Sensitivity, Uncertainty, and Robustness Analysis

EES 4760/5760

Agent-Based & Individual-Based Computational Modeling

Jonathan Gilligan

Class #25: Tuesday April 19 2018

Sensitivity, Uncertainty, Robustness

Sensitivity, Uncertainty, Robustness

  • Does ABM reproduce patterns robustly?
    • or are patterns sensitive to specific values for parameters?
  • How uncertain are the results?
    • What can the model tell us about parameters that we can’t measure?
  • Sensitivity Analysis focuses on small changes in parameters.
  • Robustness Analysis looks at large changes in parameters.

Is high sensitivity good or bad?

  • Bad: If model is testing a general theory, but is very sensitive to parameter values, that is evidence against the theory.

    • Does model work across the entire range of observed values for parameters?
  • Good: If the model is being used to evaluate parameters we can’t measure, higher sensitivity can mean less uncertainty about parameters.

Challenges: Computational Complexity

  • We would like to do global sensitivity analysis:
    • Vary all parameters over their entire ranges, in every combination.
    • Can’t: computationally unfeasible.
    • There are strategies to make global sensitivity analysis feasible, but they are complicated.
  • Instead: local sensitivity analysis:
    • Small variations around most likely values of parameters.
    • Vary one parameter at a time, or multiple parameters?
      • Interactions
    • Sampling parameter values
      • Random (Monte Carlo)
      • Systematic (e.g., Latin Hypercube)

Example: Wild Dog Model

  • Packs of wild dogs in nature preserve.
  • Goal: Keep them from going extinct in next 100 years.
  • Vary parameters:
    • Mortality rate of adult dogs in pack
    • Mortality rate of dispersers
    • Meeting rate of disperser groups
    • Carrying capacity

Analyzing data:

  • Contour plots
  • “Small multiple” plots
  • Analyze four-dimensional data set using a grid of nine plots.

Example Research Model

Example Research Model

J.J. Jordan et al., “Third-party punishment as a costly signal of trustworthiness,” Nature 530, 473 (2016). doi:10.1038/nature16981

  • Cooperation and Cheating
    • Common situation:
      • Everyone is better off if everyone cooperates than if everyone cheats.
      • Once everyone else has chosen their action, any individual is better off cheating than cooperating.
      • Nash equilibrium: Everyone making the best choice for himself produces the worst outcome for everyone.
      • Opposite of the “invisible hand” in economics.

Prisoner’s Dilemma

B Cooperates B Defects
A Cooperates 5, 5 0, 7
A Defects 7, 0 1, 1
  • No matter what player A does, player B is better off defecting
  • No matter what player B does, player A is better off defecting
  • If both players defect, both are worse off than if both cooperated.

Tragedy of the commons

  • Ten farmers share a pasture.
  • A pasture can support 100 cows.
  • If \(N_{\text{cows}} \le 100\), each cow produces \(\$1,\!000\) worth of milk per month.
  • If \(N_{\text{cows}} > 100\), each cow produces \[\$1,\!000 \times \left(1 - \frac{(N_{\text{cows}} - 100)}{100}\right)\] worth of milk per month.
  • Each farmer has 10 cows, each farmer earns \(\$10,\!000\) per month.
  • One farmer adds 1 cow: total 101.
    • Each cow produces \(\$1,\!000 \times (1 - (101 - 100)/100) = \$990\).
    • First farmer earns \(11 \times \$990 = \$10,\!890\),
    • Everyone else earns \(\$9,\!900\).
  • Each farmer adds 1 cow: total 110.
    • Each cow produces \(\$900\). Each farmer earns \(\$9,\!000\).

Iterated games

  • If only playing once, best strategy is to cheat, because it is rational for everyone else to cheat.
  • If playing multiple turns, threat of punishment in future rounds promotes cooperation.
  • It is generally costly to punish people.
  • If someone cheats against you, it’s often worthwhile to punish them.
  • If you see someone cheating against another person and you aren’t affected, is it worth your while to punish the cheater, even if it costs you?
  • Does tragedy of commons inhibit people from punishing?

Theory

  • Punishment sends a signal:
    • Deters cheaters.
    • Signals that you are trustworthy.

Game

Game

  • Player has two roles: Signaler and Chooser
  • Signaler can be either Trustworthy or Exploitative.
  • Two kinds of signals: Helping or Punishing a third party.
  • Two stages:
    1. Signalers can pay costs to send signals.
    2. Choosers decide whether to accept Signalers as partners.
  • Cost of signaling can be either small (\(s\)) or large (\(\ell\))

Payoffs

Payoffs after second stage are:

Trustworthy Signaller Exploitative Signaller
Chooser Accepts \(m\), \(r\) \(-e\), \(r\)
Chooser Rejects 0, 0 0, 0
  • \(m\) is benefit of mutual cooperation,
  • \(r\) is reward for being trustworthy,
  • \(e\) is harm from exploitation.

Rational strategies

  • \(b\) is expected benefit from trustworthy Signalers
  • \(c\) is expected cost from exploitative Signalers
  • \(I_{SH}\) is informativeness of small helping costs
  • \(I_{SP}\) is informativeness of small punishment costs

Agent-based model

Agent-based model

  • Signaler randomly chosen to be Trustworthy or Exploitative.
  • Chooser does not know Signaler type
  • Evolution of strategies:
    • Each agent plays a certain number of turns (a generation)
    • Agents have probability of reproducing based on earnings from game.
      • Offspring inherit strategy with some random “mutations”

Outcome of evolution

Human Game

Human Game

  • Amazon Mechanical Turk (Internet)
  • Human players assigned to one of three games:
    • Signaler can only punish.
    • Signaler can only help.
    • Signaler can help and punish.

Trust Game

  • To check whether signals are interpreted accurately by Chooser agents, run a second game:
    • Chooser gets some money \(M\).
      • Chooses how much to send to Signaler (\(x\)).
    • Money sent to Signaler is tripled (Signaler get \(3x\))
    • Signaler decides how much of the \(3x\) to return to Chooser.

Perception of signaling:

  • Chooser shows trust by sending more money to Signalers who punish and who help.
  • Helping is a more powerful signal to Chooser than punishing.
  • This matches theory of rational behavior.

Actual signaliing

  • Signalers who punish are more trustworthy: return more money to Chooser,
    • But only if helping is not an option.
  • Helping is indeed a more accurate signal of trustworthiness.

Signaler Choice

  • Signaler is less likely to punish when helping is an option.

Conclusions

  • Evolved strategies of agents match both pure theory (rational strategy) and experimental results.
  • It is advantageous for third parties to carry out costly punishments when the punishments can signal trustworthiness to others in the community.
  • When there are less costly or more effective ways to signal trustworthiness, third parties are less likely to punish.

General ideas about agent-based modeling

  • Model interactions between individuals
    • Direct: individual-individual
    • Indirect: individual-environment, environment-individual
  • Focus on emergent properties
    • Patterns or phenomena that were not deliberately programmed in, but arise spontaneously from interactions of agents with each other and with environment.
  • Pattern-oriented modeling:
    • Start simple, but aim to build in enough complexity to produce multiple patterns seen in nature, or predicted in theory.
    • As you design model think about what kinds of “currency” you will use to assess its value.