[@HeY04]) tells us that if the random search points $x^{(0)},x^{(1)},x^{(2)},\dots$ generated by a search heuristic form a homogeneous absorbing Markov chain, then there always exists a potential function such that condition (\[eq:additivedrift\]) in Theorem \[thm:additivedrift\] holds with equality; namely the function that attributes to each search point the expected optimization time of the algorithm starting in that point.

\[lem:idealpotential\] Let $\mathcal{S}$ be a finite search space and $\{x^{(t)}\}_{t\in\mathbb{N}}$ the search points generated by a homogeneous absorbing Markov chain on $\mathcal{S}$. Let $T$ be the random variable that denotes the fist point in time $t\in\mathbb{N}$ such that $x^{(t)}$ is optimal.

Then the drift on the potential function $g\colon\mathcal{S}\to\mathbb{R}$ with $$g(x):={\ensuremath{\operatorname{E}}}[T\mid x^{(0)}=x]$$ satisfies $${\ensuremath{\operatorname{E}}}\big[g(x^{(t)})-g(x^{(t+1)}){\,\big|\,}T>t\big]=1.$$

In a way, ${\ensuremath{\operatorname{E}}}[T\mid x^{(0)}=x]$ is an “ideal” potential function for Theorem \[thm:additivedrift\]. It satisfies the additive drift condition (\[eq:additivedrift\]) with equality and results in precise upper bound on ${\ensuremath{\operatorname{E}}}[T\mid x^{(0)}=x]$. However, the previous theorem is not directly helpful in the runtime analysis of randomized search heuristics. In order to apply the previous theorem, we need to know the exact expected optimization time of a algorithm starting from every point in the search point. But with all this known, Theorem \[thm:additivedrift\] does not provide new information.

Still, the 