Part IV. Games on networks
IV-5. Analysis of these models
1. Introduction
In Part II, we learned to implement and analyze models in well-mixed populations. These are models where every agent has the same probability of interacting with every other agent. In such models, the state of the system can be defined by the distribution of strategies in the population; in other words, the strategy distribution contains all the information we need to (probabilistically) predict the evolution of the system as accurately as it is possible. Because of this, we saw that there are various mathematical techniques that we can use to analyze and characterize the dynamics of these models impressively well (see chapter II-5). For well mixed populations, the usefulness of mathematical analysis cannot be overstated.
In Part III, we learned to implement and analyze models on spatial grids. In these models, the strategy distribution is not enough to predict the evolution of the system anymore because each agent has its own set of neighbors (see chapter III-5). Nonetheless, in grid models there is still some symmetry in the sense that all agents have the same number of neighbors (except, maybe, at the boundaries). Because of this, for some of these models, we could still derive some analytical approximations such as the pair approximation for regular networks. These approximations can be useful because they are sometimes able to predict overall trends, but in general there is no guarantee that they will work well. For this reason, the relative usefulness of the computer simulation approach as a tool for exploration and analysis (vs the mathematical approach) is significantly greater in spatial models than in well-mixed populations.
Finally, here in Part IV, we have learned to implement models where agents are embedded on arbitrary networks. In these models, each agent has its own set of neighbors and, in general, there is no symmetry whatsoever, so the state of the system must be defined at the individual level. Because of this, the mathematical approach loses much of its usefulness and we are bound to resort to computer simulation as the main tool for exploration and analysis.
The challenge is high, because heterogeneity in both agent types and connectivity structure breaks down the symmetry of agents, and thus requires a dramatic change of perspective in the description of the system from the aggregate level to the agent level. The resulting huge increase in the relevant system variables makes most standard analytical techniques, operating with differential equations, fixed points, etc., largely inapplicable. What remains is agent based modeling, meaning extensive numerical simulations and analytical techniques going beyond the traditional mean-field level. Szabó and Fáth (2007, p. 102)
In this chapter, we are going to illustrate the importance of four best practices that we should try to follow when using the computer simulation approach:
- Avoid errors
- Use informat