We study the problem of fairly allocating a set of m goods among n agents in the asymptotic setting, where each item's value for each agent is drawn from an underlying joint distribution. Prior works have shown that if this distribution is well-behaved, then an envy-free allocation exists with high probability when m=Ω(n log n). Under the stronger assumption that item values are independently and identically distributed (i.i.d.) across agents, it is known that this requirement improves to m=Ω(n log n / log log n), which is tight. However, these results rely on non-strategyproof mechanisms, such as maximum-welfare allocation or the round-robin algorithm, limiting their applicability in settings with strategic agents. In this work, we extend the theory to a broader, more realistic class of joint value distributions, allowing for correlations among agents, atomicity, and unequal probabilities of having the highest value for an item. We show that envy-free allocations continue to exist with a high probability when m=Ω(n log n). More importantly, we give a new randomized mechanism that is truthful in expectation, efficiently implementable in polynomial time, and outputs envy-free allocations with high probability, answering an open question from the literature. We further extend our mechanism to settings with asymptotic weighted fair division and multiple agent types and good types, proving new results in each case.

We study fair mechanisms for the classic job scheduling problem on unrelated machines with the objective of minimizing the makespan. This problem is equivalent to minimizing the egalitarian social cost in the fair division of chores. The two prevalent fairness notions in the fair division literature are envy-freeness and proportionality. Prior work has established that no envy-free mechanism can provide better than an Ω(log m / log log m)-approximation to the optimal makespan, where m is the number of machines, even when payments to the machines are allowed. In strong contrast to this impossibility, our main result demonstrates that there exists a proportional mechanism (with payments) that achieves a 3/2-approximation to the optimal makespan, and this ratio is tight. To prove this result, we provide a full characterization of allocation functions that can be made proportional with payments. Furthermore, we show that for instances with normalized costs, there exists a proportional mechanism that achieves the optimal makespan. We conclude with important directions for future research concerning other fairness notions, including relaxations of envy-freeness. Notably, we show that the technique leading to the impossibility result for envy-freeness does not extend to its relaxations.

We study best-of-both-worlds guarantees for the fair division of indivisible items among agents with subadditive valuations. Our main result establishes the existence of a random allocation that is simultaneously ex-ante 1/2-envy-free, ex-post 1/2-EFX and ex-post EF1, for every instance with subadditive valuations. We achieve this result by a novel polynomial-time algorithm that randomizes the well-established envy cycles procedure in a way that provides ex-ante fairness. Notably, this is the first best-of-both-worlds fairness guarantee for subadditive valuations, even when considering only EF1 without EFX.

We consider the problem of fair division, where a set of indivisible goods should be distributed fairly among a set of agents with combinatorial valuations. To capture fairness, we adopt the notion of shares, where each agent is entitled to a fair share, based on some fairness criterion, and an allocation is considered fair if the value of every agent (weakly) exceeds her fair share. A share-based notion is considered universally feasible if it admits a fair allocation for every profile of monotone valuations. A major question arises: is there a non-trivial share-based notion that is universally feasible? The most well-known share-based notions, namely the proportional share and the maximin share, are not universally feasible, nor are any constant approximations of them.

We show Maker wins the Maker-Breaker perfect matching game in n/2 + o(n) turns when the bias is at least n/ln n − f(n)n/(ln n)5/4, for any f going to infinity with n and n sufficiently large (in terms of f).

We show that there is a constant C such that for any b < n/ln n − Cn/(ln n)3/2, Maker can win the Maker-Breaker Hamilton cycle game in n + Cn/√ln n steps.

We study the fair division of a collection of mindivisible goods amongst a set of nagents. Whilst envy-free allocations typically do not exist in the indivisible-goods setting, envy-freeness can be achieved if some amount of a divisible good (money) is introduced. Specifically, Halpern and Shah (SAGT 2019, pp.374-389) showed that, given additive valuation functions where the marginal value of each good is at most one dollar for each agent, there always exists an envy-free allocation requiring a subsidy of at most (n-1)·m dollars. The authors also conjectured that a subsidy of $n-1$ dollars is sufficient for additive valuations. We prove this conjecture. In fact, a subsidy of at most one dollar per agent is sufficient to guarantee the existence of an envy-free allocation. Further, we prove that for general monotonic valuation functions an envy-free allocation always exists with a subsidy of at most 2(n-1) dollars per agent. In particular, the total subsidy required for monotonic valuations is independent of the number of goods.