Part V. Agent-based models vs ODE models
V-1. Introduction
Many models in Evolutionary Game Theory are described as systems of Ordinary Differential Equations (ODEs). The most famous example is the Replicator Dynamics (Taylor and Jonker, 1978), which reads:
where:
- is the fraction of -strategists in the population.
- describes how the fraction of -strategists changes in time.
- is the expected payoff of strategy . That is, if is the payoff that an -strategist obtains against a -strategist, then . This is also the average payoff that an -strategist would obtain if he played with the whole population.
- is the average payoff in the population.
ODE models are very different from the agent-based models we have considered in this book. Our goal in this Part V is to clarify the relationship between these two kinds of models. We will see that most ODE models in Evolutionary Game Theory can be seen as the mean dynamic of an agent-based model where agents follow a certain decision rule in a well-mixed population.[1] This implies that those ODE models provide a good deterministic approximation to the dynamics of the corresponding agent-based model over finite time spans when the number of agents is sufficiently large (Benaïm & Weibull, 2003; Sandholm, 2010a, chapter 10; Roth & Sandholm, 2013). For this reason, ODE models in Evolutionary Game Theory are often called infinite-population models (because they describe dynamics of populations whose size tends to infinity), while agent-based models are sometimes called finite-population models.[2]
It is also clear that the ODE models represent a higher level of abstraction than the agent-based models, in the sense that the variables in the ODE models are population-level aggregates, while the agent-based models are defined at the individual level (fig. 1).
In fact, in many cases, there is a wide range of different agent-based models which share the same mean dynamic. For instance, consider the replicator dynamics, which is often derived as the infinite-population limit of a certain model of biological evolution (see derivations in e.g. Weibull (1995, section 3.1.1), Vega-Redondo (2003, section 10.3.1) or Alexander (2023, section 3.2.1)). In the following chapters we will see that the replicator dynamics is also the mean dynamic of the following disparate agent-based models:[3]
- A model where agents in a well-mixed population follow the imitative pairwise-difference rule using expected payoffs (Helbing,1992; Schlag, 1998; Sandholm, 2010a, example 5.4.2; Sandholm, 2010b, example 1).
- A model where agents in a well-mixed population play the game just once with a random agent and follow the imitative pairwise-difference rule (Izquierdo et al., 2019, example A.2).
- A model where agents in a well-mixed population follow the so-called imitative linear-attraction rule using expected payoffs (Hofbauer, 1995a; Sandholm, 2010a, example 5.4.4; Sandholm, 2010b, example 1).
- A model where agents in a well-mixed populatio