Replicator Cyclic Payoff Spirals
Rock beats scissors, yet cyclic competition can spiral inward, circle forever or spiral outward depending on the cost of losing.
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Replicator Cyclic Payoff Spirals
In Evolutionary Game Theory, the Replicator Dynamic models how populations of strategies evolve when successful strategies reproduce faster. Rock-Paper-Scissors provides a classic example of cyclic dominance where no single strategy is unconditionally best.
Cyclic Dominance
Unlike the Hawk-Dove game which models a scalar trait, Rock-Paper-Scissors has three interacting frequencies x_R, x_P, x_S that sum to 1. Each type beats one opponent and loses to another. Let L be the cost of losing, while winning yields 1. The payoff matrix is:
A = [ [ 0, -L, 1], [ 1, 0, -L], [-L, 1, 0] ]
The replicator equation is given by x_i = x_i [ (Ax)_i - x^T Ax ]. Because the population fractions must sum to 1, the dynamics are confined to a 2D simplex. Cyclic dominance causes trajectories to circle the central mixed equilibrium. Depending on the value of L, these trajectories can form closed neutral orbits, spiral inward, or spiral outward.
Worked Example
Consider a Biased population with fractions x = [0.5, 0.25, 0.25]^T and loss magnitude L = 1.
- Calculate expected payoffs Ax:
- Rock payoff: (0)(0.5) + (-1)(0.25) + (1)(0.25) = 0
- Paper payoff: (1)(0.5) + (0)(0.25) + (-1)(0.25) = 0.25
- Scissors payoff: (-1)(0.5) + (1)(0.25) + (0)(0.25) = -0.25
- Calculate mean payoff x^T Ax: (0.5)(0) + (0.25)(0.25) + (0.25)(-0.25) = 0
- Calculate derivatives x_i:
- Rock derivative: 0.5(0 - 0) = 0
- Paper derivative: 0.25(0.25 - 0) = 0.0625
- Scissors derivative: 0.25(-0.25 - 0) = -0.0625
Experiment
Explore the interactive module to see how the loss magnitude and initial population affect the replicator flow. Notice that absent types remain absent on the boundaries, and the interior linearization predicts whether orbits are attracting, neutral, or repelling.
- Set Loss magnitude to a value (e.g., 0.5 for an attracting spiral).
- Set Population to a starting configuration.
- Observe the resulting vector field and interior equilibrium type in the interactive module.
Limitations
The replicator dynamic models continuous deterministic population change, assuming infinite population size and immediate, uniform interaction mixing. It does not account for demographic stochasticity in small populations or spatial/network structure where agents only play against neighbors.
References
For more details on evolutionary game theory and replicator dynamics, refer to the MIT lecture slides:
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Replicator Cyclic Payoff Spirals module.
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