Fair Prophets

Abstract

We initiate the study of $\alpha$-fair prophet inequalities. This interpolates between utilitarian welfare $(\alpha=0)$, Nash welfare $(\alpha=1)$, and Rawlsian max-min fairness $(\alpha\to\infty)$. Given the non-linearity of the objective, it matters when the expectation is applied. For instance, for the Rawlsian objective, it matters whether we aim to maximize $\min \mathbb{E}[u_i]$ or $\mathbb{E}[\min u_i]$. We refer to the former as the ex-ante model, and the latter as the ex-post model. For ex-ante fairness, full distributional knowledge yields a tight competitive ratio of exactly $1/2$ for every $\alpha\ge 0$. Under sample access, $O(n\log n)$ samples per distribution suffice for a constant competitive ratio when $\alpha\in(0,1]$. In contrast, for every $\alpha>1$, no finite number of samples improves upon the trivial $1/n$ guarantee. Thus, unlike in the utilitarian setting, full-information and sample-access prophet inequalities become fundamentally separated. For ex-post fairness, under full information, we obtain a uniform constant ratio for all $\alpha\in(0,1)$, while for every $\alpha>1$ the competitive ratio collapses to $1/n$. In the sample-access model, one sample per distribution suffices for each fixed $\alpha<1$, but no sample budget depending only on $n$ yields a uniform constant guarantee as $\alpha\to 1$. Beyond these phase transitions for $\alpha$-fairness, our results open the door to a broader theory of prophet inequalities for non-linear welfare objectives.