1 Introduction
A fundamental problem that often arises in several settings is that of allocating items, resources, or tasks in a fair manner. There are several notions of fairness that have been considered in the literature. Among them, two of the strongest ones are envyfreeess (no agent should envy another agent’s outcome) and equitability (every agent should get the same utility). When monetary transfers are not allowed, there may not exist any outcome that is envyfree or equitable. This leads to the question: under what conditions fairness can be achieved via monetary transfers?
Contributions
We consider the situation where we use monetary transfers to achieve both properties simultaneously. Our results include the following.
We first present a sufficient condition for allocations that can lead to equitability and envyfreeness by monetary transfers. In contrast to most of the related results that focus on additive or positive valuations, the statement holds for any superadditive valuations whether they are positive or negative. It also leads to a natural and simple algorithm to achieve envyfreeness and equitability. The result holds if we replace the payment balance conditions with the condition that agents get subsidies.
For the domain of additive valuations, we provide a complete characterization of allocations that can lead to equitability and envyfreeness by monetary transfers.
We use our insights to design a polynomialtime distributed algorithm that finds an allocation and payment such that the new allocation achieves as much social welfare as a given allocation and the outcome satisfies envyfreeness, equitability, and payment balance. Finally, we discuss issues around computation and bounds for minimal payments to achieve fairness.
2 Related Work
In the fair division literature (see, e.g. (Aziz, 2020; Bouveret et al., 2016; Brams and Taylor, 1996)), envyfreeness (Foley, 1967) and equitability (Freeman et al., 2019; Robertson and Webb, 1997) are wellknown fairness properties. When the items are divisible goods, an equitable and envyfree allocation is guaranteed to exist (Alon, 1987). On the other hand, when considering indivisible goods, neither of the two properties are guaranteed to be achievable.
In this paper, we consider achieving these properties with the help of monetary transfers. Fair allocation with money is wellestablished, especially in the context of roomrent division. A feature of most of the work in the area is that each agent has demand for exactly one item (room)(Aragones, 1995; Klijn, 2000; Maskin, 1987; Su, 1999; Svensson, 1983). More general models where envyfreeness is achieved via sidepayments have been considered by Haake et al. (2002) and Meertens et al. (2002). Chevaleyre et al. (2017) consider the distributed allocation of goods and focussed on convergence to envyfree and efficient outcomes via trades among agents.
More recently, there has been focus on computing envyfree allocations when agents have demands for multiple items and monetary transfers are allowed (Haake et al., 2002). In particular, Halpern and Shah (2019) popularized the problem of finding allocations for which minimal subsidies will result in envyfreeness.^{1}^{1}1The problem of computing minimal subsidies to achieve fairness can be viewed under the framework of “control of fair division” (Aziz et al., 2016) whereby fairness is achieved by minimal modification to the original problem. In followup work, the computational of minimal subsidies has been considered in further depth both from the perspectives of exactly minimal subsidies (Brustle et al., 2019) and approximately minimal subsidies (Caragiannis and Ioannidis, 2020). In our model, the valuations can be positive or negative and we additionally target equitability. In particular, we use a simple formula for the payment given to each agent.
3 Setup
We consider the setting in which there is a set of agents and a set of tasks . Each agent has a valuation function . The function specifies a value for a given bundle . The value can be positive or negative. We assume that for all .
The valuation function of an agent is supermodular if for each , and , . The valuation function of an agent is additive if for each , and such that , the following holds: . The valuation function of an agent is superadditive if for each , and such that , the following holds: . Note that supermodularity and additivity are stronger conditions than superaddivity. We assume that valuations satisfy the weaker notion of superaddivity.
An allocation is a partitioning of the tasks into bundles where is the bundle allocated to agent . For an allocation , the social welfare is .
An outcome is a pair consisting of the allocation and the payments made by the agents. Formally, an outcome is a pair where is the allocation that specifies bundle for agent and specifies the payment made by agent . If is negative, it means agent gets money. We say that is balanced if .
An agent ’s utility for a bundlepayment pair is . In other words, we assume quasilinear utilities. An outcome is envyfree if for all , it holds that . An outcome is equitable if for all , . An allocation is envyfreeable if there exists a payment function such that is envyfree. An allocation is equitableconvertible if there exists a payment function such that is equitable. An allocation is EFEQconvertible if there exists a payment function such that is both equitable and envyfree.
For any given allocation , the corresponding envygraph is a complete directed graph with vertex set . For any pair of agents the weight of arc is the envy agent has for agent under the allocation : . For any path or cycle in the graph, the weight of the is the sum of weights of arcs along .
4 Sufficient and necessary conditions to achieve fairness
We note that every allocation is trivially equitableconvertible: each agent can be given money so that their utility is equal to . On the other hand, not every allocation is envyfreeable or EFEQconvertible.
We say that an allocation is reassignmentstable, if it maximizes the social welfare across all reassignments of its bundles to agents. Halpern and Shah (2019) assumed positive additive utilities and presented the following elegant characterization of envyfreeable allocations.
Theorem 1.
Under positive additive utilities, the following conditions are equivalent for a given allocation:

the allocation is envyfreeable

the allocation is reassignmentstable

for the allocation, there is no positive weight cycle in the corresponding envygraph
The equivalence between the first two conditions has been proved previously (see e.g., Haake et al. (2002) and Mu’alem (2009)). Reassignment stability was referred to as localefficiency by Mu’alem (2009).
We explore the conditions under which an allocation is EFEQconvertible. Firstly, we show that even for positive additive utility, reassignmentstability is not sufficient to simultaneously achieve envyfreeness and equitability via payments.
Example 1.
Even for positive additive valuations and a given envyfreeable allocation, there may not exist any payments to the agents to achieve both envyfreeness and equitability. Consider an instance with the following additive utilities. We consider an allocation indicated with the squares in which agent 1 gets and 2 gets .
200  100  
2  1 
The allocation is envyfreeable because it is reassignmentstable. We show that there exist no payments to achieve both envyfreeness and equitability simultaneously. Without loss of generality suppose that agents are paid money. The minimum amount needed to obtain equitability is to pay 199 to agent 2. We can maintain equitability by giving equal amounts of money to both the agents. Note however, that the outcome will continue having envy. Agent 1 envies agent 2: ∎
The example above shows that reassignmentstability is not sufficient to achieve envyfreeness and equitability. Reassignmentstability was the key technique used by Halpern and Shah (2019) and Brustle et al. (2019) in their algorithmic results to achieve envyfreeness. In our quest to achieve both envyfreeness and equitability via monetary transfers, we focus on allocations that are transferstable. We say that an allocation is transferstable if there exist no such that . We note that under additive valuations, transferstability is stronger than the reassignmentstability property.
Lemma 1.
Under additive valuations, if an allocation is transferstable, then it is reassignmentstable.
Proof.
Suppose there exists a reassignment which increases total welfare. This means that the movement of at least one bundle to some agent increases the social welfare, which implies that the allocation is not transferstable. ∎
Since transferstability is a stronger property than reassignmentstability, a natural question is whether it can be used to achieve stronger fairness guarantees. We answer the question in the affirmative in the following lemma. The lemma applies to the class of superadditive valuations.
Lemma 2.
For a transferstable allocation , suppose each agent makes a payment equal to . Then if agent valuations are superadditive, the outcome is envyfree and equitable.
Proof.
We first want to prove envyfreeness: for all , it holds that . By transferstability of allocation ,
Since is superadditive, it follows that
By combining the two inequalities above, we get
The last inequality indicates that agent is not envious of and hence satisfies envyfreeness.
Next, we argue that the outcome satisfies equitability. Each agent gets utility . Since each agent has the same utility , the outcome satisfies equitability. ∎
The payment function used in the lemma is not new. It is referred to as the Knaster payments (Knaster, 1946) and is inspired by the idea that each agent should get utility that is at least the proportionality guarantee that was popularized by Steinhaus (1948). In the literature on fair allocation with money, Knaster payments have typically been applied on welfare maximizing allocations. Raith (2000) discusses them prominently in the context of 2 agents and additive valuations. We show that it is sufficient to consider superadditive valuations and transferstable allocations for Knaster payments to achieve both equitability and envyfreeness.
Our insights also show that any social welfare maximizing allocation is EFEQconvertible.
Corollary 1.
For superadditive utilities, a social welfare maximizing allocation is EFEQconvertible.
Proof.
A social welfare maximizing allocation is transferstable. By Lemma 2, it is EFEQconvertible. ∎
In Lemma 2, we have shown that for (super)additive valuations, transferstability is a sufficient condition to simultaneously achieve equitability and envyfreeness via payments. Next, we show that transferstability is also a necessary condition.
Lemma 3.
Under additive utilities, if an allocation is EFEQconvertible, then it is transferstable.
Proof.
Suppose an allocation is not transferstable. Then there exist agents such that
The inequality is depicted in Figure 1.
If is not envyfreeable, we are done so we assume that is envyfreeable. Then it must be that
or we can swap the allocations of and to get a welfare improvement which means that is not envyfreeable which implies that it is not EFEQconvertible. The case is depicted in Figure 2.
By the characterization result of Halpern and Shah (2019), we know that does not admit an envycycle. Therefore, either
or
We first consider the case which is depicted in Figure 3.
Since , it follows that
Since gets a strictly higher value than from her allocation, we need to pay money to agent to ensure equitability. In particular, agent is paid amount . In that case agent
’s estimation of agent
’s outcome is where we know that . Therefore agent is envious of agent . Hence is not EFEQconvertible.We distinguish between two cases (a) and (b) .
We already considered case (a) in the previous analysis (Figure 3). Therefore, we now consider case (b) and assume that which is depicted in Figure 5.
We distinguish between two further final cases: case and the case .

which is depicted in Figure 6.
Since , agent is envious of agent and needs money to remove the envy. On other hand, we know that so agent needs more money to achieve equitability. Both the properties cannot be met.

which is depicted in Figure 7.
Since , agent is envious of agent and needs money to remove the envy. The exact amount needed to remove envy is . But then the new utility of agent is which we know (see Figure 7) is more than so equitability is violated.
We have proved that in all the cases, if an allocation is not transferstable, then it is not EFEQconvertible. ∎
We obtain the following result: transferstability characterizes EFEQconvertible allocations.
Theorem 2.
Under additive utilities, an allocation is EFEQconvertible if and only if it is transferstable.
Corollary 2.
Assuming that we have access to an oracle that gives the utility of an agent for a bundle in constant time, then for additive valuations, there exists a algorithm to check whether a given allocation is EFEQconvertible.
Proof.
By Theorem 2, we need to check whether the allocation is transferstable or not. ∎
5 An algorithm to achieve fairness with payments
The following lemma shows that a greedy distributed approach can achieve a transferstable allocation.
Lemma 4.
Suppose there exists an oracle that computes the value of an agent for a bundle of tasks in time . Then, for any given allocation , a transferstable allocation can be computed in such that .
Proof.
We take any pair of agents and check if . This can be checked in time for a pair of agents and in for all pairs of agents. If , we give the allocation of to agent which results in agent getting an empty bundle. With each such operation the total social welfare increases. Hence, we the process terminates. Next, we prove that the process terminates in a polynomial number of steps.
With each operation, one of the two cases occurs. The first case is that an additional agent completely loses her bundle. When a bundle going to another agent who has a nonempty bundle, then the number of agents who have an empty bundle increases. Such operations can happen at most times. Now suppose that the the number of agents who have an empty bundle does not increase. This is only possible in the case that that agent had an empty allocation who gets the bundle . Since each transfer of a bundle is welfare improving, it cannot happen that a bundle is returned to an agent . Therefore such operations can happen at most times until the bundle will not move to any agent with an empty allocation.
Hence these operations can happen at most times until no more transfers are possible. ∎
Lemma 2 and Lemma 4 give us an easy constructive method to achieve envyfreeness and equitability. The method is presented as Algorithm 1. We use the algorithm in the proof of Lemma 4 to obtain a transferstable allocation.^{2}^{2}2In practice in several domains such as routingbased task allocation, one would expect most reasonable and balanced allocations to be transferstable. After that we use the payment function specified in Lemma 2 to achieve envyfreeness and equitability. Algorithm 1 leads to the following theorem.
Theorem 3.
Suppose agents have superadditive valuations. Then for a given allocation , an allocation and payment function can be computed in polynomial time such that

the outcome is equitable and envyfree,

, and

is balanced.
Note that our result allows for some payments to be positive, i.e., some agents need to pay money. If we insist on simply using subsidies from a third party to achieve envyfreeness, then we can find the largest payment made by an agent and give each agent an additional amount of so that agents only get money and do not need to give money. To be precise, if the balanced payment is , we can get negative or zero payments as follows: is set to . By doing this, we obtain the next theorem.
Theorem 4.
Suppose agents have superadditive valuations. Then for a given allocation , an allocation and payment function can be computed in polynomial time such that

the outcome is equitable and envyfree,

, and

for each , .
Note that if we are not given an initial allocation, then we can achieve an outcome satisfying envyfreeness and equitability in an even simpler way. We bundle all the tasks together and then give the bundle to an agent for which value is the highest. By construction, the allocation is transferstable. We then implement the payment function as specified in Algorithm 1.
6 Minimal payments to achieve fairness
When using payments to achieve fairness, one may want to use the minimal exchange of money or subsidy to achieve fairness. The problem has been explored by Halpern and Shah (2019) and Brustle et al. (2019) when the goal is envyfreeness.
Suppose a given allocation is EFEQconvertible. Then there is a lineartime algorithm to compute the minimal payments to achieve both envyfreeness and equitability. The key insight is that for EFEQconvertible allocations, it is sufficient to simpy focus on achieving equitability. Any additional and uniform payment for all agents does not affect envyfreeness. Therefore, we can give agents sufficient money to ensure that each agent has utility equal to . Next, we consider the problem in which we can choose a suitable allocation so as to require minimal payments to acheive fairness.
Theorem 5.
Computing the minimum payments to simultaneously achieve envyfreeness and equitability is strongly NPhard. Unless , there exists no deterministic polynomialtime algorithm that approximates within any given positive factor the minimum payments to simultaneosly achieve envyfreeness and equitability.
The proof follows from the fact that checking whether there exists an envyfree allocation (that requires zero payments to achieve envyfreeness) is NPcomplete if the agents have identical valuations. Under identical valuations, envyfreenss also implies equitability. The inapproximability result follows from the fact than even checking whether zero payment is required is NPhard.
In order to achieve reasonable bounds on the maximum subsidy to achieve envyfreeness, Halpern and Shah (2019) and Brustle et al. (2019) assume valuations in which for all and . Our goal is to achieve envyfreeness and equitability simultaneously. We first note that we inherit any lower bounds on subsidies required to get envyfreeness. Therefore, when an EFEQconvertible allocation is given, the minimum subsidy required is in the worst case. We also observe that at least payment may be required even when an EFEQconvertible allocation is not given and we can choose an EFEQconvertible allocation intelligently. The reason is that in order to achieve transferstability, it may be the case that all the items need to be given to the same agent.
7 Conclusions
Achieving fairness via payments is an interesting reseach direction. In this paper, we focussed on envyfreeness and equitability and presented a characterization of allocations that are EFEQconvertible allocation.
Acknowledgements
Aziz is supported by a UNSW Scientia Fellowship, and Defence Science and Technology (DST) under the project “Auctioning for distributed multi vehicle planning” (DST 9190). He thanks Ioannis Caragiannis, Alex Lam and Bo Li for helpful comments and pointers.
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