Learning and Replicator Dynamics

Replicator Dynamics

This lecture covers:

1. What is Replicator Dynamics
2. Fisher's Theorem
3. Fisher's Theorem vs Six Sigma

1. Replicator Dynamics

Tells us how a population changes/evolves over time as a function of payoffs and proportions.

    Set of types {1,2,3,...,N}

    Payoff of each type, Py[i]

    Proportion of each type, Pr[i]

Rational agents will choose the highest payoff.
Rule-based agents will copy someone else.

The Model

Define the weight of a strategy as:

     weight = Py[i] * Pr[i]  
This is reasonable because if Pr[i] = 0, then there the strategy cannot replicate.

The dynamics of the process is:

    Pr[t+1][i] = Pr[t][i] * Py[i] / Sum( Pr[t][i] * Py[i])

In other words, the proportion of a strategy in the next time step is the ratio of its weight over the sum of all weights.

Application of Replicator Dynamics

Shake-bow game.
Replicator dynamics also explains how it leads to an equilibrium in the shake-bow game.

SUV-Compact game
Replicator dynamics lead to sub-optimal Nash Equilibrium.

2. Fisher's Theorem

  "The change in average fitness due to selection will be proportional to the variance."

Consider the replicator dynamics in ecology:

    Set of types {1,2,3,...,N}

    Fitness of each type, Py[i]

    Proportion of each type, Pr[i]
The fitness wheel is a good metaphor where:

• the size of slice represents fitness
• the number of slices represent proportion

Fisher's Theorem is a combination of:

1. Model 1: There is no cardinal
2. Model 2: Rugged Landscape
3. Model 3: Replicator dynamics

The role of variation plays in adaptation.

Fisher's Theorem explain why you should be the worst musician in the best band.
For the same average, larger variation will result in greater gain or greater adaptation.

3. Variation or Six Sigma

Opposite Proverbs.
Models have assumptions but proverbs don't.

Context for Six Sigma
- Fixed Landscape (Equilibrium world)

Context for Fishers Fundamental Theorem
- Dynamic/Dancing Landscape (Cyclic, Random or Complex world).