Theorem 4.2, p. 17 — on a low-moreishness, near-constant-value manifold, E[S(ω_T)] ≥ E[S(ω_S)] − 2βE[T(ω_S)] − α(v₀ + β)/(1 − α)

Let $W>0$ be the outside option and let $\mathcal M\subseteq\Omega=[0,1)\times[0,1)\times\mathbb R_+$ be a content manifold containing at least one point with strictly positive expected utility, $g_S(\omega)>0$. Let $\omega_S=(p_S,q_S,\bar v_S)$ be a maximizer of $g_S$ over $\mathcal M$ (3), and let $\omega_T=(p_T,q_T,\bar v_T)$ be a maximizer of $g_T$ over $\{\omega\in\mathcal M: g_S(\omega)\ge0\}$ with $g_S(\omega_T)\ge0$ (4). Suppose that

1. $p\le\alpha$ for all $(p,q,\bar v)\in\mathcal M$, where $\alpha<1$;
2. for some $v_0$, $|\bar v-v_0|<\beta$ for all $(p,q,\bar v)\in\mathcal M$.

Then the user's utility at $\omega_T$ is not much less than its utility at $\omega_S$:

Here $\mathbb E[S(\omega)]=\max(g_S(\omega),0)$ is system 2's expected utility (1), and $\mathbb E[T(\omega)]$ is the expected session length (2), which is $0$ when $g_S(\omega)<0$. When $\alpha$ and $\beta$ are close to $0$, maximizing engagement is near-optimal for the user.

Formalization Note Three hypotheses implicit in the paper are explicit: $W>0$ (§2.1 normalizes $W=1$), $\alpha<1$ (without it condition 1 is vacuous on $\Omega$ and the bound fails), and the paper's standing assumption (p. 16) that $\mathcal M$ contains a point with $\mathbb E[S]>0$. "For some $v_0$" is expressed by taking $v_0$ as a parameter satisfying condition 2, which is equivalent because $v_0$ appears in the conclusion. The theorem holds for every pair of maximizers, so no tie-breaking rule and no closedness of $\mathcal M$ are needed.