1 Introduction
The allocation of scarce resources among interested agents is a problem that arises frequently and plays a major role in our society. We often want to ensure that the allocation that we select is fair to the agents—the literature of fair division, which dates back to the design of cakecutting algorithms over half a century ago [Steinhaus1948, Dubins and Spanier1961], provides several ways of defining what fair means. An issue orthogonal to fairness is efficiency, or social welfare, which refers to the total happiness of the agents. A fundamental question is therefore how much efficiency we might lose if we want our allocation to be fair.
This question was first addressed by Caragiannis et al. [2012], who introduced the price of fairness concept to capture the efficiency loss due to fairness constraints. For any fairness notion and any given resource allocation instance with additive valuations, they defined the price of fairness of the instance to be the ratio between the maximum social welfare over all allocations and the maximum social welfare over allocations that are fair according to the notion. The overall price of fairness for this notion is then defined as the largest price of fairness across all instances. Caragiannis et al. considered the classical fairness notions of envyfreeness, proportionality and equitability, and presented a series of results on the price of fairness with respect to these notions. As an example, they showed that for the allocation of indivisible goods among agents, the price of proportionality is , meaning that the efficiency of the best proportional allocation can be a linear factor away from that of the best allocation overall.
Property  Price of  Strong price of  

General  General  
Envyfreeness up to one good (EF1) 



Envyfreeness up to any good (EFX)  
Roundrobin (RR)  
Balancedness (BAL)  
Maximum Nash welfare (MNW) 



Maximum egalitarian welfare (MEW)  for  
Leximin (LEX)  
Pareto optimality (PO) 
Caragiannis et al.’s work sheds light on the tradeoff between efficiency and fairness in the allocation of both divisible and indivisible resources. However, a significant limitation of their study is that while an allocation satisfying each of the three fairness notions always exists when goods are divisible, this is not the case for indivisible goods. Indeed, none of the notions can be satisfied in the simple instance with at least two agents and a single good to be allocated. Caragiannis et al. circumvented this issue by ignoring instances in which the fairness notion in question cannot be satisfied. As a result, their price of fairness analysis, which is meant to capture the worstcase efficiency loss, fails to cover certain scenarios that may arise in practice.^{1}^{1}1From the above example, one may think that such scenarios are rare exceptions. However, for envyfreeness, these scenarios are in fact common if the number of goods is not too large compared to the number of agents [Dickerson et al.2014, Manurangsi and Suksompong2019]. In addition, the fact that certain instances are not taken into account in the price of fairness have seemingly contradictory consequences. For example, since envyfree allocations are always proportional when valuations are additive, it may appear at first glance that the price of envyfreeness must be at least as high as the price of proportionality. This is not necessarily the case, however, because there are instances that admit proportional but no envyfree allocations.^{2}^{2}2Indeed, the instance that Caragiannis et al. used to show that the price of proportionality is at least admits no envyfree allocation. Thus, it is still possible that the price of envyfreeness is lower than the price of proportionality.
To address these limitations, in this paper we study the price of fairness for indivisible goods with respect to fairness notions that can be satisfied in every instance. Among other notions, we consider envyfreeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin.^{3}^{3}3See Section 2 for the definitions of these notions. In addition to deriving bounds on the price of fairness for these notions, we also introduce the concept of strong price of fairness, which captures the efficiency loss in the worst fair allocation as opposed to that in the best fair allocation. The relationship between the price of fairness and the strong price of fairness is akin to that between the price of stability and the price of anarchy for equilibria. While the strong price of fairness is too demanding to yield any nontrivial guarantee for some fairness notions, as we will see, it does provide meaningful guarantees for other notions.
1.1 Our Results
The majority of our results can be found in Table 1; we highlight a subset of these next. For the price of EF1, we provide a lower bound of and an upper bound of . We then show that two common ways to obtain an EF1 allocation—the roundrobin algorithm and MNW—have a price of fairness of linear order (for roundrobin the price is exactly ), implying that these methods cannot be used to improve the upper bound for EF1. We also show that improving this upper bound would yield a corresponding improvement on the price of envyfreeness gap for divisible goods left open by CaragiannisKaKa12 CaragiannisKaKa12. On the other hand, if we allow dependence on the number of goods , the price of EF1 is —this means that the lower bound is almost tight unless the number of goods is huge compared to the number of agents. For MNW, maximum egalitarian welfare (MEW), and leximin, we prove an asymptotically tight bound of on the price of fairness. Moreover, with the exception of EF1 and MNW, we establish exactly tight bounds in the case of two agents for all fairness notions.
On the strong price of fairness front, we show via a simple instance that the strong price of EF1 and balancedness are infinite, meaning that there are arbitrarily bad EF1 and balanced allocations. Nevertheless, a roundrobin allocation, which satisfies these two properties, always has welfare within a factor of the optimal allocation, and this factor is exactly tight. For MNW and leximin, the strong price of fairness, like the price of fairness, is of linear order. However, while the price of MEW is also , the strong price of MEW is infinite for (and for ). Finally, we consider Pareto optimality, for which the price of fairness is trivially . We show that the strong price of Pareto optimality is .
1.2 Related Work
The price of fairness was introduced independently by BertsimasFaTr11 BertsimasFaTr11 and CaragiannisKaKa12 CaragiannisKaKa12. Bertsimas et al. studied the concept for divisible goods with respect to fairness notions such as proportional fairness and maxmin fairness. Caragiannis et al. presented a number of bounds for both goods and chores (i.e., items that yield negative utility), both when these items are divisible and indivisible. The price of fairness has subsequently been examined in several other settings, including for contiguous allocations of divisible goods [Aumann and Dombb2015], indivisible goods [Suksompong2019], and divisible chores [Heydrich and van Stee2015], as well as in the context of machine scheduling [Bilò et al.2016].
Typically, the price of fairness study focuses on quantifying the efficiency loss solely in terms of the number of agents. A notable exception to this is the work of Kurz14 Kurz14, who remarked that certain constructions used to establish worstcase bounds for indivisible goods require a large number of goods. As a result, Kurz investigated the dependence of the price of fairness on both the number of agents and the number of goods, and found that the price indeed improves significantly if we limit the number of goods.
2 Preliminaries
Denote by the set of agents and the set of goods. Each agent has a nonnegative utility for each good . The agents’ utilities are additive, meaning that for every agent and subset of goods . Following CaragiannisKaKa12 CaragiannisKaKa12, we normalize the utilities across agents by assuming that for all . We refer to a setting with agents, goods, and utility functions as an instance. An allocation is a partition of into bundles such that agent receives bundle . The (utilitarian) social welfare of an allocation is defined as . The optimal social welfare for an instance , denoted by , is the maximum social welfare over all allocations for this instance.
A property is a function that maps every instance to a (possibly empty) set of allocations . Every allocation in is said to satisfy property .
We are now ready to define the price of fairness concepts.
Definition 2.1.
For any given property of allocations and any instance, we define the price of P for that instance to be the ratio between the optimal social welfare and the maximum social welfare over allocations satisfying :
The overall price of is then defined as the supremum price of fairness across all instances.
Similarly, the strong price of for a given instance is the ratio between the optimal social welfare and the minimum social welfare over allocations satisfying :
The overall strong price of is then defined as the supremum price of fairness across all instances.
We will only consider properties such that is nonempty for every instance , so the (strong) price of fairness is always welldefined. With the exception of Theorem 3.7, we will be interested in the price of fairness as a function of , and assume that can be arbitrary.
Next, we define the fairness properties that we consider. The first two properties are relaxations of the classical envyfreeness notion.
Definition 2.2 (Ef1).
An allocation is said to satisfy envyfreeness up to one good (EF1) if for every pair of agents , there exists a set with such that .
Definition 2.3 (Efx).
An allocation is said to satisfy envyfreeness up to any good (EFX) if for every pair of agents and every good , we have .
It is clear that EFX imposes a stronger requirement than EF1. An EF1 allocation always exists [Lipton et al.2004], while for EFX the existence question is still unresolved [Caragiannis et al.2016]. As such, we will only consider EFX in the case of two agents, for which existence is guaranteed [Plaut and Roughgarden2018].
The roundrobin algorithm, which we describe below, always computes an EF1 allocation (see, e.g., [Caragiannis et al.2016]).
Definition 2.4 (Rr).
The roundrobin algorithm works by arranging the agents in some arbitrary order, and letting the next agent in the order choose her favorite good from the remaining goods.^{4}^{4}4In case there are ties between goods, we may assume worstcase tie breaking, since it is possible to obtain an instance with infinitesimal difference in welfare and any desired tiebreaking between goods by slightly perturbing the utilities. An allocation is said to satisfy roundrobin (RR) if it is the result of applying the algorithm with some ordering of the agents.
Our next property is balancedness, which means that the goods are as spread out among the agents as possible. Balancedness and similar cardinality constraints have been considered in recent work [Biswas and Barman2018]. In addition to satisfying EF1, an allocation produced by the roundrobin algorithm is also balanced.
Definition 2.5 (Bal).
An allocation is said to be balanced (BAL) if for any .
Next, we define a number of welfare maximizers.
Definition 2.6 (Mnw).
The Nash welfare of an allocation is defined as . An allocation is said to be a maximum Nash welfare (MNW) allocation if it has the maximum Nash welfare among all allocations.^{5}^{5}5In the case where the maximum Nash welfare is 0, an allocation is a MNW allocation if it gives positive utility to a set of agents of maximal size and moreover maximizes the product of utilities of the agents in that set.
Definition 2.7 (Mew).
The egalitarian welfare of an allocation is defined as . An allocation is said to be a maximum egalitarian welfare (MEW) allocation if it has the maximum egalitarian welfare among all allocations.
Definition 2.8 (Lex).
An allocation is said to be leximin (LEX) if it maximizes the lowest utility (i.e., the egalitarian welfare), and, among all such allocations, maximizes the second lowest utility, and so on.
Finally, we define Pareto optimality. While this is an efficiency notion rather than a fairness notion, we also consider it as it is a fundamental property in the context of resource allocation.
Definition 2.9 (Po).
Given an allocation , another allocation is said to be a Pareto improvement if for all with at least one strict inequality. An allocation is Pareto optimal (PO) if it does not admit a Pareto improvement.
CaragiannisKuMo16 CaragiannisKuMo16 showed that a MNW allocation always satisfies EF1 and Pareto optimality. It is clear from the definition that any leximin allocation is Pareto optimal and maximizes egalitarian welfare. The problem of computing a MEW allocation has been studied by BezakovaDa05 BezakovaDa05 and BansalSr06 BansalSr06. Leximin allocations were studied by BogomolnaiaMo04 BogomolnaiaMo04 and shown to be applicable in practice by KurokawaPrSh15 KurokawaPrSh15.
3 EnvyFreeness
In this section, we consider envyfreeness relaxations and the roundrobin algorithm, which always produces an EF1 allocation. We begin with a lower bound on the price of EF1.
Theorem 3.1.
The price of EF1 is .
Proof.
Let , and assume that the utilities are as follows:

For : for , and otherwise.

for , and otherwise.

For : for all .
Consider the allocation that assigns goods to agent for and the remaining goods to agent . The social welfare of this allocation is . On the other hand, in any EF1 allocation, each of the agents must receive at least one good—otherwise some agent would receive at least two goods and agent would envy her. This means that the social welfare is at most . Hence the price of EF1 is at least . ∎
For two agents, we establish an almost tight bound on the price of EF1 and a tight bound on the price of EFX.
Theorem 3.2.
For , the price of EF1 is at least and at most .
Proof.
Lower bound: Let and , and assume that the utilities are as follows:
The optimal social welfare is , achieved by assigning the first good to agent 1 and the last two goods to agent 2. However, in any EF1 allocation the last two goods cannot both be given to agent 2. Hence the social welfare of an EF1 allocation is at most . Taking , we find that the price of EF1 is at least .
Upper bound: Consider an arbitrary instance. Sort the goods so that ; goods such that are put at the front and those with at the back, with arbitrary tiebreaking within each group of goods. (Goods that yield zero value to both agents can be safely ignored since they have no effect on the optimal welfare or the maximum welfare of an EF1 allocation.) For ease of notation, for any we write and . We also define .
Let for some and . It is easy to see that . If , both agents have identical valuations and the price of EF1 is , so we may assume that . The allocation is an optimal allocation, and the optimal social welfare is . Without loss of generality, assume that . Note that we must have , since otherwise both and are smaller than and switching and would yield a higher social welfare. We can further assume that , because otherwise is also an EF1 allocation and the price of fairness is .
Next, we describe how to obtain a particular EF1 allocation . Let be the smallest index such that and . Clearly, . In the allocation , we assign the goods to agent 1, and to agent 2.
Allocation satisfies EF1.
The EF1 condition is satisfied for agent 1, because by definition.
For agent 2, since is the smallest index such that and , we have either or .
If , then coincides with the optimal allocation , and . Clearly EF1 is satisfied.
Else, , and we have . Note also that . Therefore,
where we take a fraction to be infinite if it has denominator . (None of the fractions can have both numerator and denominator .) This implies that
Thus,
implying that EF1 is again satisfied.
The price of EF1 for this instance is at most .
Now we analyze the social welfare of the allocation and compare it to the optimal social welfare.
If , the price of EF1 is . Assume from now on that . We have and . Since , we also have . Moreover, . Thus,
Therefore the ratio between the optimal social welfare and the social welfare of is
We further analyze the last expression. First, taking its partial derivative with respect to gives
which is always positive when . This shows that the last expression is monotone increasing in . Thus
Finally, this expression is maximized when and yields a value of , completing the proof. ∎
Theorem 3.3.
For , the price of EFX is .
Proof.
Lower bound: Let and , and assume that the utilities are as follows:

.

.
The optimal social welfare is , achieved by assigning the first two goods to agent 1 and the last good to agent 2. On the other hand, in any EFX allocation, no agent can get both of the goods that they positively value. Hence, the social welfare of an EFX allocation is at most . Taking , we find that the price of EFX is at least .
Upper bound: Consider an arbitrary instance. If in an optimal allocation both agents get utility at least , this allocation is also envyfree and hence EFX, so the price of EFX is . Otherwise, the maximum welfare is at most . Now we show that there always exists an EFX allocation with social welfare at least ; this immediately yields the desired bound.
Let the first agent partition the goods into two bundles such that her values for the bundles are as equal as possible. Denote by and the values of the two bundles, where . Suppose that all goods of zero value, if any, are in the second bundle. Let be the corresponding values for the second agent, and assume without loss of generality that . Consider the partition of the first agent, and assume that the two bundles yield value and to the second agent, respectively. If , by assigning the first bundle to the first agent and the second bundle to the second agent, we have an envyfree allocation with welfare at least . Else, . By definition of , we also have . We assign the first bundle to the second agent and the second bundle to the first agent. The second agent is clearly envyfree. If the first agent still has envy after removing some good from the first bundle, then by moving good to the second bundle, we create a more equal partition, a contradiction. Hence the allocation is EFX to the first agent. The social welfare of this allocation is . ∎
Next, we give a simple instance showing that EF1 and EFX allocations can have arbitrarily bad welfare.
Theorem 3.4.
The strong price of EF1 is . For , the strong price of EFX is .
Proof.
Let , and assume that for all and otherwise. The allocation that assigns good to agent for every has social welfare . On the other hand, the allocation that assigns good to agent for and good to agent is EF1 and EFX, but has social welfare . The conclusion follows. ∎
We now turn our attention to the roundrobin algorithm. We show that it is always possible to order the agents to obtain a welfare of .
Lemma 3.5.
For any instance, there exists an ordering of the agents such that the roundrobin algorithm implemented with this ordering produces an allocation with social welfare at least , and this bound is tight.
Proof.
We claim that if we choose the ordering of the agents uniformly at random, the expected social welfare is at least . The desired bound immediately follows from this claim.
To prove the claim, consider an arbitrary agent , and assume without loss of generality that . Note that if the agent is ranked th in the ordering, her utility is at least , where . Hence, the agent’s expected utility is at least
It follows from linearity of expectation that the expected social welfare is at least , as claimed.
The tightness of the bound follows from the instance where every agent has utility for the same good. ∎
Lemma 3.5 yields a linear price of fairness for roundrobin.
Theorem 3.6.
The price of roundrobin is . Consequently, the price of EF1 is at most .
Proof.
Upper bound: Consider an arbitrary instance. Since every agent receives utility at most , the optimal social welfare is at most . On the other hand, by Lemma 3.5, there exists an ordering of the agents such that the roundrobin algorithm yields welfare at least . Hence the price of roundrobin is at most .
Lower bound: Let for some large that is divisible by , and assume that the utilities are such that for each agent , for and otherwise.
Consider the allocation that assigns goods to agent 1, and to agent for every . In this allocation, agent 1 gets utility 1, while each remaining agent gets utility . The social welfare is therefore . This converges to for large .
On the other hand, consider the roundrobin algorithm with an arbitrary ordering of the agents, and assume without loss of generality that agents always break ties in favor of goods with lower numbers. Hence, regardless of the ordering, the goods get chosen in the order . As a result, every agent gets exactly of their valued goods, so her utility is , and the social welfare is . Hence the price of roundrobin is . ∎
The argument for the lower bound in Theorem 3.6 works even if we can choose a new ordering of the agents in every round. This means that the fixed order is not a barrier to obtaining a better price of fairness, but rather the “each agent picks exactly once in every round” aspect of the algorithm.
One may notice that the lower bound construction uses an exponential number of goods. This is in fact necessary to obtain an instance with a high price of roundrobin. As we show next, the lower bound on the price of EF1 is almost tight as long as is not too large compared to .
Theorem 3.7.
The price of roundrobin is . Consequently, the price of EF1 is .
Proof.
Consider any instance . We claim that there exists an ordering for which the roundrobin algorithm produces an allocation with social welfare at least . First, observe that if , then Lemma 3.5 immediately yields the desired claim. Henceforth, we will only focus on the case where .
Fix an optimal allocation , and let . For each , let us partition into , where is defined by
Furthermore, define and .
Let . We have
However, since agent values each item in at most , we have . This implies that , which is no more than . Hence,
(1) 
Thus, it suffices to show the existence of an ordering such that roundrobin produces an allocation with social welfare at least .
Observe that (1) implies that . We now consider two cases, based on . Since for each , we have .
Case 1: . In this case, we will show that the roundrobin algorithm with arbitrary ordering yields an allocation with social welfare at least .
To see this, let us consider the roundrobin procedure with arbitrary ordering, and consider the set of goods that are picked in the first rounds; let denote this set. Now, observe that
This implies that
Since , there must be more than agents such that . Let denote the set of such agents.
We claim that, in each of the first rounds, every agent must receive an item she values at least . The reason is that agent picks her favorite good, which she must value at least as much as the good(s) left unpicked in . Moreover, she values the latter at least , so this must also be a lower bound of her utility for the former.
From the claim in the previous paragraph, we can conclude that the social welfare of the allocation produced is at least
as desired. Note that we use the assumption to conclude that in the first inequality above.
Case 2: . In this case, we will show that if we choose the ordering in a careful manner, then the social welfare obtained in the first round alone already suffices.
Similarly to Case 1, observe that since , there are more than agents whose is nonempty. Let denote the set of such agents.
We will construct the ordering stepbystep as follows. For , we let be any agent such that (1) is not yet in the ordering and (2) not all goods in are already picked by . Note that such an agent exists because, at each step , at most two candidate agents become invalid: the agent , and the agent whose good in is picked by . Since we start with valid candidates, even after steps, there are still valid candidate agents to be chosen from.
The remainder of the ordering can be chosen arbitrarily. We now argue that the resulting roundrobin allocation has the desired social welfare. To see this, for , observe that agent must pick a good that is worth at least to her in the first round, since not all goods in have been picked. As a result, the social welfare is at least
where the first inequality follows from . ∎
While Theorem 3.7 shows that the price of EF1 is close to unless the number of goods is huge, if we are only interested in the dependence on the number of agents, the gap still remains between and . In fact, CaragiannisKaKa12 CaragiannisKaKa12 left exactly the same gap on the price of envyfreeness for divisible goods. In Section 7, we exhibit an interesting connection between the indivisible and divisible goods settings by showing that the price of EF1 for indivisible goods is always at least the price of envyfreeness for divisible goods. This implies that improving the upper bound on the price of EF1 would also yield a corresponding improvement on the price of envyfreeness.
We end this section by establishing an exact bound on the strong price of roundrobin.
Theorem 3.8.
The strong price of roundrobin is .
Proof.
Upper bound: Consider an arbitrary instance. Since every agent receives utility at most , the optimal social welfare is at most . On the other hand, in the roundrobin algorithm, the first agent gets to choose an item ahead of all other agents in every round and therefore does not envy any other agent in the resulting allocation. This implies that her utility, and hence the social welfare, is at least . It follows that the strong price of roundrobin is at most .
Lower bound: Let be a large number divisible by , and assume that the utilities are as follows:

for all .

For : , and otherwise.
Consider the allocation that assigns good to agent for every , and the remaining goods to agent 1. In this allocation, every agent receives utility . Agent 1 receives utility , which converges to for large . Therefore the social welfare converges to .
On the other hand, consider the roundrobin algorithm with the ordering of the agents , and assume without loss of generality that agents always break ties in favor of goods with lower numbers. The first agent gets utility exactly , while the remaining agents get zero utility since their only valuable good is “stolen” by the agent before them in the first round. Hence the social welfare is . This means that the strong price of roundrobin is , as desired. ∎
4 Balancedness
In this section, we consider balancedness. We begin by establishing an asymptotically tight bound on the price of balancedness.
Theorem 4.1.
The price of balancedness is .
Proof.
Lower bound: Consider the instance in Theorem 3.1. The social welfare can be as high as , while a similar argument shows that the social welfare of any balanced allocation is at most . The conclusion follows.
Upper bound: We claim that for any instance , the maximum social welfare of a balanced allocation is always within a factor of the optimal social welfare; this claim implies the desired upper bound. If , the claim follows immediately from Lemma 3.5. We therefore assume that . We will show that there is a balanced allocation such that ; this suffices for our claim because . We consider two cases.
Case 1: . Fix an optimal allocation, and let be the set of agents who receive at least goods in the optimal allocation, and the complement set of agents. Since there are at most agents in , they contribute at most to , so the agents in contribute at least . We let each agent in keep her most valuable goods (or all of her goods, if she has fewer than this number of goods). This yields a total utility of at least . Since due to the assumption , the remaining goods can be reallocated to obtain a balanced allocation, which has social welfare at least .
Case 2: . Fix an optimal allocation, and let be the set of agents who receive at least goods in the optimal allocation, and the complement set of agents. Since there are at most agents in , they contribute at most to , so the agents in contribute at least . We let each agent in keep her most valuable good (if she receives at least one good). This yields a total utility of at least . The remaining goods can be reallocated to obtain a balanced allocation, which has social welfare at least . ∎
For two agents, we give an exact bound on the welfare that can be lost due to imposing balancedness.
Theorem 4.2.
For , the price of balancedness is .
Proof.
Lower bound: Let be a large even number, and assume that the utilities are as follows:

and otherwise.

for all .
Consider the allocation that assigns the first good to the first agent and the remaining goods to the second agent. The social welfare is , which converges to for large . On the other hand, in any balanced allocation, the first agent gets utility at most while the second agent gets utility , so the social welfare is at most . Hence the price of balancedness is at least .
Upper bound: Consider an arbitrary instance. If
is odd, we may add a dummy good that yields zero utility to both agents—this does not change the optimal social welfare or the maximum social welfare of a balanced allocation. We may therefore assume that
is even.Sort the goods so that . Let be the last good such that , and assume without loss of generality that . An optimal allocation assigns the set of goods to the first agent and the complement set to the second agent, yielding social welfare , where . On the other hand, consider the balanced allocation that assigns goods to the first agent and the remaining goods to the second agent. Note that at most half of the goods in are reallocated to the second agent, and these are the goods with the lowest difference in utility between the two agents. Hence, the utility loss going from the first to the second allocation is at most , implying that the social welfare of the second allocation is at least . The price of balancedness is therefore at most
This ratio is increasing in and reaches the maximum at , where its value is , completing the proof. ∎
Finally, the same construction as in Theorem 3.4 shows that balanced allocations can have arbitrarily bad welfare.
Theorem 4.3.
The strong price of balancedness is .
5 Welfare Maximizers
In this section, we consider allocations that maximize different measures of welfare. To start with, we show that every MNW and leximin allocation yields a decent welfare.
Lemma 5.1.
For any instance, every MNW allocation and every leximin allocation has social welfare at least , and both bounds are tight.
Proof.
We first establish the bound for MNW. Consider any MNW allocation where agent receives bundle , and assume for contradiction that . Fix any agent . Since , there exists such that . Construct a directed graph with vertices , and add an edge from to if . Since every vertex has at least one outgoing edge, the graph consists of a directed cycle. For every edge in the cycle, we give to agent instead of agent . If we consider the change in the multiset of the utilities between the old and new allocations, at least one number increases while others remain the same. This means that either we have decreased the number of agents who get zero utility, or keep this number fixed and increase the product of utilities of the agents who get nonzero utility. Either case contradicts the definition of an MNW allocation.
To show the bound for leximin, we apply the same argument. An improvement in the multiset of utilities as described in the last step contradicts the definition of leximin.
Finally, the tightness of the bounds follows from the instance where every agent has utility for the same good. ∎
Lemma 5.1 allows us to show that the price of MNW and the strong price of MNW are both of linear order. Similar techniques can be used for the price of MEW and both prices of leximin, as we establish in the two subsequent theorems.
Theorem 5.2.
The price of MNW and the strong price of MNW are .
Proof.
It suffices to show that the price of MNW is and the strong price of MNW is .
Lower bound: Let and , and assume that the utilities are as follows:

and otherwise.

For : , , and otherwise.
Consider the allocation that assigns good to agent for , and good to agent 1. The social welfare of this allocation is . On the other hand, the unique MNW allocation assigns good to agent for every . The social welfare of this allocation is . Taking , we find that the price of MNW is .
Upper bound: Consider an arbitrary instance. Since every agent receives utility at most , the optimal social welfare is at most . On the other hand, by Lemma 5.1, the social welfare of any MNW allocation is at least . The conclusion follows. ∎
Theorem 5.3.
The price of MEW is .
Proof.
Lower bound: Consider the instance in Theorem 5.2. For , the social welfare of the optimal allocation approaches , while the social welfare of the unique MEW allocation approaches .
Upper bound: First, we claim that for any instance, there exists a MEW allocation with social welfare at least . To prove this claim, we apply the same argument as in Lemma 5.1, but starting with a MEW allocation with maximum social welfare. An improvement in the multiset of utilities as described in the argument does not decrease the egalitarian welfare and strictly increases the social welfare, which gives us the desired contradiction.
Combined with the observation that the optimal social welfare is at most in any instance, this claim immediately yields the desired upper bound. ∎
Theorem 5.4.
The price of leximin and the strong price of leximin are .
Proof.
Since all leximin allocations have the same social welfare for any given instance, it suffices to show the statement for the price of leximin.
Lower bound: Consider the instance in Theorem 5.2. For , the social welfare of the optimal allocation approaches , while the social welfare of the unique leximin allocation approaches .
Upper bound: Consider an arbitrary instance. Since every agent receives utility at most , the optimal social welfare is at most . On the other hand, by Lemma 5.1, the social welfare of any leximin allocation is at least . The conclusion follows. ∎
Surprisingly, MEW allocations can be arbitrarily bad when there are at least three agents.
Theorem 5.5.
For , the strong price of MEW is infinite.
Proof.
Let , and assume that the utilities are as follows:

and otherwise.

For : and otherwise.
Observe that in any allocation, some agent does not get a desired good. This means that every allocation has egalitarian welfare 0, and all allocations are MEW. Now, there exists an allocation with social welfare 0, for example the allocation that assigns good to agent for , and assigns good 1 to agent . Since there also exists an allocation with positive social welfare, the strong price of MEW is infinite. ∎
We now turn to the case of two agents. For MNW, we establish almost tight bounds on both prices of fairness.
Theorem 5.6.
For , the price of MNW and the strong price of MNW are at least and at most .
Proof.
It suffices to show that the price of MNW is at least and the strong price of MNW is at most .
Lower bound: Let and , and assume that the utilities are as follows:

, , .

, , .
The optimal social welfare is , obtained by assigning the first two goods to the first agent and the last good to the second agent. On the other hand, one can check that the maximum Nash welfare is , obtained (uniquely) by assigning the first good to the first agent and the last two goods to the second agent. This allocation yields social welfare . Taking , we find that the price of MNW is at least .
Upper bound: Consider an arbritrary instance. Suppose that the optimal social welfare is . If , then Lemma 5.1 immediately implies that the price of MNW of this instance is at most .
We now focus on the case where . Let us assume further that, in an optimal allocation, the first agent has utility and the second has utility , where and . Since , we have .
Next, consider any MNW allocation. Suppose that in this allocation the first agent has utility and the second has utility . Since the Nash welfare of this allocation must be at least that of the optimal allocation, we have . As a result, the social welfare of this allocation is , where the first inequality follows from . Thus, the price of MNW of this instance is at most
where the inequality follows from . ∎
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