Replicator Dynamics Visualizer

Interactive exploration of evolutionary game dynamics on the simplex

ẋᵢ = xᵢ[(Ax)ᵢ - xᵀAx]

Replicator dynamics describes how successful strategies spread through a population. Strategies with above-average payoff grow, while those below average decline.

Select a Game

Simplex Visualization

Click on the simplex to set initial conditions. The triangle represents all possible strategy distributions (x+y+z=1). Each vertex is a pure strategy, interior points are mixed strategies.

Strategy 1 (x)
Strategy 2 (y)
Strategy 3 (z)
Nash Equilibrium
ESS

Control Panel

Payoff Matrix A

Edit the 3×3 payoff matrix. Each entry Aᵢⱼ represents the payoff of strategy i against strategy j.

Current State

Strategy 1 (x) 0.333
Strategy 2 (y) 0.333
Strategy 3 (z) 0.334
Sum 1.000
Payoff 1 0.00
Payoff 2 0.00
Payoff 3 0.00
Average 0.00

Phase Portrait

Vector field showing the direction of evolution at each point in the simplex. Arrow length indicates speed of change.

Time Evolution

Watch how strategy frequencies evolve over time according to the replicator equation.

Equilibrium Analysis

Learn More

Replicator dynamics is a mathematical model used in evolutionary game theory to describe how successful strategies spread through a population over time.

The Replicator Equation:

ẋi = xi[(Ax)i - xTAx]

Where:

  • xᵢ = frequency of strategy i in the population
  • A = payoff matrix where Aᵢⱼ is payoff of i vs j
  • (Ax)ᵢ = expected payoff of strategy i
  • xᵀAx = average payoff in the population

The simplex is a geometric representation of all possible population states. For three strategies, it forms a triangle where every point represents a valid distribution.

The Constraint:

x + y + z = 1, where x, y, z ≥ 0

Vertices:

  • (1, 0, 0): Pure strategy 1 (entire population plays strategy 1)
  • (0, 1, 0): Pure strategy 2
  • (0, 0, 1): Pure strategy 3

Interior points represent mixed populations where multiple strategies coexist.

Equilibrium points are states where the population composition stops changing (ẋ = 0). These can be stable (attractors) or unstable (repellors).

Types of Equilibria:

  • Pure Strategy Equilibria: Population consists of a single strategy
  • Mixed Strategy Equilibria: Multiple strategies coexist in stable proportions
  • Interior Equilibria: All strategies have positive frequency

Evolutionarily Stable Strategy (ESS):

A strategy is an ESS if, when adopted by a population, it cannot be invaded by any rare mutant strategy. All ESS are Nash equilibria, but not all Nash equilibria are ESS.

Rock-Paper-Scissors

A cyclic dominance game where each strategy beats one and loses to another. The interior equilibrium is a center with closed orbits around it.

Hawk-Dove

Models conflict between aggressive (Hawk) and passive (Dove) strategies. Typically has a stable mixed equilibrium.

Coordination

Players benefit from choosing the same strategy. Multiple stable pure equilibria exist.

Stag Hunt

A tension between safety and cooperation. Two pure equilibria: one risk-dominant, one payoff-dominant.

Biology

Evolution of animal behavior, predator-prey dynamics, and the evolution of cooperation.

Economics

Learning in games, market dynamics, and the spread of economic behaviors.

Social Science

Cultural evolution, spread of innovations, and social norm formation.

Computer Science

Multi-agent learning, algorithmic game theory, and distributed optimization.