A collection of one or more functions, called *metrics*, of the form $p\colon X_1\times X_2\times\cdots\times X_n\rightarrow R$, where

- each $i$ with $1\leq i\leq n$ is a
*domain role* - an element $x\in X_i$ is an
*entity*(playing the domain role $i$) - a tuple $(x_1,\ldots,x_n)\in X_1\times\cdots\times X_n$ is an
*interaction* - the value $p(x_1,\ldots,x_n)\in R$ is an
*outcome*(of the interaction) - the ordered set $R$ is the
*outcome set*

These are really only interesting from the perspective of coevolutionary algorithms when $n\geq 2$. When $n=1$, you have one or more single-variable functions, meaning something that looks like an optimization problem or a multi-objective optimziation problem as opposed to a co-optimization problem.

It is important to recognize that an interactive domain does not specify a solution concept–in other words, what one might want to find–only the structure of its information. As an analogy, a function $f\colon S\rightarrow\mathbb{R}$ does not specify enough information to be optimized; you'd also have to know whether you're trying to minimize or maximize, whether you're seeking one or all solutions, whether you're looking for an argument or a value, etc. The function is like an interactive domain; the arg max (for instance) is the solution concept.

## References

- The book chapter titled Coevolutionary Principles in the Handbook of Natural Computing goes into depth about interactive domains and their role in coevolutionary algorithms