Measures of diversity typically quantify differences in
weights or behavior of agents but ignore performance. Effective diversity measures the variety of effective agents
(agents with support under Nash):

Definition

Notation from Transitive cyclic decomposition.

Given population $B$ , let $A_{ij} = \phi(w_i,w_j)$ be the strategy payout matrix of the population.

let $p$ be a Nash equilibria on $A$ (note that this is just a vector of probabilities).

Let

\lfloor x \rfloor_+ = \left\{ \begin{array}{lr} 0 & \text{ if } x \le 0 \\ x & \text{ if } x > 0 \end{array} \right\}

denote the relu rectifier

Then the effective diversity of the population is

d(B) := p^T \cdot \lfloor A \rfloor_+ p

Note than the rectification of $\lfloor A \rfloor_+$ implies that we only add interactions which are good for the first player (but players are symmetric).

Algorithm to solve

The Response to rectified Nash algorithm directly uses this as its objective and proceeds greedily.

Citation

Citation in Transitive cyclic decomposition