On Interim Envy-Free Allocation Lotteries
Ioannis Caragiannis, Panagiotis Kanellopoulos, Maria Kyropoulou
aa r X i v : . [ c s . G T ] F e b On Interim Envy-Free Allocation Lotteries
IOANNIS CARAGIANNIS,
Aarhus University, Denmark
PANAGIOTIS KANELLOPOULOS,
University of Essex, United Kingdom
MARIA KYROPOULOU,
University of Essex, United KingdomWith very few exceptions, recent research in fair division has mostly focused on deterministic allocations.Deviating from this trend, we study the fairness notion of interim envy-freeness (iEF) for lotteries over alloca-tions, which serves as a sweet spot between the too stringent notion of ex-post envy-freeness and the veryweak notion of ex-ante envy-freeness. iEF is a natural generalization of envy-freeness to random allocationsin the sense that a deterministic envy-free allocation is iEF (when viewed as a degenerate lottery). It is alsocertainly meaningful as it allows for a richer solution space, which includes solutions that are provably betterthan envy-freeness according to several criteria. Our analysis relates iEF to other fairness notions as well, andreveals tradeoffs between iEF and efficiency. Even though several of our results apply to general fair divisionproblems, we are particularly interested in instances with equal numbers of agents and items where alloca-tions are perfect matchings of the items to the agents. Envy-freeness can be trivially decided and (when it canbe achieved, it) implies full efficiency in this setting. Although computing iEF allocations in matching alloca-tion instances is considerably more challenging, we show how to compute them in polynomial time, whilealso maximizing several efficiency objectives. Our algorithms use the ellipsoid method for linear program-ming and efficient solutions to a novel variant of the bipartite matching problem as a separation oracle. Wealso study the extension of interim envy-freeness notion when payments to or from the agents are allowed.We present a series of results on two optimization problems, including a generalization of the classical rentdivision problem to random allocations using interim envy-freeness as the solution concept.
Plenty of situations arise in the real world every day, where assets need to be distributed amongindividuals. Making sure that everyone gets what they are entitled to is an imperative, yet vague,aspiration that is open to interpretation. Fair division is a research area that deals with problemsof distributing assets in a way that is considered fair. Fair allocation problems, that focus on indi-visible items, have received considerable attention from the EconCS community recently.Among the fairness notions that have been proposed to capture the necessity for impartialityand justice, envy-freeness [23, 39] is, without doubt, the prevailing one. Envy-freeness requiresthat each individual, or agent, prefers their own share to anyone else’s. However natural andintuitive, though, envy-freeness may not be possible to achieve. In addition, the universality of fairdivision disputes justifies many different definitions of fairness. Some popular fairness notions inthe literature include proportionality and max-min fair share, among others.The vast majority of the related literature focuses on deterministic allocations. The few recentexceptions (e.g., [3, 7, 24]) that consider random allocations (lotteries or probability distributionsover allocations) are either too liberal or too conservative in the fairness concepts they consider.For example, ex-ante envy-freeness compares the random bundle allocated to an agent, in termsof expected valuation, to the random bundle allocated to any other agent. Ex-ante envy-freeness isvery weak as a fairness guarantee. Indeed, a lottery that allocates all items to an agent selected uni-formly at random is ex-ante envy-free; clearly, such a lottery can hardly be considered fair. On theother extreme, the notion of ex-post envy-freeness requires that every outcome of a random allo-cation is envy-free. Ex-post envy-freeness is very strict and essentially invalidates the advantagesof randomness.The notion of interim envy-freeness [34] serves as middle ground between ex-post and ex-anteenvy-freeness, balancing between the stringency of the constraint and the substance of the fairnessguarantee. In particular, consider an instance where a set of indivisible items are to be allocated to oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou
2a set of agents, and a fixed, publicly known, lottery over allocations. Let an outcome of the lotterybe realized and each agent observe only their own allocation. Each agent, then, compares theirwealth to the random bundle allocated to any other agent, conditioned upon their own realizedallocation. Let, for example, the lottery have many possible outcomes (allocations in its support),but only two of which allocate the bundle 𝑎 of items to agent 1; let agent 2 obtain bundle 𝑏 in thefirst of these outcomes and bundle 𝑐 in the second one. The interim envy-freeness constraint foragent 1 with respect to agent 2 and bundle 𝑎 , is satisfied if the value that agent 1 has for 𝑎 is at leastas high as her average value for 𝑏 and 𝑐 , according to the probability that agent 2 receives them(the relative probability of the two outcomes in the lottery). If such a constraint is satisfied forevery agent, with respect to every other agent and any possible bundle, i.e., everyone’s allocatedbundle is always worth to them at least what they can estimate anyone else is receiving, then thelottery is said to be interim envy-free.Interim envy-freeness can be naturally extended to accommodate for payments, similarly to therecent fair allocation literature [26, 29]. It is a known fact that payments can help eliminate envyin the deterministic allocation case, both in the form of subsidies paid to the agents to compensatefor an unsatisfactory bundle, and in the form of rent payments paid by the agents to make theirallocation look less desirable to others. Rent division [2] is a fundamental fair allocation probleminvolving payments, where the input consists of a total rent amount, a set of agents, and an equalnumber of rooms on which the preferences of agents are expressed. The goal is to assign a price toeach room so that the room prices sum up to the total rent, and to match agents to rooms so thateveryone prefers their own allocation and rent share. Using the interim envy-freeness concept, weconsider natural extensions of problems with payments to the random allocation case, both in therent division and in the subsidy distribution context.Matching allocation instances, as in the rent division setting just discussed, are relevant in manyapplications; hence, we partially focus on this case. An important technical advantage is that suchinstances allow for an easy computation of (deterministic) envy-free allocations, as opposed togeneral allocation instances, for which relevant problems are typically NP-hard. However, allow-ing randomization seems to make the situation considerably more complex. Interestingly, as wewill see, the added complication still allows for positive computational results related to interimenvy-free lotteries. To the best of our knowledge, interim envy-freeness (iEF) has not received any attention by theEconCS community. We justify its importance as a fairness notion for lotteries of allocations, bydemonstrating a rich menu of interesting properties it enjoys. First, we relate it to the most im-portant fairness properties for deterministic allocations and lotteries. In terms of strength as afairness property, iEF is proved to lie between proportionality and envy-freeness in the followingway. Clearly, when viewed as a degenerate lottery, any envy-free allocation is iEF. Also, everyiEF lottery is defined over proportional allocations. These implications are shown to be strict ina strong sense. We show that there are allocation instances that admit proportional allocationsbut no iEF lottery, and instances that admit iEF lotteries but no envy-free allocation. Comparedto fairness properties for lotteries, iEF lies between ex-ante envy-freeness (which can be alwaysattained trivially) and ex-post envy-freeness (which is too restrictive). These findings and obser-vations appear in Section 3.Our next goal is to explore the trade-offs between iEF and economic efficiency (Section 4).We pay special attention to matching allocation instances where envy-freeness implies Pareto-efficiency and optimal utilitarian, egalitarian, and average Nash social welfare. A careful interpre-tation of these facts reveals that envy-freeness is a very restrictive fairness property. In contrast, as oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou Θ ( 𝑛 ) times far from optimal, where 𝑛 is thenumber of agents.Bounds on the price of iEF give only rough estimates of the best social welfare of iEF lotteries.We present polynomial-time algorithms for computing iEF lotteries that maximize the utilitarian,egalitarian, and log-Nash social welfare. Our algorithms follow a general template that can bebriefly described as follows. The problem of computing an iEF lottery of maximum social welfareis formulated as a linear program. This linear program has exponentially many variables; to solveit, we exploit the execution of the ellipsoid method to its dual. As the dual linear program hasexponentially many constraints, the ellipsoid method needs access to polynomial-time separationoracles that check whether the dual variables violate the dual constraints or not. We design suchseparation oracles by exploiting connections to maximum edge-pair-weighted bipartite perfectmatching (2EBM), a novel (to the best of our knowledge) combinatorial optimization problem thatinvolves perfect matchings in bipartite graphs. We show how to solve 2EBM in polynomial timeby exploiting an elegant lemma by Cruse [21] on decompositions of doubly-stochastic centro-symmetric matrices. We believe that 2EBM is a natural combinatorial optimization problem ofindependent interest and with applications in other contexts. These computational results appearin Section 5 and constitute the most technically intriguing results in the paper.Finally, we extend the definition of interim envy-freeness to accommodate for settings wheremonetary transfers (payments) are allowed. We define and study two related optimization prob-lems. In subsidy minimization, which is motivated by a similar problem for deterministic alloca-tions that was studied recently, we seek iEF pairs of lotteries and payments to the agents so that thetotal expected amount of payments is minimized. In utility maximization, which extends the well-known rent division problem, we seek iEF pairs of lotteries and payments that are collected fromthe agents and contribute to a fixed rent; the objective is to maximize the minimum expected agentutility. We consider different types of payments depending on whether the payments are agent-specific, bundle-specific, or unconstrained (i.e., specific to agents and allocations). iEF is proved tobe considerably more powerful than envy-freeness, allowing for much better solutions to the twooptimization problems compared to their deterministic counterparts. We also showcase the impor-tance of both agent-specific and bundle-specific payments by showing that they are incomparableto each other, in the context of the two optimization problems. By applying our computational tem-plate, we present efficient algorithms that compute optimal solutions to subsidy minimization andutility maximization using unconstrained payments, violating the iEF condition only marginally.These results are presented in Section 6. We believe that they attest to the significance of interimenvy-freeness too and will motivate further study. Previous work on randomness in allocation problems is clearly related to ours. Aziz [6] discussesthe benefits of randomization in social choice settings, including fair division, thus supportingrelevant studies despite the related challenges. Gajdos and Tallon [25] study ex-ante and ex-postnotions of fairness in a setting with an inherent uncertainty imposed by the environment. Morerecently, Aleksandrov et al. [3] analyze the ex-post and ex-ante envy-freeness guarantees of par-ticular algorithms for online fair division. Freeman et al. [24] study the possibility of achievingex-ante and ex-post fairness guarantees simultaneously in the classical fair allocation setting. Forexample, they show that there always exists an ex-ante envy-free lottery, with all allocations inits support satisfying a relaxed fairness property known as envy-freeness up to one item (EF1; oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou [ , ] . They conjecturethat subsidies of 𝑛 − An instance of an allocation problem consists of a set N of 𝑛 agents and a set I of 𝑚 items. Agent 𝑖 ∈ N has valuation 𝑣 𝑖 ( 𝑗 ) for item 𝑗 ∈ I . By abusing notation, we use 𝑣 𝑖 ( 𝑆 ) to denote the valuationof agent 𝑖 for the set (or bundle) of items 𝑆 . We assume that valuations are non-negative andadditive, i.e., 𝑣 𝑖 ( 𝑆 ) = Í 𝑗 ∈ 𝑆 𝑣 𝑖 ( 𝑗 ) . We remark, though, that several of our results (including theconcept of interim envy-freeness, which we define later in this section) carry over to more generalvaluations. Furthermore, even though this is rarely required for our positive statements, in ourexamples we use normalized valuations satisfying Í 𝑗 ∈I 𝑣 𝑖 ( 𝑗 ) = 𝑖 ∈ N .An allocation 𝐴 = ( 𝐴 , 𝐴 , ..., 𝐴 𝑛 ) of the items in I to the agents of N is simply a partition ofthe items of I into 𝑛 bundles 𝐴 , 𝐴 , ..., 𝐴 𝑛 , with the understanding that agent 𝑖 ∈ N gets thebundle of items 𝐴 𝑖 . An allocation 𝐴 = ( 𝐴 , 𝐴 , ..., 𝐴 𝑛 ) is envy-free (EF) if 𝑣 𝑖 ( 𝐴 𝑖 ) ≥ 𝑣 𝑖 ( 𝐴 𝑘 ) for everypair of agents 𝑖 and 𝑘 . In words, the allocation 𝐴 is envy-free if no agent prefers the bundle ofitems that has been allocated to some other agent to her own. The allocation 𝐴 is proportional if 𝑣 𝑖 ( 𝐴 𝑖 ) ≥ 𝑛 𝑣 𝑖 (I) . An instance may not admit any envy-free or proportional allocation; to see why,consider an instance in which all agents have a positive valuation of 1 for a single item (and zerovalue for any other item). It is well-known that, due to additivity, an envy-free allocation is alwaysproportional. oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝐴 and a vector p consisting of a payment 𝑝 𝑖 to each agent 𝑖 ∈ N is envy-free withpayments if 𝑣 𝑖 ( 𝐴 𝑖 ) + 𝑝 𝑖 ≥ 𝑣 𝑖 ( 𝐴 𝑘 ) + 𝑝 𝑘 for every pair of agents 𝑖 and 𝑘 . The term envy-freeable refersto an allocation that can become envy-free with an appropriate payment vector. Depending on thesetting, payments can be restricted to be non-negative (e.g., representing subsidies that are givento the agents [14, 17, 29]) or non-positive (e.g., when payments are collected from the agents, likein the rent division problem [26]).In addition to their fairness properties, allocations are typically assessed in terms of their effi-ciency. We say that an allocation 𝐴 = ( 𝐴 , 𝐴 , ..., 𝐴 𝑛 ) is Pareto-efficient if there is no other alloca-tion 𝐴 ′ = ( 𝐴 ′ , 𝐴 ′ , ..., 𝐴 ′ 𝑛 ) with 𝑣 𝑖 ( 𝐴 ′ 𝑖 ) ≥ 𝑣 𝑖 ( 𝐴 𝑖 ) for every agent 𝑖 ∈ N , with the inequality beingstrict for at least one agent of N . The term social welfare is typically used to assign a cardinalscore that characterizes the efficiency of an allocation. Among the several social welfare notions,the utilitarian, egalitarian, and Nash social welfare are the three most prominent. We use the no-tation U ( 𝐴 ) , E ( 𝐴 ) , avN ( 𝐴 ) , and lgN ( 𝐴 ) to refer to the utilitarian, egalitarian, average Nash, andlog-Nash social welfare, respectively, of an allocation 𝐴 = ( 𝐴 , ..., 𝐴 𝑛 ) ; the corresponding efficiencyscores are defined as follows: U ( 𝐴 ) = Õ 𝑖 ∈N 𝑣 𝑖 ( 𝐴 𝑖 ) , E ( 𝐴 ) = min 𝑖 ∈N 𝑣 𝑖 ( 𝐴 𝑖 ) , avN ( 𝐴 ) = Ö 𝑖 ∈N 𝑣 𝑖 ( 𝐴 𝑖 ) ! / 𝑛 , lgN ( 𝐴 ) = Õ 𝑖 ∈N ln 𝑣 𝑖 ( 𝐴 𝑖 ) . The price of fairness , introduced independently in [10] and [18], refers to a class of notions thataim to quantify trade-offs between fairness and efficiency. For example, the price of envy-freenesswith respect to the utilitarian social welfare for an allocation instance (that admits at least oneenvy-free allocation) is the ratio of the optimal utilitarian social welfare of the instance over theutilitarian social welfare of the best envy-free allocation. Different price of fairness notions followby selecting different fairness concepts and social welfare definitions.We are particularly interested in matching allocation instances , in which the number of agentsis equal to the number of items. Our assumptions (e.g., for non-negative valuations) imply that theonly reasonably fair (e.g., proportional) allocations should then assign (or match) exactly one itemto each agent. We refer to such allocations as matchings . Notice that, an envy-free matching mustallocate to each agent her most-valued item. As such, whenever an envy-free allocation existsin a matching instance, it is Pareto-efficient and maximizes the social welfare, according to alldefinitions of social welfare mentioned above. Hence, the price of envy-freeness is trivially 1 inthis case (with respect to all the social welfare definitions given above).
Random allocations and interim envy-freeness
We consider random allocations that are produced according to lotteries (or probability distribu-tions). The lottery Q over allocations of the items of I to the agents of N is ex-ante envy-freeif E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑖 )] ≥ E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )] for every pair of agents 𝑖, 𝑘 ∈ N . Q is ex-post envy-free if anyallocation it produces with positive probability is envy-free (or, in other words, if all allocations inthe support of Q are envy-free).We now provide the formal definition of the central concept of this paper. We say that a lottery Q over allocations is interim envy-free (iEF) if for any pair of agents 𝑖, 𝑘 ∈ N and any possiblebundle of items 𝑆 that agent 𝑖 can get in a random allocation produced by Q , it holds 𝑣 𝑖 ( 𝑆 ) ≥ E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] . We typically use the small letter 𝑏 to denote a matching, instead of the usual notation of 𝐴 for allocations in generalinstances.oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑖 has when she gets a bundle 𝑆 is at least as highas the average value she has for the bundle that agent 𝑘 gets, conditioned on 𝑖 ’s allocation.We extend the notion of interim envy-freeness to pairs of lotteries over allocations and pay-ments to/from the agents, in an analogous way that recent work has defined envy-freeness withpayments. In fact, we differentiate between different payment schemes with respect to whetherpayments are per agent (A-payments), per bundle (B-payments), or per allocation and agent (C-payments). We similarly extend the notion of price of fairness and envy-freeability to the case ofiEF. We postpone providing formal definitions for the corresponding sections that these notionsare being examined. In this section we compare interim envy-freeness with other fairness notions, with the aim toidentify possible fairness implications. The first implication follows easily by the definitions andhas been observed before in more general contexts than ours (e.g., see [40]).
Lemma 3.1.
Any iEF lottery Q is ex-ante envy-free. Proof.
Indeed, using the definitions of iEF, ex-ante envy-freeness and well-known propertiesof random variables, we have E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑖 )] = Õ 𝑆 ⊆I 𝑣 𝑖 ( 𝑆 ) · Pr 𝐴 ∼ Q [ 𝐴 𝑖 = 𝑆 ] ≥ Õ 𝑆 ⊆I E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] · Pr 𝐴 ∼ Q [ 𝐴 𝑖 = 𝑆 ] = E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )] , for every pair of agents 𝑖 and 𝑘 . (cid:3) An even simpler observation is that any lottery that deterministically produces an envy-freeallocation 𝐴 is trivially iEF. Indeed, 𝐴 𝑖 is the only bundle that can be given to agent 𝑖 , who weaklyprefers it to the bundle 𝐴 𝑘 that is allocated to agent 𝑘 . Trivially, E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] = 𝑣 𝑖 ( 𝐴 𝑘 ) and the iEF condition is identical to the envy-freeness condition 𝑣 𝑖 ( 𝐴 𝑖 ) ≥ 𝑣 𝑖 ( 𝐴 𝑘 ) . We can slightlyextend this argument to obtain the following implication. Lemma 3.2.
Any ex-post envy-free lottery Q is iEF. However, the opposite is not true; we show below that the existence of an iEF allocation doesnot imply the existence of an EF allocation. This is important as it indicates that the set of iEFallocations is larger than those of EF ones.
Lemma 3.3.
There exist allocation instances with an iEF lottery but with no EF allocation.
Proof.
Consider the matching allocation instance at the left of Table 1 and the lottery Q whichreturns matchings 𝑎 - 𝑏 - 𝑐 and 𝑎 - 𝑐 - 𝑏 , with probability 1 / Q , EF and, consequently, iEF conditionsfor them are satisfied. To see that the iEF condition is satisfied for agent 1, observe that she isallocated item 𝑎 in both matchings in the support of Q , for which she has a value of 1 /
3. Agent2 (and, similarly, agent 3) gets item 𝑏 with probability 1 / 𝑐 with probability 1 /
2. Agent1’s average value for the item agent 2 (or agent 3) gets is 2 / · / + · / = /
3. Hence, theiEF condition for agent 1 with respect to agent 2 (and, similarly, for agent 1 with respect to 3) issatisfied. The proof that the lottery Q is iEF is complete.In the same example, it can be easily seen that there is no EF allocation. Indeed, as the onlyagent who has positive value for item 𝑎 is agent 1, agent 1 should get this item and be envious ofthe agent who gets item 𝑏 . The proof of the lemma is complete. (cid:3) oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑎 𝑏 𝑐 / / / /
23 0 1 / / 𝑎 𝑏 𝑐 / / / /
33 1 / / / Table 1. The two matching instances that are used in the proofs of Lemmas 3.3 and 3.5 to distinguish betweeniEF, EF, and proportionality. Throughout the paper, we consider several examples with three agents and items 𝑎 , 𝑏 , and 𝑐 . A concise notation like 𝑎 - 𝑐 - 𝑏 is used to represent the matching in which agents , , and getitems 𝑎 , 𝑐 , and 𝑏 , respectively. Our next lemma relates iEF to proportionality and is used extensively in our proofs.
Lemma 3.4.
Any allocation in the support of an iEF lottery is proportional.
Proof.
Consider an iEF lottery Q and any agent 𝑖 ∈ N . By the definition of iEF, we have thatfor any allocation in the support of Q where agent 𝑖 gets the bundle of items 𝑆 , it holds that 𝑣 𝑖 ( 𝑆 ) ≥ E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] for each other agent 𝑘 . By summing up over all other agents we get ( 𝑛 − ) 𝑣 𝑖 ( 𝑆 ) ≥ Õ 𝑘 ≠ 𝑖 E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] = E 𝐴 ∼ Q [ Õ 𝑘 ≠ 𝑖 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] = 𝑣 𝑖 (I \ 𝑆 ) . By adding 𝑣 𝑖 ( 𝑆 ) to both sides of the above inequality and rearranging, we get 𝑣 𝑖 ( 𝑆 ) ≥ 𝑛 𝑣 𝑖 (I) ,implying that any allocation in the support of Q is proportional. (cid:3) However, iEF is a stronger property than proportionality as the next lemma shows.
Lemma 3.5.
There exist allocation instances with a proportional allocation but with no iEF lottery.
Proof.
Consider the instance at the right of Table 1. In this instance, allocation 𝑎 - 𝑐 - 𝑏 is theonly proportional allocation. Hence, by Lemma 3.4, to show that no iEF lottery exists, it suffices toconsider only the (lottery that deterministically returns) allocation 𝑎 - 𝑐 - 𝑏 . In this allocation, agents1 and 2 are envious of agent 3, contradicting the iEF requirement. (cid:3) Our next lemma exploits Lemma 3.5 to quantify the disparity between envy-freeness and interimenvy-freeness.
Lemma 3.6.
The maximum envy at any allocation in the support of an iEF lottery, when the agentvaluations are normalized, can be as high as − 𝑛 and this is tight. Proof.
Consider the matching instance of Table 2 with 𝑛 agents/items. First, observe that aniEF lottery should always give item 𝑔 to agent 1. Indeed, any other agent has value 0 for 𝑔 andcould not satisfy the iEF condition if she got it. Hence, in any allocation in the support of an iEF (ifone exists), agent 1 has envy 1 − 𝑛 for the agent who gets item 𝑔 . It remains to show that such aniEF lottery does exist. Indeed, any lottery which gives item 𝑔 to agent 1 and in which the agents2 , , ..., 𝑛 are assigned uniformly at random the items 𝑔 , 𝑔 , ..., 𝑔 𝑛 is iEF. The agents 2, 3, ..., 𝑛 areclearly not envious of each other or of agent 1, as each of them gets the maximum possible value.Agent 1 has expected value 1 / 𝑛 for the bundle of each of the remaining agents, which is equal toher value for the item ( 𝑔 ) she gets.To see that 1 − / 𝑛 is the maximum agent envy in any allocation in the support of an iEF lottery,recall that Lemma 3.4 implies that the value of each agent 𝑖 is at least 1 / 𝑛 and, then, the averagevalue 𝑖 has for the bundle allocated to another agent 𝑘 cannot exceed 1 − / 𝑛 . (cid:3) oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑔 𝑔 𝑔 . . . 𝑔 𝑛 / 𝑛 ( 𝑛 − )/ 𝑛 . . .
02 0 1 /( 𝑛 − ) /( 𝑛 − ) . . . /( 𝑛 − ) ... ... ... ... ... ...𝑛 /( 𝑛 − ) /( 𝑛 − ) . . . /( 𝑛 − ) Table 2. The instance used in the proof of Lemma 3.6.
We now compare iEF to two fairness properties for deterministic allocations: min-max sharefairness (mMS) and epistemic envy-freeness (EEF). mMS and EEF were considered by Bouveretand Lemaitre [13] and Aziz et al. [9], respectively, who characterized instances in terms of theproperties of the allocations they admit.Given an allocation instance with a set of agents N and a set of items I , let A be the set of allallocations of the items to the agents. Denote by 𝜏 𝑖 the min-max share of agent 𝑖 , which is definedas 𝜏 𝑖 = min 𝐴 ∈A max 𝑗 ∈N 𝑣 𝑖 ( 𝐴 𝑗 ) . In words, the min-max share 𝜏 𝑖 is the minimum over all allocations of the maximum value agent 𝑖 has for some bundle. An allocation 𝐴 = ( 𝐴 , 𝐴 , ..., 𝐴 𝑛 ) is mMS (or satisfies the min-max sharefairness criterion) if 𝑣 𝑖 ( 𝐴 𝑖 ) ≥ 𝜏 𝑖 for every agent 𝑖 ∈ N .An allocation 𝐴 = ( 𝐴 , 𝐴 , ..., 𝐴 𝑛 ) is EEF if for every agent 𝑖 ∈ N there exists an allocation 𝐵 ∈ A with 𝐵 𝑖 = 𝐴 𝑖 such that 𝑣 𝑖 ( 𝐴 𝑖 ) ≥ 𝑣 𝑖 ( 𝐵 𝑗 ) for every agent 𝑗 ∈ N . In words, there is a redistributionof the items that agent 𝑖 does not get to the other agents, so that agent 𝑖 is not envious.By this definition, it is clear that an instance that admits an EF allocation also admits an EEFallocation [9]. Any EEF allocation is also mMS [9], while any mMS allocation is proportional [13].These implications are strict [9, 13]. There are instances that admit proportional allocations buthave no mMS allocation, instances with an mMS allocation but no EEF allocation, and instanceswith an EEF allocation but with no EF allocation.Where does iEF stand in this hierarchy of properties? So far, we have shown that EF implies iEFwhich in turn implies proportionality. We have furthermore seen (in Lemmas 3.3 and 3.5) that theseimplications are strict. Interestingly, iEF is incomparable to both mMS and EEF, as the followingtwo statements show. Lemma 3.7.
There exist matching instances with an iEF lottery that do not admit any mMS (and,subsequently, any EEF) allocation.
Proof.
The proof uses the instance at the left of Table 1 which does have an iEF as explained inthe proof of Lemma 3.3. By definition, in any mMS (and, by the implication shown in [9], any EEF)allocation, the value of each agent should be at least her mMS share. The corresponding shares are2 /
3, 1 /
2, and 1 / (cid:3) Lemma 3.8.
There exist allocation instances with an EEF (and, subsequently, an mMS) allocationbut with no iEF lottery. Min-max share fairness should not be confused with max-min share (MmS) fairness, a weaker-than-proportionality fair-ness property that has received much attention recently [32]. See [13] for a taxonomy of many fairness properties that wehave mentioned here.oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou Proof.
Consider the instance shown in Table 3. We first observe that the allocation in whichagent 1 gets item 𝑎 , agent 2 gets items 𝑏 and 𝑐 , and agent 3 gets item 𝑑 is EEF. Indeed, agent 2 isclearly non-envious in this distribution. Agent 1 does not envy the other agents in the redistribu-tion where agent 2 gets item 𝑏 and agent 3 gets items 𝑐 and 𝑑 . Similarly, agent 3 does not envy theother agents in the redistribution where agent 2 gets item 𝑐 and agent 1 gets items 𝑎 and 𝑏 . 𝑎 𝑏 𝑐 𝑑 /
10 3 /
10 3 /
10 02 4 /
10 3 /
10 3 /
10 03 0 3 /
10 3 /
10 4 / Table 3. The instance used in the proof of Lemma 3.8.
We now show that no iEF lottery exists. As the proportionality threshold is 1 / ({ 𝑎 } , { 𝑏, 𝑐 } , { 𝑑 }) and ({ 𝑏, 𝑐 } , { 𝑎 } , { 𝑑 }) . For any lottery over these two allocations, among agents 1 and 2, the one whogets item 𝑎 is envious of the other who gets items 𝑏 and 𝑐 . (cid:3) We now explore tradeoffs between interim envy-freeness and efficiency. Two well-studied refine-ments of Pareto-efficiency are relevant for lotteries of allocations: ex-ante and ex-post Pareto-efficiency. A lottery Q over allocations is ex-ante Pareto-efficient if there exists no other lottery Q ′ such that E 𝐴 ∼ Q ′ [ 𝑣 𝑖 ( 𝐴 𝑖 )] ≥ E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑖 )] for every agent 𝑖 ∈ N , with the inequality being strictfor at least one agent of N . A lottery is ex-post Pareto-efficient if all allocations in its support arePareto-efficient. It is well-known that ex-ante Pareto-efficiency implies ex-post Pareto-efficiency.For allocation instances with two agents, the allocations in the support of an iEF lottery are envy-free and, thus (as observed in Section 2), Pareto-efficient. This is due to the fact that any allocationin the support of an iEF lottery is proportional (by Lemma 3.4) and hence envy-free, since thereare only two agents. This implies that an iEF lottery is ex-post and ex-ante Pareto-efficient. In thefollowing, we show that this may not be the case in instances with more agents.
Theorem 4.1.
There exist matching instances with 𝑛 ≥ agents in which no iEF lottery is ex-post(and, consequently, ex-ante) Pareto-efficient. Proof.
Consider the matching instance of Table 4. We will show that the only iEF lottery Q returns the allocations 𝑎 - 𝑏 - 𝑐 , 𝑎 - 𝑐 - 𝑏 , and 𝑏 - 𝑐 - 𝑎 equiprobably. Observe that allocation 𝑎 - 𝑐 - 𝑏 (whereall agents have value 1 /
3) is Pareto-dominated by both 𝑎 - 𝑏 - 𝑐 and 𝑏 - 𝑐 - 𝑎 (in which two agents getvalue 1 / /
3) and, hence, Q is not ex-post Pareto-efficient.It remains to show that Q is the only iEF lottery. First, observe that allocations 𝑎 - 𝑏 - 𝑐 , 𝑏 - 𝑐 - 𝑎 , and 𝑎 - 𝑐 - 𝑏 are the only proportional allocations and, by Lemma 3.4, the only ones that can be used inthe support of another iEF lottery Q ′′ , assuming that one exists. Now, assume that Q ′′ does nothave one of these allocations in its support. We distinguish between three cases: • If 𝑎 - 𝑏 - 𝑐 is not part of the support, then agent 3 gets item 𝑏 whenever agent 1 gets item 𝑎 . TheiEF condition is violated for agent 1 with respect to agent 3 and bundle { 𝑎 } . This is our only example with a non-matching instance. This is necessary. It can be easily seen that, in matching instances,EF is equivalent to EEF. Consequently, EEF implies iEF in these instances. It can be easily seen that ex-post and ex-ante Pareto-efficiency coincide for matching allocation instances with two agents.This is not true in general.oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑎 𝑏 𝑐 / / / /
33 1 / / / Table 4. The instance in the proof of Theorem 4.1. • If 𝑏 - 𝑐 - 𝑎 is not part of the support, then agent 3 gets item 𝑏 whenever agent 2 gets item 𝑐 . TheiEF condition is violated for agent 2 with respect to agent 3 and bundle { 𝑐 } . • If 𝑎 - 𝑐 - 𝑏 is not part of the support, then agent 2 gets item 𝑏 whenever agent 1 gets item 𝑎 . TheiEF condition is violated for agent 1 with respect to agent 2 and bundle { 𝑎 } .Hence, the three allocations 𝑎 - 𝑏 - 𝑐 , 𝑏 - 𝑐 - 𝑎 , and 𝑎 - 𝑐 - 𝑏 all appear in the support of Q ′′ . Since Q ′′ isdifferent than Q , we distinguish between two cases. First, assume that the allocations 𝑎 - 𝑏 - 𝑐 and 𝑎 - 𝑐 - 𝑏 have different probabilities in Q ′′ . Then, whenever agent 1 gets item 𝑎 , some of agents 2 and3, call her 𝑖 , gets item 𝑏 with probability strictly higher than 1 /
2. Hence, the expected value ofagent 1 for the bundle of agent 𝑖 is strictly higher than 1 / 𝑖 and bundle { 𝑎 } . The other case in which the allocations 𝑏 - 𝑐 - 𝑎 and 𝑎 - 𝑐 - 𝑏 havedifferent probabilities in Q ′′ is symmetric. We conclude that Q ′′ is not iEF.We complete the proof by showing that the lottery Q is indeed iEF. Agent 3 is never envious.The iEF condition for agent 1 (or agent 2) clearly holds when she gets item 𝑏 (since her value foritem 𝑏 is the highest). When agent 1 gets item 𝑎 (respectively, when agent 2 gets item 𝑐 ), she has avalue of 1 /
3. Then, each of the other two agents 2 and 3 (respectively, 1 and 3) gets equiprobablyitems 𝑏 and 𝑐 (respectively, 𝑎 and 𝑏 ). The expected value agent 1 (respectively, agent 2) has for theitem allocated to agent 2 or agent 3 (respectively to agent 1 or agent 3) is equal to 1 /
3, and the iEFcondition holds with equality. (cid:3)
To assess the impact of fairness in random allocations to social welfare, we need to extend theprice of fairness definition to lotteries. We do so implicitly here by defining the price of iEF (onecan similarly define, e.g., the price of ex-ante envy-freeness). We say that the price of iEF withrespect to a social welfare measure is the worst-case ratio over all allocation instances with atleast one iEF lottery, of the optimal social welfare in the instance over the expected social welfareof the best iEF lottery (where “best” is defined with respect to this social welfare measure).In our next theorems, we bound the price of iEF with respect to different social welfare notions.In the proof of our upper bounds, we consider normalized valuations. This is a typical assumptionin the related literature as well, e.g., see [18]. This assumption is not necessary for average Nashsocial welfare.
Theorem 4.2.
The price of iEF with respect to the utilitarian, egalitarian, and average Nash socialwelfare is at most 𝑛 , when the agent valuations are normalized. Proof.
The proof follows by Lemma 3.4, which implies that the valuation of each agent inany allocation in the support of an iEF lottery is at least 1 / 𝑛 . Then, the utilitarian, egalitarian,and average Nash social welfare is at least 1, 1 / 𝑛 , and 1 / 𝑛 , respectively, while the correspondingoptimal values are at most 𝑛 , 1, and 1, respectively. (cid:3) The next two statements indicate that our price of iEF upper bounds with respect to utilitarianand egalitarian social welfare are asymptotically tight.
Theorem 4.3.
The price of iEF with respect to the utilitarian social welfare is at least Ω ( 𝑛 ) . oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou Proof.
Let 𝜖 > 𝑘 ≥ 𝑛 = 𝑘 agents and items. For 𝑖 = , , ..., 𝑘 , agent 𝑖 has value 𝑘𝑘 + for item 𝑖 , value 𝑘 + for item 𝑖 + 𝑘 , and value 0 for any other item. For 𝑖 = 𝑘 + , 𝑘 + , ..., 𝑘 , agent 𝑖 has value 𝑘 + 𝜖 foritems 1 , , ..., 𝑘 and value 𝑘 − 𝜖 for items 𝑘 + , ..., 𝑘 .An optimal allocation has utilitarian social welfare (at least) 𝑘 𝑘 + + − 𝑘𝜖 . To see why, consider theallocation in which agent 𝑖 gets item 𝑖 for 𝑖 = , , ..., 𝑘 . We now claim that no iEF lottery Q overallocations has welfare higher than 𝑘𝑘 + + + 𝑘𝜖 . The lower bound on the price of iEF will follow bythe relation between 𝑛 and 𝑘 (and by taking 𝜖 to be sufficiently small). Indeed, by Lemma 3.4, thesupport of Q should consist of allocations in which agents 𝑘 + , 𝑘 + , ..., 𝑘 get items 1 , , ..., 𝑘 fora total value of + 𝑘𝜖 . Then, the maximum value each of the agents 1 , , ..., 𝑘 gets from the items 𝑘 + , ..., 𝑘 is 𝑘 + .It remains to present such a lottery Q . It suffices to assign item 𝑖 + 𝑘 to agent 𝑖 for 𝑖 = , , ..., 𝑘 andassign uniformly at random the items 1 , , ..., 𝑘 to the agents 𝑘 + , ..., 𝑘 . Clearly, agents 𝑘 + , ..., 𝑘 are not envious. For 𝑖 = , ..., 𝑘 , agent 𝑖 has value 𝑘 + . Her expected value for the item of anotheragent ℓ is 0 if ℓ is one of the 𝑘 first agents besides 𝑖 and 𝑘 + if ℓ is one of the 𝑘 last agents. Noticethat in the latter case, agent ℓ gets item 𝑖 (for which agent 𝑖 has value 𝑘𝑘 + ) with probability 1 / 𝑘 ,while she gets items for which agent 𝑖 has no value otherwise. (cid:3) Theorem 4.4.
The price of iEF with respect to the egalitarian social welfare is at least Ω ( 𝑛 ) . Proof.
Consider the following matching instance with 𝑛 agents and items. Agent 1 has value1 / / 𝑖 = , , ..., 𝑛 −
2, agent 𝑖 has value 1 / 𝑖 , 𝑖 +
1, and 𝑛 . Agent 𝑛 − / 𝑛 hasvalue 𝑛 − for item 1 and 1 − 𝑛 − for item 𝑛 . All other agent valuations are 0.An optimal allocation has egalitarian social welfare at least 1 /
3. To see why, consider the alloca-tion where agent 𝑖 gets item 𝑖 , for 𝑖 = , , ..., 𝑛 . Furthermore, we claim that any iEF lottery Q overallocations has egalitarian social welfare 𝑛 − . Indeed, by Lemma 3.4, any allocation in the supportof Q has agent 1 getting either item 1 or item 2. In the former case, agent 2 must get item 2 and,hence, agent 1 envies agent 2. So, agent 1 gets item 2 and, again by Lemma 3.4, item 1 must beassigned to agent 𝑛 .Consider now the following lottery. Agent 1 gets item 2, agent 𝑛 gets item 1, while the remainingitems are assigned in the following way. Let ℓ be selected equiprobably from { , . . . , 𝑛 − } . Then,agent ℓ gets item 𝑛 , each agent 𝑖 , for 𝑖 = , ..., ℓ −
1, gets item 𝑖 + 𝑖 , for 𝑖 = ℓ + , ..., 𝑛 − 𝑖 . Note that all agents, besides agent 𝑛 , get an item of maximum value to them. Agent 𝑛 obtains value 𝑛 − from item 1; she does not envy another agent, as, apart from item 1, she onlyvalues item 𝑛 positively and any other agent gets item 𝑛 with probability at most 𝑛 − . (cid:3) A very similar proof to the one of Theorem 4.3 yields our best (albeit not known to be tight)lower bound for the price of iEF with respect to the average Nash social welfare.
Theorem 4.5.
The price of iEF with respect to the average Nash social welfare is at least Ω (cid:0) √ 𝑛 (cid:1) . Proof.
The proof uses the same instance with the one in the proof of Theorem 4.3 and a verysimilar reasoning. By considering again the allocation in which agent 𝑖 gets item 𝑖 for 𝑖 = , , ..., 𝑘 ,we get that the optimal allocation has average Nash social welfare at least (cid:18) (cid:16) 𝑘𝑘 + (cid:17) 𝑘 (cid:0) 𝑘 − 𝜖 (cid:1) 𝑘 (cid:19) 𝑘 .As we argued in the proof of Theorem 4.3, the support of any iEF lottery Q consists of allocationsin which agents 𝑘 + 𝑘 +
2, ..., 2 𝑘 get distinct items among items 1, 2, ..., 𝑘 ; each of these agents oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑘 + 𝜖 . Then, for 𝑖 = , , ..., 𝑘 , agent 𝑖 gets item 𝑘 + 𝑖 and has value 𝑘 + . The average Nashsocial welfare of such allocations (which do exist as we have also shown) is (cid:16) (cid:0) 𝑘 + (cid:1) 𝑘 (cid:0) 𝑘 + 𝜖 (cid:1) 𝑘 (cid:17) 𝑘 .By setting 𝜖 = 𝑘 , we obtain that the price of iEF with respect to the average Nash social welfareis (at least) © « (cid:16) 𝑘𝑘 + (cid:17) 𝑘 (cid:0) 𝑘 − 𝜖 (cid:1) 𝑘 (cid:0) 𝑘 + (cid:1) 𝑘 (cid:0) 𝑘 + 𝜖 (cid:1) 𝑘 ª®®¬ 𝑘 = r 𝑘 = √ 𝑛 . The theorem follows. (cid:3)
We devote this section to proving the following statement.
Theorem 5.1.
For matching instances, an iEF lottery of maximum expected utilitarian, egalitarian,or log-Nash social welfare can be computed in polynomial time in terms of the number of agents.
Our algorithms use linear programming. Let M be the set of all possible perfect matchingsbetween the agents in N and the items in I (more formally, M is the set of all perfect matchingsin the complete bipartite graph 𝐺 = (N , I , N × I) ). For agent 𝑖 ∈ N and item 𝑗 ∈ I , denoteby M 𝑖 𝑗 the set of matchings from M in which item 𝑗 is assigned to agent 𝑖 . Also, for a matching 𝑏 ∈ M and an agent 𝑘 ∈ N , 𝑏 ( 𝑘 ) denotes the item of I to which agent 𝑘 is matched in 𝑏 . Then, aniEF lottery can be computed as the solution to the following linear program.maximize Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · SW ( 𝑏 ) subject to Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) ≥ , 𝑖 ∈ N , 𝑗 ∈ I , 𝑘 ∈ N \ { 𝑖 } Õ 𝑏 ∈M 𝑥 ( 𝑏 ) = 𝑥 ( 𝑏 ) ≥ , 𝑏 ∈ M (1)The variables of the linear program are the probabilities 𝑥 ( 𝑏 ) , for every matching 𝑏 ∈ M , withwhich the lottery produces matching 𝑏 . Together with the non-negativity constraints on x , thesecond constraint Í 𝑏 ∈M 𝑥 ( 𝑏 ) = x = ( 𝑥 ( 𝑏 )) 𝑏 ∈M definesa lottery over all matchings of M . The notation SW ( 𝑏 ) is used here to refer generally to the socialwelfare of matching 𝑏 . We will specifically replace SW by U , E , and lgN later. The objective ofthe linear program is to maximize the expected social welfare E 𝑏 ∼ x [ SW ( 𝑏 )] or, equivalently, thequantity Í 𝑏 ∈M 𝑥 ( 𝑏 ) · SW ( 𝑏 ) .The first set of constraints represent the iEF conditions. Indeed, the constraint is clearly true forevery agent 𝑖 ∈ N and item 𝑗 ∈ I that is never assigned to agent 𝑖 under x (i.e., when Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] = 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] > 𝑘 ∈ N \ { 𝑖 } , interim envy-freeness requires that 𝑣 𝑖 ( 𝑗 ) ≥ E 𝑏 ∼ x [ 𝑣 𝑖 ( 𝑏 ( 𝑘 ))| 𝑏 ( 𝑖 ) = 𝑗 ] . (2)By multiplying the left-hand-side of (2) with Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] , we get 𝑣 𝑖 ( 𝑗 ) · Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] = Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑗 ) , oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou E 𝑏 ∼ x [ 𝑣 𝑖 ( 𝑏 ( 𝑘 ))| 𝑏 ( 𝑖 ) = 𝑗 ] · Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] = Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑘 )) . Hence, inequality (2) is equivalent to the first constraint of the linear program (1).The linear program (1) has exponentially many variables, i.e., one variable for each of the 𝑛 !different matchings of M . To solve it efficiently, we will resort to its dual linear programmaximize 𝑧 subject to 𝑧 + Í ( 𝑖,𝑗 ) ∈ 𝑏𝑘 ∈N\{ 𝑖 } ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) + SW ( 𝑏 ) ≤ , 𝑏 ∈ M 𝑦 ( 𝑖, 𝑗, 𝑘 ) ≥ , 𝑖 ∈ N , 𝑗 ∈ I , 𝑘 ∈ N \ { 𝑖 } (3)The dual linear program (3) has polynomially many variables and exponentially many constraints.Fortunately, we will be able to solve it using the ellipsoid method [27, 35]. To do so, all we needis a polynomial-time separation oracle, which takes as input values for the dual variables 𝑧 and 𝑦 ( 𝑖, 𝑗, 𝑘 ) for all triplets ( 𝑖, 𝑗, 𝑘 ) consisting of agent 𝑖 ∈ N , item 𝑗 ∈ I , and agent 𝑘 ∈ N \ { 𝑖 } , andeither computes a matching 𝑏 ∗ for which a particular constraint is violated, or correctly concludesthat no constraint of the dual linear program (3) is violated.Let us briefly remind the reader how solving the dual linear program using the ellipsoid methodcan give us an efficient solution to the primal linear program as well; a more detailed discussioncan be found in [27, 35]. To solve the dual linear program, the ellipsoid method will make onlypolynomially many calls to the separation oracle. This is due to the fact that, among the exponen-tially many constraints, the ones that really constrain the variables of the dual linear program arevery few; the rest are just redundant. Then, after having kept track of the execution of the ellip-soid method on the dual linear program, the primal linear program can be simplified by settingimplicitly to 0 all variables that correspond to dual constraints that were not returned as violatedones by the calls of the separation oracle during the execution of the ellipsoid method. As a finalstep, the solution of the simplified primal linear program (which is now of polynomial size) willgive us the solution x ; this will have only polynomially-many matchings in its support.In the rest of this section, we will show how to design such separation oracles for the dual linearprogram (3) when we use the utilitarian, egalitarian, or log-Nash definition of the social welfare.Our separation oracles essentially solve instances of a novel variation of the maximum bipartitematching problem. We believe that this can be of independent interest, with applications in manydifferent contexts. The maximum edge-pair-weighted perfect bipartite matching
Instances of the maximum edge-pair-weighted perfect bipartite matching problem (or, 2EBM, forshort) consist of the complete bipartite graph 𝐺 = (N , I , N × I) with a weighting function 𝜓 thatassigns weight 𝜓 ( 𝑒 , 𝑒 ) to every ordered pair of non-incident edges 𝑒 and 𝑒 from N × I . Theobjective is to compute a perfect matching 𝑏 ∈ M so that the total weight over all edge-pairs of 𝑏 ,denoted by Ψ ( 𝑏 ) = Õ ( 𝑖,𝑗 ) ∈ 𝑏 Õ ( 𝑘,ℓ ) ∈ 𝑏 : 𝑘 ≠ 𝑖 𝜓 ( ( 𝑖, 𝑗 ) , ( 𝑘, ℓ )) , (4) oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou Let X be the set of quadruples ( 𝑖, 𝑗, 𝑘, ℓ ) where 𝑖, 𝑘 ∈ N and 𝑗, ℓ ∈ I , with 𝑖 ≠ 𝑘 and 𝑗 ≠ ℓ .We can view such a quadruple as the ordered pair of edges ( 𝑖, 𝑗 ) and ( 𝑘, ℓ ) in the input graph 𝐺 .Essentially, the quadruples of X correspond to all possible ordered pairs of different edges in theinput graph. To compute a perfect matching 𝑏 ∈ M of maximum total edge-pair weight, we will usethe following integer linear program with Θ ( 𝑛 ) variables and constraints (where 𝑛 is the numberof agents and items). We remark that, from now on, we simplify the notation 𝜓 (( 𝑖, 𝑗 ) , ( 𝑘, ℓ )) anduse 𝜓 ( 𝑖, 𝑗, 𝑘, ℓ ) instead.maximize Í ( 𝑖,𝑗,𝑘,ℓ ) ∈X 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) · 𝜓 ( 𝑖, 𝑗, 𝑘, ℓ ) subject to Í 𝑗,ℓ ∈I : ( 𝑖,𝑗,𝑘,ℓ ) ∈X 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) = , 𝑖 ∈ N , 𝑘 ∈ N \ { 𝑖 } Í 𝑖,𝑘 ∈N : ( 𝑖,𝑗,𝑘,ℓ ) ∈X 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) = , 𝑗 ∈ I , ℓ ∈ I \ { 𝑗 } 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) = 𝑡 ( 𝑘, ℓ, 𝑖, 𝑗 ) , ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ X 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ { , } , ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ X (5)For a quadruple ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ X , the variable 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) indicates whether both edges ( 𝑖, 𝑗 ) and ( 𝑘, ℓ ) belong to the perfect matching ( 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) =
1) or not ( 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) = 𝑖 and 𝑘 , exactly one has both its edges inthe matching. Similarly, the second constraint indicates that among all edge pairs with endpointsat item nodes 𝑗 and ℓ , exactly one has both its edges in the matching. The third constraint ensuressymmetry of the variables so that they are consistent to our interpretation.We relax the integrality constraint of (5) and replace it by 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) ≥ , ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ X . (6)Then, we compute an extreme solution of the resulting linear program (e.g., again, using the ellip-soid method [27, 35]). We claim (in Lemma 5.4) that this solution is integral, i.e., all variables havevalues either 0 or 1, and are, hence, solutions to the integer linear program (5) and, consequently,to our maximum edge-pair-weighted perfect bipartite matching problem.To prove this, we can view the solution t of the relaxation of the linear program (5) as a squarematrix 𝑇 . In this matrix, each row corresponds to a pair of different agents and each column to apair of different items. Then, the first and the second set of constraints indicate that 𝑇 is doublystochastic . Here, ideally, we would like to use the famous Birkhoff-von Neumann theorem [11],which states that any doubly stochastic matrix is a convex combination of permutation matrices(i.e., square binary matrices with exactly one 1 at each row and each column) and conclude that theextreme solutions of the relaxation of the linear program (5) are integral and, hence, correspond toperfect matchings. Unfortunately, the linear program (5) has the additional symmetry constraintthat does not allow for such a use of the Birkhoff-von Neumann theorem.Fortunately, we can use an extension due to Cruse [21] which applies to centro-symmetric ma-trices. Definition 5.2. An 𝑁 × 𝑁 matrix 𝑇 = ( 𝑇 𝑢,𝑣 ) 𝑢,𝑣 ∈[ 𝑁 ] is called centro-symmetric if it satisfies 𝑇 𝑢,𝑣 = 𝑇 𝑁 + − 𝑢,𝑁 + − 𝑣 for all 𝑢, 𝑣 ∈ [ 𝑁 ] . We remark that the problem of computing a perfect matching of maximum total edge weight in an edge-weighted completebipartite graph with edge weight 𝑤 ( 𝑒 ) for each edge 𝑒 , is equivalent to 2EBM by defining the edge-pair weights of thelatter as 𝜓 ( 𝑒, 𝑒 ′ ) = 𝑤 ( 𝑒 ) 𝑛 − for every ordered pair of edges 𝑒 and 𝑒 ′ .oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou Theorem 5.3 (Cruse [21]). If 𝑁 is even, then any 𝑁 × 𝑁 centro-symmetric doubly stochasticmatrix is the convex combination of centro-symmetric permutation matrices. We use Theorem 5.3 in the proof of the next lemma.
Lemma 5.4.
Any extreme solution of the relaxation of the linear program (5) is integral.
Proof.
We will define an alternative representation of a feasible solution t of the relaxation ofthe linear program (5) as a doubly stochastic matrix 𝑇 with 𝑁 rows and columns. To do so, wewill use a particular mapping of each pair of different agents (respectively, of each pair of differentitems) to particular rows (respectively, columns) of the matrix 𝑇 . This particular mapping will allowus to argue that the matrix 𝑇 is centro-symmetric. As 𝑁 is clearly even, Theorem 5.3 will give usthat 𝑇 is a convex combination of centro-symmetric permutation matrices, which correspond tointegral solutions.As both sets N and I contain 𝑛 elements each, we may view them as integers from [ 𝑛 ] . We definethe bijection 𝜋 from ordered pairs of different integers from [ 𝑛 ] to integers of [ 𝑁 ] as follows. Forevery ordered pair ( 𝑖, 𝑘 ) of different integers from [ 𝑛 ] , let 𝜋 ( 𝑖, 𝑘 ) = 𝑖 − Õ ℎ = ( 𝑛 − ℎ ) + 𝑘 − 𝑖 if 𝑖 < 𝑘 , and 𝜋 ( 𝑖, 𝑘 ) = 𝑛 ( 𝑛 − ) + − 𝜋 ( 𝑘, 𝑖 ) otherwise. By this definition, we have 𝜋 ( 𝑖, 𝑘 ) + 𝜋 ( 𝑘, 𝑖 ) = 𝑛 ( 𝑛 − ) + . (7)Note that, for 𝑖 = , , ..., 𝑛 − 𝑘 = 𝑖 + , ..., 𝑛 , 𝜋 ( 𝑖, 𝑘 ) takes all distinct integer values from 1to 𝑛 ( 𝑛 − )/
2. Then, for the remaining pairs ( 𝑖, 𝑘 ) with 𝑖 = , ..., 𝑛 and 𝑘 = , ..., 𝑖 − 𝜋 ( 𝑖, 𝑘 ) takesall distinct integer values from 𝑛 ( 𝑛 − ) down to 𝑛 ( 𝑛 − )/ +
1. Hence, since 𝑁 = 𝑛 ( 𝑛 − ) , eachdistinct ordered pair of different integers from [ 𝑛 ] is mapped to a different integer of [ 𝑁 ] under 𝜋 . Hence, 𝜋 is indeed a bijection. Now, for every quadruple ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ X , we store the value of 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) in the entry 𝑇 𝜋 ( 𝑖,𝑘 ) ,𝜋 ( 𝑗,ℓ ) of matrix 𝑇 . By the properties of 𝜋 , the matrix 𝑇 is well-defined.We will complete the proof by showing that 𝑇 is centro-symmetric. Indeed, let 𝑢 and 𝑣 be anyintegers in [ 𝑁 ] and assume that 𝑢 = 𝜋 ( 𝑖, 𝑘 ) and 𝑣 = 𝜋 ( 𝑗, ℓ ) for pairs of distinct integers ( 𝑖, 𝑘 ) and ( 𝑗, ℓ ) . We have 𝑇 𝑢,𝑣 = 𝑇 𝜋 ( 𝑖,𝑘 ) ,𝜋 ( 𝑗,ℓ ) = 𝑡 ( 𝑖, 𝑗, 𝑘, ℓ ) = 𝑡 ( 𝑘, ℓ, 𝑖, 𝑗 ) = 𝑇 𝜋 ( 𝑘,𝑖 ) ,𝜋 ( ℓ,𝑗 ) = 𝑇 𝑁 + − 𝜋 ( 𝑖,𝑘 ) ,𝑁 + − 𝜋 ( 𝑗,ℓ ) = 𝑇 𝑁 + − 𝑢,𝑁 + − 𝑣 , i.e., 𝑇 is indeed centro-symmetric. The first and sixth equalities follow by the definition of 𝑢 and 𝑣 . The second and fourth equalities follow by the definition of the entries of matrix 𝑇 . The thirdequality is the symmetry constraint of linear program (5). The fifth equality follows by (7). (cid:3) Hence, the execution of the ellipsoid algorithm on the relaxation of the linear program (5) willreturn an integral solution that corresponds to a solution of 2EBM. The next statement summarizesthe above discussion.
Theorem 5.5.
We are ready to show how solutions to appropriately defined instances of 2EBM can be usedas separation oracles for solving the linear program (3) when SW is the utilitarian (Section 5.1),egalitarian (Section 5.2), and log-Nash (Section 5.3) social welfare. oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou By the definition of the utilitarian social welfare, we have U ( 𝑏 ) = Õ ( 𝑖,𝑗 ) ∈ 𝑏 𝑣 𝑖 ( 𝑗 ) = Õ ( 𝑖,𝑗 ) ∈ 𝑏 Õ ( 𝑘,ℓ ) ∈ 𝑏 : 𝑘 ≠ 𝑖 𝑣 𝑖 ( 𝑗 ) + 𝑣 𝑘 ( ℓ ) ( 𝑛 − ) , and, using SW ( 𝑏 ) = U ( 𝑏 ) , the constraint of the dual linear program (3) corresponding to a match-ing 𝑏 ∈ M is equivalent to Õ ( 𝑖,𝑗 ) ∈ 𝑏 Õ ( 𝑘,ℓ ) ∈ 𝑏 : 𝑘 ≠ 𝑖 (cid:18) ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( ℓ )) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) + 𝑣 𝑖 ( 𝑗 ) + 𝑣 𝑘 ( ℓ ) ( 𝑛 − ) + 𝑧𝑛 ( 𝑛 − ) (cid:19) ≤ . (8)So, consider the instance of 2EBM with edge weights defined as 𝜓 ( 𝑖, 𝑗, 𝑘, ℓ ) = ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( ℓ )) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) + 𝑣 𝑖 ( 𝑗 ) + 𝑣 𝑘 ( ℓ ) ( 𝑛 − ) + 𝑧𝑛 ( 𝑛 − ) . Then, for a matching 𝑏 ∈ M , we have that the objective function of 2EBM, Ψ ( 𝑏 ) , shown in (4),is equal to the left-hand-side of inequality (8) and, consequently, to the left-hand-side of the con-straint of the dual linear program (3), when the utilitarian definition of the social welfare is used.Now, the separation oracle for the dual linear program (3) works as follows. It solves the instanceof 2EBM just described and computes a matching 𝑏 ∗ ∈ M that maximizes the quantity Ψ ( 𝑏 ) , i.e., 𝑏 ∗ ∈ arg max 𝑏 ∈M Ψ ( 𝑏 ) . If Ψ ( 𝑏 ∗ ) >
0, the constraint corresponding to the matching 𝑏 ∗ in the duallinear program (3) is returned as a violating constraint. Otherwise, it must be Ψ ( 𝑏 ) ≤ 𝑏 ∈ M and the separation oracle correctly returns that no such violating constraintexists. Let 𝐿 denote the different values the valuations 𝑣 𝑖 ( 𝑗 ) of an agent 𝑖 for item 𝑗 can get, i.e., 𝐿 = { 𝑣 𝑖 ( 𝑗 ) : 𝑖 ∈ N , 𝑗 ∈ I} . For 𝑒 ∈ 𝐿 , denote by M 𝑒 the set of perfect matchings so that for any agent 𝑖 thatis assigned to item 𝑗 , it holds that 𝑣 𝑖 ( 𝑗 ) ≥ 𝑒 . Observe that the perfect matching 𝑏 ∈ M belongs toset M 𝑒 for every 𝑒 ≤ E ( 𝑏 ) . Then, the constraints of the dual linear program (3) for the egalitariandefinition of the social welfare are captured by the following set of constraints: Õ ( 𝑖,𝑗 ) ∈ 𝑏 Õ ( 𝑘,ℓ ) ∈ 𝑏 : 𝑘 ≠ 𝑖 (cid:18) ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( ℓ )) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) + 𝑒 + 𝑧𝑛 ( 𝑛 − ) (cid:19) ≤ , 𝑒 ∈ 𝐿, 𝑏 ∈ M 𝑒 (9)Indeed, for every matching 𝑏 ∈ M , the set of constraints (9) contains the constraint correspondingto 𝑏 in the dual linear program (3) with SW = E and, possibly, the redundant constraints 𝑧 + Õ ( 𝑖,𝑗 ) ∈ 𝑏𝑘 ∈N\{ 𝑖 } ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) + 𝑒 ≤ , for 𝑒 ∈ 𝐿 with 𝑒 < E ( 𝑏 ) (if any). So, to design the separation oracle for the dual linear program (3),it suffices to design a separation oracle for the set of constraints (9), for each of the O( 𝑛 ) differentvalues of 𝑒 ∈ 𝐿 . We now show how to do so.For 𝑒 ∈ 𝐿 , let X 𝑒 be the subset of X such that 𝑣 𝑖 ( 𝑗 ) ≥ 𝑒 and 𝑣 𝑘 ( ℓ ) ≥ 𝑒 . Essentially, the quadruplesof X 𝑒 correspond to all possible (ordered) pairs of different edges in a perfect matching of M 𝑒 . Now,for every 𝑒 ∈ 𝐿 , consider the instance of 2EBM with weights 𝜓 ( 𝑖, 𝑗, 𝑘, ℓ ) = ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( ℓ )) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) + 𝑒 + 𝑧𝑛 ( 𝑛 − ) oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ X 𝑒 . Then, for a matching 𝑏 ∈ M 𝑒 , the objective function of 2EBM, Ψ ( 𝑏 ) ,shown in (4), is equal to the left-hand-side of inequality (9). Now, for each 𝑒 ∈ 𝐿 , the separationoracle computes the matching 𝑏 ∗ 𝑒 that maximizes the quantity Ψ ( 𝑏 ) among all matchings of M 𝑒 .A violating constraint (corresponding to matching 𝑏 ∗ 𝑒 ) is then found if Ψ ( 𝑏 ∗ 𝑒 ) > 𝑒 ∈ 𝐿 .Otherwise, the separation oracle concludes that no constraint is violated. Observe that the definition of the log-Nash social welfare implies lgN ( 𝑏 ) = Õ ( 𝑖,𝑗 ) ∈ 𝑏 ln 𝑣 𝑖 ( 𝑗 ) = Õ ( 𝑖,𝑗 ) ∈ 𝑏 Õ ( 𝑘,ℓ ) ∈ 𝑏 : 𝑘 ≠ 𝑖 ln 𝑣 𝑖 ( 𝑗 ) + ln 𝑣 𝑘 ( ℓ ) ( 𝑛 − ) . Hence, the only modification that is required in the approach we followed for the utilitarian socialwelfare is to replace the term 𝑣 𝑖 ( 𝑗 )+ 𝑣 𝑘 ( ℓ ) ( 𝑛 − ) by ln 𝑣 𝑖 ( 𝑗 )+ ln 𝑣 𝑘 ( ℓ ) ( 𝑛 − ) in the definition of 𝜓 . In this section, we extend the notion of interim envy-freeness by accompanying lotteries overallocations with payments to/from the agents. In this case, the definition of iEF uses both thevalue an agent has for item bundles and the payment she receives or contributes. We distinguishbetween three different types of payments. A vector of agent-dependent payments or
A-payments consists of a payment 𝑝 𝑖 for each agent 𝑖 ∈ N . More refined payments are defined using additionalinformation for an allocation instance. We say that the agents receive bundle-dependent payments or B-payments when each agent is associated with a payment of 𝑝 ( 𝑆 ) when she receives the bundleof items 𝑆 . Finally, we say that the agents get allocation-dependent payments or C-payments wheneach agent 𝑖 is associated with payment 𝑝 𝑖 ( 𝐴 ) in allocation 𝐴 .We now extend the notion of interim envy-freeness to pairs of lotteries and payments by distin-guishing between the three payment types. Definition 6.1.
We say that a pair of a lottery Q and a vector of A-payments p is iEF if for everypair of agents 𝑖, 𝑘 ∈ N and every bundle of items 𝑆 agent 𝑖 can get under Q , it holds 𝑣 𝑖 ( 𝑆 ) + 𝑝 𝑖 ≥ E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] + 𝑝 𝑘 . Definition 6.2.
We say that a pair of a lottery Q and a vector of B-payments p per bundle of itemsis iEF if for every pair of agents 𝑖, 𝑘 ∈ N and every bundle of items 𝑆 agent 𝑖 can get under Q , it holds 𝑣 𝑖 ( 𝑆 ) + 𝑝 ( 𝑆 ) ≥ E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 ) + 𝑝 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] . For C-payments, we give a more general definition that allows for marginal violations of iEF.The notion of 𝜖 -iEF will be useful later in Section 6.3. Definition 6.3.
Let 𝜖 ≥ . We say that a pair of a lottery Q and a vector of C-payments p peragent and allocation is 𝜖 -iEF if for every pair of agents 𝑖, 𝑘 ∈ N and every bundle of items 𝑆 agent 𝑖 can get under Q , it holds 𝑣 𝑖 ( 𝑆 ) + E 𝐴 ∼ Q [ 𝑝 𝑖 ( 𝐴 )| 𝐴 𝑖 = 𝑆 ] ≥ E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 ) + 𝑝 𝑘 ( 𝐴 )| 𝐴 𝑖 = 𝑆 ] − 𝜖 . The term iEF with C-payments is used alternatively to 0-iEF. We remark that the payments areadded to the value agents have for bundles in the above definitions. So, in general, the paymentsare assumed to be received by the agents. To represent payments that are contributed by the agents,we can allow negative entries in the payment vectors. We also remark that the definitions referto general allocation instances. Indeed, the result we present in Section 6.1 applies to generalinstances. Then, in Sections 6.2 and 6.3, we restrict our attention to matching instances and adaptthe definitions accordingly. oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou Q that can be complemented with vectors of A-payments p ,so that the pair Q , p is iEF with A-payments. There, our focus is on the existence of payments, withno additional restrictions on them. In Sections 6.2 and 6.3, we specifically consider two particularoptimization problems that involve iEF allocations with payments; we define these problems inthe following.In both optimization problems, we are given an allocation instance and the objective is to com-pute a lottery Q over allocations and a payment vector p so that the pair Q , p is iEF with payments.In subsidy minimization , the payments are subsidies given to the agents by an external authority. So,the corresponding entries in the payment vector p are constrained to be non-negative. The goal ofsubsidy minimization is to find an iEF allocation and accompanying payments, such that the totalexpected amount of subsidies is minimized; the objective is equal to Í 𝑖 ∈N 𝑝 𝑖 , E 𝐴 ∼ Q [ Í 𝑖 ∈N 𝑝 ( 𝐴 𝑖 )] ,or E 𝐴 ∼ Q [ Í 𝑖 ∈N 𝑝 𝑖 ( 𝐴 )] , depending on whether p is an A-, B-, or C-payment vector, respectively. Sub-sidy minimization is the generalization of the problem that was recently studied for deterministicallocations and envy-freeness in [14, 17, 29].Our second optimization problem is called utility maximization and can be thought of as anextension of rent division [26] to lotteries and iEF. There is a rent 𝑅 and the payments are contri-butions from the agents that, in expectation, should sum up to 𝑅 . So, the entries in the paymentvector p are constrained to be non-positive. The goal is to find an iEF allocation and accompany-ing payments, such that the minimum expected utility over all agents is maximized. Dependingon whether the problem asks for A-, B-, or C-payments, the utility of agent 𝑖 ∈ N from allocation 𝐴 is 𝑣 𝑖 ( 𝐴 𝑖 ) + 𝑝 𝑖 , 𝑣 𝑖 ( 𝐴 𝑖 ) + 𝑝 ( 𝐴 𝑖 ) , and 𝑣 𝑖 ( 𝐴 𝑖 ) + 𝑝 𝑖 ( 𝐴 ) , respectively.As we will see in Section 6.2, both subsidy minimization and utility maximization admit muchbetter solutions compared to their versions with deterministic allocations and envy-freeness thathad been previously studied in the literature. In addition, the quality of solutions depends onthe type of payments. We demonstrate that there is no general advantage of A- or B-paymentsagainst each other; this justifies the importance of both types of payments. Clearly, C-paymentsallow for the best possible solutions as they generalize both A- and B-payments. In Section 6.3, werestrict our focus on matching instances and show how to solve subsidy minimization and utilitymaximization efficiently, by exploiting the machinery we developed in Section 5. We now extend the notion of envy-freeability from recent works focusing on the use of subsidiesin fair division settings (e.g., see [14, 17, 29]), and earlier studies in rent division (e.g., see [4, 38]).Given an allocation instance and a lottery Q over allocations, we say that Q is iEF-able with A-payments if there is a vector p of A-payments so that the pair Q , p is iEF. Even though we will notneed these terms here, we can define the term iEF-ability with B- or C-payments analogously.Our main result in this section (Theorem 6.5) will be a characterization of the lotteries that areiEF-able with A-payments. The notion of the interim envy graph will be very useful; it extends thenotion of the envy-graph that is central in the characterization of envy-freeable allocations (see,e.g., [29]). Definition 6.4.
Given a lottery Q over allocations of the items in set I to the agents in set N ,the interim envy graph iEG ( Q , N , I) is a complete directed graph with 𝑛 nodes corresponding to theagents of N , and edge weights defined as 𝑤 ( 𝑖, 𝑘 ) = max 𝑆 ⊆I :Pr 𝐴 ∼ Q [ 𝐴 𝑖 = 𝑆 ] > (cid:8) E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] − 𝑣 𝑖 ( 𝑆 ) (cid:9) oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou for every directed edge ( 𝑖, 𝑘 ) . Our characterization follows; it extends well-known characterizations for deterministic envy-freeable allocations, e.g., see [26, 29]. A quick comparison reveals that the second condition inTheorem 6.5 is much less restrictive than an analogous condition for envy-freeability, which assertsthat envy-freeable allocations locally maximize the utilitarian social welfare among all possibleredistributions of the bundles to the agents. This justifies our claim that the space of iEF-ablelotteries is quite rich.
Theorem 6.5.
For a lottery Q over allocations, the following statements are equivalent:(i) Q is iEF-able with A-payments.(ii) For every agent 𝑖 ∈ N , let 𝑆 𝑖 be any bundle of items that is allocated to agent 𝑖 with positiveprobability according to Q . Also, let 𝜎 : N → N be any permutation of agents. Then, Õ 𝑖 ∈N 𝑣 𝑖 ( 𝑆 𝑖 ) ≥ Õ 𝑖 ∈N E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝜎 ( 𝑖 ) )| 𝐴 𝑖 = 𝑆 𝑖 ] . (iii) The interim envy graph iEG ( Q , N , I) has no cycle of positive weight. Proof. (i) ⇒ (ii). First, assume that statement (i) holds, i.e., that lottery Q is iEF-able with A-payments. This implies that there exists a vector p of A-payments so that the pair Q , p is iEF. Byapplying the iEF condition (Definition 6.1) for agent 𝑖 ∈ N , agent 𝜎 ( 𝑖 ) ∈ N , and bundle 𝑆 𝑖 , we get: 𝑣 𝑖 ( 𝑆 𝑖 ) + 𝑝 𝑖 ≥ E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝜎 ( 𝑖 ) )| 𝐴 𝑖 = 𝑆 𝑖 ] + 𝑝 𝜎 ( 𝑖 ) . By rearranging and summing these inequalities over all agents 𝑖 ∈ N , we get Õ 𝑖 ∈N (cid:0) 𝑣 𝑖 ( 𝑆 𝑖 ) − E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝜎 ( 𝑖 ) )| 𝐴 𝑖 = 𝑆 𝑖 ] (cid:1) ≥ Õ 𝑖 ∈N (cid:0) 𝑝 𝜎 ( 𝑖 ) − 𝑝 𝑖 (cid:1) . (10)Since 𝜎 is a permutation over the agents, it is Í 𝑖 ∈N 𝑝 𝜎 ( 𝑖 ) = Í 𝑖 ∈N 𝑝 𝑖 and the RHS of (10) is equalto 0. Then, inequality (10) is equivalent to statement (ii). (ii) ⇒ (iii). We now show that if the interim envy graph iEG ( Q , N , I) had a cycle of positiveweight, this would violate statement (ii) for a particular permutation, 𝜎 ∗ , of the agents and a se-lection of bundles 𝑆 ∗ , 𝑆 ∗ , ..., 𝑆 ∗ 𝑛 such that bundle 𝑆 ∗ 𝑖 is given to agent 𝑖 with positive probability foreach agent 𝑖 ∈ N under Q . Let 𝐶 be such a positive-weight cycle, consisting of the agents 𝑖 , 𝑖 , ..., 𝑖 | 𝐶 |− . Define 𝜎 ∗ to be the permutation of agents defined as 𝜎 ∗ ( 𝑖 ) = 𝑖 for each agent 𝑖 who does notbelong to cycle 𝐶 and as 𝜎 ∗ ( 𝑖 𝑗 ) = 𝑖 𝑗 + | 𝐶 | for each agent 𝑖 𝑗 (with 𝑗 = , , ..., | 𝐶 | −
1) in the cycle 𝐶 . Observe that E 𝐴 ∼ Q [ 𝑣 𝐴 𝜎 ∗( 𝑖 ) | 𝐴 𝑖 = 𝑆 𝑖 ] = 𝑣 𝑖 ( 𝑆 𝑖 ) for every agent 𝑖 not belonging to 𝐶 and every set 𝑆 𝑖 allocated to 𝑖 with positive probability. Hence, Õ 𝑖 ∈N\ 𝐶 𝑣 𝑖 ( 𝑆 𝑖 ) = Õ 𝑖 ∈N\ 𝐶 E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝜎 ∗ ( 𝑖 ) )| 𝐴 𝑖 = 𝑆 𝑖 ] . (11)Furthermore, define the bundles 𝑆 ∗ 𝑖 for each agent 𝑖 ∈ N as follows. For each agent 𝑖 that does notbelong to 𝐶 , let 𝑆 ∗ 𝑖 be any bundle that 𝑖 gets with positive probability according to Q . For agent 𝑖 𝑗 (with 𝑗 = , , ..., | 𝐶 | −
1) of the cycle 𝐶 , define 𝑆 ∗ 𝑖 𝑗 = arg max 𝑆 ⊆I :Pr 𝐴 ∼ Q [ 𝐴 𝑖𝑗 = 𝑆 ] > (cid:8) E 𝐴 ∼ Q [ 𝑣 𝑖 𝑗 ( 𝐴 𝑖 𝑗 + mod | 𝐶 | )| 𝐴 𝑖 𝑗 = 𝑆 ] − 𝑣 𝑖 𝑗 ( 𝑆 ) (cid:9) . oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou ( Q , N , I) is positive, we have(using the definition of edge weight in the interim envy graph in Definition 6.4)0 < | 𝐶 |− Õ 𝑗 = max 𝑆 ⊆I :Pr 𝐴 ∼ Q [ 𝐴 𝑖𝑗 = 𝑆 ] > (cid:8) E 𝐴 ∼ Q [ 𝑣 𝑖 𝑗 ( 𝐴 𝑖 𝑗 + | 𝐶 | )| 𝐴 𝑖 𝑗 = 𝑆 ] − 𝑣 𝑖 𝑗 ( 𝑆 ) (cid:9) = | 𝐶 |− Õ 𝑗 = (cid:16) E 𝐴 ∼ Q [ 𝑣 𝑖 𝑗 ( 𝐴 𝑖 𝑗 + | 𝐶 | )| 𝐴 𝑖 𝑗 = 𝑆 ∗ 𝑖 𝑗 ] − 𝑣 𝑖 𝑗 ( 𝑆 ∗ 𝑖 𝑗 ) (cid:17) = Õ 𝑖 ∈ 𝐶 (cid:0) E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝜎 ∗ ( 𝑖 ) )| 𝐴 𝑖 = 𝑆 ∗ 𝑖 ] − 𝑣 𝑖 ( 𝑆 ∗ 𝑖 ) (cid:1) , and, equivalently, Õ 𝑖 ∈ 𝐶 𝑣 𝑖 ( 𝑆 ∗ 𝑖 ) < Õ 𝑖 ∈ 𝐶 E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝜎 ∗ ( 𝑖 ) )| 𝐴 𝑖 = 𝑆 ∗ 𝑖 ] . (12)Now, by summing equation (12) with equation (11) for 𝑆 𝑖 = 𝑆 ∗ 𝑖 , we get a contradiction to statement(ii) for the particular permutation 𝜎 ∗ and the given selection of sets 𝑆 ∗ 𝑖 for 𝑖 ∈ N . (iii) ⇒ (i). Finally, assuming that the interim envy graph iEG ( Q , N , I) has no positive cycle,we will show that Q is iEF-able with A-payments. For every agent 𝑖 ∈ N , define her payment 𝑝 𝑖 to be the total weight in the longest (in terms of total length) path from node 𝑖 in the interim envygraph (this is well defined because iEG ( Q , N , I) has no positive cycles). Then, for every pair ofagents 𝑖, 𝑘 ∈ N , and for every bundle 𝑆 ′ agent 𝑖 gets with positive probability under Q , we have(using the definition of the edge weights in the interim envy graph) 𝑝 𝑖 ≥ 𝑝 𝑘 + 𝑤 ( 𝑖, 𝑘 ) = 𝑝 𝑘 + max 𝑆 ⊆I :Pr 𝐴 ∼ Q [ 𝐴 𝑖 = 𝑆 ] > (cid:8) E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ] − 𝑣 𝑖 ( 𝑆 ) (cid:9) ≥ 𝑝 𝑘 + E 𝐴 ∼ Q [ 𝑣 𝑖 ( 𝐴 𝑘 )| 𝐴 𝑖 = 𝑆 ′ ] − 𝑣 𝑖 ( 𝑆 ′ ) . By rearranging we get that the iEF condition for agent 𝑖 with respect to agent 𝑘 and the bundle 𝑆 ′ is satisfied. (cid:3) We now attempt a comparison between different types of payments. First, we remark that iEFlotteries with A-payments (or B-payments) can yield much lower total expected subsidies andmuch higher minimum expected utility for utility maximization instances, compared to envy-freeallocations with payments. This should be clear given Lemma 3.3; we give explicit bounds onsubsidy minimization and utility maximization with the next example.
Example 6.6.
Consider the instance at the left of Table 5. By the characterization of Halpernand Shah [29], we know that in matching instances only the most efficient allocation of items toagents is envy-freeable. Therefore, 𝑎 - 𝑏 - 𝑐 and 𝑎 - 𝑐 - 𝑏 are the only envy-freeable allocations and thisis possible with a payment of 1 / 𝑎 ) and no payments to the other two agents(or, to the other two items). In contrast, the lottery that has both allocations in its support withequal probability is iEF without any payments.Now, consider the matching instance at the right of Table 5 and let 𝑅 =
1. Again, 𝑎 - 𝑏 - 𝑐 and 𝑎 - 𝑐 - 𝑏 are the only envy-freeable allocations. The rent shares that make each of them EF are 0, 1 / / oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑎 𝑏 𝑐 / / / /
23 0 1 / / 𝑎 𝑏 𝑐 / / / /
23 0 1 / / Table 5. An instance of subsidy minimization (left) and utility maximization (right) with three agents whereenvy-freeness is inferior to iEF with A- and B-payments. iEF with payments 1 /
6, 5 /
12, and 5 /
12 by agents 1, 2, and 3, (or, to items 𝑎 , 𝑏 , 𝑐 ) which sum up to1. The (expected) utility of each agent is then 1 / (cid:3) We now compare A-payments to B-payments in terms of the quality of the solutions they yieldin the two optimization problems. The proof of Theorem 6.7 shows two instances, of subsidy min-imization and utility maximization, respectively. For subsidy minimization, the expected amountof subsidies achieved with B-payments is arbitrarily close to 0, while A-payments need a constantamount of subsidies. Similarly, in the utility maximization instance, the minimum expected agentutility is arbitrarily close to 0 with A-payments and considerably higher with B-payments.
Theorem 6.7.
The solution of subsidy minimization and utility maximization with B-paymentscan be strictly better than the solution of the corresponding problems with A-payments.
Proof.
For subsidy minimization, consider the matching instance at the left of Table 6. Under B-payments, consider the lottery Q that selects among allocations 𝑎 - 𝑏 - 𝑐 and 𝑎 - 𝑐 - 𝑏 equiprobably andthe corresponding payment vector p such that 𝑝 ( 𝑎 ) = 𝜖 , 𝑝 ( 𝑏 ) =
0, and 𝑝 ( 𝑐 ) = 𝜖 . It can be verifiedthat the pair Q , p is iEF with a total payment amount of 3 𝜖 . Any iEF pair with A-payments, however,would require a total payment of at least 1 / + 𝜖 . To see why, consider an iEF pair Q ′ , p ′ with asmaller total payment amount. If agent 1 gets item 𝑐 with positive probability, then Q ′ necessarilyincludes an allocation where two agents get their unique least valuable item. This leads to a cyclewith positive weight in the interim envy graph and, by Theorem 6.5, Q ′ cannot be part of any iEFpair. Therefore, agent 1 gets either item 𝑎 or 𝑏 in Q ′ . Note that when 2 gets 𝑎 it is always the casethat 1 gets 𝑏 and, so that agent 2 does not envy agent 1, it must be 𝑝 ≥ / + 𝜖 + 𝑝 > / + 𝜖 ; whenagent 3 gets 𝑎 the analysis is symmetric. So, agent 1 always gets item 𝑎 in Q ′ . If Q ′ randomizes overallocations 𝑎 - 𝑏 - 𝑐 and 𝑎 - 𝑐 - 𝑏 , then we must have both 1 / − 𝜖 + 𝑝 ≥ / + 𝜖 + 𝑝 and 1 / − 𝜖 + 𝑝 ≥ / + 𝜖 + 𝑝 which cannot occur. So, it remains to examine the case of the deterministic allocation 𝑎 - 𝑏 - 𝑐 (or 𝑎 - 𝑐 - 𝑏 which is symmetric). Indeed, the allocation 𝑎 - 𝑏 - 𝑐 with agent-payment vector ( / , , 𝜖 ) is an iEF pair with a total payment of 1 / + 𝜖 , as desired. 𝑎 𝑏 𝑐 / / / + 𝜖 / − 𝜖 / + 𝜖 / − 𝜖 𝑎 𝑏 𝑐 / / / + 𝜖 / − 𝜖 / + 𝜖 / − 𝜖 Table 6. An instance of subsidy minimization (left) and utility maximization (right) with three agents whereB-payments are superior, for an arbitrarily small 𝜖 > . For utility maximization and 𝑅 =
1, consider the instance at the right of Table 6. Under B-payments, consider the lottery Q that selects allocations 𝑎 - 𝑏 - 𝑐 and 𝑎 - 𝑐 - 𝑏 equiprobably and thepayment vector p such that 𝑝 ( 𝑎 ) = − / 𝑝 ( 𝑏 ) = − / − 𝜖 , and 𝑝 ( 𝑐 ) = − / + 𝜖 . One can again oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou Q , p is iEF where the expected utility of each agent is 1 /
12. Any iEF pair withA-payments, however, would lead to a minimum expected utility of at most 𝜖 /
3. Indeed, let Q ′ and p ′ be an iEF pair with minimum expected utility larger than 𝜖 /
3. Again, Theorem 6.5 suggests thatit cannot be that, in Q ′ , at least two agents get an item that they value at 0 as this would createa cycle of positive weight in the interim envy graph. So, agent 1 can only be assigned items 𝑎 or 𝑏 . Let agent 2 get item 𝑎 with positive probability (the argument for agent 3 is symmetric). Thenagent 1 must get item 𝑏 and, hence, the allocation 𝑏 - 𝑎 - 𝑐 is in Q ′ . It must be 𝑝 ≥ / + 𝜖 + 𝑝 and 𝑝 ≥ / − 𝜖 + 𝑝 , so that agent 2 does not envy 1 or 3. By summing the two inequalities and adding 𝑝 to both sides, and since 𝑝 + 𝑝 + 𝑝 = −
1, we get 𝑝 ≥
0. Since payments are non-positive, weobtain 𝑝 = 𝑝 = − / − 𝜖 and 𝑝 = − / + 𝜖 and the minimum expected utility is atmost 0. So, agent 1 is always assigned item 𝑎 in Q ′ . If Q ′ randomizes over allocations 𝑎 - 𝑏 - 𝑐 and 𝑎 - 𝑐 - 𝑏 , it must be 1 / − 𝜖 + 𝑝 ≥ / + 𝜖 + 𝑝 and 1 / − 𝜖 + 𝑝 ≥ / + 𝜖 + 𝑝 which cannot occur. So, thereremains the deterministic allocation 𝑎 - 𝑏 - 𝑐 (or 𝑎 - 𝑐 - 𝑏 which is symmetric). Indeed, the allocation 𝑎 - 𝑏 - 𝑐 forms an iEF pair with the agent-payment vector (− 𝜖 / , − / − 𝜖 / , − / + 𝜖 / ) with aminimum expected utility of 𝜖 /
3, as desired. (cid:3)
Perhaps surprisingly, there are also instances where A-payments are preferable to B-payments.
Theorem 6.8.
The solution of subsidy minimization and utility maximization with A-paymentscan be strictly better than the solution of the corresponding problems with B-payments.
Proof.
For subsidy minimization, consider the instance at the left of Table 7. For A-payments,consider the lottery Q that selects among allocations 𝑎 - 𝑏 - 𝑐 , 𝑎 - 𝑐 - 𝑏 , and 𝑏 - 𝑐 - 𝑎 equiprobably, togetherwith the payment vector p = ( , , 𝜖 ) . It can be verified that the pair Q , p is iEF with a total paymentof 3 𝜖 . An iEF pair with B-payments, however, would require payments at least 6 𝜖 . Indeed, consideran iEF pair Q ′ , p ′ with a smaller payment amount. If in Q ′ an agent gets with positive probabilityan item she values at 0, the total payment is at least 2 /
5. Hence Q ′ randomizes over (at most) 𝑎 - 𝑏 - 𝑐 , 𝑎 - 𝑐 - 𝑏 , and 𝑏 - 𝑐 - 𝑎 . If 𝑎 - 𝑏 - 𝑐 is not in Q ′ , then agent 2 deterministically gets item 𝑐 . Then, Q ′ cannotinclude both 𝑎 - 𝑐 - 𝑏 and 𝑏 - 𝑐 - 𝑎 as, then, both 2 / + 𝑝 ≥ / + 𝑝 and 1 / − 𝜖 + 𝑝 ≥ / + 𝜖 + 𝑝 musthold so that agents 1 and 3 do not envy each other. Regardless of whether Q ′ contains just 𝑎 - 𝑐 - 𝑏 orjust 𝑏 - 𝑐 - 𝑎 , the payment is at least 1 / 𝑏 - 𝑐 - 𝑎 is not in Q ′ ,then agent 1 deterministically gets item 𝑎 and the analysis is symmetric, leading to a payment of atleast 1 / 𝑎 - 𝑐 - 𝑏 is not in Q ′ , then a payment of at least 1 / Q ′ consists of all three allocations, this yields 1 / − 𝜖 + 𝑝 ( 𝑎 ) ≥ / + 𝜖 + 𝑝 ( 𝑏 ) and 1 / − 𝜖 + 𝑝 ( 𝑐 ) ≥ / + 𝜖 + 𝑝 ( 𝑏 ) so that agent 3 is not envious. This gives a lower bound of6 𝜖 on the payment. Indeed, the lottery that selects equiprobably among allocations 𝑎 - 𝑏 - 𝑐 , 𝑎 - 𝑐 - 𝑏 ,and 𝑏 - 𝑐 - 𝑎 together with payments 𝑝 ( 𝑎 ) = 𝜖 , 𝑝 ( 𝑏 ) =
0, and 𝑝 ( 𝑐 ) = 𝜖 is an iEF pair with a totalpayment of 6 𝜖 . 𝑎 𝑏 𝑐 / / / /
53 1 / − 𝜖 / + 𝜖 / − 𝜖 𝑎 𝑏 𝑐 / − 𝜖 / + 𝜖
02 0 11 / + 𝜖 / − 𝜖 /
10 2 / / Table 7. An instance of subsidy minimization (left) and utility maximization (right) with three agents whereA-payments are preferable, for an arbitrarily small 𝜖 > . For utility maximization and 𝑅 =
1, consider the instance at the right of Table 7. For A-payments,consider the lottery Q that selects allocation 𝑎 - 𝑏 - 𝑐 with probability 3 / 𝑎 - 𝑐 - 𝑏 with probability 2 / oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑏 - 𝑐 - 𝑎 with probability 3 /
10, together with the payment vector p = (− / − 𝜖 / , − / − 𝜖 / , − / + 𝜖 ) . It can be verified that the pair Q , p is iEF with minimum expected utility of 11 / + 𝜖 . An iEF pair with B-payments leads to a minimum expected utility of at most 2 𝜖 /
3. Assumeotherwise and consider a pair Q ′ , p ′ with minimum expected utility greater than 2 𝜖 / Q ′ cannotinclude 𝑐 - 𝑎 - 𝑏 as then it should hold 𝑝 ( 𝑎 ) > 𝑝 ( 𝑐 ) and 𝑝 ( 𝑐 ) > 𝑝 ( 𝑎 ) . Furthermore, Q ′ cannot includejust the allocation 𝑎 - 𝑐 - 𝑏 as then it must hold 𝑝 ′ ( 𝑎 ) − 𝑝 ′ ( 𝑏 ) ≥ / + 𝜖 and 𝑝 ′ ( 𝑎 ) − 𝑝 ′ ( 𝑏 ) ≤ / Q ′ includes 𝑎 - 𝑐 - 𝑏 , then, since 𝑐 - 𝑎 - 𝑏 is not in Q ′ , it must be 𝑝 ′ ( 𝑎 ) = 𝑝 ′ ( 𝑐 ) = − /
10 and 𝑝 ′ ( 𝑏 ) = − / Q ′ randomizes over allocations 𝑎 - 𝑏 - 𝑐 and 𝑏 - 𝑐 - 𝑎 ; any such choice yields 𝑝 ( 𝑎 ) = − / − 𝜖 / 𝑝 ′ ( 𝑏 ) = 𝑝 ′ ( 𝑐 ) = − / + 𝜖 / 𝜖 / (cid:3) 𝜖 -iEF allocations with C-payments In this section, we show how to solve efficiently subsidy minimization and utility maximizationwhen we are allowed to use C-payments (and sharp approximations of iEF). Our algorithms uselinear programming and the machinery we developed in Section 5.
Our result for subsidy minimization is the following.
Theorem 6.9.
Let 𝜖 > . Consider an instance of subsidy minimization with C-payments and letOPT be the value of its optimal solution. Our algorithm computes a lottery Q and a C-payment vector p of expected value OPT, so that the pair Q , p is 𝜖 -iEF with C-payments. To prove the theorem, we begin by defining a linear program for computing an iEF pair of lotteryand C-payment vector. We use the variable vector x to denote the lottery. By Definition 6.3, theiEF constraint for agent 𝑖 ∈ N who is assigned item 𝑗 ∈ I with positive probability under lottery x and another agent 𝑘 ∈ N \ { 𝑖 } is 𝑣 𝑖 ( 𝑗 ) + E 𝑏 ∼ x [ 𝑝 𝑖 ( 𝑏 )| 𝑏 ( 𝑖 ) = 𝑗 ] ≥ E 𝑏 ∼ x [ 𝑣 𝑖 ( 𝑏 ( 𝑘 )) + 𝑝 𝑘 ( 𝑏 )| 𝑏 ( 𝑖 ) = 𝑗 ] . (13)By multiplying the left-hand-side of (13) with the positive probability Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] , we get 𝑣 𝑖 ( 𝑗 ) · Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] + E 𝑏 ∼ x [ 𝑝 𝑖 ( 𝑏 )| 𝑏 ( 𝑖 ) = 𝑗 ] · Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] = Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑗 ) + Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · 𝑝 𝑖 ( 𝑏 ) . (14)By multiplying the right-hand-side of (13) again with Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] , we obtain E 𝑏 ∼ x [ 𝑣 𝑖 ( 𝑏 ( 𝑘 )) + 𝑝 𝑘 ( 𝑏 )| 𝑏 ( 𝑖 ) = 𝑗 ] · Pr 𝑏 ∼ x [ 𝑏 ( 𝑖 ) = 𝑗 ] = Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑘 )) + Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · 𝑝 𝑘 ( 𝑏 ) . (15)Hence, using (14) and (15), (13) yields Õ 𝑏 ∈M 𝑖𝑗 ( 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) + 𝑥 ( 𝑏 ) · 𝑝 𝑖 ( 𝑏 ) − 𝑥 ( 𝑏 ) · 𝑝 𝑘 ( 𝑏 )) ≥ . (16)Notice the products 𝑥 ( 𝑏 ) · 𝑝 𝑖 ( 𝑏 ) and 𝑥 ( 𝑏 ) · 𝑝 𝑘 ( 𝑏 ) in the above expression. In such terms, both factorsare unknowns that we have to compute. As 𝑝 𝑖 ( 𝑏 ) always appears multiplied with 𝑥 ( 𝑏 ) in the aboveexpressions, we can avoid non-linearities by introducing the variable 𝑡 𝑖 ( 𝑏 ) for every agent 𝑖 ∈ N oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑏 ∈ M to be thought of as equal to 𝑥 ( 𝑏 ) · 𝑝 𝑖 ( 𝑏 ) . With this interpretation in mind,equation (16) becomes Õ 𝑏 ∈M 𝑖𝑗 ( 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) + 𝑡 𝑖 ( 𝑏 ) − 𝑡 𝑘 ( 𝑏 )) ≥ . Furthermore, our objective is to minimize Í 𝑏 ∈ 𝑀 Í 𝑖 ∈N 𝑡 𝑖 ( 𝑏 ) since Õ 𝑏 ∈M Õ 𝑖 ∈N 𝑡 𝑖 ( 𝑏 ) = Õ 𝑏 ∈M Õ 𝑖 ∈N 𝑥 ( 𝑏 ) · 𝑝 𝑖 ( 𝑏 ) = E 𝑏 ∼ x "Õ 𝑖 ∈N 𝑝 𝑖 ( 𝑏 ) . Summarizing, our linear program for subsidy minimization isminimize Õ 𝑏 ∈M Õ 𝑖 ∈N 𝑡 𝑖 ( 𝑏 ) subject to Õ 𝑏 ∈M 𝑖𝑗 ( 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) + 𝑡 𝑖 ( 𝑏 ) − 𝑡 𝑘 ( 𝑏 )) ≥ ,𝑖 ∈ N , 𝑗 ∈ I , 𝑘 ∈ N \ { 𝑖 } Õ 𝑏 ∈M 𝑥 ( 𝑏 ) = 𝑥 ( 𝑏 ) ≥ , 𝑏 ∈ M 𝑡 𝑖 ( 𝑏 ) ≥ , 𝑏 ∈ M , 𝑖 ∈ N (17) Lemma 6.10.
The linear program (17) can be solved in polynomial time.
Proof.
We follow our approach from Section 5 and solve the linear program (17) by applyingthe ellipsoid method to its dual linear program:maximize 𝑧 subject to 𝑧 + Õ ( 𝑖,𝑗 ) ∈ 𝑏𝑘 ∈N\{ 𝑖 } ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) ≤ , 𝑏 ∈ M Í 𝑘 ∈N\{ 𝑖 } ( 𝑦 ( 𝑖, 𝑏 ( 𝑖 ) , 𝑘 ) − 𝑦 ( 𝑘, 𝑏 ( 𝑘 ) , 𝑖 )) − ≤ , 𝑖 ∈ N , 𝑏 ∈ M 𝑦 ( 𝑖, 𝑗, 𝑘 ) ≥ , 𝑖 ∈ N , 𝑗 ∈ I , 𝑘 ∈ N \ { 𝑖 } (18)The first set of constraints of the dual linear program (18) is very similar to the constraints of thelinear program (3) for the problem without payments. The separation oracle will check whether agiven assignment of values to the dual variables satisfies the first set of constraints by solving anappropriately defined instance of 2EBM. The second set of constraints is simpler; the separationoracle will perform several classical bipartite matching computations.In particular, consider the instance of 2EBM consisting of the complete bipartite graph 𝐺 (N , I , N × I) with edge-pair weight 𝜓 ( 𝑖, 𝑗, 𝑘, ℓ ) = ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( ℓ )) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) + 𝑧𝑛 ( 𝑛 − ) for each quadruple ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ X . Then, the total weight Ψ ( 𝑏 ) over all edge-pairs of matching 𝑏 ∈M is equal to the LHS of the constraint corresponding to matching 𝑏 among those in the first set ofconstraints. Furthermore, for agent 𝑖 ∈ N , consider the complete bipartite graph 𝐺 𝑖 (N , I , N × I) with weight 𝑐 ( 𝑖, ℓ ) = Í 𝑘 ∈N\{ 𝑖 } 𝑦 ( 𝑖, ℓ, 𝑘 ) − 𝑛 for every ℓ ∈ I and 𝑐 ( 𝑘, ℓ ) = − 𝑦 ( 𝑘, ℓ, 𝑖 ) − 𝑛 , for every 𝑘 ∈ N \ { 𝑖 } and ℓ ∈ I . Then, the total weight of edges in a perfect matching 𝑏 ∈ M is equal tothe LHS of the second constraint of the dual linear program (18) associated with matching 𝑏 ∈ M and agent 𝑖 ∈ N . oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑏 ∗ in 𝐺 . Then, it computes a max-imum edge-weighted perfect matching 𝑏 ∗ 𝑖 in graph 𝐺 𝑖 for 𝑖 ∈ N . If Ψ ( 𝑏 ∗ ) >
0, then it reports theconstraint associated with matching 𝑏 ∗ ∈ M from the first set of constraints of the dual linearprogram (18) as violating. If the total edge-weight of some matching 𝑏 ∗ 𝑖 is strictly positive, thenit reports the constraint associated with agent 𝑖 ∈ N and matching 𝑏 ∗ ∈ M from the second setof constraints of (18) as violating. Otherwise, i.e., if all perfect matchings have non-positive totaledge-pair or edge weight, the separation oracle reports that no constraint is violated. (cid:3) A solution of the linear program (17) naturally gives an iEF pair of lottery x and C-paymentsvector p when 𝑥 ( 𝑏 ) = 𝑏 ∈ M implies that 𝑡 𝑖 ( 𝑏 ) = 𝑖 ∈ N .Unfortunately, we have not excluded the case that 𝑥 ( 𝑏 ) = 𝑡 𝑖 ( 𝑏 ) > Lemma 6.11.
For every 𝜖 > , given any extreme solution to the linear program (17) of objectivevalue OPT, an 𝜖 -iEF lottery x ′ with C-payments p ′ of total expected value OPT can be computed inpolynomial time. Proof.
Let us assume that we have an extreme solution to linear program (17). Since the numberof non-trivial constraints is polynomial (at most 𝑛 ) and the rest just require that the variables 𝑥 ( 𝑏 ) and 𝑡 𝑖 ( 𝑏 ) are non-negative, we conclude that the number of variables that have non-zero valuesare at most polynomially many (i.e., at most 𝑛 ).Let 𝑉 max = max 𝑖 ∈N ,𝑗 ∈I 𝑣 𝑖 ( 𝑗 ) and set 𝛿 = 𝜖 𝑉 max . Denote by K the set of matchings 𝑏 ∈ M with 𝑥 ( 𝑏 ) > K the set of matchings 𝑏 such that 𝑥 ( 𝑏 ) = 𝑡 𝑖 ( 𝑏 ) > 𝑖 ∈ N .Our transformation sets 𝑥 ′ ( 𝑏 ) = ( − 𝛿 ) 𝑥 ( 𝑏 ) for every matching 𝑏 ∈ K and 𝑥 ′ ( 𝑏 ) = 𝛿 /|K | forevery matching 𝑏 ∈ K . The remaining variables are left intact.We first prove that x ′ is a lottery. Indeed, we have Õ 𝑏 ∈M 𝑥 ′ ( 𝑏 ) = Õ 𝑏 ∈K 𝑥 ′ ( 𝑏 ) + Õ 𝑏 ∈K 𝑥 ′ ( 𝑏 ) = ( − 𝛿 ) Õ 𝑏 ∈K 𝑥 ( 𝑏 ) + Õ 𝑏 ∈K 𝛿 |K | = . To see this, note that Í 𝑏 ∈K 𝑥 ( 𝑏 ) =
1. Furthermore, observe that Õ 𝑏 ∈M 𝑖𝑗 𝑥 ′ ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) = ( − 𝛿 ) Õ 𝑏 ∈M 𝑖𝑗 ∩K 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) + Õ 𝑏 ∈M 𝑖𝑗 ∩K 𝛿 |K | · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 )))≥ Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) − 𝛿 Õ 𝑏 ∈M 𝑖𝑗 ∩K 𝑥 ( 𝑏 ) · 𝑉 max − 𝛿 · 𝑉 max ≥ Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) − 𝛿 · 𝑉 max = Õ 𝑏 ∈M 𝑖𝑗 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) − 𝜖. oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝑥 ′ , we have (using the factthat the lottery x satisfies the constraint) Õ 𝑏 ∈M 𝑖𝑗 ( 𝑥 ′ ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) + 𝑡 𝑖 ( 𝑏 ) − 𝑡 𝑘 ( 𝑏 ))≥ Õ 𝑏 ∈M 𝑖𝑗 ( 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) + 𝑡 𝑖 ( 𝑏 ) − 𝑡 𝑘 ( 𝑏 )) − 𝜖 ≥ − 𝜖. To construct the payment vector p that makes x ′ 𝜖 -iEF, it suffices to set 𝑝 𝑖 ( 𝑏 ) = 𝑡 𝑖 ( 𝑏 )/ 𝑥 ′ ( 𝑏 ) forevery 𝑏 ∈ K ∪ K and 𝑝 𝑖 ( 𝑏 ) = (cid:3) In this section, we prove the following result for utility maximization.
Theorem 6.12.
Let 𝜖 > . Consider an instance of utility maximization with C-payments and letOPT be the value of its optimal solution. Our algorithm computes a lottery Q and a C-payment vector p of expected value at least OPT − 𝜖 , so that the pair Q , p is 𝜖 -iEF with C-payments. Again, our algorithm uses the solution of an appropriate linear program defined as follows:maximize 𝑞 subject to 𝑞 − Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑖 )) + Õ 𝑏 ∈M 𝑡 𝑖 ( 𝑏 ) ≤ 𝑖 ∈ N Õ 𝑏 ∈M 𝑖𝑗 ( 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) − 𝑡 𝑖 ( 𝑏 ) + 𝑡 𝑘 ( 𝑏 )) ≥ ,𝑖 ∈ N , 𝑗 ∈ I , 𝑘 ∈ N \ { 𝑖 } Õ 𝑏 ∈M 𝑥 ( 𝑏 ) = Õ 𝑏 ∈M Õ 𝑖 ∈N 𝑡 𝑖 ( 𝑏 ) = 𝑅𝑥 ( 𝑏 ) ≥ , 𝑏 ∈ M 𝑡 𝑖 ( 𝑏 ) ≥ , 𝑏 ∈ M , 𝑖 ∈ N (19)Compared to our previous linear program (17), this one has the extra variable 𝑞 , which denotes theminimum expected utility over all agents. Note that, here, we require 𝑡 𝑖 ( 𝑏 ) to be non-negative andinterpret 𝑡 𝑖 ( 𝑏 ) as − 𝑥 ( 𝑏 ) · 𝑝 𝑖 ( 𝑏 ) ; recall that payments in utility maximization are non-positive. Theobjective is to maximize 𝑞 and the first set of constraints guarantees that 𝑞 is at least the expectedutility E 𝑏 ∼ x [ 𝑣 𝑖 ( 𝑏 ( 𝑖 ) + 𝑝 𝑖 ( 𝑏 )] for every agent 𝑖 ∈ N . Indeed, observe that E 𝑏 ∼ x [ 𝑣 𝑖 ( 𝑏 ( 𝑖 )) + 𝑝 𝑖 ( 𝑏 )] = Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · ( 𝑣 𝑖 ( 𝑏 ( 𝑖 )) + 𝑝 𝑖 ( 𝑏 )) = Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑖 )) − Õ 𝑏 ∈M 𝑡 𝑖 ( 𝑏 ) . The second set of constraints captures the iEF requirements. Due to the definition of the paymentvariables, the signs of 𝑡 𝑖 ( 𝑏 ) and 𝑡 𝑘 ( 𝑏 ) in the second constraint are different compared to the linearprogram (17). The third set of constraints restricts x to be a lottery and the fourth one guaranteesthat the payments contribute to the rent.The proof of Theorem 6.12 follows by the next two lemmas. Lemma 6.13 uses again the ap-proach we developed in Section 5 to compute a solution to the linear program (19), and Lemma 6.14modifies the solution of the linear program to come up with a pair of lottery and correspondingC-payments that marginally violate iEF and achieve a marginally lower objective value. Lemma 6.13.
The linear program (19) can be solved in polynomial time. oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou Proof.
Once again, we solve the linear program (19) by applying the ellipsoid method to itsdual linear program, which is now the following:maximize 𝑧 + 𝑅𝑔 subject to 𝑧 + Õ 𝑖 ∈N 𝑤 𝑖 · 𝑣 𝑖 ( 𝑏 ( 𝑖 )) + Õ ( 𝑖,𝑗 ) ∈ 𝑏𝑘 ∈N\{ 𝑖 } ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( 𝑏 ( 𝑘 ))) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) ≤ , 𝑏 ∈ M Õ 𝑘 ∈N\{ 𝑖 } ( 𝑦 ( 𝑘, 𝑏 ( 𝑘 ) , 𝑖 ) − 𝑦 ( 𝑖, 𝑏 ( 𝑖 ) , 𝑘 )) − 𝑤 𝑖 + 𝑔 ≤ , 𝑖 ∈ N , 𝑏 ∈ M Õ 𝑖 ∈N 𝑤 𝑖 ≥ 𝑦 ( 𝑖, 𝑗, 𝑘 ) ≥ , 𝑖 ∈ N , 𝑗 ∈ I , 𝑘 ∈ N \ { 𝑖 } (20)Our separation oracle will check whether a given assignment of values to the dual variables satis-fies the first set of constraints by solving an instance of 2EBM consisting of a complete bipartitegraph 𝐺 (N , I , N × I) with edge-pair weight 𝜓 ( 𝑖, 𝑗, 𝑘, ℓ ) = ( 𝑣 𝑖 ( 𝑗 ) − 𝑣 𝑖 ( ℓ )) · 𝑦 ( 𝑖, 𝑗, 𝑘 ) + 𝑤 𝑖 · 𝑣 𝑖 ( 𝑗 ) 𝑛 − + 𝑧𝑛 ( 𝑛 − ) for each quadruple ( 𝑖, 𝑗, 𝑘, ℓ ) ∈ X . Indeed, the total edge-pair weight Ψ over all edge-pairs ofmatching 𝑏 ∈ M is equal to the LHS of the corresponding first constraint of the dual linear program(20).For the second set of constraints, the separation oracle will solve classical maximum bipartitematching problems. In particular, for agent 𝑖 ∈ N , it will compute a perfect matching of maximumtotal edge weight for the instance consisting of the complete bipartite graph 𝐺 𝑖 (N , I , N × I) withedge weight 𝑐 ( 𝑖, ℓ ) = − Õ 𝑘 ∈N\{ 𝑖 } 𝑦 ( 𝑖, ℓ, 𝑘 ) − 𝑤 𝑖 𝑛 + 𝑔𝑛 for every ℓ ∈ I and 𝑐 ( 𝑘, ℓ ) = 𝑦 ( 𝑘, ℓ, 𝑖 ) − 𝑤 𝑖 𝑛 + 𝑔𝑛 , for every 𝑘 ∈ N \ { 𝑖 } and ℓ ∈ I . Again, the total weight of edges in a perfect matching 𝑏 ∈ M isequal to the LHS of the second constraint of the dual linear program (20) associated with matching 𝑏 ∈ M and agent 𝑖 ∈ N .Finally, checking whether the dual variables satisfy the third constraint is trivial. (cid:3) Lemma 6.14.
For every 𝜖 > , given an extreme solution to the linear program (19) of objectivevalue OPT, an 𝜖 -iEF lottery x ′ with C-payments p ′ of total expected value at least OPT − 𝜖 can becomputed in polynomial time. Proof.
We repeat the proof of Lemma 6.11 to compute the lottery x ′ and the payment vector p so that the pair x ′ , p is 𝜖 -iEF. Set 𝛿 = 𝜖𝑉 max . The expected value of agent 𝑖 ∈ N for her item is E 𝑏 ∼ x ′ [ 𝑣 𝑖 ( 𝑏 ( 𝑖 ))] = Õ 𝑏 ∈M 𝑥 ′ ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑖 )) ≥ ( − 𝛿 ) Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑖 ))≥ Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑖 )) − 𝛿 Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · 𝑉 max ≥ Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑖 )) − 𝜖. The first inequality is due to the definition of x ′ (recall that 𝑥 ′ ( 𝑏 ) = ( − 𝛿 ) 𝑥 ( 𝑏 ) for 𝑏 ∈ K while 𝑥 ′ ( 𝑏 ) ≥ 𝑥 ( 𝑏 ) = 𝑏 ∈ M \ K ), the second one follows by the definition of 𝑉 max , and oannis Caragiannis, Panagiotis Kanellopoulos, and Maria Kyropoulou 𝛿 and since Í 𝑏 ∈M 𝑥 ( 𝑏 ) =
1. Hence, by the definition ofvariable 𝑞 in the solution of the linear program (19), we have that the minimum expected utility ismin 𝑖 ∈N { E 𝑏 ∼ x ′ [ 𝑣 𝑖 ( 𝑏 ( 𝑖 ) + 𝑝 𝑖 ( 𝑏 )]} ≥ min 𝑖 ∈N ( Õ 𝑏 ∈M 𝑥 ( 𝑏 ) · 𝑣 𝑖 ( 𝑏 ( 𝑖 )) − Õ 𝑏 ∈M 𝑡 𝑖 ( 𝑏 ) ) − 𝜖 ≥ 𝑞 − 𝜖 = 𝑂𝑃𝑇 − 𝜖, as desired. (cid:3) We believe that interim envy-freeness can be a very influential fairness notion and can play forlotteries of allocations the role that envy-freeness has played for deterministic allocations. Ourwork aims to reinvigorate the study of this notion; we hope this will further intensify the studyof fairness in random allocations overall. At the conceptual level, iEF can inspire new fairnessnotions, analogous to known relaxations of envy-freeness, such as envy-freeness up to one (EF1;see [15]) and up to any item (EFX; see [19]), that have become very popular recently. Even thoughit is very tempting, we refrain from proposing additional definitions here.An appealing feature of iEF lotteries is that they are efficiently computable in matching al-location instances. Of course, besides the importance of the ellipsoid algorithm in theory (see,e.g., [27, 35]), our methods have apparent limitations. Combinatorial algorithms for solving thecomputational problems addressed in Sections 5 and 6.3 are undoubtedly desirable. An intermedi-ate first step would be to design a combinatorial algorithm for the maximum edge-pair-weightedperfect bipartite matching problem (2EBM). This could be useful elsewhere, as 2EBM is a verynatural problem with possible applications in other contexts.At the technical level, there is room for several improvements of our results. Our bounds onthe price of iEF with respect to the average Nash social welfare have a gap between Ω (√ 𝑛 ) (The-orem 4.5) and O( 𝑛 ) (Theorem 4.2). Also, in Section 5, we show how to compute iEF lotteries thatmaximize the expected log-Nash social welfare. Although maximizing log-Nash and average Nashsocial welfare are equivalent goals for deterministic allocations, this is not the case for lotteries.Computing iEF lotteries of maximum expected average Nash welfare is elusive at this point.We left for the end the many open problems that are related to iEF with payments. Our charac-terization in Section 6.1 has been used only in the proof of Theorem 6.7. It would be interesting tosee whether it has wider applicability and, in particular, whether it can lead to efficient algorithmsfor computing iEF pairs of lotteries with A-payments. This is not clear to us, as our characteri-zation seems to be much less restrictive than existing ones for envy-freeability (which, e.g., havegiven rise to transforming the rent division problem to a bipartite matching computation [26]).Furthermore, characterizations of iEF-ability with B- or C-payments are currently elusive. Our re-sults in Section 6.2 reveal gaps on the quality of solutions for the two optimization problems thatthe different types of payments allow. Our analysis in the proofs of Theorems 6.7 and 6.8 is nottight; determining the maximum gap between A-, B, and C-payments on the quality of solutionsfor subsidy minimization and utility maximization is open. Finally, from the computational pointof view, our solutions to subsidy minimization and utility maximization yield pairs of lotteries andC-payments that are only approximately iEF. Can these problems be solved exactly? Also, solvingboth subsidy minimization and utility maximization with A- or B-payments will be of practicalimportance. What is the complexity of these problems? ACKNOWLEDGEMENTS
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