Stochasticity

EES 4760/5760

Agent-Based & Individual-Based Computational Modeling

Jonathan Gilligan

Class #19: Wednesday Mar. 21 2018

Model to Download

Vignette: Participatory Agent-Based Modeling for Decision Support

Managing Groundwater in Chicago

  • C. Hoch et al., Planning Theory & Practice 16, 319 (2015),
  • J. Radinsky et al., J. Environ. Planning & Management 60, 1296 (2017).
  • Regional groundwater plan
    • Requires coordination among independent suburbs
    • Stakeholders: homeowners, developers, lenders, realtors, businesses, elected officials
  • ABM as a tool to explore sustainability under different policies
    • Teach planners to use models
    • Method: Explore impacts on groundwater levels of different policy interventions for land and water use.
    • Research Question: How does interaction with models affect thinking of participants, especially when they discover that their favorite policies do not appear to be sustainable?

Response to Working with Model

  • Model did help planners identify and grapple with misleading assumptions, but they were reluctant to discard those assumptions and instead raised objections about model.
    • If model predictions conflicted with preconceptions, tendency to criticize model.
    • Ease of working with model gave impression it was a “toy”
  • Cognitive bias is prevalent and difficult to overcome.
  • Best practices with ABMs:
    • Moving from talking about tools to using tools to talking about planning
    • Moving from one-dimentional to multi-dimensional/multi-level thinking
    • Moving from one pattern of argumentation to flexible exploration of multiple lines.
  • Many challenges
  • Some promising results
  • Further research needed

Stochasticity

Stochasticity:

Why do we use random numbers?

  • To “inject ignorance” into a model:
    • We want to represent some kind of variability but
    • We do not want all the details of what causes the variability

      ask patches [set profit 1000 + (random 1000)]
      ask turtles [ if random-float 1.0 < mortality-prob [die] ]

Common uses of stochasticity

  • Initialization

    set fish-length random-normal 50 length-std-dev          
  • In submodels

    ifelse random-float 1.0 < q
    [ uphill elevation ]
    [ move-to one-of neighbors ]

Guidance for Stochasticity

  • Do use stochasticity to initialize model differently on different runs
    • Makes sure that effects you see are not artifacts of a specific initialization
  • Do use stochasticity to simplify representation of very complex processes
    • If wild dogs live an average of 5 years:
      • instead of a detailed submodel that determines exactly when each dog will die,
      • let dogs die at random with a 20% probability of dying each tick.
  • Don’t use too much stochasticity
    • If you put too many different sources of randomness into your models every run may be so different you can’t discover any general properties.

Distributions

What is a Distribution?

What is a Distribution?

  • In simulation programming, an algorithm that produces (pseudo)random numbers that fit a particular statistical distribution.

    let x1 random-normal 1.0  0.25
    let x2 random-gamma  2.0  4.0

Randomness and Pseudo-Randomness

Randomness and Pseudo-Randomness

  • We could get true random numbers from physical processes:
    • Radioactive decay
    • Thermal noise in electronic components
    • Chaotic systems (e.g., lava lamps)

      Lava Lamps for random number generation at CloudFlare Close-up of lava lamps
  • But often you want reproducible random numbers:
    • Pseudo-random number generators
    • Begin with a “seed” number and use it to generate a series of numbers that
      • look random to all appearances,
      • but can be reproduced by starting from the same seed.

Challenges of pseudo-random numbers

  • Because pseudo-random number generators use deterministic equations, there will be patterns to their output, but hopefully those patterns won’t be very pronounced.
    • Output will eventually repeat exactly, but hopefully not for a very long time.
  • RANDU random number generator (IBM, 1960s) turned out to be horrible:
    • Plot triplets of consecutive numbers in three-dimensions

triplets of consecutive output from RANDU

  • When told of this problem, an IBM programmer replied that the numbers were guaranteeded to be random when taken one at a time, but not if taken in groups.
  • NetLogo has a high-quality random number generator.

Controlling randomness

Controlling randomness

  • random-seed number
    • As long as number is the same, you get the same sequence of random numbers

      to setup
          clear-all
          random-seed 32149
          ...
      end

Controlling randomness

  • with-local-randomness [ commands ]
    Runs without changing sequence of random numbers in other parts of the model

    to move
      with-local-randomness
      [
        random-seed 63592
        ...
      ]
    end

How can we see a distribution?

  • Histograms

    to plot-histogram-normal
      clear-all
      set-plot-pen-mode 1 ; bar mode
      set-plot-pen-interval 0.1
      set-plot-x-range -1 3
      let x (list)
      ; fill x with 5000 random numbers from a normal distribution        
      repeat 5000 [ set x fput (random-normal 1.0 0.25) x]      
      histogram x
    end

histogram

Uniform distributions

  • Integer: random n gives an integer \(i\): \(0 \le i < n\)
    • From 0 to \((n - 1)\)
  • Continuous: random-float z gives a number \(x\): \(0 \le x < z\)
    • Should we worry that \(x < z\)?

      to test
        let num_draws 10000
        let max-rand 0
        repeat num_draws
        [
          let x random-float 1000
          if x > max-rand [ set max-rand x ]
        ]
        show max-rand
      end
      observer> test
      observer: 999.9869678378017

Poisson distribution

  • For countable things that happen at a small rate.
    • On every turn a random number of agents turn red,
      with an average of 5% of agents

      ask n-of (random-poisson (0.05 * count turtles)) turtles
      [
        set color red
      ]

      or

      let n random-poisson (0.05 * count turtles)
      ask n-of n turtles [set color red]

Normal distribution

  • For measurable things with an average value

    set weight random-normal 150 20  ; weight in pounds       
    set height random-normal 70 2    ; height in inches

Stochastic Business Investors

Stochastic Business Investors

Model: ees4760.jgilligan.org/models/class_19/business_investor_class_19.nlogo

Original model:

Investors move to neighbor with highest expected utility (including own patch)

Average over 10,000 runs:

Alternative Frequency
Higher profit, lower risk 83.3%
Higher profit, higher risk 5.4%
Lower profit, lower risk 4.9%
Lower profit, higher risk 0%
Don’t move 92.7%
  • Mean wealth = $128,400
  • Total wealth = $12,000,000

Stochastic Model

Original model:

Alternative Frequency
Higher profit, lower risk 83.3%
Higher profit, higher risk 5.4%
Lower profit, lower risk 4.9%
Lower profit, higher risk 0%
Don’t move 92.7%

Stochastic model

  • If there are neighbors with higher profit and lower risk:
    • 83.3% probability of moving to one of them
  • Otherwise, if there are neighbors with higher profit and higher risk:
    • 5.4% probability of moving to one of them
  • etc.

Compare models:

Original model:

Alternative Frequency
Higher profit, lower risk 83.3%
Higher profit, higher risk 5.4%
Lower profit, lower risk 4.9%
Lower profit, higher risk 0%
Don’t move 92.7%
  • Mean wealth = $128,400
  • Total wealth = $12,000,000

Stochastic model:

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