds the same way to incentives or information
Local interactions dominate: Who you know matters more than population averages
- Job opportunities through networks, not random matching
- Disease spread through actual contact patterns
Path dependence: History and timing matter
- Early adopters can shift entire market dynamics
- Small initial differences can lead to dramatically different outcomes
Emergence: Simple individual rules create complex social patterns
- Segregation can emerge without strong individual preferences (as we’ll see with Schelling)

ABMs in Julia¶

We need a way to represent these three components in Julia
Agents: represent as a Julia struct
- Struct fields record agent properties
- Our custom agent type can have methods that ascribe behavior to agents
Environments: either explicitly as Julia struct or implicitly in the update rules
Rules: julia functions
- Key function is step! which will allow our agents to make decisions and have the environment and agent properties update in response

Schelling Segregation Model¶

We will work on learning how to represent our agents, rules, and environment in Julia
To make that discussion more concrete, it will be helpful to have a model to implement
The “hello world” of ABMs may just be the Schelling segregation model
References include QuantEcon and Agents.jl tutorial

Schelling’s Work¶

Thomas Schelling won a nobel price in economics for his study of racial segregation
At the heart of his study, was a model proposed in 1969 for how racial segregation can occur in urban areas
One theme of this work (and ABMs in general) is that local interactions (like decisions of individual agents) can lead to surprising aggregate results

The Model¶

Environment: 25x25 grid of single family dwellings
Agents with properties:
- location (x,y) coordinate for current home
- type: orange or blue. Fixed over time. 250 of each
- happiness: 0 if less than $N$ of neighbors are of same type, 1 otherwise
Rules:
- Agents choose to move to unoccupied grid point if unhappy

Note neighbors for a particular cell are the the 8 other cells surrounding the cell of interest. Corner or edge cells have less than 8 neighbors