How states are represented

ludics represents the population as an ordered vector of individual strategies, one entry per player:

where \(a_i \in \{0, 1, \ldots, k-1\}\) is the strategy currently played by player \(i\), \(N\) is the population size, and \(k\) is the number of strategies.

For example, with \(N = 3\) players and \(k = 2\) strategies (0 = defect, 1 = cooperate):

State	Meaning
[0, 0, 0]	all three players defect
[0, 0, 1]	players 0 and 1 defect; player 2 cooperates
[1, 1, 0]	players 0 and 1 cooperate; player 2 defects
[1, 1, 1]	all three players cooperate

The full state space contains \(k^N\) states: for \(N=3\), \(k=2\) this is 8.

Why individual-indexed states?

This representation tracks who is playing what, not just how many players use each strategy. It is more general than a frequency-based representation: it supports heterogeneous players (different contribution levels, different aspiration levels) and asymmetric payoff functions where the identity of the player matters, not just the aggregate counts.

The cost is a larger state space. For symmetric games where only the count of each strategy matters, many states are payoff-equivalent: [0, 1, 0] and [0, 0, 1] produce the same fitness values under a homogeneous PGG. ludics does not collapse these equivalent states; it works with the full \(k^N\) space.

Absorbing states

In games with two strategies, the two fully-homogeneous states (all-0 and all-1) are typically absorbing under extrinsic dynamics (Moran, Fermi), since there is no fitness difference to drive a change. Their absorption probabilities are the fixation probabilities central to evolutionary game theory.