Technical Stuff: the replicator dyanamics

• The simulators for the Skyrms models for chapters 1, 2, and 5 are driven by the "discrete replicator dynamics". Replicator dynamics are a simple mathematical formulation of the dynamics of competitive growth. A population consists of a "large" number of individuals playing different strategies. How well each strategy does against each other is given by a table of payoffs, or "payoff matrix". The growth rate of each strategy is derived from the expected payoffs given the current distribution of strategies in the population.

• So how does this work (mathematically)? The discrete replcator dynamics, which are a version of the math that population geneticists use to model evolution. [See Refs. 2&3] The replicator dynamics modifies the relative frequency of the various strategies (two in this case) as their expected payoff is greater or lesser than the mean expected payoff. In other words,

  p_i' = p_i * (S_j p_j * payoff( i | j ) / S_k,j p_j * p_k * payoff( k | j ))

  • Notice that these dynamics run in discrete cycles, rather than continuous ones. This makes both the math and the programming easier, and the simulations run faster.

• INTERPRETATION: there are two basic kinds of processes that behave this way. If strategies are inherited via biological reproduction and the payoffs are given in terms of fitness, then the replicator dynamics describes the way that proportions of strategies change over time in the biological population. But they also describe the case where strategies propogate via imitation, where the payoffs are what drives the rate of imitation, reinforcement, and inhibition of behaviors. So evolutionary models of the propogation of strategies may apply either to simple instinctive behaviors, or to more complex learned behavior where imitation and reinforcement are the primary mechanisms of learning.

• The models for chapters 1 and 5 incorporate a "viscosity" parameter to model certain spatial effects, where individuals do not move far from where they are born (or from where they learned the strategy). the net result is that players are somewhat more likely to play against themselves.  So what the Viscosity parameter decreases the likelihood of running into somone who plays a different strategy. If Viscosity is zero, then encounters are random. If Viscosity is 1, then you always play against your own strategy. If Pr( i | j ) is the likelihood of i running into j, and p_j is the relative frequency of j, then
  • Pr( i | j ) = p_j + ( 1 - Viscosity) * p_j [when i = j]
  • Pr( i | j ) = p_j - p_j * Viscosity [when i and j are different]
  • The replicator dynamics then becomes

    p_i' = p_i * (S_j Pr( i | j ) * payoff( i | j ) / S_k,j p_k * Pr( k | j ) * payoff( k | j ))

References:

1. Harms, W. F., 1997, Evolution and Ultimatum Bargaining, Theory and Decision, v. 42, 147-175.
2. Harms, W. F., 1997, "Reliability and Novelty: information gain in multi-level selection systems", Erkenntnis, v. 46, 335-363.
3. Hofbauer, J., and Sigmund, K., 1988 The Theory of Evolution and Dynamical Systems: Cambridge University Press.
4. Schuster, P., and Sigmund, K., 1983, "Replicator Dynamics", Journal of Theoretical Biology, v. 100, 533-538.
5. Skyrms, B., 1996, Evolution of the Social Contract: Cambridge University Press. [This is the source for the Viscosity formula.]

Bill Harms