SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\text {s u b j e c t} \sum_ {1 \leq i \leq n} f _ {i, j} - f _ {i, j} ^ {\prime} = s _ {j} \quad \forall_ {j}
$$
$$
0 \leq f _ {i, j} ^ {\prime} \leq f _ {i, j} \quad \forall_ {i, j} \tag {2}
$$
Generally, it is hard to deal with the above optimization program. Here, we use an important property of  $C_{\mathcal{V}}^{\mathcal{F}}(\cdot)$  to obtain our results.
Lemma 2.5. Let  $\mathcal{F}$  be an arbitrary fractional allocation and assume that for every agent  $i$ ,  $v_{i}(\cdot)$  is  $\mathsf{XOS}$ . Then, for every partition of the items into fractional sets  $S_{1}, S_{2}, \ldots, S_{t}$ , we have  $\sum_{1 \leq k \leq t} C_{\mathcal{V}}^{\mathcal{F}}(S_{k}) \leq \sum_{1 \leq i \leq n} v_{i}(\mathcal{F})$ .
For a randomized allocation, we define two types of fairness guarantees, namely ex-ante and ex-post as in Definitions 2.6 and 2.7.
Definition 2.6 (ex-ante). Given a randomized allocation  $\mathcal{R}$ , we say  $\mathcal{R}$  is  $\alpha$ -MMS ex-ante, if for every agent  $i$ , we have  $v_{i}(\mathcal{R}) \geq \alpha$ . Similarly,  $\mathcal{R}$  is  $\alpha$ -proportional, if for every agent  $i$ ,  $v_{i}(\mathcal{R}) \geq \alpha \pi_{i}$ .
Definition 2.7 (ex-post). An allocation  $\mathcal{R}$  is  $\alpha$ -MMS ex-post, if for every allocation  $\mathcal{A} \in D(\mathcal{R})$ , we have that allocation  $\mathcal{A}$  is  $\alpha$ -MMS. Also, we say  $\mathcal{R}$  is  $\alpha$ -proportional ex-post if every allocation in the support of  $\mathcal{R}$  is  $\alpha$ -proportional.
One tool that we refer to in this paper is the result of Babaioff et al. (2022) for converting a fractional allocation into a faithful randomized allocation.
Theorem 2.8 (Proved by Babaioff et al. (2022)). Assume that the valuations are additive and let  $\mathcal{F}$  be a fractional allocation. Then there exists a randomized allocation  $\mathcal{R}$  such that the ex-ante utility of the agents for  $\mathcal{R}$  is the same as the utility of the agents in  $\mathcal{F}$ , and for every allocation  $\mathcal{A}$  in the support of  $\mathcal{R}$  the following holds:  $\forall i, v_i(\mathcal{A}_i) \geq v_i(\mathcal{R}) - \max_{j : f_i, j \notin \{0,1\}} v_i(\{b_j\})$ .
In this paper, we use a more delicate analysis of the method used in Theorem 2.8 to convert fractional allocations into randomized ones. This helps us improve our ex-post approximation guarantees.
3 Ex-ante Guarantees
In this section, our goal is to explore the possibility of designing randomized allocation that is  $\alpha$ -MMS ex-ante or  $\alpha$ -proportional ex-ante. Note that, in contrast to the additive case, for XOS valuations there is no meaningful correspondence between proportionality and maximin-share; proportional share can be larger or smaller than maximin-share. Recall that for the additive case, we always have  $\pi_i \geq \mathrm{MMS}_i$  and therefore, maximin-share is implied by proportionality. For fractionally subadditive valuations, however,  $\pi_i$  can be as small as  $\mathrm{MMS}_i / n$ .
For the additive setting, a simple fractional allocation that allocates a fraction  $1 / n$  of each item to each agent guarantees proportionality and consequently maximin-share. Using Theorem 2.8 one can convert this allocation to a randomized allocation that is proportional ex-ante. In Observation 3.1 we show that proportionality can be guaranteed ex-ante for XOS valuations.
Observation 3.1. Every randomized allocation that allocates each item with probability  $\frac{1}{n}$  to each agent is proportional ex-ante.
In contrast to the additive setting, finding an allocation that guarantees maximin-share is not trivial. Indeed, the simple fractional allocation that guarantees proportionality in Observation 3.1 can be as bad as  $O(1/n)$ -MMS.
Observation 3.2. Given any instance  $\mathcal{I}$ , let  $\mathcal{F}(\mathcal{I})$  be the fractional allocation that allocates a fraction  $1/n$  of each item to each agent. Then, there exists an instance  $\mathcal{I}$  such that the maximin-share guarantee of  $\mathcal{F}(\mathcal{I})$  is  $O(1/n)$ .
Generally, there are two main challenges in the process of designing a randomized allocation that guarantees an approximation of maximin-share. In contrast to the additive setting, finding a fractional or randomized allocation that approximates maximin-share is not easy. As well as that, transforming a fractional allocation into a randomized one is not straightforward. Indeed, as we show in Lemma 3.3, neither fractional allocations nor randomized allocations can guarantee MMS. We prove an upper bound on the best approximation guarantee of each one of these allocation types.
Lemma 3.3. For XOS valuations, the best MMS guarantee for fractional allocations and the best ex-ante MMS guarantee for randomized allocation is upper bounded by  $3 / 4$ .
Before we prove our lower bound on the maximin-share guarantee for randomized allocations, we note that another challenge about XOS valuations is that in sharp contrast to additive valuations, transforming a fractional allocation to a randomized one is not easy. Indeed, we can show that for a fractional allocation  $\mathcal{F}$  there might be randomized allocations  $\mathcal{R}$  and  $\mathcal{R}'$  with different utility guarantees for the agents, such that in both  $\mathcal{R}$  and  $\mathcal{R}'$  the probability that each item  $b_{j}$  is allocated to agent  $i$  is equal to  $f_{i,j}$ . Example 3.4 gives more insight into this challenge.
Example 3.4. Consider the following instance: there are  $n^2$  items. The valuation of agent  $i$  is an XOS set function consisting of  $n$  additive valuation functions as follows: partition the items into  $n$  bundles each with  $n$  items. For each additive function  $u_{i,k}$ , the value of each item in the  $k^{\text{th}}$  bundle is  $1/n$  and the value of the rest of the items is 0. Define allocations  $\mathcal{R}$  and  $\mathcal{R}'$  as follows: (i): allocation  $\mathcal{R}$  allocates each item to each agent with probability  $1/n$ , and (ii): allocation  $\mathcal{R}'$  considers a random permutation of the bundles in the optimal MMS partition of the agents and allocates the  $i^{\text{th}}$  bundle in the permutation to agent  $i$ . In both of these allocations, each item is allocated to each agent with probability  $1/n$ . However, the maximin-share guarantee of  $\mathcal{R}$  is  $O(\log n/n)$ . On the other hand, allocation  $\mathcal{R}'$  guarantees value 1 to all agents.
Despite these hurdles, in Theorem 3.5 we show that there exists a randomized allocation that guarantees  $1/4$ -MMS to all the agents ex-ante. To prove Theorem 3.5, we first show that a fractional allocation exists that is  $1/4$ -MMS. Next, we convert it to a randomized allocation. Theorem 3.5 along with Lemma 3.3 leave a gap of  $[1/4, 3/4)$  between the best upper bound and the best lower-bound for the maximin-share guarantee of fractional allocations in the XOS setting.
Theorem 3.5. There exists a randomized allocation that is  $1/4$ -MMS ex-ante.
4 Ex-ante and Ex-post Guarantees
Unfortunately, the randomized allocation obtained by Theorem 3.5 has no ex-post fairness guarantee. The issue is that we use Theorem 2.8 to convert the fractional allocation into a randomized one. However, Theorem 2.8 only guarantees that the ex-post value of each agent is at least the value of her fractional allocation minus the value of the heaviest item which is partially (and not fully) allocated to her in the fractional allocation. However, currently, we have no upper bound on the value of the allocated items, and therefore, the ex-post value of an agent might be close to 0. To resolve this, we allocate valuable items beforehand to keep the value of the remaining items as small as possible. In Lemma 4.1 we explain a simple and very practical fact that is frequently used in the previous studies. Ghodsi et al. (2018); Seddigin and Seddigin (2022); Barman and Krishna Murthy (2017); Amanatidis et al. (2019).
Lemma 4.1. Removing one item and one agent from the instance does not decrease the maximin-share value of the remaining agents for the remaining items.
Given that our goal is to construct a randomized allocation which is  $1/4$ -MMS ex-ante, by Lemma 4.1 we can assume without loss of generality that the value of each item to each agent is less than  $1/4$ ; otherwise, we can reduce the problem using Lemma 4.1. However, a combination of this assumption and Theorem 3.5 still gives no ex-post guarantee. To improve the ex-post guarantee, we revisit the proof of Theorem 2.8 and show that for our setting, a stronger guarantee can be achieved using the matching method for converting a fractional allocation into a randomized one.
Lemma 4.2. Assume that the valuations are additive and let  $\mathcal{F}$  be a complete fractional allocation with  $f_{ij} \in \{0,1/2,1\}$  for all  $i$  and  $j$ . Then there is a randomized allocation  $\mathcal{R}$  with  $D(\mathcal{R}) = \{\mathcal{A}^1, \mathcal{A}^2\}$ , such that
- For every agent  $i$  we have  $v_{i}(\mathcal{R}) = v_{i}(\mathcal{F})$ .
- For every agent  $i$  we have
$$
\min \left\{v _ {i} (\mathcal {A} _ {i} ^ {1}), v _ {i} (\mathcal {A} _ {i} ^ {2}) \right\} \geq v _ {i} (\mathcal {R}) - \frac {\max \{v _ {i} (b _ {j}) \mid f _ {i j} = 1 / 2 \}}{2}.
$$
In Lemma 4.2, we show that we can find a fractional allocation with a special structure that makes the transformation step more efficient. These ideas together help us achieve a randomized allocation with  $1/4$ -MMS guarantee ex-ante and  $1/8$ -MMS guarantee ex-post. In order to do so, we use the following algorithm:
1. While there exists an item  $b_{j}$  with value at least  $1/4$  to an agent  $i$ , allocate  $b_{j}$  to agent  $i$  and remove  $i$  and  $b_{j}$  respectively from  $\mathcal{N}$  and  $\mathcal{M}$ .
2. For the remaining agents  $[n]$  and items proceed as follows: for every agent  $i$ , define  $\bar{v}_i$  as follows: for every subset  $S$  of items,  $\bar{v}_i(S) = \min(t, v_i(S))$ . Let  $\bar{v} = (\bar{v}_1, \dots, \bar{v}_n)$  and return a half-integral allocation  $\mathcal{F}$  that maximizes the social welfare with respect to  $\bar{v}$ , i.e.,  $\mathcal{F} = \arg \max_{F \in \Pi} \sum_{i \in \mathcal{N}'} \bar{v}_i(F_i)$  where  $\Pi$  is the set of all half-integral allocations of the remaining items to the remaining agents.
3. Convert  $\mathcal{F}$  into a randomized allocation using Lemma 4.2.
The goal in Step 2 is to find a fractional allocation that is  $1/4$ -MMS. However, we want this allocation to have a special structure that facilitates constructing the randomized allocation. Therefore, instead of directly choosing the allocation that maximizes social welfare, we consider  $\bar{v}_i$  as the valuation function of agent  $i$  and return a half-integral allocation  $\mathcal{F}$ .
Lemma 4.3.  $\mathcal{F}$  is  $1 / 4$ -MMS.
Now we are ready to prove Theorem 1.1.
Proof. The ex-ante guarantee follows from Lemmas 4.3 and 1.
Let  $\mathcal{A}^1$  and  $\mathcal{A}^2$  be the integral allocations obtained by Lemma 4.2. Consider any agent  $i$ , and let  $u_{i,i'}$  be such that  $v_i(\mathcal{R}) = \sum_j f_{ij} u_{i,i'}(b_j)$ . Then, by Lemma 4.2 for  $r \in \{1,2\}$  we have
$$
u _ {i, i ^ {\prime}} \left(\mathcal {A} _ {i} ^ {r}\right) \geq u _ {i, i ^ {\prime}} \left(\mathcal {R} _ {i} ^ {\ell}\right) - \frac {\max  \left\{u _ {i , i ^ {\prime}} \left(b _ {j}\right) \mid f _ {i , j} = 1 / 2 \right\}}{2}
$$
and since by Lemma 4.1 we know the value of each item for each agent is less than  $1/4$ , we have
$$
v _ {i} \left(\mathcal {A} _ {i} ^ {\ell}\right) \geq v _ {i} \left(\mathcal {R} _ {i}\right) - \frac {\max  _ {j} v _ {i} \left(b _ {j}\right)}{2} > \frac {1}{4} - \frac {1}{8} = \frac {1}{8}.
$$
Hence, the ex-post guarantee holds as well.
5 3/13-MMS Allocation
In this section, our goal is to improve the approximation guarantee of MMS for deterministic allocations in the fractionally subadditive setting. We show that guaranteeing a factor  $3/13 \approx 0.230769$  of maximin-share to all the agents is always possible. Before this work, the best approximation guarantee for maximin-share in the XOS setting was 0.219225-MMS Seddighin and Seddighin (2022).
Our algorithm for improving the ex-post guarantee is based on our previous algorithms plus two additional steps and a more in-depth analysis. In this algorithm, before finding the allocation that maximizes social welfare, we strengthen our upper bound on the value of items. For this, we add two more steps to our algorithm in which we satisfy some of the agents with two items and three items. In contrast to the first step (i.e., allocating single items to agents), these steps might decrease the maximin-share value of the remaining agents for the remaining items. Let  $t = 6/13$ . The goal is to find a  $t/2$ -MMS allocation. Our allocation algorithm is as follows:
1. If there exists an item  $b_{j}$  with value at least  $t / 2$  to an agent  $i$ , allocate  $b_{j}$  to agent  $i$  and remove  $i$  and  $b_{j}$  respectively from  $\mathcal{N}$  and  $\mathcal{M}$ .
2. If there exists a pair of items  $b_{j}, b_{k}$  with the total value of at least  $t / 2$  to some agent  $i$ , allocate  $\{b_{j}, b_{k}\}$  to agent  $i$  and remove  $i$  and  $\{b_{j}, b_{k}\}$  from  $\mathcal{N}$  and  $\mathcal{M}$  respectively.
3. If there exists a triple of items  $b_{j}, b_{k}, b_{s}$  with the total value of at least  $t / 2$  to some agent  $i$ , allocate  $\{b_{j}, b_{k}, b_{s}\}$  to agent  $i$  and remove  $i$  and  $\{b_{j}, b_{k}, b_{s}\}$  from  $\mathcal{N}$  and  $\mathcal{M}$  respectively.
4. For the remaining agents  $\mathcal{N}'$  and items  $\mathcal{M}'$ , proceed as follows: for every agent  $i$ , define  $\bar{v}_i$  as follows: for every subset  $S$  of items,  $\bar{v}_i(S) = \min(t, v_i(S))$ . Let  $\bar{v} = (\bar{v}_1, \dots, \bar{v}_n)$  and return an allocation  $\mathcal{A}$  that maximizes the social welfare with respect to  $\bar{v}$ , i.e.,  $\mathcal{A} = \arg \max_{A \in \Pi} \sum_{i \in \mathcal{N}'} \bar{v}_i(A_i)$  where  $\Pi$  is the set of all allocations of  $\mathcal{M}'$  to  $\mathcal{N}'$ .
In the rest of this section, we analyze the above algorithm. By Lemma 4.1, after Step 1, the MMS value of all the agents is at least 1. Let  $n$  be the number of remaining agents after Step 1. We denote by  $n_1$  and  $n_2$ , the number of agents that are satisfied in Steps 2 and 3 respectively and let  $n' = n - n_1 - n_2 = |\mathcal{N}'|$  be the number of remaining agents after Step 3. In contrast to the first step, Step 2 and 3 might decrease the maximin-share value of the remaining agents for the remaining items. However, we prove that the remaining items satisfy special structural properties.
Observation 5.1. Since no item can satisfy any remaining agent after Step 1, for every agent  $i$  and every item  $b_{j}$ , we have  $v_{i}(\{b_{j}\}) < t / 2$ .
Also, by the method that we allocate the items in Step 3, after this step Observation 5.2 holds.
Observation 5.2. Since after Step 3, no triple of items can satisfy an agent, for every different item  $b_{j}, b_{k}, b_{s}$  and every agent  $i$  we have  $v_{i}(\{b_{j}, b_{k}, b_{s}\}) < t / 2$ .
Note that since the valuations are XOS, Observation 5.2 implies no upper bound better than  $t / 2$  on the value of a single item to an agent. For example, consider the following extreme scenario: for a small constant  $\epsilon > 0$ , the value of every subset of items to agent  $i$  is equal to  $t / 2 - \epsilon$ . It is easy to check that this valuation function is XOS. For this case, the value of every triple of items is also equal to  $t / 2 - \epsilon$ , but this implies no upper bound better than  $t / 2$  on the value of a single item.
Lemma 5.3. Fix a remaining agent  $i$  and consider the  $n$  bundles with value at least 1 in an MMS partition of agent  $i$  after Step 1. Put these bundles into 4 different sets  $B_0, B_1, B_2, B_{\geq 3}$ , where for  $0 \leq \ell \leq 2$ , set  $B_{\ell}$  contains bundles that lose exactly  $\ell$  items in Steps 2 and 3, and  $\overline{B}_{\geq 3}$  contains bundles that lose more than 4 items in these steps. After Step 3, the following inequality holds:  $n' \leq |B_0| + \frac{2}{3} |B_1| + \frac{1}{3} |B_2|$ .
Finally, in Step 4, we find the integral allocation  $\mathcal{A}$  that maximizes social welfare with respect to  $\bar{v}$  for the remaining agents. Let  $Z = \sum_{i\in \mathcal{N}'}\bar{v}_i(\mathcal{A}_i)$ . Since for each remaining agent  $i$ ,  $\bar{v}_i(\mathcal{A}_i)$  is upper-bounded by  $t$ , we have  $Z\leq n't$ . If for every agent  $i$ ,  $v_{i}(\mathcal{A}_{i})\geq t / 2$  holds, then  $\mathcal{A}$  is  $t / 2$ -MMS, and we are done. Therefore, for the rest of this section, assume that for an agent  $i^{*}$ , we have  $v_{i^{*}}(\mathcal{A}_{i^{*}}) < t / 2$ .
Let  $B_0, B_1$ , and  $B_2$  be the sets defined for agent  $i^*$  in Lemma 5.3. In Lemma 5.4, we give lower bounds on the contribution of the bundles in  $B_0, B_1$  and  $B_2$  to  $\mathcal{A}$  respectively.
Lemma 5.4. After Step 3, for all bundles
-  $X \in B_0$ , there exists a partition of  $X$  into  $X^1$  and  $X^2$  such that  $C_{\bar{v}}^{\mathcal{A}}(X^1) + C_{\bar{v}}^{\mathcal{A}}(X^2) \geq t$ .
-  $X \in B_1$ , there exists a partition of  $X$  into  $X^1$  and  $X^2$  such that  $C_{\bar{v}}^{\mathcal{A}}(X^1) + C_{\bar{v}}^{\mathcal{A}}(X^2) \geq \frac{2}{3} t$ .
$X\in B_2,C_{\bar{v}}^{\mathcal{A}}(X)\geq \frac{1}{2} t.$
Proof. Let  $\mathcal{A}$  be the output of our Algorithm. Towards a contradiction, assume for agent  $i^{*}$ ,  $v_{i^{*}}(\mathcal{A}_{i^{*}}) < 3 / 13 = t / 2$ . For all agents  $i$  which are removed during the first three steps, we have  $v_{i}(\mathcal{A}_{i}) \geq t / 2 = 3 / 13$ . Therefore,  $i^{*} \in \mathcal{N}'$ . For all  $X \in B_{0}$ , let  $X^{1}$  and  $X^{2}$  be as defined in
Lemma 5.4. We have
$$
\begin{array}{l} t \left(n ^ {\prime} - \frac {1}{2}\right) > \sum_ {i \in \mathcal {N} ^ {\prime}} \bar {v} _ {i} \left(\mathcal {A} _ {i}\right) \quad (\text {f o r a l l} i \in \mathcal {N} ^ {\prime}, \bar {v} _ {i} \left(\mathcal {A} _ {i}\right) \leq t \text {a n d} \bar {v} _ {i *} \left(\mathcal {A} _ {i *}\right) <   t / 2) \\ \geq \sum_ {X \in B _ {0}} \left(C _ {\bar {v}} ^ {\mathcal {A}} \left(X ^ {1}\right) + C _ {\bar {v}} ^ {\mathcal {A}} \left(X ^ {2}\right)\right) + \sum_ {X \in B _ {1}} \left(C _ {\bar {v}} ^ {\mathcal {A}} \left(X ^ {1}\right) + C _ {\bar {v}} ^ {\mathcal {A}} \left(X ^ {2}\right)\right) + \sum_ {X \in B _ {2}} C _ {\bar {v}} ^ {\mathcal {A}} (X) (Lemma 2.5) \\ \geq t \left| B _ {0} \right| + \frac {2}{3} t \left| B _ {1} \right| + \frac {1}{2} t \left| B _ {2} \right| \geq t n ^ {\prime}. (Lemma 5.4 and Lemma 5.3) \\ \end{array}
$$
In this paper, we developed randomized and deterministic allocations that guarantee approximations of maximin-share for fractionally subadditive valuations.
For deterministic allocations, to achieve a better approximation with the same technique, one idea would be to allocate more items to agents in the first phase. We believe it is promising that this extension gives a better approximation factor. However, extending the analysis is not trivial, and new ideas will be necessary to prove its correctness. A more fundamental obstacle lies in the fact that, regardless of the number of items allocated in the first phase, the approximation factor converges to 1/4. Therefore, developing new techniques is imperative to improve the approximation factor beyond 1/4.
For randomized allocations, our goal was to guarantee a fraction of MMS ex-ante, and a smaller fraction of MMS ex-post. Several interesting questions remain open. The most straight-forward direction is to improve approximation guarantees for both ex-ante and ex-post cases. None of our results are proved to be tight. Another notable point is that the running time of none of our algorithms is polynomially bounded. Are there efficient algorithms which achieve the same guarantees?
Akrami, H. and Garg, J. (2024). Breaking the 3/4 barrier for approximate maximin share. In Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA, 2024.
Akrami, H., Garg, J., Sharma, E., and Taki, S. (2023a). Simplification and improvement of MMS approximation. CoRR, abs/2303.16788.
Akrami, H., Garg, J., and Taki, S. (2023b). Improving approximation guarantees for maximin share. CoRR, abs/2307.12916.
Aleksandrov, M., Aziz, H., Gaspers, S., and Walsh, T. (2015). Online fair division: Analysing a food bank problem. In *IJCAI* 2015, Buenos Aires, Argentina, July 25-31, 2015, pages 2540-2546. AAAI Press.
Amanatidis, G., Ntokos, A., and Markakis, E. (2019). Multiple birds with one stone: Beating 1/2 for efx and gmms via envy cycle elimination. arXiv preprint arXiv:1909.07650.
Aziz, H. (2020). A probabilistic approach to voting, allocation, matching, and coalition formation. CoRR, abs/2002.10171.
Aziz, H., Ganguly, A., and Micha, E. (2023). Best of both worlds fairness under entitlements. AAMAS.
Babaioff, M., Ezra, T., and Feige, U. (2021). Fair and truthful mechanisms for dichotomous valuations. In AAAI 2021, IAAI 2021, EAAI 2021, Virtual Event, February 2-9, 2021, pages 5119-5126. AAAI Press.
Babaioff, M., Ezra, T., and Feige, U. (2022). On best-of-both-worlds fair-share allocations. In WINE 2022, Troy, NY, USA, December 12–15, 2022, Proceedings, pages 237–255. Springer.
Nash Jr, J. F. (1950). The bargaining problem. *Econometrica: Journal of the Econometric Society*, pages 155-162.
Plaut, B. and Roughgarden, T. (2020). Almost envy-freeness with general valuations. SIAM J. Discret. Math., 34(2):1039-1068.
Procaccia, A. D. and Wang, J. (2014). Fair enough: Guaranteeing approximate maximin shares. EC '14, page 675-692, New York, NY, USA. Association for Computing Machinery.
Seddighin, M. and Seddighin, S. (2022). Improved maximin guarantees for subadditive and fractionally subadditive fair allocation problem. In AAAI 2022, IAAI 2022, EAAI 2022 Virtual Event, February 22 - March 1, 2022. AAAI Press.
Steinhaus, H. (1948). The problem of fair division. *Econometrica*, 16(1).
Uziahu, G. B. and Feige, U. (2023). On fair allocation of indivisible goods to submodular agents.
Varian, H. R. (1973). Equity, envy, and efficiency.