Part V. Agent-based models vs ODE models
V-3. Mean Dynamics
1. Introduction
Every dynamic that can be run with the NetLogo model we have implemented in the previous chapter (nxn-games-in-well-mixed-populations.nlogo) can be usefully seen as a time-homogeneous Markov chain on the space of possible strategy distributions. This means that the fraction of agents that are using each strategy, i.e., the population state, contains all the information we need to –probabilistically– predict the evolution of the agent-based dynamic as accurately as it is possible.
Let denote the fraction of agents using strategy . In games with strategies, population states are elements of the simplex . More specifically, if there are agents, population states are elements of the finite grid .
A fully parameterized model defines a discrete-time Markov chain on state space , where is the index for the ticks. Our goal in this chapter is to approximate the transient dynamics of this (stochastic) process with its corresponding (deterministic) mean dynamic.
In the next section, we explain how to derive the mean dynamic in general, and in section 3 we present the mean dynamic for every possible parameterization of NetLogo model nxn-games-in-well-mixed-populations.nlogo. Then, in section 4 we show how we can numerically solve the mean dynamic using the Euler method within NetLogo. In section 5, we present some representative simulations together with their corresponding mean dynamics solved at runtime within NetLogo. In this way, we will be able to appreciate how useful the mean dynamic can be. We conclude the chapter (and the book) emphasizing how sensitive agent-based evolutionary dynamics can be to seemingly unimportant details.
2. The mean dynamic
The mean dynamic (Benaïm & Weibull, 2003; Sandholm, 2010a, chapter 10; Roth & Sandholm, 2013) is a system of ODEs that provides a good deterministic approximation to the dynamics of the stochastic evolutionary process
- over finite time spans,
- when the number of agents is sufficiently large, and
- when the revision probability is sufficiently low.
To derive the mean dynamic, we consider the behavior of the process over the next time units, departing from population state . For convenience, we define one unit of clock time as 1/prob-revision ticks, i.e., the number of ticks over which every agent is expected to receive exactly one revision opportunity.[1] Thus, over the time interval , the number of agents who are expected to receive a revision opportunity is
Of the agents who revise their strategies over the time interval , are expected to be -strategists. Let be the conditional probability that a reviser using strategy adopts strategy (so denotes the probability that a reviser using strategy keeps using strategy after revision). Using this notation, the expected number of -revisers who adopt strategy over the time interval is Hence, the expected change in the number of agents that are using strategy over the time interval equals: