Heterogeneous Intensities

Ppopulation dynamics consider intensity parameters which control the rationality of the process. When an intensity parameter is high, players will make the rational choice with a higher probability, accepting strategies which are judged to perform better with a higher probability, whether that be through direct comparison of payoffs, or through fitness proportional selection.

In ludics, we catagorise intensities into the selection intensity \(\epsilon\) and the choice intensity \(\beta\).

Selection intensity

Selection intensity is used in fitness proportional selection to control how much a player takes payoff value into account. The Moran process, for example, uses this value as follows:

\[ \frac{1}{N} \cdot\frac{\sum_{i:a_i = b_{I(\textbf{a,b})}}{1 - \epsilon + \epsilon\pi_i(\mathbf{a})}}{\sum_{a_j}1 - \epsilon + \epsilon\pi_i(\mathbf{a})} \]

If \(\epsilon = 0\), then we have neutral drift, where the payoffs do not affect the transition probabilities. If \(\epsilon = 1\), fitness is equal to the payoff. The value of \(\epsilon\) must by chosen such that all players admit a strictly positive fitness.

Heterogeneous Selection Intensity

ludics allows heterogeneous selection intensities. That is, a matrix \(\epsilon\), with entries \(\epsilon_{ij}\). In this case, the transition probability in the Moran process becomes:

\[ \frac{1}{N} \cdot\frac{\sum_{i:a_i = b_{I(\textbf{a,b})}}{1 - \epsilon_{I(\textbf{a,b}), i} + \epsilon_{I(\textbf{a,b}), i}\pi_i(\mathbf{a})}}{\sum_{a_j}1 - \epsilon_{I(\textbf{a,b}), i} + \epsilon_{I(\textbf{a,b}), i}\pi_i(\mathbf{a})} \]

A greater \(\epsilon_{ij}\) indicates that player \(i\) has a bias towards selecting player \(j\); in other words, player \(i\) sees an inflated value for player \(j\)'s fitness. On the other hand, a lower \(\epsilon_{ij}\) indicates that player \(i\) is bias against player \(j\), seeing a decreased fitness for said player.

Choice intensity

For population dynamcis which decides whether or not to update using the Fermi imitation function \(\frac{1}{1 + e^{\beta\Delta(\pi)}}\), choice intensity \(\beta\) controls the rationality of the decision. A higher \(\beta\) leads to players caring more about the payoff difference when making decision, and as \(\beta \to \infty\), players accept the higher payoff strategy in the comparison almost certainly. Conversely, a value of \(\beta=0\) gives neutral drift.

Heterogeneous Choice Intensity

A heterogeneous value of \(\beta\) works differently depending on the process. For the three dynamics included in ludics which feature choice intensity, it is implemented as follows:

  • Fermi imitation dynamics: \(\beta_{ij}\) is the rationality with which player \(i\) accepts the strategy of player \(j\). As such, the choice intensity matrix is of shape \((N,N)\).
  • Introspection Dynamics: \(\beta_{ik}\) is the rationality with which player \(i\) accepts strategy \(k\), where \(k\) is the index of the strategy according to the representation of states. The shape of the choice intensity matrix is \((N,K)\), where \(K\) is the number of strategies.
  • Aspiration Dynamics: \(\beta_{ik}\) is the rationality with which player \(i\) changes strategy when playing strategy \(k\). This is because aspiration dynamics does not consider which strategy to change to (traditionally it was only defined for 2 strategies). Thus a player considers the rationality by which they switch away from (or do not switch away from) their current strategy. The shape of the choice intensity matrix is \((N,K)\)