Indivisible Resources Research Articles

Mixed Fair Division: A Survey

Fair division considers the allocation of scarce resources among agents in such a way that every agent gets a fair share. It is a fundamental problem in society and has received significant attention and rapid developments from the game theory and artificial intelligence communities in recent years. The majority of the fair division literature can be divided along at least two orthogonal directions: goods versus chores, and divisible versus indivisible resources. In this survey, besides describing the state of the art, we outline a number of interesting open questions and future directions in three mixed fair division settings: (i) indivisible goods and chores, (ii) divisible and indivisible goods (mixed goods), and (iii) indivisible goods with subsidy which can be viewed like a divisible good.

On Optimal Tradeoffs between EFX and Nash Welfare

A major problem in fair division is how to allocate a set of indivisible resources among agents fairly and efficiently. The goal of this work is to characterize the tradeoffs between two well-studied measures of fairness and efficiency --- envy freeness up to any item (EFX) for fairness, and Nash welfare for efficiency --- by saying, for given constants α and β, whether there exists an α-EFX allocation that guarantees a β-fraction of the maximum Nash welfare (β-MNW). For additive valuations, we show that for any α ∈ [0,1], there exists a partial allocation that is α-EFX and 1/(α+1)-MNW. This tradeoff turns out to be tight (for every α) as demonstrated by an impossibility result that we give. We also show that for α ∈ [0, φ-1 ≃ 0.618] these partial allocations can be turned into complete allocations where all items are assigned. Furthermore, for any α ∈ [0, 1/2], we show that the tight tradeoff of α-EFX and 1/(α+1)-MNW with complete allocations holds for the more general setting of subadditive valuations. Our results improve upon the current state of the art, for both additive and subadditive valuations, and match the best-known approximations of EFX under complete allocations, regardless of Nash welfare guarantees. Notably, our constructions for additive valuations also provide EF1 and constant approximations for maximin share guarantees.

Mixed Fair Division: A Survey

The fair allocation of resources to agents is a fundamental problem in society and has received significant attention and rapid developments from the game theory and artificial intelligence communities in recent years. The majority of the fair division literature can be divided along at least two orthogonal directions: goods versus chores, and divisible versus indivisible resources. In this survey, besides describing the state of the art, we outline a number of interesting open questions in three mixed fair division settings: (i) indivisible goods and chores, (ii) divisible and indivisible goods (i.e., mixed goods), and (iii) fair division of indivisible goods with subsidy.

Almost Envy-Free Allocations of Indivisible Goods or Chores with Entitlements

We here address the problem of fairly allocating indivisible goods or chores to n agents with weights that define their entitlement to the set of indivisible resources. Stemming from well-studied fairness concepts such as envy-freeness up to one good (EF1) and envy-freeness up to any good (EFX) for agents with equal entitlements, we present, in this study, the first set of impossibility results alongside algorithmic guarantees for fairness among agents with unequal entitlements. Within this paper, we expand the concept of envy-freeness up to any good or chore to the weighted context (WEFX and XWEF respectively), demonstrating that these allocations are not guaranteed to exist for two or three agents. Despite these negative results, we develop a WEFX procedure for two agents with integer weights, and furthermore, we devise an approximate WEFX procedure for two agents with normalized weights. We further present a polynomial-time algorithm that guarantees a weighted envy-free allocation up to one chore (1WEF) for any number of agents with additive cost functions. Our work underscores the heightened complexity of the weighted fair division problem when compared to its unweighted counterpart.

The existence and efficiency of PMMS allocations

Fair division of indivisible resources is a fundamental problem in many disciplines, including computer science, economy, operations research, etc. The existence of PMMS allocations of goods is a major open problem in fair division, even for additive valuations. At the state of the art, there is no identified setting where PMMS allocations are deemed impossible, and the known existing results regarding their existence are limited to highly constrained settings. In this paper, we provide an algorithm for computing a PMMS allocation for identical variant. We also use the algorithm to find the PMMS allocations for 3 agents in line graphs. Moreover, we show that a 45-PMMS allocation can be computed in polynomial time when agents agree on the ordinal ranking of the goods. A quantitative gauge for assessing the impact on social welfare is known as the price of fairness, which quantifies the maximum possible reduction in social welfare due to the imposition of fairness constraints. Originally explored in the context of divisible goods, the concept of the price of fairness continues to be a pivotal metric for evaluating the effect of fairness considerations on overall social welfare. Consequently, we study the efficiency decrease under PMMS allocations and prove a tight upper bound on the price of PMMS allocations.

A Strategic Decision Support System Using Multiplayer Non-Cooperative Games for Resource Allocation After Natural Disasters

A severe natural disaster causes concurrent emergencies in separate geographical locations. However, tackling all the emergencies simultaneously with the limited available resources is challenging. A novel single-stage, non-cooperative multiplayer game-based solution approach for resource allocation is proposed in this paper, where the crisis locations in the demand of resources are treated as the players. The game-based decision support system (DSS) is intended to be implemented by the concerned disaster management authority to obtain a practical strategy for the allocation of indivisible and divisible resources among individual players in a limited resource environment. Any feasible allocation is associated with some cost on the basis of a non-monetary cost function. A discrete strategic game is formulated to tackle indivisible resource allocation and a continuous-kernel game for divisible resources. A mathematical analysis establishes that the game with the proposed cost function possesses at least one pure strategy Nash equilibrium (PSNE) for both types of games. Based on different selection criteria, a particular PSNE is chosen from the collection of multiple PSNEs and used as a desirable resource allocation strategy. A complexity analysis of the proposed algorithm is also carried out. Case studies are given in this paper to demonstrate the results developed. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —The cost function proposed in this paper to formulate the non-cooperative game-based DSS will be useful to disaster management authorities in resource allocation problems after any natural disaster. While distributing a limited quantity of essential resources, the cost function maintains an overall balance so that none of the disaster locations is deprived or favoured unreasonably. The feasible allocations of the existing resources are determined by the solutions of a multiplayer game. The case study indicates that the disaster management authority can choose different post-disaster ground truths to compute the possible resource demands of the disaster locations. Using the demand vector (a vector comprising the demands of the players as elements) and the number of available resources as the inputs, the game-theoretic model generates the allocation vector, according to which the disaster locations can be allocated resources.

Differentially Private Fair Division

Fairness and privacy are two important concerns in social decision-making processes such as resource allocation. We study privacy in the fair allocation of indivisible resources using the well-established framework of differential privacy. We present algorithms for approximate envy-freeness and proportionality when two instances are considered to be adjacent if they differ only on the utility of a single agent for a single item. On the other hand, we provide strong negative results for both fairness criteria when the adjacency notion allows the entire utility function of a single agent to change.

The evolutionarily stable strategy, animal contests, parasitoids, pest control and sociality.

The evolutionarily stable strategy, ESS, concept was first used in biology to understand sex ratio bias and, shortly afterwards, to explore the logic of contests over essential and indivisible resources. ESS models formed the basis of much subsequent research on animal behaviour and placed game-theoretic thinking firmly within the behavioural ecology approach. Among behavioural ecologists studying parasitoids, it was those asking questions about the evolution of sex ratios who first made extensive use of the game-theoretic approach. A later growth of interest in parasitoid host defence and fighting behaviour made use of these tractable study species to explore contests and their connections to further aspects of life-history evolution plus some pest control applications. Our aims are to (i) introduce the topic of contests, which are engaged in by a very wide array of animal taxa, and the importance, both historical and conceptual, of the game-theoretic approach to their study, and (ii) review recent studies of parasitoid contests, including those that have considered the context of social evolution and the performance of parasitoids as agents of biological control. We consider that game-theoretic models are eminently testable and applicable and will likely endure as valuable tools in studies of parasitoid biology. This article is part of the theme issue 'Half a century of evolutionary games: a synthesis of theory, application and future directions'.

Job Matching with Subsidy and Taxation

Abstract In markets for indivisible resources such as workers and objects, subsidy and taxation for an agent may depend on the set of acquired resources and prices. This paper investigates how such transfer policies interfere with the substitutes condition, which is critical for market equilibrium existence and auction mechanism performance among other important issues. For environments where the condition holds in the absence of policy intervention, we investigate which transfer policies preserve the substitutes condition in various economically meaningful settings, establishing a series of characterisation theorems. For environments where the condition may fail without policy intervention, we examine how to use transfer policies to re-establish it, finding exactly when transfer policies based on scales are effective for that purpose. These results serve to inform policymakers, market designers, and market participants of how transfer policies may impact markets, so more informed decisions can be made.

Strategyproof Mechanism for Two-Sided Matching with Resource Allocation

In this work, we consider a student-project-resource matching-allocation problem, where students have preferences over projects and the projects have preferences over students. In this problem, students and indivisible resources are many-to-one matched to projects whose capacities are endogenously determined by the resources allocated to them. Traditionally, this problem is decomposed into two separate problems: (1) resources are allocated to projects based on expectations (a resource allocation problem), and (2) students are matched to projects based on the capacities determined in the previous problem (a matching problem). Although both problems are well-understood, if the expectations used in the first are incorrect, we obtain a sub-optimal outcome. Thus, this problem should be solved as a whole without dividing it into two parts. We show that no strategyproof mechanism satisfies fairness and weak efficiency requirements. Given this impossibility result, we develop a new class of strategyproof mechanisms called Sample and Deferred Acceptance (SDA), which satisfies several properties on fairness and efficiency. We experimentally compare several SDA instances as well as existing mechanisms, and show that an SDA instance strikes a good balance of fairness and efficiency when students are divided into different types according to their preferences.

Parameterized complexity of envy-free resource allocation in social networks

We consider the classical problem of allocating indivisible resources among agents in an envy-free (and, where applicable, proportional) way. Recently, the basic model was enriched by introducing the concept of a social network which allows to capture situations where agents might not have full information about the allocation of all resources. We initiate the study of the parameterized complexity of these resource allocation problems by considering natural parameters which capture structural properties of the network and similarities between agents and resources. In particular, we show that even very general fragments of the considered problems become tractable as long as the social network has constant treewidth or clique-width. We complement our results with matching lower bounds which show that our algorithms cannot be substantially improved.

A Structure-Tensor Approach to Integer Matrix Completion in Indivisible Resource Allocation

Indivisible resource allocation motivates us to study the matrix completion concerning the class of (0,1)-matrices with prescribed row/column sums and preassigned zeros. We illustrate and generalize the (0,1)-matrix completion in the following two scenarios: a demand-response application involving nonnegative integer matrices with different bounds across rows and an edge caching matching problem allowing row and column sums to vary within separately designated bounds. The applications require analytic characterizations of the supply adequacy and cause large-scale matrix completion instances. Remarkably, we derive a structure tensor and use its nonnegativity to establish a necessary and sufficient condition under which the considered matrix class is nonempty. The tensor condition can characterize the adequacy of a supply for a prescribed demand and facilitate identifying the minimum supplement to the supply so that the augmented supply becomes adequate when the adequacy gap is nonzero. Notably, we design a tensor-based combinatorial algorithm to construct a required matrix, representing a feasible resource allocation. Numerical simulations justify the efficiency of our approach.

Three-player game-theoretic allocation of indivisible resources during natural disasters

Resource allocation is an integral part of disaster management. After a natural disaster, multiple concurrent emergencies in distinct locations often make resource allocation challenging for the disaster management authority. This article focuses on disaster resource allocation based on a novel three-player, non-cooperative, single-stage, strategic game where the emergency locations are considered as the players. The decision-making authority solves a game-theoretic algorithm to determine a suitable strategy for allocating indivisible resources among three disaster locations where the available resources are insufficient to satisfy all the players simultaneously. Based on a unique non-monetary cost function, each player incurs a penalty for any possible allocation strategy. Mathematical analysis shows that, for the proposed game, at least one pure strategy Nash equilibrium (PSNE) always exists, which can be a desirable allocation to the players. Payoff dominance and utopia-point-based solutions are used to select a single PSNE from a set of multiple PSNEs.

A new fairness notion in the assignment of indivisible resources

We study the problem of assigning objects. There may be multiple copies of each object. Each agent is assigned at most one. Monetary transfer is not allowed. We require a rule to be fair and efficient. We introduce an axiom that is a natural weakening of no-envy. We call it “bounded no-envy”. It states that for each pair of agents who receive objects of the same rank in their respective preferences, each of them should find his assignment at least as desirable as the other’s. Bounded no-envy is compatible with Pareto efficiency. In particular, the immediate acceptance rules (a.k.a. the “Boston mechanisms”) satisfy both. Our main result is that the immediate acceptance rules are the only rules satisfying weak non-wastefulness, bounded no-envy, rank-respecting invariance, resource monotonicity, and bilateral consistency.

Envy-free allocations respecting social networks

Finding an envy-free allocation of indivisible resources to agents is a central task in many multiagent systems. Often, non-trivial envy-free allocations do not exist and determining whether they exist is computationally hard even under highly restricted settings. Classical envy-freeness requires that every agent likes the resources allocated to it at least as much as the resources allocated to any other agent. In many situations this assumption can be relaxed since the agents often do not even know each other. We enrich the envy-freeness concept by taking into account (directed) social networks of the agents. Thus, we require that every agent likes its own allocation at least as much as those of all its (out)neighbors. This leads to a “more local” concept of envy-freeness. We also consider a “strong” variant where every agent must like its own allocation more than those of all its (out)neighbors.We analyze the classical and the parameterized complexity of finding allocations that are complete and, at the same time, envy-free with respect to one of the variants of our new concept. To this end, we study different restrictions of the agents' preferences and of the social network structure. We identify cases that become easier (from Σ2P-hard or NP-hard to polynomial-time solvable) and cases that become harder (from polynomial-time solvable to NP-hard) when comparing classical envy-freeness with our graph envy-freeness. Furthermore, we spot cases where graph envy-freeness is easier to decide than strong graph envy-freeness, and vice versa. On the route to one of our fixed-parameter tractability results, we also establish a connection to a directed and colored variant of the classical Subgraph Isomorphism problem, thereby extending a known fixed-parameter tractability result for the latter.

Capacity constraints as a trigger for high growth

High-growth enterprises (HGEs) have a large economic impact but are notoriously hard to predict. Previous research has linked high-growth episodes to the configuration of lumpy indivisible resources inside firms, such that high capacity utilisation levels might stimulate future growth. We theorize that firms reaching critically high capacity utilisation levels reach a “trigger point” involving either broad-based investment in further growth or shrinking back to previous levels. We analyze EIBIS survey data (matched to ORBIS) which features a question on time-varying capacity utilisation. Overcapacity is a transitory state. Firms enter into overcapacity after a period of the rapid growth of sales and profits, and the years surrounding overcapacity have higher employment growth rates. Firms operating at overcapacity make incremental investments (e.g. capacity expansion, process improvements and modern machinery) rather than investing in R&D and new product development. We find support for the “fork in the road” hypothesis: for some firms, overcapacity is associated with launching into massive investments and subsequent sales growth, while for other firms, overcapacity is negatively related to both investments and sales growth.

Constraints in fair division

The fair allocation of resources to interested agents is a fundamental problem in society. While the majority of the fair division literature assumes that all allocations are feasible, in practice there are often constraints on the allocation that can be chosen. In this survey, we discuss fairness guarantees for both divisible (cake cutting) and indivisible resources under several common types of constraints, including connectivity, cardinality, matroid, geometric, separation, budget, and conflict constraints. We also outline a number of open questions and directions.

Achieving Envy-freeness and Equitability with Monetary Transfers

When allocating indivisible resources or tasks, an envy-free allocation or equitable allocation may not exist. We present a sufficient condition and an algorithm to achieve envy-freeness and equitability when monetary transfers are allowed. The approach works for any agent valuation functions (positive or negative) as long as they satisfy superadditivity. For the case of additive utilities, we present a characterization of allocations that can simultaneously be made equitable and envy-free via payments. Our study shows that superadditive valuations constitute the largest class of valuations for which an envy-free and equitable outcome exists for all instances. We then present a distributed algorithm to compute an approximately envy-free outcome for any class of valuations.

Affiliation history and age similarity predict alliance formation in adult male bottlenose dolphins.

Male alliances are an intriguing phenomenon in the context of reproduction since, in most taxa, males compete over an indivisible resource, female fertilization. Adult male bottlenose dolphins (Tursiops aduncus) in Shark Bay, Western Australia, form long-term, multilevel alliances to sequester estrus females. These alliances are therefore critical to male reproductive success. Yet, the long-term processes leading to the formation of such complex social bonds are still poorly understood. To identify the criteria by which male dolphins form social bonds with other males, we adopted a long-term approach by investigating the ontogeny of alliance formation. We followed the individual careers of 59 males for 14 years while they transitioned from adolescence (8-14 years of age) to adulthood (15-21 years old). Analyzing their genetic relationships and social associations in both age groups, we found that the vast majority of social bonds present in adolescence persisted through time. Male associations in early life predict alliance partners as adults. Kinship patterns explained associations during adolescence but not during adulthood. Instead, adult males associated with males of similar age. Our findings suggest that social bonds among peers, rather than kinship, play a central role in the development of adult male polyadic cooperation in dolphins.

What do we need to know to recognize a contest?

Animals frequently use agonistic contests as a way to solve disputes over indivisible resources. Such agonistic contests often represent interactions between an owner and a non-owner of a resource. However, some behaviors adopted by rivals during agonistic interactions are similar to behaviors adopted in other types of interactions. Thus, the possibility exists that some interactions between individuals can be misinterpreted as actual agonistic contests. Herein, we synthesize information from prior studies that present interactions that may be confounded with actual agonistic contests. We also point potential problems when different types of confoundment occur and provide suggestions of how to distinguish between agonistic contests and alternative interactions. For this, we made a distinction between completely non-agonistic interactions and quasi-agonistic interactions (i.e., interactions in which at least one rival is not motivated to fight). We also show potential biases in the understanding of how rivals decide who is the winner of a contest for studies that consider non- or quasi-agonistic interactions as actual agonistic contests.