Do changes in payoff make sense with the changes in traits?

[HakanBrian]

After running a few simulation runs the data seems to exhibit an odd behavior where the trait values for the population changes, sometimes even rapidly, with little to no change in the payoffs. The data collected for these runs was the average values for traits and payoff, so there may be hidden we aren't seeing. Regardless, there needs to be an investigation into whether these changes in payoff make sense with the changes in traits.

[HakanBrian]

For further investigation, I will complete multiple runs with the same initial parameters to identify any trends. I will also see if any possible trends make sense. One difference between the two models is the Mathematica model has a relatedness mechanism, which may one of the keys things we are missing. Basically, we need to ensure everything that we in simulation is working as expected before moving on.

Edit: see the comment below

[HakanBrian]

As far as testing what could be causing some issues identifying trends, I will list some considerations.

• Mutation Rate: Too high or Too low. If too high, the system might exhibit excessive variability. If too low, the system might not explore enough of the state space.

• Timescale: Too short or Too Long. If too short, there may not be enough time for meaningful trends to emerge. If too long, the system may reach a dynamic equilibrium where it appears random.

• Initial Conditions: Initial conditions naturally have a great influence on the dynamics of the system. Consider using conditions where we may know the expected outcome.

• Number of Individuals: number of Individuals can affect the statistical significance of observed trends. A small number of Individuals might lead to high variability, making trends harder to discern.

• Individual Interaction Rules: The rules we use to govern the actions of individuals heavily influence the dynamics of the model. If we have not correctly implemented these or even have small discrepancies, then the outcomes of the Julia and Mathematica model will be different.

• Noise: noise is a possible concern which can obscure trends. This is especially true if our system is sensitive to initial conditions and random events.

Solution?

• [ ] Experiment with mutation rate by gradually adjusting the mutation rate and observing how it influences the emergent behavior of individuals.
• [ ] Extend the timescale to see if larger trends become apparent, and periodically check for emerging patterns.
• [ ] Vary the initial conditions systematically to understand their impact on the model's behavior.
• [ ] Increase the number of individuals and observe if larger populations lead to clearer trends.
• [ ] Review and ensure that the interaction rules in our Julia model align closely with the dynamics described by our Mathematica model.
• [ ] Analyze the impact of stochastic elements by running multiple simulations with the same parameters and averaging the results.
• [ ] Systematically vary key parameters and observe how changes affect the model's output, then identify parameters that need careful calibration to match the behavior observed in the Mathematica model.

[HakanBrian]

After further testing, we have found observable trends in the payoff and action that match our Mathematica model. However, now we question of whether changes in traits (norm and punishment) reflect the trends we are seeing in the payoff and action. We know these trends are correct, so why are we not observing similar results in the punishment and norm? Perhaps there is too much noise, or we may need to tweak and play with some other part of the model to obtain clearer trends in the punishment and norm.

[HakanBrian]

This issue has been resolved by including the invests individuals make into punishment into the calculation of fitness during reproduction and by increasing the scale of the differences between payoffs, which essentially increases the strength of selection.

So in essence, we forgot to include a small calculation step in the fitness, which ended up having a large effect on the dynamics.