Multi-unit bilateral trade refers to the setting, where there is a buyer and a seller, who holds a finite number of units of an indivisible item. An automated mechanism has to decide how many units are transferred from the seller to the buyer and the corresponding payment from the buyer to the seller. The buyer and the seller have both either increasing or increasing submodular valuation functions in the number of units in possession. The (single-unit) bilateral trade problem arises as a particular case. We study the problem of social welfare maximisation by establishing the fraction (approximation ratio) of the optimal social welfare that a fixed-price mechanism can recover. Fixed-price mechanisms, understood as per-unit price in the multi-unit setting, have been characterised as the only truthful, individually rational and strongly budget balanced mechanisms. We narrow the gap on the approximation ratio of optimal fixed-price mechanisms for bilateral trade, which has been shown to lie between 0.72 and 0.7381. We show that it must lie between 0.7292 and 0.73805, which leads to improved bounds on the approximation ratio of optimal fixed-price mechanisms for multi-unit bilateral trade. In particular, we show that multi-unit bilateral trade is at least as hard as single-unit bilateral trade, and obtain several hardness results for different numbers of units.

Prophet inequalities are a central object of study in optimal stopping theory. In the iid model, a gambler sees values in an online fashion, sampled independently from a given distribution. Upon observing each value, the gambler either accepts it as a reward or irrevocably rejects it and proceeds to observe the next value. The goal of the gambler, who cannot see the future, is maximising the expected value of the reward while competing against the expectation of a prophet (the offline maximum). In other words, one seeks to maximise the gambler-to-prophet ratio of the expectations. This model has been studied with infinite, finite and unknown number of values. When the gambler faces a random number of values, the model is said to have random horizon. We consider the model in which the gambler is given a priori knowledge of the horizon's distribution. Alijani et al. (2020) designed a single-threshold algorithm achieving a ratio of $1/2$ when the random horizon has an increasing hazard rate and is independent of the values. We prove that with a single threshold, a ratio of $1/2$ is actually achievable for several larger classes of horizon distributions, with the largest being known as the $\mathcal{G}$ class in reliability theory. Moreover, we show that this does not extend to its dual, the $\overline{\mathcal{G}}$ class (which includes the decreasing hazard rate class), while it can be extended to low-variance horizons. Finally, we construct the first example of a family of horizons, for which multiple thresholds are necessary to achieve a nonzero ratio. We establish that the Secretary Problem optimal stopping rule provides one such algorithm, paving the way towards the study of the model beyond single-threshold algorithms.

Considerable research efforts have recently been made to show that a random neural network $N$ contains subnetworks capable of accurately approximating any given neural network that is sufficiently smaller than $N$, without any training. This line of research, known as the Strong Lottery Ticket Hypothesis (SLTH), was originally motivated by the weaker Lottery Ticket Hypothesis, which states that a sufficiently large random neural network $N$ contains sparse subnetworks that can be trained efficiently to achieve performance comparable to that of training the entire network $N$. Despite its original motivation, results on the SLTH have so far not provided any guarantee on the size of subnetworks. Such limitation is due to the nature of the main technical tool leveraged by these results, the Random Subset Sum (RSS) Problem. Informally, the RSS Problem asks how large a random i.i.d. sample $\Omega$ should be so that we are able to approximate any number in $[-1,1]$, up to an error of $ \epsilon$, as the sum of a suitable subset of $\Omega$. We provide the first proof of the SLTH in classical settings, such as dense and equivariant networks, with guarantees on the sparsity of the subnetworks. Central to our results, is the proof of an essentially tight bound on the Random Fixed-Size Subset Sum Problem (RFSS), a variant of the RSS Problem in which we only ask for subsets of a given size, which is of independent interest.