We must assign n agents to m posts subject to negative congestion; what assignment is fair and efficient? If congestion is anonymous (each agent adds one unit), it is always possible to assign each agent to one of the agent’s top n out of the [Formula: see text] feasible allocations. This ordinal interpretation of ex ante fairness can be adjusted if congestion is weighted (agent-specific). An assignment is competitive if I don’t want to move to an empty post or to an occupied one at its current congestion level. If it exists, the competitive assignment is essentially unique, efficient, and ex ante fair. Among agents endowed with cardinal expected utilities, we can randomize the selection of our assignment. Under anonymous congestion, every problem has a unique competitive congestion profile implemented by a mixture of deterministic assignments, rounding up or down the competitive congestion, and approximately ex ante fair, efficient, and welfare equivalent. Some of these properties are lost under weighted congestion.

We divide efficiently a pile of indivisible goods in common property, using cash transfers to ensure fairness among agents with utility linear in money. We compare three cognitively feasible and privacy preserving division rules in terms of the guarantees (worst case utility) they offer to the participants. In the first version of Divide Choose to n agents, they bid for the role of Divider then everyone bids on the shares of the Divider’s partition. In the second version each agent announces a partition and they all bid to select the most efficient one. In the Bid Sell rule the agents bid for the role of Seller: with two agents the smallest bid defines the Seller who then charges any price constrained only by her winning bid. Both rules reward subadditive utilities and penalise superadditive ones, and B S more so than both D C-s. B S is also better placed to collect a larger share of the surplus when agents play safe.

Proportionality (PROP) is one of the simplest and most intuitive fairness criteria used for allocating items among agents with additive utilities. However, when the items are indivisible, ensuring PROP becomes unattainable, leading to increased focus on its relaxations. In this paper, we focus on the relaxation of proportionality up to any item (PROPX), where proportionality is satisfied if an arbitrary item is removed from every agent's allocation. We show that PROPX is an appealing fairness notion for the allocation of indivisible chores, which approximately implies some share-based notions, such as maximin share (MMS) and AnyPrice share (APS). We further provide a comprehensive understanding of PROPX allocations, regarding the computation, approximation, and compatibility with efficiency. On top of these, we extend the study to scenarios where agents do not share equal liability towards the chores, and approximate PROPX allocations using partial information about agents' utilities.

In our context-free model of a commons, the function$\mathcal{W}$ transforms the profile of the agents' types $(x_{1},..,x_{n})$ to a freely transferable output $\mathcal{W}(x_{1},..,x_{n})$ that they must share fairly. We expand the ubiquitous concept of \textit{endogenous fair shares} to include both a lower and an upper bound on agent $i$'s share at the interim stage where $i$ only knows its own type $x_{i}$. Two functions $(g^{-},g^{+})$ form a pair of tight guarantees if 1) they satisfy the system of inequalities $% \sum_{1}^{n}g^{-}(x_{i})\leq \mathcal{W}(x)\leq \sum_{1}^{n}g^{+}(x_{i})$ for all profiles, and 2) the interval $[g^{-}(x_{i}),g^{+}(x_{i})]$ is inclusion minimal across all types. For super (resp sub) modular functions 1) the \textit{Unanimity }share% \textit{\ }$\frac{1}{n}\mathcal{W}(x_{i},x_{i},..,x_{i})$ is the unique tight upper (resp lower) guarantee, 2) two \textit{Stand Alone} shares $% g(x_{i})=\mathcal{W}(x_{i},\overbrace{x_{0},..,x_{0}})-\frac{n-1}{n}\mathcal{% W}(\overbrace{x_{0},..,x_{0}})$ (where $x_{0}$ is the smallest or largest type) bracket all tight guarantees on the other side of Unanimity, 3) serial cost sharing implements the Unanimity and Stand Alone guarantees. In applications to specific microeconomic models, tight guarantees vindicate or dismiss familiar deterministic sharing rules and suggest new ones with a clear normative interpretation. Our examples include joint production with substitute or complementary inputs, allocating an indivisible good and cash transfers, sharing the cost (or benefit) of the variance or the spread of types, the waiting cost in a queue, and more.

We evaluate the fairness of a rule allocating among agents with equal rights by computing their utility for a hypothetical worst-case share, which only depends on their own valuation and the number of agents. For indivisible goods, Budish proposed the maximin share: the least utility of a bundle in the best partition of the objects; unfortunately maximin share is not always satisfiable. Earlier, Hill proposed the worst maximin share over all utilities with the same largest possible single-object value. More conservative than maximin share, it is guaranteed to be satisfiable for any possible profile of utilities and its computation is elementary, whereas it is NP-hard to compute maximin share. We apply Hill's approach to the allocation of indivisible bads (objects with disutilities) and compute in closed form the worst-case minimax share for a given value of the worst single bad. We show that the worst-case minimax share is close to the original minimax share and that its monotonic closure is the best guaranteed share for all allocation instances with a common upper bound on the value of the worst single bad.

A mechanism guarantees a certain welfare level to its agents, if each of them can secure that level against unanimously adversarial others. How high can such a guarantee be, and what type of mechanism achieves it? In the n‐person probabilistic voting/bargaining model with p deterministic outcomes a guarantee takes the form of a probability distribution over the ranks from 1 to p. If n ≥ p, the uniform lottery is shown to be the only maximal (unimprovable) guarantee. If n < p, combining (variants of) the familiar random dictator and voting by veto mechanisms yields a large family of maximal guarantees: it is exhaustive if n = 2 and almost so if p ≤ 2 n. Voting rules à la Condorcet or Borda, even in probabilistic form, are ruled out by our worst case viewpoint.

We propose a fair and efficient solution for assigning agents to m posts subject to congestion, when agents care about both their post and its congestion. Examples include assigning jobs to busy servers, students to crowded schools or crowded classes, commuters to congested routes, workers to crowded office spaces or to team projects etc... Congestion is anonymous (it only depends on the number n of agents in a given post). A canonical interpretation of ex ante fairness allows each agent to choose m post-specific caps on the congestion they tolerate: these requests are mutually feasible if and only if the sum of the caps is n. For ex post fairness we impose a competitive requirement close to envy freeness: taking the congestion profile as given each agent is assigned to one of her best posts. If a competitive assignment exists, it delivers unique congestion and welfare profiles and is also efficient and ex ante fair. In a fractional (randomised or time sharing) version of our model, a unique competitive congestion profile always exists. It is approximately implemented by a mixture of ex post deterministic assignments: with an approxination factor equal to the largest utility loss from one more unit of congestion, the latter deliver identical welfare profiles and are weakly efficient. Our approach to ex ante fairness generalises to the model where each agent's congestion is weighted. Now the caps on posts depend only upon own weight and total congestion, not on the number of other agents contributing to it. Remarkably in both models these caps are feasible if and only if they give to each agent the right to veto all but (1/m) of their feasible allocations.

Allocating resources to individuals in a fair manner has been a topic of interest since ancient times, with most of the early mathematical work on the problem focusing on resources that are infinitely divisible. Over the last decade, there has been a surge of papers studying computational questions regarding the indivisible case, for which exact fairness notions such as envy-freeness and proportionality are hard to satisfy. One main theme in the recent research agenda is to investigate the extent to which their relaxations, like maximin share fairness (MMS) and envy-freeness up to any good (EFX), can be achieved. In this survey, we present a comprehensive review of the recent progress made in the related literature by highlighting different ways to relax fairness notions, common algorithm design techniques, and the most interesting questions for future research.

When dividing a “manna” Ω of private items (commodities, workloads, land, time slots) between n agents, the individual guarantee is the welfare each agent can secure in the worst case of other agents’ preferences and actions. If the manna is nonatomic and utilities are continuous (not necessarily monotone or convex) the minmax utility, that of our agent’s best share in the agent’s worst partition of the manna, is guaranteed by Kuhn’s generalization of divide and choose. The larger maxmin utility—of the agent’s worst share in the agent’s best partition—cannot be guaranteed even for two agents. If, for all agents, more manna is better than less (or less is better than more), the new bid and choose rules offer guarantees between minmax and maxmin by letting agents bid for the smallest (or largest) size of a share they find acceptable. Funding: A. Bogomolnaia and H. Moulin acknowledge the support from the Basic Research Program of the National Research University Higher School of Economics. H. Moulin’s research was also supported by a Chaire Blaise Pascal of the Region Ile-de-France, 2020-21.

As in public good provisions, in a public bad situation such as abatement, the non-cooperative interplay of the participants typically results in low levels of quantities (provision or abatement). In a simple class of n-person quadratic games, we show how Coarse correlated equilibria, using simple mediation devices, can significantly outperform Nash equilibrium outcomes in terms of a stated policy objective.

Ann likes oranges much more than apples; Bob likes apples much more than oranges. Tomorrow they will receive one fruit that will be an orange or an apple with equal probability. Giving one half to each agent is fair for each realization of the fruit. However, agreeing that whatever fruit appears will go to the agent who likes it more gives a higher expected utility to each agent and is fair in the average sense: in expectation, each agent prefers the allocation to the equal division of the fruit; that is, the agent gets a fair share. We turn this familiar observation into an economic design problem: upon drawing a random object (the fruit), we learn the realized utility of each agent and can compare it to the mean of the agent’s distribution of utilities; no other statistical information about the distribution is available. We fully characterize the division rules using only this sparse information in the most efficient possible way while giving everyone a fair share. Although the probability distribution of individual utilities is arbitrary and mostly unknown to the manager, these rules perform in the same range as the best rule when the manager has full access to this distribution. This paper was accepted by Ilia Tsetlin, behavioral economics and decision analysis.

The theory of algorithmic fair allocation is at the center of multi-agent systems and economics in recent decades due to its industrial and social importance. At a high level, the problem is to assign a set of items that are either goods or chores to a set of agents so that every agent is happy with what she obtains. In this survey, we focus on indivisible items, for which exact fairness as measured by envy-freeness and proportionality cannot be guaranteed. One main theme in the recent research agenda is designing algorithms that approximately achieve fairness criteria. We aim at presenting a comprehensive survey of recent progress through the prism of algorithms, highlighting the ways to relax fairness notions and common techniques to design algorithms, as well as the most interesting questions for future research.

The theory of algorithmic fair allocation is at the center of multi-agent systems and economics in recent decades due to its industrial and social importance. At a high level, the problem is to assign a set of items that are either goods or chores to a set of agents so that every agent is happy with what she obtains. In this survey, we focus on indivisible items, for which exact fairness as measured by envy-freeness and proportionality cannot be guaranteed. One main theme in the recent research agenda is designing algorithms that approximately achieve fairness criteria. We aim at presenting a comprehensive survey of recent progressthrough the prism of algorithms, highlighting the ways to relax fairness notions and common techniques to design algorithms, as well as the most interesting questions for future research.

The celebrated tragedy of the commons arises when a certain technology with increasing marginal cost is the common property of its users. Examples include the exploitation of fisheries and other natural resources, as well as queuing problems where users want a service and the externalities result from congestion.

The guarantee of an anonymous mechanism is the worst case welfare an agent can secure against unanimously adversarial others. How high can such a guarantee be, and what type of mechanism achieves it? We address the worst case design question in the n-person probabilistic voting/bargaining model with p deterministic outcomes. If n is no less than p the uniform lottery is the only maximal (unimprovable) guarantee; there are many more if p>n, in particular the ones inspired by the random dictator mechanism and by voting by veto. If n=2 the maximal set M(n,p) is a simple polytope where each vertex combines a round of vetoes with one of random dictatorship. For p>n>2, we show that the dual veto and random dictator guarantees, together with the uniform one, are the building blocks of 2 to the power d simplices of dimension d in M(n,p), where d is the quotient of p-1 by n. Their vertices are guarantees easy to interpret and implement. The set M(n,p) may contain other guarantees as well; what we can say in full generality is that it is a finite union of polytopes, all sharing the uniform guarantee.

Satellites launched by independent spacefaring agencies and firms create space congestion and collision risk. Taking as benchmark the cost of a marginal reduction of the congestion rate, we discuss tax mechanisms financing a debris removal effort. We compare the non-cooperative equilibrium traffic when there is a tax on each new launch to recover cleanup costs, with the welfare optimal traffic under a centralized tax. We find that under the latter it is twice as easy to recover cleanup costs and increase traffic than under the former. We also show that a linear tax is twice as effective as a quadratic one.

We consider fair allocation of indivisible items under additive utilities. When the utilities can be negative, the existence and complexity of an allocation that satisfies Pareto optimality and proportionality up to one item (PROP1) is an open problem. We show that there exists a strongly polynomial-time algorithm that always computes an allocation satisfying Pareto optimality and proportionality up to one item even if the utilities are mixed and the agents have asymmetric weights. We point out that the result does not hold if either of Pareto optimality or PROP1 is replaced with slightly stronger concepts.

In a public good provision or a public bad abatement situation, the non-cooperative interplay of the participants typically results in low levels of provision or abatement. In the familiar class of n-person quadratic games, we show that Coarse Correlated equilibria (CCEs) - simple mediated communication devices that do not alter the strategic structure of the game - can significantly outperform the Nash equilibrium in terms of the policy objective above.