Theorem 10, second claim — a randomized spin-block algorithm is $e/(e-1)$-competitive

Let $C>0$ be the context-switch cost of the spin-block problem. There is a randomized on-line algorithm $A$ and a constant $a$ such that for every finite sequence $\sigma$ of lock waits

that is, $A$ is $\frac{e}{e-1}$-competitive against an oblivious adversary.

This is the attainment half of Theorem 10; together with the lower bound it shows that $e/(e-1)$ is the optimal randomized competitive factor.

Formalization Note The randomized algorithm is a probability distribution over deterministic on-line algorithms whose cost on each fixed input is a measurable function of the random outcome, so the expected cost is a genuine expectation. The hypothesis $C>0$ is the paper's "some large cost $C$".