Optimal Single-Choice Prophet Inequalities from Samples

We study the single-choice Prophet Inequality problem when the gambler is given access to samples. We show that the optimal competitive ratio of $1/2$ can be achieved with a single sample from each distribution. When the distributions are identical, we show that for any constant $\varepsilon > 0$, $O(n)$ samples from the distribution suffice to achieve the optimal competitive ratio ($\approx 0.745$) within $(1+\varepsilon)$, resolving an open problem of Correa, D\"utting, Fischer, and Schewior.Comment: Appears in Innovations in Theoretical Computer Science (ITCS) 202

Unknown I.I.D. Prophets: Better Bounds, Streaming Algorithms, and a New Impossibility

A prophet inequality states, for some α ∈ [0, 1], that the expected value achievable by a gambler who sequentially observes random variables X1, . . . , Xn and selects one of them is at least an α fraction of the maximum value in the sequence. We obtain three distinct improvements for a setting that was first studied by Correa et al. (EC, 2019) and is particularly relevant to modern applications in algorithmic pricing. In this setting, the random variables are i.i.d. from an unknown distribution and the gambler has access to an additional βn samples for some β ≥ 0. We first give improved lower bounds on α for a wide range of values of β; specifically, α ≥ (1 + β)/e when β ≤ 1/(e − 1), which is tight, and α ≥ 0.648 when β = 1, which improves on a bound of around 0.635 due to Correa et al. (SODA, 2020). Adding to their practical appeal, specifically in the context of algorithmic pricing, we then show that the new bounds can be obtained even in a streaming model of computation and thus in situations where the use of relevant data is complicated by the sheer amount of data available. We finally establish that the upper bound of 1/e for the case without samples is robust to additional information about the distribution, and applies also to sequences of i.i.d. random variables whose distribution is itself drawn, according to a known distribution, from a finite set of known candidate distributions. This implies a tight prophet inequality for exchangeable sequences of random variables, answering a question of Hill and Kertz (Contemporary Mathematics, 1992), but leaves open the possibility of better guarantees when the number of candidate distributions is small, a setting we believe is of strong interest to applications

Prophet Inequalities for IID Random Variables from an Unknown Distribution

A central object in optimal stopping theory is the single-choice prophet inequality for independent, identically distributed random variables: given a sequence of random variables X1, . . . , Xn drawn independently from a distribution F , the goal is to choose a stopping time τ so as to maximize α such that for all distributions F we have E[Xτ ] ≥ α · E[maxt Xt ]. What makes this problem challenging is that the decision whether τ = t may only depend on the values of the random variables X1, . . . , Xt and on the distribution F . For a long time the best known bound for the problem had been α ≥ 1 − 1/e ≈ 0.632, but quite recently a tight bound of α ≈ 0.745 was obtained. The case where F is unknown, such that the decision whether τ = t may depend only on the values of the random variables X1, . . . , Xt , is equally well motivated but has received much less attention. A straightforward guarantee for this case of α ≥ 1/e ≈ 0.368 can be derived from the solution to the secretary problem, where an arbitrary set of values arrive in random order and the goal is to maximize the probability of selecting the largest value. We show that this bound is in fact tight. We then investigate the case where the stopping time may additionally depend on a limited number of samples from F , and show that even with o(n) samples α ≤ 1/e. On the other hand, n samples allow for a significant improvement, while O(n2) samples are equivalent to knowledge of the distribution: specifically, with n samples α ≥ 1 − 1/e ≈ 0.632 and α ≤ ln(2) ≈ 0.693, and with O(n2) samples α ≥ 0.745 − ε for any ε > 0

Prophet Inequalities Require Only a Constant Number of Samples

In a prophet inequality problem, $n$ independent random variables are presented to a gambler one by one. The gambler decides when to stop the sequence and obtains the most recent value as reward. We evaluate a stopping rule by the worst-case ratio between its expected reward and the expectation of the maximum variable. In the classic setting, the order is fixed, and the optimal ratio is known to be 1/2. Three variants of this problem have been extensively studied: the prophet-secretary model, where variables arrive in uniformly random order; the free-order model, where the gambler chooses the arrival order; and the i.i.d. model, where the distributions are all the same, rendering the arrival order irrelevant. Most of the literature assumes that distributions are known to the gambler. Recent work has considered the question of what is achievable when the gambler has access only to a few samples per distribution. Surprisingly, in the fixed-order case, a single sample from each distribution is enough to approximate the optimal ratio, but this is not the case in any of the three variants. We provide a unified proof that for all three variants of the problem, a constant number of samples (independent of n) for each distribution is good enough to approximate the optimal ratios. Prior to our work, this was known to be the case only in the i.i.d. variant. We complement our result showing that our algorithms can be implemented in polynomial time. A key ingredient in our proof is an existential result based on a minimax argument, which states that there must exist an algorithm that attains the optimal ratio and does not rely on the knowledge of the upper tail of the distributions. A second key ingredient is a refined sample-based version of a decomposition of the instance into "small" and "large" variables, first introduced by Liu et al. [EC'21]

Online Pen Testing

We study a "pen testing" problem, in which we are given $n$ pens with unknown amounts of ink $X_1, X_2, \ldots, X_n$, and we want to choose a pen with the maximum amount of remaining ink in it. The challenge is that we cannot access each $X_i$ directly; we only get to write with the $i$-th pen until either a certain amount of ink is used, or the pen runs out of ink. In both cases, this testing reduces the remaining ink in the pen and thus the utility of selecting it. Despite this significant lack of information, we show that it is possible to approximately maximize our utility up to an $O(\log n)$ factor. Formally, we consider two different setups: the "prophet" setting, in which each $X_i$ is independently drawn from some distribution $\mathcal{D}_i$, and the "secretary" setting, in which $(X_i)_{i=1}^n$ is a random permutation of arbitrary $a_1, a_2, \ldots, a_n$. We derive the optimal competitive ratios in both settings up to constant factors. Our algorithms are surprisingly robust: (1) In the prophet setting, we only require one sample from each $\mathcal{D}_i$, rather than a full description of the distribution; (2) In the secretary setting, the algorithm also succeeds under an arbitrary permutation, if an estimate of the maximum $a_i$ is given. Our techniques include a non-trivial online sampling scheme from a sequence with an unknown length, as well as the construction of a hard, non-uniform distribution over permutations. Both might be of independent interest. We also highlight some immediate open problems and discuss several directions for future research.Comment: To appear at ITCS 2023; v2 added discussion on a closely related work of Awerbuch, Azar, Fiat, and Leighton (1996