Replicator Dynamics, Fisher's Theorem & Many-Model Thinking

How evolutionary dynamics shape strategy populations, why variation drives adaptation, and how combining diverse models produces wisdom exceeding any individual.

10a. Replicator Dynamics Simulator

Strategies that perform above the population average grow in proportion; those below average shrink. Apply to classic games like Hawk-Dove, Rock-Paper-Scissors, and Coordination -- each producing fundamentally different dynamics.

Game Selection

Payoff Matrix

Population

Time0
Dominant--
EquilibriumNo
Avg Fitness--

How It Works

Replicator dynamics: dx_i/dt = x_i * (f_i - f_avg), where x_i is strategy proportion and f_i is its fitness against the current population mix.

  • Hawk-Dove: Hawks fight for resources, Doves share. Too many Hawks means costly fights -- equilibrium emerges at a mixed population.
  • Rock-Paper-Scissors: Cyclic dominance with no stable equilibrium. Populations orbit forever (or spiral inward/outward).
  • Coordination: Two equilibria (all A or all B). Which one depends on initial conditions -- path dependence matters.
  • Key: The payoff structure of the game determines whether we get stable mixing, cycling, or winner-take-all.

Real World Examples

  • 🦅 Hawk-Dove in animal territorial contests (escalation vs retreat)
  • 🏢 Aggressive vs cooperative firms competing in markets
  • 🎓 Cultural evolution of social norms over generations
  • 💊 Antibiotic resistance cycling through bacterial populations
  • 👔 Fashion cycles -- styles go in and out as they become too common
  • 🌍 Evolutionary biology -- frequency-dependent selection

Key Insight

"The structure of the game determines everything. Hawk-Dove reaches a stable mix, Rock-Paper-Scissors cycles forever, and Coordination locks in -- the same evolutionary dynamics, but three completely different outcomes depending on the payoff matrix."

10b. Fisher's Fundamental Theorem of Natural Selection

The rate of increase in mean fitness of a population equals the genetic variance in fitness. More variation means faster adaptation -- but this is not the same as Six Sigma's "reduce variation to improve quality."

Controls

FISHER'S THEOREM
d(mean fitness)/dt = Var(fitness)
-- Rate of Change
=  
-- Variance
Generation0
Mean Fitness--
Variance--
Adaptation Rate--

How It Works

Fisher's Fundamental Theorem (1930): the rate of increase in mean fitness of a population at any time equals the genetic variance in fitness at that time.

  • High variance = fast adaptation: A diverse population adapts quickly because selection has more material to work with.
  • Low variance = slow adaptation: An inbred, homogeneous population improves slowly -- even if current average fitness is high.
  • Selection reduces variance: Paradox -- adaptation consumes the very variation that drives it, slowing itself down over time.
  • vs Six Sigma: Six Sigma reduces variation in output (errors). Fisher says variation in approaches (exploration) speeds improvement. Both are correct -- vary strategies, standardize execution.

Real World Examples

  • 🧬 Diverse vs inbred populations -- cheetahs (low genetic diversity) adapt slowly to new diseases
  • 🚀 Startup ecosystems -- many diverse companies means faster innovation overall
  • 💡 Brainstorming -- the divergent thinking phase generates variation for selection
  • 🎯 A/B testing -- more variants tested means faster optimization
  • 🦠 Immune system diversity -- broader antibody repertoire fights novel pathogens better
  • 🏭 City economies -- diverse industry mix weathers shocks better (Jane Jacobs)

Key Insight

"Variation is the fuel of adaptation. Fisher's theorem says: more diversity = faster improvement. Six Sigma says: less variation = higher quality. Both are right -- you want variation in STRATEGIES (exploration) but consistency in EXECUTION (exploitation)."

10c. Many-Model Thinking / Diversity Prediction Theorem

The Diversity Prediction Theorem proves mathematically: Crowd Error = Average Individual Error - Prediction Diversity. Diverse crowds ALWAYS outperform their average member. The key is diversity of thought, not just quantity.

Controls

Models

Crowd Error--
Avg Indiv Error--
Diversity--
Improvement--
DIVERSITY PREDICTION THEOREM
Crowd Error = Avg Individual Error - Prediction Diversity
-- Crowd Error
=  
-- Avg Error
-  
-- Diversity

How It Works

The Diversity Prediction Theorem is an algebraic identity (not a statistical approximation -- it is always exactly true):

  • Crowd Error = squared error of the crowd's average prediction from truth
  • Avg Individual Error = average of each model's squared error from truth
  • Prediction Diversity = average squared distance of individual predictions from the crowd average
  • Since diversity is always >= 0, the crowd ALWAYS does at least as well as the average individual.
  • Key insight: Adding a model that thinks DIFFERENTLY helps more than adding a copy of the best model.

Real World Examples

  • 🌤 Weather forecasting -- ensemble of different atmospheric models beats any single one
  • 🎬 Netflix Prize -- winning team combined 800+ different models
  • 📈 Prediction markets -- aggregate diverse opinions into surprisingly accurate forecasts
  • 🩹 Medical diagnosis -- panels of specialists with different training outperform lone experts
  • 🗳 Election forecasting -- 538/Economist aggregate many polls with different methodologies
  • 👥 Jury decisions -- diverse juries make better decisions than homogeneous ones

Key Insight

"Diversity trumps ability. The 10th-best model that thinks differently contributes more to the crowd's accuracy than a second copy of the best model. This is mathematically guaranteed by the Diversity Prediction Theorem -- and it's why diverse teams, ensemble methods, and prediction markets consistently outperform individual experts."