Population dynamics

A population dynamic is the function by which players in a population update their action types. The most common type of population dynamic is a "birth-death process". This is a process in which one player will change their action type at a time, traditionally representing a player "dying" and another player "reproducing" - that is, creating a copy of themself to replace the player who "died".

There are 5 population dynamics considered in ludics, all of which are birth-death processes. They fall into two catagories:

  1. Extrinsic Dynamics

These are dynamics in which players decide on new strategies by copying other players who perform well in a given population. The purely extrinsic dynamics included in ludics are the Moran process (Moran, 1958) and Fermi imitation dynamics (Szabó and Tőke, 1998).

  1. Intrinsic Dynamics

These are dynamics in which players decide on new strategies entirely based on their own fitness. The purely intrinsic dynamics included in ludics are introspection dynamics (Couto, Giaimo and Hilbe, 2022) and aspiration dynamics (Du et al., 2014).

Full citations are in the bibliography.

The Moran Process

This follows the algorithm:

  1. A player \(i\) is selected to reproduce with probability proportional to their fitness in the population.
  2. A player \(j\) is selected uniformly at random to be replaced.
  3. Player \(j\) adopts the strategy of player \(i\).

The transition matrix for a Moran process is defined as follows:

\[ T_\textbf{ab} = \begin{cases} \frac{1}{N} \cdot\frac{\sum_{a_i = b_{I(\textbf{a,b})}}{f_i(a)}}{\sum_{a_j}f_j(a)} & \text{if }\textbf{b} \in \text{Neb($\textbf{a}$)}\text{, differing at position }I(\textbf{a,b})\\ 0 & \text{if }\textbf{b} \notin \text{Neb($\textbf{a}$) and $\textbf{a}$}\neq \textbf{b}\\ 1 - \sum_{\textbf{c} \in S \setminus \text{\{\textbf{a}\}}}T_{ac} & \text{if }\textbf{a}=\textbf{b} \end{cases} \]

Where we denote the fitness of player \(i\) in state \(a\) by \(f_i(a)\), and the set of states which differ from state \(a\) in exactly one position by Neb(\(a\))

Fermi imitation dynamics

This follows the algorithm:

  1. A player \(i\) is chosen at random to consider changing strategy, and another player \(j\) is chosen to have their strategy considered
  2. Player \(i\) accepts player \(j\)'s strategy with a probability according to the Fermi logit function \(\phi(\Delta(f)) = \frac{1}{1 + e^{\beta(\Delta(f))}}\) where \(\Delta(f) = f_i(a) - f_j(a)\) is the difference between player \(i\)'s fitness and player \(j\)'s fitness

Where \(\beta\) is the choice intensity of the system which defines how often a player makes the "more rational" decision, with \(\beta = 0\) resulting in completely random updates.

The transition matrix of a process operating under to Fermi imitation dynamics is defined as follows:

\[ T_\textbf{ab} = \begin{cases} \frac{1}{N(N-1)}\sum_{a_j=b_{I(\textbf{a,b})}}\phi(f_{\text{I(\textbf{a,b})}}(a) - f_{j}(\textbf{a})) & \text{if }\textbf{b} \in \text{Neb($\textbf{a}$)}\\ 0 & \text{if }\textbf{b} \notin \text{Neb($\textbf{a}$) and $\textbf{a}$}\neq \textbf{b}\\ 1 - \sum_{\textbf{c} \in S \setminus \text{\{\textbf{a}\}}}T_{ab} & \text{if }\textbf{a}=\textbf{b} \end{cases} \]

Aspiration dynamics

Aspiration dynamics is only defined for games with exactly two strategies, and follows the algorithm:

  1. A player \(i\) is picked to switch strategy at random
  2. They change strategy with probability \(\phi(f_i(a) - A_i)\), where \(A_i\) is the aspiration of player \(i\), the fitness that they wish to obtain.

The transition matrix of a process operating under aspiration dynamics is defined as follows:

\[ T_{ab} = \begin{cases} \frac{1}{N} \cdot \phi(f_{I(a,b)}(a) - A_{I(a,b)}) & \text{if $\textbf{b}$} \in \text{Neb}(\textbf{a})\\ 0 & \text{if }\textbf{b} \notin \text{Neb($\textbf{a}$) and $\textbf{a}$}\neq \textbf{b}\\ 1 - \sum_{\textbf{c} \in S \setminus \text{\{\textbf{a}\}}}T_{ab} & \text{if }\textbf{a}=\textbf{b} \end{cases} \]

Introspection Dynamics

This follows the algorithm:

  1. A player \(i\) is picked at random to reconsider their strategy
  2. A strategy \(k\) is picked at random for them to consider
  3. They accept the new strategy with a probability \(\phi(\Delta(f))\), where \(\Delta(f) = f_i(a) - f_i(b)\) is the difference between a player's current payoff and the possible payoff they could obtain by switching strategy.

The transition matrix of a process operating under introspection dynamics is defined as follows:

\[ T_{\textbf{ab}} = \begin{cases} \dfrac{1}{N(m_j - 1)} \, \phi (f_i(a) - f_i(b)) & \text{if } \textbf{b} \in \mathrm{Neb}(\textbf{a}) \text{ and } j = I(\textbf{a,b}),\\[1.2em] 0 & \text{if } \textbf{b} \notin \mathrm{Neb}(\textbf{a}) \text{and $\textbf{a}$} \neq \textbf{b},\\[0.8em] 1 - \sum_{\textbf{c} \in S \setminus \text{\{\textbf{a}\}}}T_{ab} & \text{if }\textbf{a}=\textbf{b} \end{cases} \]

Mutation

Mutation is a step which can be added to any birth-death process. It follows the algorithm:

  1. The population dynamic occurs as usual
  2. Before the chosen player \(i\) changes strategy, they may instead mutate to another action type \(k\) with probability \(\mu_{ik}\)

We can apply mutation to functions by the following transformation:

\[ T^m_{ab} = T_{ab}(1 - \sum_{k \in K}\mu_{I(ab), k}) + \frac{\mu_{I(a,b), b_{I(a,b)}}}{N} \]

where \(K\) is the set of action types.