Dynamic Fair Division with Partial Information
Abstract
We consider the fundamental problem of fairly and efﬁciently allocating T indivisible items among n agents with additive preferences.
The items become available over a sequence of rounds, and every item must be allocated immediately and irrevocably before the next one arrives.
Previous work shows that when the agents’ valuations for the items are drawn from known distributions, it is possible (under mild technical assumptions) to ﬁnd allocations that are envy-free with high probability and Pareto efﬁcient ex-post.
We study a partial-information setting, where it is possible to elicit ordinal but not cardinal information.
When a new item arrives, the algorithm can query each agent for the relative rank of this item with respect to a subset of the past items.
When values are drawn from i.i.d.
distributions, we give an algorithm that is envy-free and (1 − (cid:15) ) -welfare-maximizing with high probability.
We provide similar guarantees (envy-freeness and a constant approximation to welfare with high probability) even with minimally expressive queries that ask for a comparison to a single previous item.
For independent but non-identical agents, we obtain envy-freeness and a constant approximation to Pareto efﬁciency with high probability.
We prove that all our results are asymptotically tight.
Introduction
We consider the following fundamental problem in fair division.
A set of T indivisible items, arriving one at a time, must be allocated among a set of n agents with additive preferences.
The value v i,t that agent i has for the item in round t is realized once the item arrives.
Each item must be allocated immediately and irrevocably upon arrival and we ask that the overall allocation is fair and efﬁcient .
Previous work on this problem shows that, despite the uncertainty about future valuations, one can achieve simultaneous fairness and efﬁciency when agents’ values are stochastic.
Speciﬁcally, when each v i,t is drawn i.i.d.
from a distribution D , the simple algorithm that maximizes welfare — each item is allocated to the agent with the highest value — is envy-free with high probability and (obviously) ex-post Pareto efﬁcient [DGK + 14, KPW16].
The same guarantee holds for independent and non-identical agents ( v i,t is drawn from an agent-speciﬁc distribution D i ) for the algorithm that maximizes weighted welfare
[BG22].
Even when agents’ valuations for an item are correlated (but items are independent), Pareto efﬁciency ex-post is compatible with strong fairness guarantees (“envy-freeness with high probability or envy-freeness up-to-one item ex-post”)
[ZP20].
Despite the computational simplicity of (most of) the aforementioned algorithms, an unappealing aspect, especially from a practical perspective, is the requirement that agents report an exact numerical value for each item.
Eliciting expressive additive valuations can be impractical, e.g., due to agents’ cognitive limitations.
Motivated by such considerations, a growing body of work in AI studies what can be achieved by algorithms that only elicit ordinal information.
This idea originates from [PR06], who deﬁned the notion of distortion to measure the worst-case deterioration of an aggregate cardinal objective (e.g., utilitarian social welfare) due to only having access to preferences of limited expressiveness, particularly ordinal rankings.
Recent works prove bounds on the distortion in the context of many problems in social choice, e.g., voting [CNPS17, GKM17, MSW20, MW19, Kem20a, Kem20b, GHS20], matching [FRFZ14], and participatory budgeting [BNPS21]; see [AFRSV21] for a recent survey.
In this paper, we study the power and limits of eliciting ordinal information in dynamic fair division.
The value v i,t of agent i for item t is drawn from an unknown distribution upon arrival, and the algorithm is provided, from each agent, partial ordinal information about this item, e.g., its rank relative to the past items allocated to this agent, or even just a single past item allocated to this agent.
Under what distributional assumptions and elicitation constraints, can we simultaneously achieve qualitative fairness and efﬁciency?
We answer these questions.
1.1 Our Contribution
We start by establishing a separation between the cardinal setting and our ordinal one.
Pareto efﬁciency alone is trivial (allocate all goods to the same agent) and, in the cardinal setting, it is known that Pareto efﬁciency ex-post is compatible with envy-freeness with high probability (as long as agents are independent).
We prove (Theorem 4) that in our setting, even for the case of two i.i.d.
agents and any known distribution, envy-freeness with high probability is incompatible with even a very mild notion of (exact) Pareto efﬁciency, one-swap-Pareto efﬁciency, which requires that there is no beneﬁcial one-to-one trade of items between agents (but allows for improvements via manyto-many trades of items).
We proceed to give an essentially matching positive result.
For any number of i.i.d.
agents and an unknown distribution D , there exists an algorithm
Algorithm 1) that is envy-free with high probability and guarantees a (1 − ε ) -approximation to the optimal utilitarian social welfare (the sum of utilities), for all ε > 0 , with high probability (Theorem 5).
When an item arrives, the algorithm learns for each agent i its relative rank compared to a subset of prior items allocated to agent i (but otherwise knows nothing about the underlying numerical valuation).
Our algorithm works in epochs.
Each epoch has an exploration/sampling phase, where each agent gets a pre-determined number of items, followed by an exploitation/ranking phase, where each fresh item is given to an agent whose empirical quantile is largest.
The goal is to make a sublinear number of errors compared to the “ideal” algorithm that allocates each item to the agent with the highest true quantile.
The algorithm has to balance the need for sampling, which leads to more accurate empirical quantiles, against the number of inefﬁcient allocations made while sampling.
A signiﬁcant technical barrier is that we cannot ﬁx a target accuracy because the underlying distribution is unknown.
That is, for every ﬁxed accuracy for the empirical quantiles, there exists a distribution for which this accuracy is not good enough for even a constant approximation to the optimal welfare.
Instead, we need to make our epochs progressively longer, thereby guaranteeing progressively better bounds.
Given this strong positive result, we explore the limits of what we can achieve when further restricting the amount of information available.
What if each agent can remember only a single item previously allocated to them, and the fresh item is compared to just this one item?
Surprisingly, the aforementioned positive result can almost be recovered even in this very restrictive setting.
We prove that there exists an algorithm (Algorithm 2) that is envy-free with high probability and guarantees a (1 − 1 /e ) 2 − ε approximation to the optimal welfare with high probability, for all ε >(Theorem 9).
Our algorithm again proceeds in epochs with gradually increasing exploration and exploitation phases; this time the goal is a sublinear number of differences compared to allocating each item to a uniformly random agent with quantile at least 1 − 1 /n , which we prove is envy-free and approximately efﬁcient (Lemmas 2 and 1).
When exploring, the algorithm puts a new item in memory, estimates its quantile, and rejects it if not sufﬁciently close to 1 − 1 /n .
We need to sample enough to ensure high conﬁdence in the estimated quantile, but also account for the additional sampling since an item’s quantile value might be far from 1 − 1 /n to begin with; several technical details need to be accounted for.
We give a near-matching lower bound: no algorithm with a memory of one item can achieve a 0 .
999 − approximation to the social welfare with high probability; therefore a constant approximation (which Algorithm 2 provides) is all we can hope for.
Finally, we relax the i.i.d.
condition and study agents that are independent but not identical; each agent i ’s values are drawn from an unknown distribution D i .
Even with unbounded memory, we
[Table]
Unbounded Memory, i
[Table]
i.d. [Table]
Limited Memory
[Table]
Unbounded Memory
[Table]
show that it is impossible to get a 1+ √ 5≈ .
809 approximation to Pareto efﬁciency with probability more than 2 / 3 , even for two agents (Theorem 12).
At the same time, we prove that Algorithmsand 2 are envy-free and 1 /e approximately Pareto efﬁcient with high probability!
Note that, though both algorithms give the same formal guarantees and Algorithm 2 elicits strictly less information, one might still prefer to use Algorithm 1 since it has signiﬁcantly shorter exploration phases.
A summary of our results can be found in Table
We leave the study of correlated agents as an interesting open problem.
Finally, we note that beyond stochastic valuations, [BKPP18] show that it is possible to achieve sublinear envy by randomly allocating every item when agents’ valuations are adversarially generated (and this is optimal); however, sublinear envy is incompatible with non-trivial efﬁciency even in the cardinal setting [ZP20].
1.2 Related Work
A number of works study fair division under ordinal preferences, e.g., [AGMW15, BEL10,
BBL + 17, NNR17], but often these models do not assume an underlying cardinal model and work directly on the ordinal preferences.
[ABM16] assume underlying cardinal information and, among other results, bound the approximation ratio of truthful mechanisms that elicit rankings.
Closer to our work, [HS21] study rules that have access to the ranking of the top- k items of each agent and bound the ratio of the social welfare of the allocation returned by a rule in the worst case.
They also characterize the value of k needed to achieve prominent notions of fairness, namely envy-freeness up to one item (EF1) and approximate maximin share guarantee (MMS), as well as bound the loss in efﬁciency incurred due to fairness constraints in this setting.
Our work contributes to the growing literature in dynamic fair division [KPS14, AAGW15, FPV15,
FPV17, BKPP18, LLL18, HPPZ19, ZP20, GPT21, BKM22, GBI21, VPF21, BGGJ22, BMS22, KS22] (and we note that the welfare-maximizing algorithms of [DGK + 14, KPW16, BG22] work in the dynamic setting, even though the their settings are not explicitly dynamic).
To the best of our knowledge, we are the ﬁrst to study imperfect expressivity in a dynamic setting in fair division.
Preliminaries
A set of T indivisible items/goods, labeled by G = { 1 , 2 , · · · , T } , needs to be allocated to a set of n agents, labeled by N = { 1 , . . . , n } .
Agent i ∈ N assigns a value v i,t ∈ [0 , 1] to item t ∈ G .
We assume agents have additive valuation functions, so v i ( S )
= (cid:80) t ∈ S v i,t for S ⊆ G .
An allocation A is a partition of the items into bundles A 1 , . . .
, A n , where A i is the set of items assigned to agent i ∈ N .
Each allocation has an associated utility proﬁle v
( A ) =
( v 1 ( A 1 ) , . . .
, v n ( A n )) .
Items arrive online, one per round.
The agents’ valuations for the item in round t (the t -th item) are realized when the item arrives.
Every item is allocated immediately and irrevocably before moving on to the next round.
We write G t = { 1 , 2 , · · · , t } for the set of items that arrived in the ﬁrst t rounds, and A ti for the allocation of agent i after the t -th item was allocated.
We consider two different models which specify how values are generated.
In the i.i.d.
model , the value of agent i for item t is independently drawn from an unknown distribution D with CDF F , i.e., v i,t ∼ D for all i ∈ N and t ∈ G .
In the non-i.i.d.
model , the value of item t for agent i is independently drawn from an unknown , agent-dependent distribution D i with CDF F
i , i.e., v i,t ∼ D i for all i ∈ N and t ∈ G .
We write X i for the random variable for i ’s valuation, and X i,t for the random variable for i ’s valuation for item t .
It is often convenient to work directly with the quantile of an agent’s value rather than the value itself; let Q i
= F i ( X i ) and Q
i,t = F i ( X i,t ) respectively be the random variable denoting the quantile of agent i the associated item.
Note that all Q i and Q i,t are i.i.d.
and follow a Unif [0 , 1] distribution.
Unless explicitly stated otherwise, we assume all distributions are continuous (i.e., do not have point masses).
Ordinal information.
We assume the realizations v i,t are not available.
Instead, our algorithms have access to ordinal information.
Speciﬁcally, given current item t , the algorithm can access each agent’s ranking for S
= { t } ∪ M , M ⊆
G t − 1 .
The size of M , which we will informally refer to as the memory size , determines the complexity of eliciting information from each agent.
In one extreme, agent i compares a new item t to a single item they had previously received, i.e.,
M ⊆
A t − 1 i , | M | ≤ 1 .
In the other extreme, t is compared to all previous items she received, so
M =
A t − 1 i .
We write σ i ( S ) for the ranking of agent i for a subset S of the items, and σ − 1 i ( S, t ) for the position of item t ∈ S with respect to a subset S according to agent i .
The highest value item is in position 1 and the lowest in position | S | .
For example, if S = { 1 , 4 } , v i, 1 = 1 and v i, 4 = 0 . 1 , σ i ( S ) = (1 (cid:31) 4) , σ − 1 i ( S, 1) = 1 and σ − 1 i ( S, 4) = 2 .
Algorithms.
An algorithm A , in each step t , queries each agent for ordinal information with respect to some subset M and then makes a (possibly randomized) allocation decision based on this ordinal information and the history so far.
An instance of our problem is parameterized by the number of agents n and the (unknown) value distributions D 1 , . . .
, D n .
Let E P ( t ) be the event that some algorithm satisﬁes property P (e.g., envy-freeness or PO or ε -welfare) at time t .
We are interested in the probability that an algorithm satisﬁes certain properties in the limit, as the number of rounds tends to inﬁnity, where the randomness is over the random choices of the algorithm as well as the randomness in the valuations.
Deﬁnition 1.
An algorithm satisﬁes P with high probability if lim t →∞ Pr[ E P ( t )]
= 1 .
Note that this deﬁnition of high probability allows for dependency on n and the underlying distributions (i.e., they are treated as constants).
Efﬁciency notions.
An allocation A Pareto dominates an allocation A (cid:48) , denoted A (cid:31) A (cid:48) , when v i ( A i ) ≥ v i ( A (cid:48) i ) for all i ∈ N and there exists j ∈ N with v j ( A j ) > v j ( A (cid:48) j ) .
An allocation A is Pareto efﬁcient or Pareto optimal (PO) if there is no feasible (integral) allocation that Pareto dominates it.
An allocation A (cid:48) is in the (one) swap-neighborhood of A when it can be created from
A with a single exchange of items between one pair of agents.
Formally, there exist i, j ∈ N and items z
j ∈
A j and z i ∈
A i so that A (cid:48)
i =
( A i \{ z i } ) ∪{ z j } , A (cid:48)
j = ( A j \{ z j } ) ∪{ z i } , and A (cid:48) k
= A k for all other agents k
(cid:54)
= i, j .
An allocation A is one-swap Pareto optimal (SPO) if it is undominated by any allocation in its swap-neighborhood.
We use a notion of approximate efﬁciency deﬁned by [RF90] according to which an allocation A is α -Pareto efﬁcient when v ( A ) /α is undominated.
The social welfare of an allocation A is sw ( A )
=
(cid:80) i ∈N v i ( A i ) .
Let allocation A ∗ denote a (social) welfare optimal allocation for which sw ( A ∗ ) ≥ sw ( A ) for all feasible allocations A .
An allocation provides an α -approximation to welfare if sw ( A ) ≥ α · sw ( A ∗ ) .
Fairness notions.
We focus on a prominent notion of fairness called envy-freeness .
An allocation
A T =
( A T 1 , . . .
, A T n ) of T items is envy-free (EF) when v i ( A T i ) ≥ v i ( A T j ) for all i, j ∈ N , and c -strongly-envy-free ( c -strong-EF) when v i ( A i ) ≥ v i ( A j ) + cT .
Ideal Quantile-based Algorithms.
For our analysis, it will be useful to compare our algorithms with ideal algorithms that know exact quantile values for every item (and, in fact, several of our lower bounds apply to these stronger algorithms too).
Two ideal algorithms of interest are (1) quantile maximization, which allocates each item to the agent with the highest quantile value for it, and (2) “ q − threshold,” which allocates each item uniformly at random among agents whose quantile is at least q (and uniformly at random over all agents, if all quantile values are less than q ).
In the i.i.d.
model, quantile maximization is the same as value maximization, and thus envy-free with high probability and ex-post welfare optimal.
The property we will use is c -strong envy-freeness, for some distribution-dependent constant c , which we state as Lemma 1.
This was essentially proved by [DGK + 14]; we provide an alternate proof that also works, essentially unchanged, for the n − 1 nthreshold algorithm; it can be found in Appendix A.1.
The main idea is that an agent’s value for an item conditioned on them receiving it is strictly larger than their value for an item received by another agent.
Combined with the fact that the probability of receiving an item is equal for all agents, standard concentration results ensure each agent strictly prefers their own bundle with high probability.
Lemma 1.
[Essentially [DGK + 14].]
In the i.i.d.
and non-i.i.d.
models, the quantile maximization algorithm and the n − 1 n -threshold algorithm are c -strongly-envy-free, with high probability, where the constant c = min i ∈N ( E [ X i | Q i ≥ 1 / 2] − E
[ X i ]) / (4 n ) .
Note that c is strictly positive since our distributions are continuous.
In the i.i.d.
model, we show that the n − 1 n -threshold algorithm gives a (cid:0) 1 − 1 e (cid:1) 2 − ε approximation to welfare (Lemma 2) with high probability.
Next, we prove that the n − 1 n -threshold algorithm guarnatees a constant approximation to welfare.
The main idea for this proof is that, with constant probability, some agent has a quantile of at least n −n for each item.
Additionally, giving an item to an agent with quantile at least n − 1 n generates within a constant factor amount of welfare as giving it to the agent with the highest welfare.
Lemma 2.
In the i.i.d.
model, the n − 1 n -threshold algorithm guarantees a (cid:16)(cid:0) 1 − 1 e (cid:1) 2 − ε (cid:17) - approximation to welfare, with high probability, for all ε > 0 .
Proof.
Let F be the CDF of an arbitrary continuous distribution.
The expected contribution of an item to the welfare of the threshold algorithm is at least
[Equation]
For the ﬁrst term we have
[Equation]
where we used the fact that the maximum of n draws from U [0 , 1] follows a Beta ( n, 1) .
The expected contribution to the welfare is thus at least
[Equation]
1 − (cid:18)
[Equation]
−
[Equation] n (cid:19) n [Equation] ≥
[Equation] [Equation] 1] n
[Equation]
max
[Equation]
) (cid:21) Finally, for any ﬁxed ε > 0 , standard Chernoff bounds tell us that with high probability, the optimal welfare of T items is at most T · (1 + ε/ 2) E (cid:126)Q ∼ Unif [0 , 1] n (cid:2) F {+− 1 (max i ∈N Q i ) (cid:3) while the welfare of the threshold algorithm is at least T · (1 − ε/ 2) (cid:0) 1 − 1 e (cid:1) 
+}
[ E (cid:126)Q ∼ Unif [0 , 1] n (cid:2) F − 1 (max i ∈N Q i ) (cid:3) .
Indeed, the expected optimal welfare is equal to T · E (cid:126)Q ∼ Unif [0 , 1] n (cid:2) F − 1 (max i ∈N Q i ) (cid:3) , the sum of T i.i.d.
random variables with expectation E (cid:126)Q ∼ Unif [0 , 1] n (cid:2) F − 1 (max i ∈N Q i ) (cid:3) .
The standard multiplicative
Chernoff bound says that the sum of i.i.d.
variables exceeds (1 + (cid:15)/ 2) times its expectation µ is at most exp (cid:0) − µ(cid:15) 2 / 12 (cid:1) .
Plugging in µ = T · E (cid:126)Q ∼ Unif [0 , 1] n (cid:2) F − 1 (max i ∈N Q i ) (cid:3) , we get the desired statement.
The statement about the welfare of the threshold algorithm follows similarly.
Thus, the algorithm is a
[Equation]
approximation to welfare, with high probability.
Finally, we prove that both ideal algorithms are approximately efﬁcient.
Let P ∗ be the following property of an allocation: all items such that exactly one agent has quantile values at least 1 − 1 /n are in the bundle of this agent.
Both ideal algorithms (quantile maximization and 1 − 1 /n -threshold) satisfy P ∗ .
We prove that, in the non-i.i.d.
model, P ∗ implies an almost 1 /e approximation to efﬁciency.
Our proof uses the fact that there is a (roughly) 1 /e probability that exactly one agent has the high quantile, so the value of an agent’s bundle in an algorithm that satisﬁes P ∗ is, with high probability, a 1 /e approximation to their value for their T/n most valuable items.
Therefore, when considering an alternate allocation A (cid:48) , the agent in A (cid:48) that gets at most T/n items cannot be improved upon by more than a 1 /e factor.
The proof can be found in Appendix A.2.
Lemma 3.
In the non-i.i.d.
model, every algorithm whose allocations satisfy P ∗ is (1 /e − ε ) -Pareto optimal, with high probability, for all ε > 0 .
Unbounded Memory in the I.I.D. Model
We explore some fundamental limits of our setting.
Efﬁciency by itself is easy: allocate all items to the same agent.
However, in contrast to the cardinal setting, we ﬁnd one-swap Pareto efﬁciency is incompatible with envy-freeness with high probability, even for two i.i.d.
agents, and even when the underlying distribution is known .
Theorem 4.
In the i.i.d.
model, even for n = 2 agents, there does not exist an algorithm A which is one-swap Pareto efﬁcient and envy-free with high probability, even when values are sampled according to D , for any continuous, bounded and known value distribution D .
The proof can be found in Appendix B.1.
The main idea is the following.
As the agents are a priori identical, we can assume without loss of generality that A gives the ﬁrst item to agent 1 .
The proof shows that, with positive probability, this decision becomes an irrevocable “mistake,” in the sense that agent 2 really liked the item and agent 1 did not.
This mistake will make envy-freeness and one-swap-Pareto efﬁciency incompatible.
Theorem 4 implies that when we have access to only ordinal information, we need to settle for some approximation to envy-freeness and efﬁciency.
Our main positive result for this section is an algorithm that essentially matches the aforementioned lower bound.
Theorem 5.
In the i.i.d.
model, Algorithm 1 achieves envy-freeness and a (1 − ε ) approximation to welfare, with high probability, for all ε > 0 .
Algorithm 1 works in epochs: each epoch k has an exploration/sampling phase, where each agent i receives a pre-determined set of items, denoted G ki , irrespective of their valuation.
This is followed by an exploitation/ranking phase, where each item is given to the agent with the highest empirical quantile (with respect to items received in the preceding exploration phase, i.e. G ki ).
The key idea for the proof is that the exploration phases will ensure nearly all of the items will go to the agent that truly has the highest quantile.
Using this, the good properties of the ideal quantile maximization algorithm will carry over to Algorithm 1.
We start with a technical lemma, which gives us a bound on the length of the exploration period we need in each epoch.
The following deﬁnition will be useful.
Deﬁnition 2.
A sample of n · m items (where each agent is allocated exactly m items) is ε -accurate if, with probability at least 1 − ε , the relative rank of a fresh item (with respect to the sample) is highest for the agent with highest quantile value.
Lemma 6.
If ε, δ ∈ (0 , 1) , and m ∈ Z
+ are such that ε > 2 n (cid:113) ln(2 n/δ ) 2 m , then giving m samples to each agent is ε -accurate with probability at least 1 − δ .
Algorithm for epoch k
= do
Phase: ( n · k 4 items)
Give the j -th item in this phase to agent j (mod n ) .
) for each item g in this phase do
Elicit σ − 1 i ( G ki ∪ { g } , g ) for all
i ∈ N .
Allocate g to an agent j ∈ arg min i ∈N σ − 1 i G ki ∪ { g } , g ) .
Proof.
We will use the Dvoretzky–Kiefer–Wolfowitz (DKW) inequality [DKW56, Mas90] to show the empirical CDF of sampled quantiles is reasonably close to a uniform distribution with probability 1 − δ .
We then show this is sufﬁcient to guarantee ε -accuracy for the chosen ε .
Let ˆ F i be the empirical CDF of the sampled quantiles for agent
i , i.e., ˆ F i ( q ) for q ∈ [0 , 1] is a random variable that describes the proportion of sampled items with quantile at most q .
Note that ˆ F i exactly captures agent i ’s ranking for a new item: if a fresh item has quantile q i for agent i and q j for agent j , then i ranks it higher than j exactly when ˆ F i ( q i ) >
ˆ F j ( q j ) .
Noting that the CDF for the actual quantile distribution (i.e., the uniform distribution) is the identity on [0 , 1] , the DKW inequality states that for all γ > 0 , Pr (cid:104) sup q ∈ [0 , 1] | ˆ F i ( q ) − q | > γ (cid:105) ≤ 2 e − 2 mγ 2 .
We want this condition to hold for all n agents, simultaneously, with probability at least 1 − δ , so we pick γ such that 2 e − 2 mγ 2 ≤ δ/n and apply a union bound; it sufﬁces to choose γ = (cid:113) ln(2 n/δ ) 2 m .
We now show that the DKW condition ( sup q ∈ [0 , 1] | ˆ F i ( q ) − q | ≤ γ ) being satisﬁed for all agents i is sufﬁcient to guarantee ε -accuracy.
Consider sampling quantiles Q 1 , . .
, Q n for a fresh item.
Let i max ∈ argmax i ∈N Q i be a quantile-maximizing agent (technically a random variable).
Our goal is to show that with probability at least 1 − ε (with respect to the samples of Q 1 , . . .
, Q n ) ˆ
F i max ( Q i max ) >
ˆ F j ( Q j ) for all j (cid:54) =
i max . This ensures that i max has the highest empirical rank, and hence receives the item.
Let Q (1) , . . .
, Q ( n ) be the respective order statistics.
A key observation is that Q ( n ) − Q ( n − 1) ∼ Beta [1 , n ] [Gen19].
The PDF of a Beta [1 , n ] distribution is f ( x ) =
nx n −for x ∈ [0 , 1] .
Since f ( x ) ≤ n , Pr (cid:2) Q ( n ) − Q ( n − 1) < ρ (cid:3) <
nρ for all ρ > 0 .
Plugging in ρ = 2 γ , we have Pr (cid:2) Q ( n ) − Q ( n − 1) ≤ 2 γ (cid:3) < 2 nγ .
We will show that as long as ε > 2 nγ , ε -accuracy holds.
First, we have Pr (cid:2) Q ( n ) − Q ( n − 1) > 2 γ (cid:3) > 1 − ε .
Conditioned on Q ( n ) − Q ( n − 1) > 2 γ , the item is given to i max .
To see why, observe Q i max = Q ( n ) and Q j ≤ Q ( n − 1) for all j (cid:54)
=
i max , by deﬁnition.
Using the DKW inequality condition, it follows that
ˆ F i max ( Q i max ) ≥ Q i max − γ > Q j + γ ≥ ˆ F j ( Q j ) .
We conclude that for ε > 2 n (cid:113) ln(2 n/δ ) 2 m , ε -accuracy is satisﬁed with probability at least 1 − δ .
Using Lemma 6, we can get, for each epoch, a bound on the number of decisions where Algorithmdiffers from the quantile maximization algorithm.
Lemma 7.
The allocation of Algorithm 1 differs from that of the quantile maximization algorithm after T steps by at most f ( T ) items with high probability,
where f ( T ) ∈
O ( poly ( n ) · T 15 / 16 ) .
Proof.
We start by bounding the accuracy of Algorithm 1 in each epoch k .
In epoch k , each agent receives k 4 items during the sampling phase.
We claim that the sample in epoch k for k ≥ 3 n is ε k -accurate for ε k := 3 n/k 3 / 2 with probability at least 1 − δ k , for δ k := 2 n/e 2 k .
Indeed, ﬁrst note that by the choice of k , we have that ε k , δ k ∈ (0 , 1) .
Hence, we just need to show that these values satisfy the inequality of Lemma 6.
We have that
[Equation]
Next, ﬁx a time T .
Slightly abusing notation, let k ( t )
=
min { K ∈ N | (cid:80)
Kk =1 nk 4 + k 8 ≥ t } be the function that given an item t returns the epoch item t is in.
Notice that T ≥ (cid:80) k ( T ) − 1 k
= nk 4 + k 8 ≥
( k ( T ) − 1) 8 , and therefore k ( T ) ≤ 2 T 1 / 8 .
In any run of the algorithm, we can classify every item t ≤ T into one of four categories.
Item t was allocated in one of the ﬁrst 3 n − 1 epochs, that is, k ( t ) < 3 n . 2.
Item t was allocated in the sampling phase of epoch k ( t ) ≥ 3 n . 3.
Item t was allocated in the ranking phase of epoch k ( t ) ≥ 3 n ; the epoch was ε k ( t ) -accurate. Item t was allocated in the ranking phase of epoch k ( t ) ≥ 3 n ; the epoch was not ε k ( t ) -accurate.
We say an item t was a mistake if it was given to an agent with a non-maximum quantile for it.
We show that the number of mistakes in each category are bounded by 3 10 n 9 , 2 nT 5 / 8 , 9 nT 15 / 16 , and 158 n ln( T ) respectively, with high probability.
This implies that the total number of mistakes is at most the sum of these quantities, which is O ( poly ( n ) · T 15 / 16 ) , with high probability, via a union bound.
The number of items in the ﬁrst category is at most
[Equation]
Hence, the number of mistakes in the ﬁrst category is also at most 3 10 n 9 .
For the second category, since k ( T ) ≤ 2 T 1 / 8 , we have that the total number of items in the sampling phase is (with probability 1 ) upper bounded by
[Equation]
Each item t in the third category has probability ε k
( t ) of being a mistake.
The expected number of mistakes is therefore at most (cid:80)
[Equation]
n 3 nk 13 / 2 ≤ 3 nk ( T ) 15 / 2 ≤ 8 nT 15 / 16 .
Using Hoeffding’s inequality we get that with high probability the number of mistakes is at most 9 nT 15 / 16 , since a deviation of nT 15 / 16 occurs with probability at most exp (cid:0) − 2 n 2 T 15 / 8 /T (cid:1) =
exp (cid:0) − 2 n 2 T 7 / 8 (cid:1) .
For the fourth category, the expected number of items is at most (cid:80)
k ( T ) k =3 n δ k k 8 = 2 n (cid:80) k
( T ) k =3 n k 8 e 2 k ≤ 2 n (cid:80) ∞
k =1 k 8 e 2 k ≤ 158 n .
Using Markov’s inequality we have that the number of mistakes is at most 158 n ln( T ) with probability at least 1 − ln( T ) , i.e., with high probability.
Finally, we can prove Theorem 5 as a relatively straightforward consequence of Lemma 7, since the ideal quantile maximization algorithm satisﬁes nice properties (e.g., Lemma 1).
The full proof can be found in Appendix B.2.
Bounded Memory in the I.I.D. Model
In this section, we are interested in the more ambitious problem of designing dynamic algorithms with even more limited partial information: each agent is allowed to “remember” only a single item.
We ﬁrst show that, in this case, we need to settle for constant approximations of welfare.
The main idea of the proof is a difﬁcult value distribution.
Agents have essentially one of three values, low, medium, high.
The key idea is that no matter what quantile thresholds an algorithm has, it will either be the case that with reasonable probability, the highest value agent has medium value but is indistinguishable from all low value agents, or the highest value agent has high value, but is indistinguishable from all medium or low value agents.
In either case, the algorithm will make a mistake with constant probability, and hence cannot guarantee close to optimal welfare.
The full proof can be found in Appendix C.1.
Theorem 8.
In the for Epoch k = 1 . . .
do
Sampling Phase: ( k 9 items)
N OT W ITHIN E RROR ← N for trial
= 1 , . .
. , k 3 do for i ∈ N OT W ITHIN E RROR do Allocate the next item to agent i , and update her memory
Test k 6 − | N OT W ITHIN E RROR | number of items (for each agent) for i ∈ N OT W ITHIN E RROR do if Proportion of test items for agent i is within ± 1 /k 2 of ( n − 1) /n
then
N OT W ITHIN E RROR
← N OT W ITHIN E RROR \ { i }
Ranking Phase: ( k 18 items) for each item g in this phase do if Some agent i has high signal then Give g to a (uniformly) random such agent else Give g to an agent uniformly at random
Our positive result matches this lower bound up to a constant.
Theorem 9.
In the i.i.d.
model, given a memory of one item per agent, Algorithm 2 achieves envyfreeness and a (1 − 1 /e ) 2 − ε approximation to welfare, with high probability, for all ε > 0 .
Algorithm 2 works in epochs, similar to Algorithm 1.
In each epoch’s exploration/sampling phase, it tries to ﬁnd an item whose quantile is close to the n − 1 n -threshold algorithm.
Epoch k makes k 3 such attempts, and each candidate item is tested against k 6 fresh items to get an estimated quantile.
If everything is within the error we can tolerate, the algorithm remembers this item for this epoch; otherwise, the agent has an arbitrary item in memory during this epoch.
During the exploitation/ranking phase, Algorithm 2 tries to mimic the n − 1 n -threshold algorithm (instead of the quantile maximization algorithm as Algorithm 1 did), and, in fact, inherits its approximation factor (Lemma 2) exactly.
Our ﬁrst technical lemma, Lemma 10, gives necessary bounds on the various variables of Algorithm 2 for a sample to be ε -accurate with respect to the ideal threshold algorithm; see Deﬁnition 3.
Its proof can be found in Appendix C.2.
Deﬁnition 3.
A set of n items in memory, one for each agent, is ε -accurate with respect to q ∗ if with probability at least 1 − ε , when a fresh item is sampled, the agents with true quantile above q ∗ are exactly those that value the fresh item more than their item in memory.
Lemma 10.
For all ε, δ ∈ (0 , 1) , if (1) at least τ trials are done with τ ≥ ln(2 n/δ ) ε/ (3 n ) , and (2) at least (cid:96) test items are used per trial for (cid:96) ≥ 18 n 2 ε 2 ln (cid:0) 4 τn δ (cid:1) , and (3) the tolerance for accepting an item is ε/ (3 n ) , then the items in memory are ε -accurate (for all agents, simultaneously) with respect to q ∗ = n − 1 n , with probability at least 1 − δ .
Though Lemmas 6 and 10 resemble each other (and are used in analogous ways), the proofs require different techniques, as the sampling processes are very different.
Next, we prove an analogue to
Lemma 7: the number of disagreements between Algorithm 2 and the ideal threshold algorithm is sublinear.
The proofs of Lemmas 7 and 11 are similar, precisely because Lemma 6 matches
Lemma 10.
Theorem 9 follows from Lemma 11 as in the i.i.d. case.
The proofs of Lemma 11 and
Theorem 9 can be found in Appendices C.3 and C.4 respectively.
Lemma 11.
The allocation of Algorithm 2 differs from that of the n − 1 n -threshold algorithm
after T steps by at most f ( T ) items with high probability, where f ( T ) ∈
O ( poly ( n ) · T 17 / 18 ) .
The Non-I.I.D. Model
In this section, we study the non-i.i.d.
model.
We ﬁrst establish a strong lower bound for the non-i.i.d.
model.
The following negative result holds even for algorithms that know the associated quantile for every fresh item.
Theorem 12.
Even for 2 non-identical agents, there is no algorithm that is EF and c -PO with probability p , for c > 1+ √ 5≈ .
809 and p > 2 / 3 , for all continuous and bounded value distributions.
Here we provide a proof sketch of Theorem 12.
The full proof can be found in Appendix D.1.
Proof Sketch.
The main idea is that an algorithm, even if it can see the exact quantiles of all items, cannot distinguish between agents having potentially different kinds of distributions.
At one extreme end, agents may have nearly identical values for all items.
On the other extreme, agents may only care (i.e., have high value) for, say, their top third of items.
Consider the case of two agents.
Notice that if both agents have nearly identical values for all items, to be EF, the bundle sizes of both agents must be approximately the same.
We can then show that, no matter how an algorithm does this, it must be the case that one agent, say agent 1 , receives a signiﬁcant number of items with quantile outside of their top third does not receive a signiﬁcant number of items with quantile in their top third.
However, suppose instead the value distributions were such that agent 2 still has nearly identical values for all items, but agent 1 only has high value for their top-third items.
In the setting described above, agent 1 would be willing to trade many of their bottom two-third quantile items for the items agent 2 received in their top third.
By setting up this trade correctly, both agents total utilities go up by a signiﬁcant amount.
The existence of such a trade violates constant approximations to PO.
Algorithms 1 and 2 are envy-free with high probability, even in the non-i.i.d.
model, since envyfreeness is not an “inter-agent” property.
Our last result shows that they also give a constant approximation to Pareto efﬁciency, by combining Lemma 3 with Lemmas 7 and 11.
Its proof can be found in Appendix D.2.
Theorem 13.
In the non-i.i.d.
model, both Algorithm 1 (unbounded memory) and Algorithm 2 (oneitem memory) are EF and (1 /e − ε ) -PO, with high probability, for all ε > 0 .
Interestingly, the guarantees for Algorithm 2 in the non-i.i.d.
model are only marginally worse compared to the i.i.d.
model; the approximation ratio decreases from (1 − 1 /e ) 2 ≈ 0 .
4 to 1 /e
≈ 0 .
Finally, we note that, even though the formal guarantees in Theorem 13 are the same for the two algorithms, and even though Algorithm 2 uses memory size of one, Algorithm 1 has the beneﬁt of much shorter epoch lengths (in addition to better guarantees under the i.i.d. model).
Conclusion
To conclude, we have analyzed the online fair division problem when agents only reveal partial information.
In multiple settings, we have proved tight bounds on what properties are attainable.
As for future work, one direction would be to move past the assumption that agent values are drawn independently, as we may not expect this to hold in practice.
Additionally, there are many other forms of partial information that could potentially achieve different properties.
For example, if agents can compare not just single items but small subsets of items, then it may be possible to achieve stronger results such as arbitrarily good approximations to PO even in the non-i.i.d.
setting.
Acknowledgements
Alexandros Psomas is supported in part by an NSF CAREER award CCF-2144208, a Google Research Scholar Award, and by the Algorand Centres of Excellence program managed by Algorand
Foundation.
Any opinions, ﬁndings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reﬂect the views of Algorand Foundation.
This material is based upon work supported by the National Science Foundation Graduate Research
Fellowship Program under Grant No. DGE1745303.
Appendix A Proofs Missing from SectionA.1 Proof of Lemma
We focus on quantile maximization.
The same proof goes through essentially unchanged for the threshold algorithm; we explain the differences whenever it’s appropriate.
Fix distributions D 1 , . . .
, D n with CDFs F 1 , . .
. , F n .
Fix two agents i and j .
We will show with high probability, i does not envy j (in a strong sense).
Union bounding over all (cid:0) n 2 (cid:1) pairs yields the lemma statement.
As in [DGK + 14], we compare the expected contribution of an item t to i ’s bundle and its expected contribution to j ’s bundle.
Let A be the random variable denoting the agent that received the item.
We want to consider the difference E
[ v i,t · I [ A = i ]] − E
[ v i,t · I [ A = j ]] .
Let H i be the event that F i ( v i,t ) ≥ n − 1 n , and L i be the compliment.
We split each of the two terms into conditional expectations depending on the signal, beginning with the ﬁrst one:
[Equation]
Note that, under quantile maximization, v i,t is positively correlated with I [ A = i ] : for any ﬁxed value v i,t , A
= i with probability F ( v i,t ) n − 1 , which is increasing in v i,t .
Therefore, the expectation of the product is greater than or equal to the product of the expectations: E
[ v i,t · I [ A = i ] | H i ] ≥ E [ v i,t | H i ] · Pr[ A = i
| H i ] and E
[ v i,t · I [ A = i ] | L i ] ≥ E [ v i,t | L i ] · Pr[ A = i
| L i ] .
Therefore
E [ v i,t · I [ A = i ]] ≥ E [ v i,t | H i ] · Pr[ A = i | H i ] · Pr[ H i ] + E [ v i,t | L i ] ·
Pr[ A = i | L i ] · Pr[ L i ] =
E [ v i,t | H i ] · Pr[ A = i and H i ] + E [ v i,t | L i ] · Pr[ A = i and L i ] .
For the threshold algorithm, we have equality above, since conditioned on either H i or L i , v i,t is independent of I
A = i ] , as the allocation depends only on the high vs low signal.
On the other hand, v i,t is negatively correlated with I [ A = j ] .
Therefore
E [
[Equation]
[Equation]
[Equation]
[Equation]
and H
[Equation]
[Equation]
E [
[Equation]
[Equation]
[Equation]
[ A
[Equation]
and L i ] .
Again, for the threshold algorithm, we have equality.
Combined, we have
[Equation]
We analyze (1), Pr[ A = i and H i ] − Pr
[ A = j and H i ] .
Let H j be the event that F j ( v j,t ) ≥ n − 1 n
Let L j be its complement.
We have:
[Equation]
Notice that Pr[ A = j and H i and L j ] = 0 because if agent i has a high quantile and j has a low quantile, j cannot receive the item (in either algorithm).
Additionally, by symmetry, Pr[ A = i and H i and H j ] =
Pr[ A = j and H i and H j ] .
Therefore, (1) simpliﬁes to
Pr[ A = i and H i and L j ] .
Finally, we note that Pr[ A = i and H i and L j ] ≥ 1 n − 1 , again, for both algorithms.
We analyze (2),
Pr[ A = j and L i ] − Pr
[ A = i and L i ] .
Let E low be the event that all agents other than i have quantile lower then n − 1 n .
Let E low be its complement, the probability that at least one agent other than i has a high quantile.
We have:
(cid:16) Pr (cid:2) A = j and L i and E low (cid:3) + Pr (cid:104)
A =
j and L i and E low (cid:105)(cid:17) − (cid:16) Pr (cid:2) A = i and L i and E low (cid:3) + Pr (cid:104) A = i and L i and E low (cid:105)(cid:17) .
Notice that Pr (cid:104)
[-= 
-]A
{+= i and L i and E low (cid:105) = 
+}
because if agent i has a low quantile and at least one other agent has a high quantile, i cannot receive the item.
Additionally, by symmetry,
Pr (cid:2)
A = j and L i and E low (cid:3) =
Pr (cid:2)
A = i and L i and E low (cid:3) , since when all agents have low quantiles, i and j are equally likely to receive the item.
Hence, the probability in (2) simpliﬁes to Pr (cid:104)
A = j and L i and E low (cid:105) =
Pr (cid:104) A =
j | L i and E low (cid:105) ·
Pr (cid:104) L i and E low (cid:105) .
The ﬁrst term is equal to 1 / ( n − 1) , since j is equally likely to receive the item compared to any agent.
The second term is n − 1 n · (cid:16) 1 − (cid:0) n − 1 n (cid:1) n − 1 (cid:17) .
Observing that (cid:0) n − 1 n (cid:1) n − 1 = (cid:0) 1 − 1 n (cid:1) n − 1 ≥ 1 e , we have that the probability in (2) is at most 1 n (1 − 1 e ) .
Overall, we have shown that
[Equation]
It remains to show that the value of i for A
Ti is at least her value for A Tj plus 14 n ( E [ X i | H i ] − E
[ X i | L i ]) with high probability.
Towards this, notice that the value of i for A Ti minus her value for A Tj is the sum of T i.i.d.
random variables, supported in [ − 1 , 1] , whose expectation is at least 12 n ( E [ X i |
H
i ] − E
[ X i | L i ]) , as we’ve established so far.
Hoeffding’s inequality then implies that the probability that this difference is less than b = 14 n ( E [ X i | H i ] − E [ X i | L i ]) is at most 2 exp (cid:16) − b 2 T 2 (cid:17) , i.e., exponentially small, since b is a constant.
Observing that E [ X i | H i ] ≥
E
[ X i | Q i ≥ 1 / 2] and E [ X i | L i ] ≤ E [ X i ] concludes the proof.
A.2 Proof of Lemma
Fix an ε ∈ (0 , 1) , and choose ε (cid:48) such that 1 − ε (cid:48) (1+ ε (cid:48) ) 2 · 1 e > 1 e − ε (using ε (cid:48) =
ε/ 3 will do).
Fix distributions with CDFs F 1 , . .
. , F n for each agent i ∈ N , and a time T .
Suppressing the superscript, for ease of notation, let A i
=
A Ti be the bundle allocated at time T to each agent i by an algorithm that satisﬁes P ∗ .
Let A top i be the set of the T/n most valuable items for each agent i .
Let A high i = { t ∈ G T | F i ( v i,t ) ≥ 1 − 1+ ε (cid:48) n } be the set of items that agent i has “high” value for, in the sense that they come from the top 1+ ε (cid:48) n portion of their distribution.
We show the following 3 n events, E ij for i ∈ N and j ∈ { 1 , 2 , 3 } , occur simultaneously with high probability (in T ).
E i 1 : v i ( A top i ) ≤ v i ( A high i ) .
E i 2 : v i ( A high i ) ≤ T · (1+ ε (cid:48) )
2 n E Q ∼ Unif [1 − 1 /n, 1] [ F − 1 ( Q )] .
E i 3 : v i ( A i ) ≥ T · 1 − ε (cid:48) en E Q ∼ Unif [1 − 1 /n, 1] [ F − 1 ( Q )] .
Each of these individually will follow from a straightforward application of Hoeffding’s inequality or Chernoff bounds, showing they each individually occur with probability exponentially close toin T .
This implies that they all occur simultaneously with high probability.
Finally, we will show that conditioned on all 3 n occurring, the allocation is (1 /e − ε ) -PO.
Let us begin with E i 1 for each agent i .
The event occurs when there are at least T/n items t ∈ G T such that F i ( v i,t ) ≥ 1 − 1+ ε (cid:48) n .
Each item independently satisﬁes this property ( F i ( v i,t ) ≥ 1 − 1+ ε (cid:48) n ) with probability 1+ ε (cid:48) n .
Hence the probability this does not occur is at most 2 exp (cid:0) − 2 ε (cid:48)
2 T (cid:1) .
Next, consider E i 2 for each agent i .
The expected contribution of each item to v i ( A high i ) is
[Equation]
We now use the following multiplicative version of the Chernoff bound,
[Equation]
to conclude that the probability that v i ( A high i ) exceeds T · (
[Equation]
ε
[Equation]
) · E [ v i ( A high i )] is at most exp
[Equation]
Finally, consider E i 3 for each agent i .
We will show that the expected contribution of each item to v i ( A i ) is at least 1 en · E Q ∼ Unif [1 − 1 n , 1] [ F − 1 i ( Q )] .
Indeed, consider an item such that the quantile for agent i is Q i > 1 − 1 /n while Q j < 1 − 1 /n for all agents j (cid:54) =
i . This occurs with probability 1 n · (cid:0) 1 − 1 n (cid:1) n − 1 ≥ 1 en , and when this occurs, since the algorithm satisﬁes P ∗ , it must allocate the item to i .
Further, when this does occur, the expected value of such an item is E
Q ∼ Unif [1 − 1 n , 1] [ F − 1 i ( Q )] , since it is independent of the other agent’s values.
Hence the expectation is at least 1 en E Q ∼ Unif [1 − 1 n , 1] [ F − 1 i ( Q )] .
Finally, we again use a multiplicative Chernoff bound to show that
Pr
[Equation]
Now, suppose that E ij hold for all i ∈ N and j ∈ { 1 , 2 , 3 } .
We show that this implies the allocation
A 1 , . . .
, A n is (1 /e − ε ) -PO.
Fix an arbitrary allocation A (cid:48) 1 , . . .
, A (cid:48) n .
We show there exists an agent i ∈ N such that v i ( A (cid:48) i ) <
v i
( A i ) 1 /e − ε .
First, there must be some agent i such that | A (cid:48)
i | ≤ T/n .
Since A (cid:48) i can be at most as valuable as the most-valuable T/n items, we have
[Equation]
as needed.
B Proofs missing from SectionB.1 Proof of Theorem
Fix an arbitrary, continuous value distribution D and an algorithm A .
As the agents are a priori identical, we can assume without loss of generality that A gives the ﬁrst item to agent 1 .
We will show that, with a positive probability, this decision becomes an irrevocable “mistake,” in the sense that agent 2 really liked the item and agent 1 did not.
This mistake will make envy-freeness and one-swap PO incompatible.
First, we ﬁnd values to make this mistake sufﬁciently bad.
Let g : [0 , 1] → [0 , 1] be the function g ( q ) =
E [ X | X ≤ F − 1 ( q )] / E
[ X ] , which maps a quantile q to the ratio of the expected value of an item below quantile q to the expected value of an arbitrary item.
g is a continuous increasing function with g (1) = 1 , so there is some quantile ˆ q < 1 such that g (ˆ q ) ≥ 0 . 9 .
Let q ∗ 2 = max(ˆ q, 0 . 9) .
Since g is increasing, g ( q ∗ 2 ) ≥ g (ˆ q ) ≥ 0 .
Let q ∗ 1 = 0 .
1 , v ∗ 1 = F − 1 ( q ∗ 1 ) and v ∗ 2 = F − 1 ( q ∗ 2 ) .
Let
E mistake be the event that X 1 , 1 < v ∗ 1 and X 2 , 1 > v ∗ 2 .
Deﬁne c :=
Pr[ E mistake ] = (1 − q ∗ 2 ) · q ∗ 1 to be the probability that E mistake occurs.
D is continuous, so c > 0 .
Our lower bound on the probability that the allocation at step t violates either envy-freeness or one-swap PO will only depend on c .
Let E j be the event that for item j we have that both X 1 ,j ≥ v ∗ 1 and X 2 ,j ≤ v ∗ 2 .
If E mistake occurs, the only way to maintain one-swap Pareto efﬁciency is to allocate item j to agent 1 every time E j occurs; otherwise, swapping items 1 and j between the two agents yields a Pareto improvement.
This constraint will make envy-freeness unlikely.
Let E manyhigh ( t ) be the event (cid:80) tj =2 X 2 ,j · I [ E j ] ≥ ( t − 1) · 0 .
7 · E [ X ] .
In other words, E manyhigh ( t ) occurs when agent 2 has a high value for items j , 2 ≤ j ≤ t , for which E j occurs (i.e., the items that must be given to agent 1 in order to satisfy one-swap PO).
Let E normalval ( t ) denote the event that (cid:80) tj =2 X 2 ,j ≤ ( t − 1) · 1 . 1 · E [ X ] .
We ﬁrst show that for sufﬁciently large t , the probability that both E manyhigh ( t ) and E normalval ( t ) occur is at least 1 / 2 .
To do so, we prove each event occurs with probability at least 3 / 4 , and then apply a union bound.
First, since each X 1 ,j and X 2 ,j are independent, Pr[ E j ] ≥ 0 .
9 · 0 . 9 = 0 .
81 , and E
[ X 2 ,j |E j ] =
E [ X 2 ,j | X 2 ,j ≤ v ∗ 2 ] .
Also, from the deﬁnition of g (ˆ q ) and the choice of q ∗ 2 , E
[ X 2 ,j | X 2 ,j ≤ v ∗ 2 ] ≥ 0 .
9 · E [ X ] .
It follows that E
[ X 2 ,j · I [ E j ]]
=
E [ X 2 ,j |E j ] · Pr [ E j ] ≥ 0 .
729 · E [ X ] .
A straightforward
Chernoff bound establishes that Pr[ E manyhigh ( t )] ≥ 3 / 4 for t at least 6 E [ X ] .
Let Y j
= X 2 ,j · I [ E j ] for all j .
Then, E [ Y j ] ≥ 0 .
729 · E [ X ] , and E [ (cid:80)
Tj =2 Y j ] ≥ ( t − 1) · 0 .
· E [ X ] .
We are interested in the probability that {+(cid:80) tj =
+}
Y j is at least ( t − 1) · 0 .
7 · E [ X ] , i.e., the probability that {+(cid:80) tj =
+}
Y j is at least 0 .
729 its expectation.
We use the following Chernoff bound: Let Y 1 , . . .
, Y n be independent random variables that take values in [0 , 1] , and let Y be their sum.
Then, for all δ ∈ [0 , 1) , Pr[ Y ≤ (1 − δ ) E
[ Y ]] ≤ e − E
[ Y ] δ 2.
Continuing our derivation:
[Equation]
E [ t (cid:88) j = Y j ]   = 1 − Pr   t (cid:88) j =
Y j < 0 .
79 E [ t (cid:88) j =
Y j ]   ≥ 1 − Pr   t (cid:88) j =
Y j ≤ 0 .
89 E [ t (cid:88) j =
Y j ]   ≥ 1 − exp (cid:32) −
E [ (cid:80) tj =2 Y j ](0 . 89) (cid:33) , which is at least 3 / 4 when E [ (cid:80) tj =2 Y j ](0 . 89) 2is at least ln(4) , or, equivalently, if t ≥ 1 + 2ln(4) 0 .
7 · (0 . 89) 2 · E [ X ] .
Since 2ln(4) 0 .
7 · (0 . 89) 2 < 5 and E [ X ] < 1 , so t ≥ 6 E [ X ] sufﬁces.
Pr[ E normalval ( t )] ≥ 3 /follows similarly.
Next, observe that E manyhigh ( t ) ∩ E normalval ( t ) is independent of E mistake , since the two events depend on disjoint sets of independent random variables.
Therefore, Pr[ E mistake ∩ E manyhigh ( t ) ∩ E normalval ( t )] =
Pr[ E mistake ] · Pr[ E manyhigh ( t ) ∩ E normalval ( t )] ≥ c · 1 / 2 for t ≥ 6 / E [ X ] .
Let E SPO ( t ) and E EF ( t ) be the events that the allocation at step t is one-swap PO, and envy-free, respectively.
When E mistake ∩ E manyhigh ( t ) ∩ E normalval ( t ) occur, the allocation cannot be both one-swap
PO and envy-free, i.e. Pr (cid:104) E SPO ( t ) ∩ E EF ( t ) | E mistake ∩ E manyhigh ( t ) ∩ E normalval ( t ) (cid:105)
= 1 .
To see this, notice that ﬁrst, due to E mistake , the only way to remain one-swap PO is to give each item j to agent
1 every time E j occurs.
Second, E manyhigh ( t ) ensures that agent 2 ’s value for these items, and hence agent 2 ’s value for agent 1 ’s bundle, is at least 0 .
7 · ( t − 1) · E [ X ] + v 2 , 1 .
Third, E normalval ( t ) ensures that agent 2’s value for all items is at most 1 . 1 · ( t − 1) · E [ X ]+ v 2 , 1 , which is strictly less than twice her value for agent 1 ’s bundle.
We conclude that the allocation at step t cannot be proportional, and is hence not envy-free.
Overall, we have that
[Equation]
Therefore, for t ≥ 6 / E [ X ] , at least one of Pr (cid:104) E SPO ( t ) (cid:105) and Pr (cid:104) E EF ( t ) (cid:105) is at least c/ 4 .
We conclude that no algorithm can be both envy-free and one-swap PO with high probability.
B.2 Proof of Theorem
Fix a distribution D with CDF F and let X be a random variable with distribution D .
Fix some ε to be (1 − ε ) -welfare-maximizing.
Let E T 1 be the event that the maximum social welfare at time T is at least 1 / 2 · E [ X ] · T , let E T 2 be the event that quantile maximization is c -strongly-EF for c =
( E [ X | F ( X ) ≥ 1 / 2] − E
[ X ]) 4 n , and let E T 3 be the event that Algorithm 1 differs from quantile maximization on at most f ( T ) items from Lemma 7.
We ﬁrst claim that E T 1 ∩E T 2 ∩E T 3 occurs with high probability in T .
Note that Lemmas 1 and 7 tell us E T 2 and E T 3 each occur with high probability, respectively.
For
E T 1 , the maximum value for each item is in expectation at least the expected value for a single agent
E [ X ] .
Hence, a Chernoff bound tells us E T 1 occurs with probability at least 1 − exp (cid:16) − E
[ X ] T(cid:17) , i.e., with high probability.
The claim holds because the intersection of a ﬁnite number of high probability events occurs with high probability.
Next, note that for sufﬁciently large T , since f ( T ) ∈
o ( T ) , f ( T ) ≤ ( E
[ X | F ( X ) ≥ 1 / 2] − E [ X ]) 8 n · T and f ( T ) ≤ ε/ 2 · E [ X ] · T (for any ﬁxed ε that does not depend on T ).
Fix such a sufﬁciently large T .
We show that, conditioned on E T 1 ∩ E T 2 ∩ E T 3 , both EF and (1 − ε ) -welfare hold.
Let
A QM =
( A QM, . . .
, A QMn ) be the allocation of quantile maximization and A =
( A 1 , . . .
, A n ) be the allocation of Algorithm 1.
Beginning with envy-freeness, we have that for all pairs of agents i and j ,
[Equation]
4 n
[Equation]
so the allocation is envy-free.
Further, noting that sw ( A QM ) is the maximum social welfare, we have the welfare approximation is at least
[Equation]
≥ ( E T 1 ) 1 − f ( T ) 1 / 2 · E [ X ] · T ≥ ( E T 3 ) 1 − ε/ 2 · E [ X ] · T 1 / 2 · E [ X ] · T = 1 − ε, as needed.
C Missing Proofs from SectionC.1 Proof of Theorem
We prove that this negative result holds even for an even stronger class of algorithms in which, at each step t , the algorithm selects quantile thresholds q t 1 , . . .
, q tn ∈ [0 , 1] for each agent, and once an item arrives the algorithm observes, for each agent, whether the quantile of their sampled value Q i,t is above or below the threshold q ti .
Note that this provides at least as much information about the fresh item as comparing it to any single prior item, since there is some uncertainty about the values and quantiles of all prior items.
We ﬁrst focus on the algorithm for a single time-step and show there is a distribution of values such that, regardless of the quantile thresholds selected and allocations made, it cannot do well.
Fix a number of agents n and assume n ≥ 3 .
We handle the special case of n = 2 at the end of this proof, as it requires a different distribution.
For simplicity we consider a distribution that takes values larger than 1 ; re-scaling (speciﬁcally, dividing all values by 2 + ε ) gives a distribution upped bounded by 1 and does not affect any of our arguments.
Consider the value distribution X , with
X ∼  
Unif [0 , ε ] with probability 1 − 1 n ,
Unif [1 , 1 + ε ] with probability 23 n , and
Unif [2 , 2 + ε ] with probability 13 n for some small ε > 0 to be ﬁxed later.
Intuitively, X is a continuous version of a discrete distribution which takes low value (near 0 ) with probability 1 − 1 n , medium value (near 1 ) with probability 23 n , and high value (near 2 ) with probability 13 n .
Let F X be its CDF.
Trivially, the maximum social welfare of T items when all agents have this value distribution is at most T · (2 + ε ) .
We show that regardless of what quantile thresholds the algorithm chooses at step t and which decision it makes given the resulting signals, the expected value of the agent receiving item t is at least (1 − ε ) · 1 144 e away from optimal.
To that end, ﬁx arbitrary thresholds q 1 , . . . , q n .
First, we partition the agents depending on whether their quantile q i is above or below 1 − 2 n 3 .
We let
N below = { i ∈ [ n ] | q i < 1 − 2 n 3 } and N above = { i ∈ [ n ] | q i ≥ 1 − 2 n 3 } .
Either | N below | ≥ (cid:100) n/ 2 (cid:101) or | N above | ≥ (cid:100) n/ 2 (cid:101) ; we analyse each case separately.
Since n ≥ 3 , we have (cid:100) n/ 2 (cid:101) ≥ 2 .
Case I: | N below | ≥ (cid:100) n/ 2 (cid:101) .
In this case, it will be difﬁcult for the algorithm to distinguish between agents in N below with medium value and those with high value.
Consider the event E that one agent
i max ∈ N below has quantile Q i max > 1 − 13 n , one agent i smax ∈ N below has quantile Q i smax ∈ (1 − 23 n , 1 − 13 n ) , and all other agents i ∈ N \ { i max , i smax } have quantile Q i < 1 − 1 n .
First, we show that Pr[ E ] ≥ 172 e , a constant.
To compute this probability, note that there are at least (cid:100) n/ 2 (cid:101) · ( (cid:100) n/ 2 (cid:101) − 1) choices of i max and i smax .
Once these have been selected, the probability of E occurring for this pair of agents is
[Equation]
Since (cid:100) n/ 2 (cid:101) · ( (cid:100) n/ 2 (cid:101) − 1) ≥ n 2 / 8 , we can that conclude Pr[ E ] ≥ 1 72 e .
Conditioned on E occurring, i max has high value, i smax has medium value, and all other agents have low value.
However, from the perspective of the algorithm, two agents ( i max and i smax ) give a high signal, and it’s equally likely that each of them is the agent with the high value (note that we condition on E ).
The algorithm must therefore allocate the item to an agent with at most medium value (upper bounded by 1 + ε ) with probability at least 1 / 2 , even though an agent with value at least 2 exists.
Hence, in this timestep, the algorithm has an additive error (compared to the optimum welfare) of at least (1 − ε ) with probability at least 1 144 e .
Case II: | N above | ≥ (cid:100) n/ 2 (cid:101) .
In this case, it will be difﬁcult for the algorithm to distinguish between agents in N above that have medium value and those with low value.
Consider the event E that one agent
i max ∈ N above has quantile Q i max ∈ (1 − 1 n , 1 − 23 n ) and all other agents i ∈ N \
{ i max } have quantile Q i < 1 − 1 n .
First, we show that Pr[ E ] ≥ 16 e .
Indeed, there are at least n/ 2 choices for i max .
For a ﬁxed choice of i max , the probability of E occurring is 13 n · (cid:0) 1 − 1 n (cid:1) n − 1 ≥ 13 en , and there are at least n/ 2 choices for i max ,
so Pr[ E ] ≥ 16 e .
Agent i max and the other members of N above (there is at least one more) are indistinguishable to the algorithm as they all have a low signal, so the algorithm must give it to an agent with value at most ε with probability at least 1 / 2 even though an agent with value at least 1 exists.
Hence, in this timestep, the algorithm has an additive error (compared to the optimum welfare) of at least (1 − ε ) with probability at least 112 e .
In either case, for every time step, the algorithm has an additive error of at least (1 − ε ) with probability at least 1 144 e , irrespective of the past allocations.
As time steps are independent, standard tail bounds give that, for sufﬁciently small ε > 0 , the error is at least 1 − ε 1000 T with high probability.
The optimal social welfare is at most (2 + ε ) ·
T ; we conclude the algorithm can be no more than an 0 .
999 − approximation to welfare.
The case of two agents.
Finally, we handle the case of two agents.
Assume values are drawn from a Unif [0 , 1] distribution.
Let q 1 , q 2 be the quantile thresholds selected by the algorithm and, without loss of generality, suppose that 0 ≤ q 1 ≤ q 2 ≤ 1 .
At least one of the differences q 1 − 0 , q 2 − q 1 , 1 − qmust be at least 1 / 3 .
Suppose q 2 − q 1 ≥ 1 / 3 (the other cases are symmetric).
We investigate the event that both agents have Q i ∈
[ q 1 , q 2 ] , so that agent 1 signals high and agent 2 signals low, which occurs with probability at least 1 / 9 .
Conditioned on this event, the signals do not provide any additional information, so the algorithm chooses the agent with smaller value at least half of the time.
In this case, the expected difference between the larger and smaller values is 1 / 9 .
Hence, the expected difference of the value from the algorithm versus the maximum social welfare is at least 19 · 12 · 19 = 1 / 162 on each item.
The maximum social welfare is at most T , and we expect the difference to be at least T/ 1000 due to concentration, so the algorithm cannot guarantee more than a . 999 approximation, as needed.
C.2 Proof of Lemma
Fix such an ε , δ , τ , and (cid:96) .
We claim that a sufﬁcient condition for ε -accuracy is that all agents accept an item with quantile within q ∗ ± ε/ (2 n ) .
Indeed, note that any sampled quantile outside this range will be classiﬁed (as high vs low) correctly.
With such an error tolerance, the probability a speciﬁc agent’s quantile (for a fresh item) falls within this range is at most ε/n .
Via a union bound over all n agents, the probability that no agent has a quantile (for a fresh item) within this range is at least 1 − ε .
Hence, all that needs to be shown is that with probability 1 − δ , all agents accept an item and the accepted item has quantile within the allowed range.
Since there are τ trials, there are at most nτ items tested across all agents.
We show that (cid:96) is large enough such that with probability 1 − δ/ 2 , all these tests are within ± ε/ (6 n ) of the true value.
Using
Hoeffding’s inequality, the probability any speciﬁc test fails is at most
2 exp [Table]
a union bound over all nτ tests yields the required probability.
Note that under the condition that all the tests are this accurate, since the threshold for acceptance is ± ε/ (3 n ) , any accepted item will be within ± ε/ (2 n ) of q ∗ , as needed.
What remains to be shown is that each agent will, with reasonable probability, accept an item.
To that end, we need to show that with probability 1 − δ/ 2 , all agents will test an item that is within ± ε/ (6 n ) of q ∗ .
If such an item is tested and the test is accurate, the empirical estimate of its quantile is within ± ε/ (3 n ) , and the item would hence be accepted.
A union bound will then tell us that both of these events would occur with probability 1 − δ .
Towards proving that each agent will test an item within ± ε/ (6 n ) of q ∗ with probability 1 − δ/ 2 , we use a union bound, showing that each agent individually will not sample such an item with probability at most δ/ (2 n ) .
In each of τ trials, the probability such an item is sampled is ε/ (3 n ) .
Hence, the probability no such item is sampled is (cid:0) 1 − ε 3 n (cid:1) τ .
We then have that
[Equation]
as needed.
C.3 Proof of Lemma
First, we prove that for all k ≥ 10 n , epoch k is ε k -accurate with probability δ k for ε k = 3 n/kand δ k
= 2 ne − k .
Since k > 3 n these are valid values between 0 and 1 .
Hence, we simply need to check that the τ and (cid:96) inequalities hold for the number of trials and number of test items speciﬁed in Algorithm 2.
For arbitrary epoch k , ln(2 n/
[Equation]
so the number of trials is sufﬁciently large.
Further,
[Equation]
Recall that k ( t ) is deﬁned as the epoch of item t .
As in the proof of Lemma 7, we characterize deviations from the ideal algorithm in four ways.
Item t was allocated in one of the ﬁrst 10 n − 1 epochs; that is, k ( t ) <
10 n . 2.
Item t was allocated during the sampling phase of epoch k ( t ) ≥ 10 n . 3.
Item t was allocated during the ranking phase of epoch k ( t ) ≥ 10 n , which was ε k ( t ) -accurate. Item t was allocated during the ranking phase of epoch k ( t ) ≥ 10 n , which was not ε k ( t ) -accurate.
We say an item t is incorrect (incorrectly allocated) when it is given to an agent with non-maximum quantile for it.
We show that the number of mistakes in each category are bounded by 10 20 n 19 , 2 T 5 / 9 , 7 nT 17 / 18 and 1 . 3 · 10 16 n respectively, with high probability.
This implies, via a union bound, that the total number of mistakes is at most the sum of these quantities, or O ( poly ( n ) · T 17 / 18 ) , with high probability.
The number of items in category 1, is at most
[Equation]
Notice that T ≥ (cid:80) k ( T ) − 1 k
=
k 9 + k 18 ≥ ( k ( T ) − 1) 18 , and therefore k ( T ) ≤ 2 T 1 / 18 .
For the second category, since k ( T ) ≤ 2 T 1 / 18 , the total number of items in the sampling phase is (with probability 1 ) upper bounded by k ( [Equation] ≤
[Equation]
For the third category, note that each item t in this category has probability ε k
( t ) of being incorrect.
The expected number of mistakes is at most
[Equation]
Using Hoeffding’s inequality we get that with high probability the number of mistakes is at most 7 nT 17 / 18 , since a deviation of nT 17 / 18 occurs with probability at most exp (cid:0) − 2 n 2 T 17 / 9 /T (cid:1) =
exp (cid:0) − 2 n 2 T 8 / 9 (cid:1) .
For the fourth category, the expected number of items in this category is at most
[Equation]
Using Markov’s inequality we have that the number of mistakes is at most 1 . 3 · 10 16 n ln( T ) with probability at least 1 − ln( T ) , i.e., with high probability.
C.4 Proof of Theorem
The proof is nearly identical to the proof of Theorem 5.
Fix a distribution D with CDF F and let
X be a random variable with distribution D .
Fix some ε to be (1 − 1 /e )
− ε welfare maximizing.
Let E T 1 be the event that the maximum social welfare at time T is at least 1 / 2 · E [ X ] · T , let E T 2 be the event the ideal threshold algorithm is c -strongly-EF for c
= ( E [ X | F ( X ) ≥ 1 / 2] − E
[ X ]) 4 n , let E T 3 be the event that the ideal threshold algorithm is a (1 − 1 /e ) 2 − ε/ 2 approximation to welfare, and let
E T 4 be the event that Algorithm 2 differs from the ideal threshold algorithm on at most f ( T ) items from Lemma 11.
We ﬁrst claim that E T 1 ∩ E T 2 ∩ E T 3 ∩ E T 4 occurs with high probability in T .
Note that Lemmas 2, 1 ,and 11 tell us each of E T 2 , E T 3 , and E T 4 occur with high probability.
For E T 1 , the maximum value for each item is in expectation at least the expected value for a single agent E [ X ] .
Hence, a Chernoff bound tells us E T 1 occurs with probability at least 1 − exp (cid:16) − E [ X ] T(cid:17) , i.e., with high probability.
The claim holds because the intersection of a ﬁnite number of high probability events occurs with high probability.
Next, note that for sufﬁciently large T , since f ( T ) ∈
o ( T ) , f ( T ) ≤ ( E
[ X | F ( X ) ≥ 1 / 2] − E [ X ]) 8 n · T and f ( T ) ≤ ε/ 4 · E [ X ] · T .
Fix such a sufﬁciently large T .
We show that conditioned on E T 1 ∩
E T 2 ∩ E T 3 ∩ E T 4 , both EF and ((1 − 1 /e ) 2 − ε )
-welfare hold.
Recall that a “difference” between Algorithm 2 and the ideal threshold algorithm refers to different distributions over the agents that get some item (i.e., a different randomized allocation).
In order to make statements about envyfreeness and efﬁciency we need a way to argue about the differences between the algorithms expost.
However, notice that without loss of generality we can couple the decision made by the two algorithms when randomized allocation is the same; that is, when Algorithm 2 does not differ from the ideal threshold algorithm we can assume without loss of generality that the agent who gets the item is the same.
Let A IT =
( A IT 1 , . . .
, A ITn ) be the allocation of the ideal threshold algorithm and
A =
( A 1 , . . .
, A n ) be the allocation of Algorithm 2.
Beginning with envy-freeness, we have that for all pairs of agents i and j ,
[Equation]
[Equation]
[Equation]
so the allocation is envy-free.
Let A ∗ be a welfare-maximizing algorithm.
For the welfare approximation, we then have
[Equation]
as needed.
D Missing Proofs from SectionD.1 Proof of Theorem
Suppose for contradiction that there is an algorithm A so that for all bounded continuous distributions ( X 1 , X 2 ) there exists a T ∗ =
T ∗ ( X 1 , X 2 ) where for all t ≥ T ∗ , A is envy-free and c -PO with probability p with p > 2 / 3 for some constant c > 1+ √ 5 4 .
Hence, there is some ε such that p > 2 / 3 + ε and 1 /c < 4 1+ √ 5 − ε = √ 5 − 1 − ε .
Consider two distributions D F and D S ; we describe these later in the proof.
Consider the three instances I
=
( D F , D F ) , I 1 =
( D S , D F ) and I 2 =
( D F , D S ) .
Let E A ,t j be the event that A is envy-free and c -PO on instance
I j at time t for j ∈ { 0 , 1 , 2 } .
By construction, Pr (cid:104) E A ,t j (cid:105) ≥ 2 / 3 + ε for all j ∈ { 0 , 1 , 2 } and t ≥ T ∗ .
Let z be a parameter we will ﬁx later in the proof, and let Z ti
=
I { Q i,t ≥ 1 − z } for i = { 1 , 2 } .
Observe that Z t 1 · Z t 2 is 1 with probability z 2 and 0 otherwise.
The following events characterize a speciﬁc notion of a “nice” sample, in which the number of items with high quantiles for both agents is near its expectation: E T
= I {| 1 T (cid:80) Tt
{+=1 Z t 1 · Z t 2 − z 2 | < δ } , E T 
 = I {| 1 T (cid:80) Tt+}
=1 Z t 1 − z | < δ } , and E T
= I {| 1 T (cid:80) Tt
=1 Z t 2 − z | < δ } for some δ > 0 .
By Hoeffding’s inequality, Pr (cid:2) ¯ E T 1 (cid:3) =
Pr (cid:104) | 1 T (cid:80)
{+Tt =1 Z t 1 · Z t 2 − z 2 | ≥ δ (cid:105) ≤ 2 exp (cid:0) − 
 Tδ 2 (cid:1) .+}
It follows that for T ≥ log(2 /ε ) / (2 δ 2 ) ,
Pr (cid:2) ¯ E T 1 (cid:3) ≤ ε .
Similarly, for T ≥ log(2 /ε ) / (2 δ 2 ) , it holds that Pr (cid:2) ¯ E T 2 (cid:3) ≤ ε , and Pr (cid:2) ¯ E T 3 (cid:3) ≤ ε .
Consider an arbitrary T > T max = max { T 0 , T 1 , T 2 , log(2 /ε ) /
(2 δ 2 ) } .
Applying a union bound,
Pr (cid:104) ¯ E A ,T∪ ¯ E A ,T∪ ¯ E A ,T∪ ¯ E [Equation] ∪
[Equation]
E
[Equation]
∪
[Equation]
E
[Equation]
(cid:105) [Equation] It follows that Pr (cid:104) E A ,T∩ E A ,T∩ E A ,T∩ E T 1 ∩ E T 2 ∩ E T 3 (cid:105) > 0 . Therefore, there must exist a sequence of T items whose quantiles satisfy all of E T 1 , E T 2 , and E T 3 , and, since A does not have access to the items’ values, there must exist an allocation A T for these T items (in the support of A ) that is EF and c -PO, no matter which of I 0 , I
or I 2 the values were taken from.
Let q T =
{ ( q 1 ( t ) , q 2 ( t )) }
Tt =1 be these items’ quantiles.
Let H B = { t ∈
[ T ] : q 1 ( t ) ≥ 1 − z and q 2 ( t ) ≥ 1 − z } be the items for which Z t 1 · Z t 2 = 1 , and H 1 = {
t ∈
[ T ] : q 1 ( t ) ≥ 1 − z } the items for which Z t 1 = 1 .
Set distributions D F = Unif [1 − w, 1] and D S , under which each item is Unif [0 , w ] with probability z and at Unif [1 − w, 1] with probability 1 − z , for some small positive w that we ﬁx later in the proof.
We have that some agent receives at most half the items in H B ; without loss of generality this is agent 2 , i.e., | A T 2 ∩ H B | ≥ | H B | / 2 .
We show that there exists a feasible more than 1 /c Pareto-improvement under the values in I 1 .
To that end, we compare A T to the allocation ˆ A
where ˆ A 1 = H 1 and ˆ
A 2 = ¯ H 1 .
We next bound the utilities of each agent under A T and ˆ A .
Beginning with agent 1 , we have
[Equation]
and
[Equation]
[Equation]
δ/ 2) .
Together, these imply
[Equation]
Next, we consider agent 2 .
We have
[Equation]
By E A 0 , A T is envy-free on I 0 .
It follows that | A T 1 | ≥ (1 − w ) | A T 2 | .
Since | A T 1 | + | A T 2 | =
T , we have that | A T 2 | ≤ 1 2 − w T .
Hence, u 2 ( A T 2 ) ≤ | A T 2 | ≤ 1 2 − w T .
Combining these, we have u 2 ( ˆ A 2 ) u
[Equation]
Choose z
= 3 −√ 5 2 .
Note that z 2 = 7 − 3 √ 5.
Choose δ, w < ε/ 25 .
We then have,
[Equation] [Equation]
and
[Equation]
so this is more than a 1 /c Pareto Imrovement.
D.2 Proof of Theorem
The proof of envy-freeness for each algorithm is nearly identical to Theorems 5 and 9 respectively; we show it here for completeness.
We focus on Algorithm 1.
The proof for Algorithm 2 goes through identically with all occurances of quantile maximization replaced with the ideal threshold algorithm and all occurances of Lemma 7 replaced with Lemma 11.
Fix a distributions D 1 , . . .
, D n with CDFs F 1 , . .
. , F n and let X i be a random variable with distribution D i .
Fix some ε to be (1 /e − ε ) -PO.
Let E = min i ∈N E
[ X i ] be the minimum expected value for all agents.
Let E T 1 be the event that each agent i ’s value for their bundle at time T is at least 1 / (2 n ) · E · T , let E T 2 be the even that quantile maximization is c -strongly-EF for c
= min i ∈N ( E [ X i | F i ( X i ) ≥ 1 / 2] − E
[ X i ]) 4 n , let E T 3 be the event that quantile maximization is a (1 /e − ε/ 2) -PO, and let E T 4 be the event that Algorithm 1 differs from quantile maximization on at most f ( T ) items from Lemma 7.
We ﬁrst claim that E T 1 ∩ E T 2 ∩ E T 3 ∩ E T 4 occurs with high probability in T .
Note that Lemmas 1, 3, and 7 tell us E T 2 , E T 3 , and E T 4 each occur with high probability, respectively.
For E T 1 , note that under quantile maximization, the probability each agent i receives an item is exactly 1 /n and the expected value conditioned on receiving the item is at least E [ X i ] ≥ E .
Hence, the expected contribution of each item to v i ( A i ) is at least 1 /n · E .
A Chernoff bound then tells us E T 1 holds for agent i with probability at least 1 − exp (cid:0) − ET 8 n (cid:1) .
A union bound over all agent’s tells us this occurs simultaneously for all agents with probability at least 1 − n exp (cid:0) − ET 8 n (cid:1) , i.e., with high probability.
The claim holds because the intersection of a ﬁnite number of high probability events occurs with high probability.
Next, note that for sufﬁciently large
T , since f ( T ) ∈ o
( T ) , f ( T ) ≤ min i ∈N ( [Equation] [Equation]
[Equation]
≥
[Equation]
[Equation] [Equation] [Equation] i ]
[Equation] n · T and f ( T ) ≤ ε/ (4 n ) · ET .
Fix such a sufﬁciently large T .
We show that conditioned on E T 1 ∩ E T 2 ∩ E T 3 ∩ E T 4 , both EF and (1 /e − ε ) -PO hold.
Let
A QM =
( A QM, . . .
, A QMn ) be the allocation of quantile maximization and A =
( A 1 , . . .
, A n ) be the allocation of Algorithm 2.
Beginning with envy-freeness, we have that for all pairs of agents i and j , v i
A i ) ≥ ( E T 4 ) v i ( A QMi )
− f ( T ) ≥ ( E T 2 ) v i ( A QM j ) − f ( T ) + min i ∈N ( E [ X i | F i ( X i )
[Equation]
[Equation] E [ X i ]) 4 n ≥ ( E T 4 ) v i ( A j ) − 2 f ( T ) + min i ∈N ( E [ X i | F i ( X i ) ≥ 1 / 2] − E [ X i ]) 4 n ≥ v i ( A j ) , so the allocation is envy-free.
Next, we show that for all agents v i ( A i ) v i ( A QMi ) ≥ 1 − ε/ 2 .
Since A QM is (1 /e − ε/ 2) -PO under E T 3 , this implies that A is a (1 /e − ε/ 2)(1 − ε/ 2) ≥ 1 /e − ε approximation to PO as well.
To that end, for each agent i we have
[Equation]
as needed.