Recent Advances in Fair Resource Allocation Rupert Freeman and - - PowerPoint PPT Presentation
Recent Advances in Fair Resource Allocation Rupert Freeman and Nisarg Shah Microsoft Research New York University of Toronto EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation Rupert Freeman and Nisarg
Rupert Freeman and Nisarg Shah
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 1
Microsoft Research New York University of Toronto
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 2
Ø Assume any prior knowledge of fair division Ø Walk you through tedious, detailed proofs Ø Claim to present a complete overview of the entire fair division realm Ø Present unpublished results
Ø Focus mostly on the case of “additive preferences” for coherence
Ø Please email nisarg@cs.toronto.edu or Rupert.Freeman@microsoft.com
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 3
Ø Proportionality Ø Envy-freeness Ø Maximin share guarantee Ø Groupwise fairness
Ø Price of fairness Ø Interplay with strategyproofness
and Pareto optimality
Ø Restricted cases
Ø Cake-cutting Ø Homogeneous divisible goods Ø Indivisible goods
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 4
Ø May be finite or infinite
Ø Valuation of agent 𝑗 is 𝑤! ∶ 2" → ℝ Ø Range is ℝ# when resources are goods, and ℝ$ when they are bads
Ø 𝐵 = 𝐵%, … , 𝐵& ∈ Π& 𝑁 is a partition of resources among agents
Ø A partial allocation 𝐵 may have ∪!∈( 𝐵! ≠ 𝑁
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 5
Ø The entire cake is allocated (full allocation) Ø Each 𝐵! ∈ , where is the set of finite
unions of disjoint intervals
Ø Each agent is allocated a single interval Ø Cuts cake at 𝑜 − 1 points
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 6
function 𝑔
/∈0 𝑔
1 2 𝑔
Ø Without loss of generality
𝛽 𝜇𝛽 𝛽 β β
𝛽 + 𝛾
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 7
following properties
0,1 = 1
∃𝑨 ∈ [𝑦, 𝑧] s.t. 𝑤- [𝑦, 𝑨] = 𝜇𝑤-([𝑦, 𝑧])
𝑤- 𝐽 + 𝑤- 𝐽′ = 𝑤- 𝐽 ∪ 𝐽′
𝛽 𝜇𝛽 𝛽 β β
𝛽 + 𝛾
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 8
Ø Infinitely many bits may be needed to fully
represent the input
Ø Query complexity is more useful
Ø Eval!(𝑦, 𝑧) returns 𝑤!
𝑦, 𝑧
Ø Cut!(𝑦, 𝛽) returns 𝑧 such that 𝑤!
𝑦, 𝑧 = 𝛽
𝑦 𝑧
𝛽
eval output cut output
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 9
2 3
Ø Each agent should receive her “fair share” of the utility.
Ø No agent should wish to swap her allocation with another agent.
Ø All agents should have the exact same value for their allocations. Ø No agent should be jealous of what another agent received.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 10
2 5] uniformly
and does not want anything else
uniformly
6 5 , 1] uniformly
and does not want anything else
1
1 2 3
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 11
2 7 ⇒ 𝑤2 𝐵2 = ⁄ 2 5
⁄
2 7 , ⁄ 8 7 ⇒ 𝑤6 𝐵6 = ⁄ 9 7
⁄
8 7 , 1 ⇒ 𝑤5 𝐵5 = ⁄ 2 5
but not envy-free or equitable
1
1 2 3
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 12
2 : ⇒ 𝑤2 𝐵2 = ⁄ 2 6
⁄
2 : , ⁄ ; : ⇒ 𝑤6 𝐵6 = ⁄ 6 5
⁄
; : , 1 ⇒ 𝑤5 𝐵5 = ⁄ 2 6
and envy-free, but not equitable
1
1 2 3
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 13
2 ; ⇒ 𝑤2 𝐵2 = ⁄ 5 ;
⁄
2 ; , ⁄ < ; ⇒ 𝑤6 𝐵6 = ⁄ 5 ;
⁄
< ; , 1 ⇒ 𝑤5 𝐵5 = ⁄ 5 ;
envy-free, and equitable
1
1 2 3
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 14
Ø Summing 𝑤! 𝐵! ≥ 𝑤! 𝐵1 over all 𝑘 gives proportionality
Ø Hence, they are equivalent.
Ø E.g. if each agent has value 0 for her own allocation and 1 for the other agent’s
allocation, it is equitable but not proportional or envy-free.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 15
Suppose the value density function 𝑔
the 𝑜6 − 𝑜 + 1 intervals into 𝑜 pieces 𝐵2, … , 𝐵3 such that 𝑤- 𝐵4 = M 1 𝑜 , ∀𝑗, 𝑘 ∈ 𝑂
Ø It is trivially envy-free (thus proportional) and equitable
Robertson-Webb model
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 17
Ø Agent 1 cuts the cake at 𝑦 such that 𝑤% 0, 𝑦
= 𝑤% 𝑦, 1 = ⁄ 1 2
Ø Agent 2 chooses the piece that she prefers.
Ø Proportional (equivalent to envy-freeness for 2 agents) Ø Needs only one cut and one eval query (optimal)
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 18
Ø Referee starts a knife at 0 and moves the knife to the right. Ø Repeat: When the piece to the left of the knife is worth 1/𝑜 to an agent, the
agent shouts “stop”, receives the piece, and exits.
Ø When only one agent remains, she gets the remaining piece.
Ø When [𝑦, 1] is left, ask each remaining agent 𝑗 to cut at 𝑧! so that 𝑤!
𝑦, 𝑧! = 1/𝑜, and give agent 𝑗∗ ∈ arg min! 𝑧! the piece [𝑦, 𝑧!∗].
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 19
Ø Interval [𝑦, 𝑧], number of agents 𝑜 (assume a power of 2 for simplicity)
Ø If 𝑜 = 1, give [𝑦, 𝑧] to the single agent. Ø Otherwise:
𝑦, 𝑨! = 𝑤! 𝑨!, 𝑧
⁄ 𝑜 2 &' mark from the left.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 20
Ø Any protocol returning a proportional allocation needs Ω(𝑜 log 𝑜) queries in
the Robertson-Webb model.
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 22
Gets complex pretty quickly!
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 23
Ø The first finite (but unbounded) protocol for any number of agents
Ø The first bounded protocol for 4 agents (at most 203 queries)
Ø A simplified version of the above protocol for 4 agents (at most 171 queries)
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 24
Ø There exists a bounded protocol for computing an envy-free allocation with 𝑜
agents, which requires 𝑃(𝑜&"""" ) queries
Ø After 𝑃 𝑜3 queries, the protocol can output a partial allocation that is
both proportional and envy-free
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 25
Any protocol for finding an envy-free allocation requires Ω(𝑜6) queries.
There is no finite (even unbounded) protocol for finding a simple envy-free allocation for 𝑜 ≥ 3 agents. Open Problem Bridge the gap between 𝑃(𝑜3!!!! ) upper bound and Ω 𝑜6 lower bound for envy-free cake-cutting
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 27
Ø Suppose we cut the cake at 𝑦 to form pieces [0, 𝑦] and [𝑦, 1] Ø Let 𝑔 𝑦 = 𝑤% 0, 𝑦
− 𝑤3 𝑦, 1
Ø By the intermediate value theorem: ∃𝑦∗ such that 𝑔 𝑦∗ = 0 Ø Allocation 𝐵% = [0, 𝑦∗] and 𝐵3 = [𝑦∗, 1] is equitable
Ø Using binary search for 𝑦∗, we can find an 𝜗-equitable allocation for 2 agents
with 𝑃 ln ⁄
% 5
queries.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 28
Ø This technique can be extended to 𝑜 agents to find an 𝜗-equitable allocation in
𝑃 𝑜 ln ⁄
% 5
queries.
Ø There exists a protocol for 𝑜 agents which finds an 𝜗-equitable allocation in
𝑃 ⁄
% 5 ln ⁄ % 5
queries.
Ø Intuition:
( ), use above protocol for finding an equitable 𝜗-equitable allocation.
( ), use a variant of the Evan-Paz algorithm to find an anti-proportional allocation
where 𝑜* = ⁄
( ) agents get value at most 1/𝑜′, and the rest receive nothing.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 29
Any protocol for finding an 𝜗-equitable allocation must require Ω
OP ⁄
" #
OP OP ⁄
" #
queries.
There is no finite (even if unbounded) protocol for finding an equitable allocation.
Ø Non-existence of bounded protocols follows from the previous result. Ø But their proof works for non-existence of unbounded protocols as well.
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 31
a fairness property 𝑌
Ø Denoted 𝑡𝑥 𝐵 = ∑!∈( 𝑤! 𝐵!
feasible allocations satisfying property 𝑌 𝑄𝑝𝐺
0 = sup Q",…,Q!
max
S∈ℱ 𝑡𝑥(𝐵)
max
S∈ℱ$ 𝑡𝑥(𝐵)
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 32
For cake-cutting, the price of proportionality is Θ 𝑜 , and the price
For cake-cutting, the price of envy-freeness is also Θ 𝑜 . This is achieved by an allocation maximizing the Nash welfare Π- 𝑤- 𝐵- .
Ø Fun fact: The price of EF in cake-cutting was mentioned as an open question in
a previous version of this tutorial, and was also believed to be open by many groups of researchers until recently.
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 34
Ø Allocation 𝐵 is weakly Pareto optimal if there is no allocation 𝐶 such that
𝑤! 𝐶! > 𝑤!(𝐵!) for all 𝑗 ∈ 𝑂.
Ø “Can’t make everyone happier”
Ø Allocation 𝐵 is Pareto optimal if there is no allocation 𝐶 such that 𝑤! 𝐶! ≥
𝑤! 𝐵! for all agents 𝑗 ∈ 𝑂, and at least one inequality is strict.
Ø “Can’t make someone happier without making someone else less happy” Ø Easy to achieve in isolation (e.g. “serial dictatorship”)
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 35
With two agents, there always exists an allocation that is envy-free (thus proportional), equitable, and Pareto
Ø Their algorithm has similarities to the more
popular “adjusted winner” algorithm, which we will see later in the tutorial.
impossible
1
1 2
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 36
Ø At equilibrium: there is an additive price function 𝑄 on the cake, and each
agent gets to buy their best piece from a budget of one unit of fake currency
Ø WCE: ∀𝑗 ∈ 𝑂, 𝑎 ⊆ 0,1 : 𝑄 𝑎 ≤ 𝑄 𝐵! ⇒ 𝑤! 𝑎 ≤ 𝑤!(𝐵!) Ø EI: ∀𝑗 ∈ 𝑂: 𝑄 𝐵! = 1
For cake-cutting, a CEEI always exists. Every CEEI is both envy-free and weakly Pareto optimal.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 37
Ø A positive slice 𝑎 is a subset of the cake valued positively by at least one agent Ø Allocation 𝐵 is called s-CEEI allocation if there exists an additive price function
𝑄 satisfying
Ø 𝑄 𝑎 > 0 iff 𝑎 is a positive slice Ø SCE: ∀𝑗 ∈ 𝑂, and positive slices 𝑎 ⊆ [0,1] and 𝑎! ⊆ 𝐵!: 6#(8#)
:(8#) ≥ 6#(8) :(8)
Ø EI: ∀𝑗 ∈ 𝑂: 𝑄 𝐵! = 1
For cake-cutting, an s-CEEI allocation always exists. Every s-CEEI allocation is envy-free and Pareto optimal.
Maximum bang-per-buck
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 39
Ø Agent 1 gets 𝑦 fraction of [0, ⁄
3 4]
Ø Agent 2 gets 1 − 𝑦 fraction of [0, ⁄
3 4] and
all of [ ⁄
3 4 , 1]
Ø 𝑤% 𝐵% = 𝑦 Ø 𝑤3 𝐵3 = 1 − x ⋅ ⁄
3 4 + ⁄ % 4 =
c
(4$3;) 4
M
(5W6X) 5 ⇒ 𝑦 = ⁄ 5 <
Ø Nash-optimal allocation:
( + , 𝑤( 𝐵( = ⁄ , -
⁄
( + , 1 , 𝑤+ 𝐵+ = ⁄ ( +
1
1 1.5 𝑦 ∶ 1 − 𝑦 Allocated to agent 1 Allocated to agent 2
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 40
Ø Agent 1 buys 𝑦 fraction of [0, ⁄
3 4]
Ø Agent 2 buys 1 − 𝑦 fraction of [0, ⁄
3 4] and
all of [ ⁄
3 4 , 1]
6 5
= 𝑏, 𝑄 ⁄
6 5 , 1
= 𝑐
Ø Spending: 𝑏 ⋅ 𝑦 = 1, 𝑏 ⋅ 1 − 𝑦 + 𝑐 = 1
1
1 1.5 Allocated to agent 1 Allocated to agent 2 𝑦 ∶ 1 − 𝑦 Price = 𝑏 Price = 𝑐
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 41
Ø Agent 2 buys parts of both pieces Ø MBB:
] 1 3 𝑐 = ] 2 3 𝑏 ⇒ 𝑏 = 2𝑐 ⇒ 𝑏, 𝑐 = ] 4 3 , ] 2 3
Ø Substituting in 𝑏 ⋅ 𝑦 = 1, we get 𝑦 = ⁄
4 C
1
1 1.5 𝑦 ∶ 1 − 𝑦 Allocated to agent 1 Allocated to agent 2 Price = 𝑏 Price = 𝑐
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 42
Ø Since 𝑏 ⋅ 𝑦 = 1, 𝑏 ⋅ 1 − 𝑦 + 𝑐 = 1, we get
that 𝑏 = 𝑐 = 1
Ø Agent 2 buys the second piece, so by MBB:
] 1 3 𝑐 ≥ ] 2 3 𝑏 ⇒ 𝑏 ≥ 2𝑐
Ø Contradiction! Ø So there is no s-CEEI with 𝑦 = 1
1
1 1.5 𝑦 ∶ 1 − 𝑦 Allocated to agent 1 Allocated to agent 2 Price = 𝑏 Price = 𝑐
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 44
Ø A direct-revelation mechanism ℎ takes as input all the valuation functions
𝑤%, … , 𝑤&, and returns an allocation 𝐵
Ø Notation: ℎ 𝑤%, … , 𝑤& = 𝐵, ℎ! 𝑤%, … , 𝑤& = 𝐵!
Ø A direct-revelation mechanism ℎ is called strategyproof if
∀𝑤%, … , 𝑤&, ∀𝑗, ∀𝑤!
D ∶ 𝑤! ℎ! 𝑤%, … , 𝑤&
≥ 𝑤!(ℎ! 𝑤%, … , 𝑤!
D, … , 𝑤& )
Ø That is, no agent 𝑗 can achieve a higher value by misreporting her valuation,
regardless of what the other agents report
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 45
Ø Technically, referred to as “truthfulness-in-expectation”
Ø A randomized direct-revelation mechanism ℎ is called strategyproof if
∀𝑤%, … , 𝑤&, ∀𝑗, ∀𝑤!
D ∶ 𝐹 𝑤! ℎ! 𝑤%, … , 𝑤&
≥ 𝐹 𝑤! ℎ! 𝑤%, … , 𝑤!
D, … , 𝑤&
Ø That is, no agent 𝑗 can achieve a higher expected value by misreporting her
valuation, regardless of what the other agents report
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 46
No non-wasteful deterministic SP mechanism is (even approximately) proportional.
Ø Since EF is at least as strict as Prop, SP+EF is also impossible subject to non-
wastefulness.
Ø Non-wastefulness can be replaced by a requirement of “connected pieces”,
and the impossibility result still holds.
Open Problem Does the SP+Prop impossibility hold even without the non-wastefulness assumption?
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 47
Ø E.g. serial dictatorship
Ø We saw that even EQ+PO allocations may not exist
Open Problem Does there exist a direct revelation, deterministic SP+EQ mechanism?
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 48
subset of {EQ,EF,PO}, and be SP in expected utilities
Ø Hence, we can only hope for SP+PO+EF or SP+EF+EQ Ø The first is an open problem, but the second combination is achievable!
Open Problem Does there exist a randomized SP mechanism which always returns a PO+EF allocation?
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 49
There is a randomized SP mechanism that always returns an EF+EQ allocation.
Ø Recall: In a perfect partition 𝐶, 𝑤! 𝐶E = ⁄
% & for all 𝑗, 𝑙 ∈ 𝑂
Ø Algorithm: Compute a perfect partition and return allocation 𝐵 which
randomly assigns the 𝑜 pieces to the 𝑜 agents
Ø SP: Regardless of what the agents report, agent 𝑗 receives each piece of the
cake with probability 1/𝑜, and thus has expected value exactly 1/𝑜
Ø EF: Assuming agents report truthfully (due to SP), agent 𝑗 always receives a
cake she values at 1/𝑜, and according to her, so do others.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 50
SP+PO+EF+EQ Rand SP+PO+EF Det Rand SP+PO+EQ Rand Rand SP+EF+EQ Det Rand PO+EF+EQ Det SP+PO Rand SP+EF Det Rand SP+EQ Det PO+EF Det EF+EQ Det Rand PO+EQ = Impossibility = Possibility
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 52
1
Piecewise constant density function Piecewise uniform density function
1 Special case of piecewise constant
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For piecewise uniform valuations, there exists a deterministic SP mechanism which returns an EF+PO allocation.
Ø Recall that for general valuations, even deterministic SP+EF is impossible.
For piecewise constant valuations, an s-CEEI (i.e. Nash-optimal) allocation can be computed in polynomial time.
Ø Recall that this is EF (thus Prop) and PO. Ø But this is not SP.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 54
If an algorithm computes an envy-free allocation for 𝑜 agents with piecewise uniform valuations with at most (𝑜) queries, then it can also compute an envy-free allocation for 𝑜 agents with general valuations with at most (𝑜) queries.
Ø Let the same algorithm interact with general valuations 𝑤%, … , 𝑤& via CUT and
EVAL queries and return an allocation 𝐵
Ø The proof constructs piecewise uniform valuations 𝑣%, … , 𝑣& which would have
resulted in the same responses and 𝑣! 𝐵1 = 𝑤! 𝐵1 for all 𝑗, 𝑘 ∈ 𝑂
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 55
Ø An allocation 𝐵 is called non-wasteful if no piece of the cake that is valued
positively by at least one agent is assigned to an agent who has zero value for it
Ø PO implies non-wastefulness
No finite protocol in the Robertson-Webb model can always produce a non-wasteful allocation, even for piecewise uniform valuations.
discussing PO
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 57
have non-positive valuation for every piece of the cake
Ø 𝑔
! 𝑦 ≤ 0, ∀𝑦 ∈ [0,1]
Ø Hence, 𝑤! 𝑌 ≤ 0, ∀𝑌 ∈
Ø Simply use −𝑔
! and −𝑤!
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 58
For burnt cake division, there exists a finite (but unbounded) protocol for finding an envy-free allocation with 𝑜 agents.
Ø Builds upon the Brams-Taylor protocol for dividing a good cake Ø But certain operations require non-trivial transformations to the world of
chores
Open Problem Is there a bounded envy-free protocol for burnt cake division?
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 60
Ø 𝐵! = 𝐵!,1 1∈[H] Ø ∀𝑗, 𝑘: 𝐵!,1∈ [0,1] Ø ∀𝑘: ∑! 𝐵!,1 ≤ 1
! " # 1 2 3
[Brams and Taylor 1996]
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 61
⁄ 𝑤2() 𝑤6().
receives goods 4.2, … , b for some 𝑘, and 𝑤2 𝐵2 = 𝑤6 𝐵6
Ø 1 is divided between the agents, if necessary
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 20 30 30 10 5 5 𝒃𝟑 10 15 20 15 10 30
⁄ 𝑤!() 𝑤#() high ⁄ 𝑤!() 𝑤#() low
15 10
[Brams and Taylor 1996]
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 62
Ø The adjusted winner procedure is envy-free (and therefore proportional),
equitable and Pareto optimal
Ø As in cake cutting, EF + EQ + PO is impossible, what about two of the three? Ø EF+EQ: Divide each good equally among agents (“perfect partition”) Ø EQ + PO: Impossible Ø EF + PO: Can achieve with CEEI
𝒉𝟐 𝒉𝟑 𝒃𝟐 1 𝒃𝟑 1 𝒃𝟒 1
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call just CEEI) becomes simpler.
Ø Assume for simplicity that ∀𝑘 ∃𝑗 with 𝑤! 1 > 0
Q%(c&) d&
≥
Q%(c') d'
for all 𝑙
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 64
[1959] convex program:
max h
!∈$
log 𝑣! 𝑡. 𝑢. ∀𝑗: 𝑣! ≤ h
.!∈/
𝐵!,"𝑤! " ∀𝑘: h
!∈$
𝐵!," ≤ 1 ∀𝑗, 𝑘: 𝐵!," ≥ 0
Ø The Eisenberg-Gale convex program can be solved in strongly polynomial time.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 65
Ø It is strategyproof in the large (SP-L) [Azevedo and Budish 2018] though
Ø No strategyproof mechanism that always outputs a complete allocation can
achieve better than a ⁄
% H approximation to the optimal social welfare for large
enough 𝑜.
Ø There is a strategyproof partial allocation mechanism that provides every agent
with a 1/𝑓 fraction of their CEEI utility.
Ø Allocation is envy-free but not proportional
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 66
each good.
Ø What if we also want PO?
Ø It is impossible to achieve SP + Prop + PO. Ø SP + PO: Serial dictatorship.
Ø Additionally achieves strict SP: agents always achieve strictly higher utility by
reporting their beliefs truthfully than by lying.
Ø Exploits a correspondence between fair division and wagering mechanisms
[Lambert et al. 2008] to utilize proper scoring rules (e.g. Brier score)
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 68
Ø 𝐵! = 𝐵!,1 1∈[H] Ø ∀𝑗, 𝑘: 𝐵!,1∈ [0,1] Ø ∀𝑘: ∑! 𝐵!,1 ≤ 1
4)
Ø However, 𝑤!(𝑝
1) can be positive, zero, or negative
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 69
Ø There always exists a CEEI allocation, which is envy-free and Pareto optimal. Ø The CEEI solution is “welfarist”, i.e., the set of feasible utility profiles is enough
to identify the set of CEEI utility profiles.
Ø The CEEI utility profile is given by the following:
then maximizing the Nash welfare gives the unique CEEI utility profile.
utility profile.
each agent.
Ø Their actual result is stronger and in a more general model
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 70
Ø Algorithms for maximizing social welfare subject to fairness constraints
Ø Possibility and impossibility results for 𝑜 − 1 cuts
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 72
𝑁s ⊆ 𝑁.
Ø 𝑤! ∶ 2" → ℝ$ in the case of bads, 𝑤! ∶ 2" → ℝ for both goods and bads
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 73
Ø Equivalently: 𝑤! 𝑌 = ∑U∈V 𝑤!() Ø Value for a good independent of other goods received
Ø Equivalently: ∀𝑌, 𝑍 with 𝑌 ⊆ 𝑍: 𝑤! 𝑌 ∪
− 𝑤! 𝑌 ≥ 𝑤! 𝑍 ∪ − 𝑤!(𝑍)
diminishing returns.
Most results for additive valuations unless stated
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[Lipton et al 2004, Budish 2011]
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there exists a good ∈ 𝐵4 for which 𝑤- 𝐵- ≥ 𝑤-(𝐵4 ∖ )
removing a single good from 𝑘’s bundle.”
Ø Note: We don’t consider 𝐵1 = ∅ a violation of EF1.
good.
Animation Credit: Ariel Procaccia
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 78
Ø One at a time, allocate a good to an agent that no one envies Ø While there is an envy cycle, rotate the bundles along the cycle.
removes envy towards them.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 79
there exists an item 𝑝 ∈ 𝐵- ∪ 𝐵4 for which 𝑤- 𝐵- ∖ 𝑝 ≥ 𝑤-(𝐵4 ∖ 𝑝 )
𝒑𝟐 𝒑𝟑 𝒑𝟒 𝒑𝟓 𝒃𝟐 2 1
𝒃𝟑 2
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 80
𝑃. = {𝑝 ∈ 𝑃: ∃𝑗 ∈ 𝑂, 𝑤- 𝑝 > 0} denote all objects that are a good for some agent.
Ø Suppose that |𝑃$| = 𝑏𝑜 for some 𝑏 ∈ ℕ. If not, add dummy bads with 𝑤! 𝑝 =
0 for all 𝑗 ∈ 𝑂.
Ø Phase 1: 𝑃$ is allocated by round robin in order (1, 2, … , 𝑜 − 1, 𝑜) Ø Phase 2: 𝑃# is allocated by round robin in order (𝑜, 𝑜 − 1, … , 2, 1) Ø Agents can choose to skip their turn in phase 2
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 81
Ø The double round robin algorithm outputs an allocation that is EF1 for
combinations of goods and bads in polynomial time.
Ø Proof idea: Let 𝑗 < 𝑘. Agent 𝑗 can envy 𝑘 up to one item in phase 1 (but not vice
versa), and agent 𝑘 can envy 𝑗 up to one item in phase 2 (but not vice versa) 𝒑𝟐 𝒑𝟑 𝒑𝟒 𝒑𝟓 𝒃𝟐 2 1
𝒃𝟑 2
𝑃$ 𝑃#
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 82
the geometric mean of agent utilities (more on this later). 𝐵 = arg max z
2/3
Ø This is just Nash-optimality from earlier
Ø Find an allocation that maximizes |{𝑤! 𝐵! > 0}|, and subject to that
maximizes ˆ
!:6# X# YZ
𝑤! 𝐵!
%/&
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 83
Ø The MNW allocation satisfies EF1 and PO. Ø PO: A Pareto-improving allocation would have higher geometric mean of
utilities for agents with non-zero utility or more agents with non-zero utility.
Ø EF1: Let !
∗ = arg max U∈X# 𝑤!(). Not-too-hard proof shows 𝑤1(𝐵1) ≥ 𝑤1(𝐵! ∖ ! ∗)
for all 𝑘. 𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 2 1 3 1 2 𝒃𝟑 10 1 1 1 2 5 𝒃𝟒 3 1 3 5 2
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 84
X3C).
Ø Actually, it’s APX-hard [Lee 2017].
Ø MNW allocation can be computed in polynomial time [Darmann and Schauer
2015, Barman et al. 2018].
Ø However, round robin already guarantees EF1 + PO in this setting.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 85
Ø There exists a pseudo-polynomial time algorithm for computing an allocation
satisfying EF1 + PO
Ø Algorithm uses local search (sequence of item swaps and price rises) to
compute an integral competitive equilibrium that is price envy-free up to one good.
Ø Price envy-free up to one good: ∀𝑗, 𝑙, ∃𝑘: 𝑞 𝐵! ≥ 𝑞(𝐵E ∖ 1 ) Ø Need different entitlements because CEEI might not exist with indivisibilities
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 86
Open Problem: Complexity of computing an EF1 + PO allocation Open Problem: Does there always exist an EF1 + PO allocation for submodular valuation functions?
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 87
Ø When items can be either goods or bads and 𝑜 = 2, an EF1 + PO allocation
always exists and can be found in polynomial time
Open Problem: Does an EF1 + PO allocation always exists for bads?
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[Conitzer et al. 2017]
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agent 𝑗, there exists a good for which 𝑤- 𝐵- ∪ ≥ 𝑤- 𝑁 𝑜 𝑤2 𝐵2 ∪ 6 = 4 ≥ 7 2 = 𝑤-(𝑁) 𝑜
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒃𝟐 1 3 3 𝒃𝟑 1 3 3
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Ø MNW Ø Barman et al. [2018] algorithm
Ø An allocation satisfying Prop1 + PO can be computed in strongly polynomial
time.
Ø In contrast, there exist instances in which no rounding of the fractional CEEI
allocation will give EF1 [Caragiannis et al., 2016].
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[Caragiannis et al. 2016]
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agents 𝑗, 𝑘, and every ∈ 𝐵4 with 𝑤- > 0, 𝑤- 𝐵- ≥ 𝑤- 𝐵4 ∖ .
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒃𝟐 10 5 5 𝒃𝟑 10 𝜗 𝜗
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Ø First, maximize the minimum utility any agent receives. Subject to this,
maximize the second-minimum utility. Then the third-minimum utility, etc. 𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 2 1 3 1 2 𝒃𝟑 10 1 1 1 2 5 𝒃𝟒 3 1 3 5 2
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 94
Ø The Leximin allocation satisfies EFX + PO for agents with (general) identical
valuations.
Ø The Leximin allocation satisfies EFX + PO for two agents with (normalized)
additive valuations.
Open Problem: Does there always exist a complete allocation satisfying EFX?
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒃𝟐 4 1 2 2 𝒃𝟑 4 1 2 2 𝒃𝟒 4 1 2 2
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Ø Easy! We can just throw all goods away and take the empty allocation.
Ø There exists a partial allocation that satisfies EFX and achieves a 2-
approximation to the optimal Nash welfare.
Ø No (complete or partial) EFX allocation can achieve a better approximation.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 96
Existence Computation Without PO With PO Without PO With PO Envy-Freeness No No NP-hard NP-hard EFX Open Open Open Open EF1 Yes Yes Polytime Open Prop1 Yes Yes Polytime Polytime
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 98
chosen bundle, how much utility can I guarantee myself?”
u(𝑇) =
max
(v",…,v')∈w'(x) min 2y4yu 𝑤-(𝑄 4)
3(𝑁)
3(𝑁) ≤ Q%(t) 3 , so Proportionality implies MMS
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 99
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 2 1 3 1 2 𝒃𝟑 10 1 1 1 2 5 𝒃𝟒 3 1 3 5 2
𝑁𝑁𝑇2
3(𝑁) = min 3, 3, 3 = 3
𝑁𝑁𝑇6
3(𝑁) = min 10, 5, 5 = 5
𝑁𝑁𝑇5
3(𝑁) = min 4, 5, 5 = 4
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 100
Ø There exist instances for which no allocation satisfies MMS.
Ø c-MMS: allocation for which 𝑤! 𝐵! ≥ 𝑑 ⋅ 𝑁𝑁𝑇!
&(𝑁)
Ø Guarantee 𝑤! 𝐵! ≥ 𝑁𝑁𝑇!
E(𝑁) for some 𝑙 > 𝑜
Ø There always exists an allocation that satisfies 𝑤! 𝐵! ≥ 𝑁𝑁𝑇!
&#% (𝑁) for
every agent 𝑗.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 101
Ø A (2/3)-MMS allocation always exists.
Ø A (2/3-𝜗)-MMS allocation can be computed in polynomial time.
Ø A (3/4)-MMS allocation always exists and a (3/4-𝜗)-MMS allocation can be
computed in polynomial time.
Ø A (3/4 + 1/(12n))-MMS allocation always exists and a (3/4)-MMS allocation can
be computed in polynomial time.
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Additive Submodular Subadditive Lower bound (existence) 3 4 + 1 12𝑜 1 3 1 10 log 𝑛 Lower bound (polynomial algorithm) 3 4 1 3
1−
! (!)!
3 4 1 2
Open Problem: Close the gaps!
[Ghodsi et al. 2018]
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Ø A 2-MMS allocation always exists and can be computed in polynomial time
when dividing bads.
Ø A (4/3)-MMS allocation always exists and can be computed in polynomial time
when dividing bads.
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u should be guaranteed for all groups 𝐾 of agents of size 𝑙
and set of goods ∪-∈z 𝐵-
5(𝑁) but 𝑤5 𝐵5 < 𝑁𝑁𝑇5 6(𝐵2 ∪ 𝐵5)
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 5 5 5 + 𝜗 5 − 𝜗 5 + 𝜗 5 − 𝜗 𝒃𝟑 5 5 5 + 𝜗 5 − 𝜗 5 + 𝜗 5 − 𝜗 𝒃𝟒 10 10 𝜗 𝜗
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∀𝑗: 𝑤- 𝐵- ≥ max
z⊆| 𝑁𝑁𝑇- z (∪4∈z 𝐵4)
Ø When valuations are additive, a 0.5-GMMS allocation exists and can be found
in polynomial time.
Ø Algorithm: Select an agent who is not envied by any other agent, and allocate
her her most preferred unallocated good.
Ø Small refinement of EF1 algorithm from earlier
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∀𝑗, 𝑘 ∈ 𝑂: 𝑤- 𝐵- ≥ 𝑤4(𝐵4)
∀𝑗, 𝑘 ∈ 𝑂, ∃ ∈ 𝐵4: 𝑤- 𝐵- ≥ 𝑤4(𝐵4 ∖ {})
∀𝑗, 𝑘 ∈ 𝑂, ∀ ∈ 𝐵4: 𝑤- 𝐵- ≥ 𝑤4(𝐵4 ∖ {})
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Ø Allocate to the lowest-utility agent the unallocated good that she values the
most.
Ø Compare to EFX, existence still unknown
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Ø An allocation satisfying EQ1 and PO may not exist. Ø Compare to EF1 + PO always exists
Ø When valuations are strictly positive, the Leximin allocation is EQX + PO
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 1 1 1 𝒃𝟑 1 1 1 𝒃𝟒 1 1 1
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Agents Allocation 𝐵 Allocation B 𝑇 𝑈
Envy-Free up to One Good (EF1)
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every partition (𝐶-)-∈x of ∪4∈} 𝐵4,
x }
⋅ (𝑤- 𝐶- )-∈x does not Pareto dominate (𝑤- 𝐵- )-∈x
group T amongst group S in such a way that every member of group S is (weakly, with at least one strictly) better off, with utilities adjusted for group sizes”
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 113
Ø “It should not be possible to redistribute the goods allocated to group T
amongst group S in such a way that every member of group S is (weakly, with at least one strictly) better off, even when one good is removed from each agent in S, with utilities adjusted for group sizes”
Partition B Agents Allocation 𝐵
𝑇 𝑈
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Ø “It should not be possible to redistribute the goods allocated to group T, with
member of group S is (weakly, with at least one strictly) better off, with utilities adjusted for group sizes”
Partition B Agents Allocation 𝐵
𝑇 𝑈
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Ø “It should not be possible to redistribute the goods allocated to group T, with
member of group S is (weakly, with at least one strictly) better off, with utilities adjusted for group sizes”
Partition B Agents Allocation 𝐵
𝑇 𝑈
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by moving a single good. ∀𝑗, 𝑘, ∈ 𝐵4: 𝑤4 > 0 and 𝑤- 𝐵- ⋅ 𝑤4 𝐵4 ≥ 𝑤- 𝐵- + ⋅ 𝑤4(𝐵4 − )
Ø Any locally Nash-optimal allocation satisfies GF1A and GF1B. Ø Can be computed in pseudo-polynomial time by local search Ø When valuations are identical, an allocation is locally Nash-optimal iff it is EFX/EQX.
Open Problem: Can we compute a locally Nash-optimal allocation in polynomial time?
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 117
Agents Allocation 𝐵 𝑇 𝑈
Open Problem: Can we give stronger guarantees when 𝑇 and 𝑈 are fixed in advance?
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 119
Ø GF1A/B (⇒ EF1) + PO Ø Scale-free Ø Natural fairness/efficiency tradeoff
Ø Computing an allocation that maximizes the geometric mean of agent utilities
under additive valuation functions is APX-hard.
Ø Approximating to within a factor of 1.00008 is NP-hard.
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Ø There exists a polynomial time algorithm that approximates the MNW
Ø There exists a polynomial time algorithm that approximates the MNW
Open Problem: Close the gap between the 1.00008 lower bound and 1.45 upper bound.
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the exact solution.
Ø There exists a polynomial time algorithm that approximates the MNW
(1/2n)-MMS and PO.
2-approximation to MNW objective [Caragiannis et al 2019].
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EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 123
welfare?
Ø The price of fairness of fairness property P is defined as the ratio of the
maximum possible social welfare and the maximum social welfare of an allocation that satisfies P.
Ø The strong price of fairness of fairness property P is defined as the ratio of the
maximum possible social welfare and the minimum social welfare of an allocation that satisfies P.
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Ø The price of fairness for proportionality, envy-freeness and equitability are:
Indivisible Goods Cake Cutting Proportionality Θ(𝑜) Θ( 𝑜) Envy-freeness Θ(𝑜) Θ( 𝑜) Equitability ∞ Θ 𝑜
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Ø Bounds on the (strong) price of fairness for indivisible goods
Price of P Strong Price of P EF1 LB: Ω 𝑜 , UB: 𝑃(𝑜) ∞ Round Robin 𝑜 𝑜3 Max Nash Welfare Θ(𝑜) Θ(𝑜) Leximin Θ(𝑜) Θ(𝑜) Pareto optimality 1 Θ(𝑜3)
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 126
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 127
rich
Ø Impossibilities from the divisible realm carry over
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒃𝟐 1 𝑦 𝒃𝟑 𝑧 1
[Amanatidis et al. 2017]
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 128
Ø Partition 𝑁 = 𝑂% ∪ 𝑂3 Ø Agent 1 receives a subset of offers 𝑃% ⊆ 2(%. Let 𝑇% = arg max
e∈f% 𝑤%(𝑇).
Ø Agent 2 receives a subset of offers 𝑃3 ⊆ 2(&. Let 𝑇3 = arg maxe∈f& 𝑤3(𝑇). Ø 𝐵% = 𝑇% ∪ (𝑂3 ∖ 𝑇3) and 𝐵3 = 𝑇3 ∪ (𝑂% ∖ 𝑇%)
2, 6 , 6, 5 , < , 𝑃6 = { ; , : }
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 3 5 5 10 4 2 𝒃𝟑 2 3 6 1 5 3
[Amanatidis et al. 2017]
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 129
Ø Partition 𝑁 = 𝐹% ∪ 𝐹3 Ø Set of exchange deals D = { 𝑇%, 𝑈
% , … , 𝑇E, 𝑈E }, where each 𝑇, 𝑈 ⊆ (𝐹%, 𝐹3)
Ø Agent 𝑗 receives allocation 𝐹! by default, with exchanges performed if they are
mutually beneficial
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒃𝟐 6 2 3 7 1 𝒃𝟑 1 6 1 4 7
[Amanatidis et al. 2017]
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 130
𝑂6 ⊆ 𝑁 and an exchange mechanism on 𝐹2 ∪ 𝐹6 ⊆ 𝑁, where 𝑂2 ∪ 𝑂6 ∪ 𝐹2 ∪ 𝐹6 = 𝑁 and 𝑂2, 𝑂6, 𝐹2, 𝐹6 are pairwise disjoint.
Ø Up to tiebreaking technicalities…
[Amanatidis et al. 2017]
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 131
Ø For 𝑜 = 2 an allocation mechanism that allocates all goods is strategyproof if
and only if it is a picking-exchange mechanism
Ø For 𝑜 = 2, any strategyproof mechanism that allocates all goods does not
achieve any positive approximation of the minimum envy or best proportionality guarantee.
Ø For 𝑜 = 2 and 𝑛 ≥ 5, no strategyproof mechanism can allocate all items and
satisfy EF1.
Ø For 𝑜 = 2, no strategyproof mechanism guarantees better than
% H/3 -MMS.
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 132
Open Problem: What is the structure of strategyproof mechanisms for 𝑜 > 2? Open Problem: What is the structure of strategyproof mechanisms for 𝑜 = 2 when not all goods have to be allocated?
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 133
2018]
Ø Stronger than EF1 and guaranteed to exist Ø Existence of EFL + PO allocations is an open question
Ø Agent social network structure Ø Connectivity constraints when items lie on a graph
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 134
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 135
items
Ø E.g. 3 ≽! 4 ≽! % ≽! C
Ø 𝐵 = 𝐵!,1 !∈ & ,1∈[H] Ø Can be interpreted as lotteries over integral allocations
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 136
extension 𝐵 ≽-
x~ 𝐶
iff ∀𝑙: ∑4≽%u 𝐵-,4 ≥ ∑4≽%u 𝐶-,4
Ø Upper/downward lexicographic [Cho 2012] Ø Pairwise comparison [Aziz et al. 2014] Ø Bilinear dominance [Aziz et al. 2014]
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 137
Ø Select a random ordering of the agents. Agents select their favorite 𝑛/𝑜 goods
in order.
Ø Agents “eat” at a constant (equal) rate. At any time, agents eat their most
preferred good that is not completely consumed.
≽6: 6 ≽6 5 ≽6 2 ≽6 <
! # " * 𝑏! 1 1/2 1/2 𝑏# 1/2 1 1/2 ! # " * 𝑏! 1 1/2 1/2 𝑏# 1 1/2 1/2
Random Priority Probabilistic Serial
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 138
agents weakly prefer and some agent strictly prefers.
Ø Probabilistic Serial satisfies SD-efficiency
Ø ≽%: % ≽% 3 ≽% 4 ≽% C
≽3: 3 ≽3 % ≽3 C ≽3 4
! # " * 𝑏! 1/2 1/2 1/2 1/2 𝑏# 1/2 1/2 1/2 1/2
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 139
allocation by misreporting their preferences.
Ø Random Priority is SD-strategyproof.
Ø ≽%: % ≽% 3 ≽% 4 ≽% C
≽3: 3 ≽3 4 ≽3 % ≽3 C
! # " * 𝑏! 1 1/2 1/2 𝑏# 1 1/2 1/2
3 ≽% %
! # " * 𝑏! 1 1/2 1/2 𝑏# 1/2 1 1/2
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 140
Ø No mechanism satisfies SD-efficiency, SD-strategyproofness, and equal
treatment of equals
Ø SD-envy-freeness: ∀𝑗, 𝑘: ∑1g%
H 𝐵!,11 ≽! eh ∑1g% H 𝐶!,11
Ø Probabilistic Serial is SD-envyfree
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 141
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 142
€, … , 𝑑u( € }
€: 𝐵€ → ℝ.
Issue 1 Issue 2 … Issue T 𝒅𝟐
𝟐
𝒅𝟑
𝟐
𝒅𝟒
𝟐
𝒅𝟐
𝟑
𝒅𝟑
𝟑
𝒅𝟒
𝟑
𝒅𝟐
𝑼
𝒅𝟑
𝑼
𝒅𝟒
𝑼
𝒃𝟐 3 1 2 5 1 6 5 5 𝒃𝟑 2 2 1 3 4 1 2 4 3 𝒃𝟒 4 4 3 2 5 4 5
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 143
€, … , 𝑑u( € }
€: 𝐵€ → ℝ.
Monday Tuesday … Sunday 𝒃𝟐 3 1 2 5 1 6 5 5 𝒃𝟑 2 2 1 3 4 1 2 4 3 𝒃𝟒 4 4 3 2 5 4 5
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 144
€ 𝑏4
= Š𝑤- € if 𝑗 = 𝑘 if 𝑗 ≠ 𝑘
𝒉𝟐 𝒉𝟑 𝒉𝟒
𝒃𝟐
𝒃𝟑 𝒃𝟒
𝒃𝟐
𝒃𝟑 𝒃𝟒
𝒃𝟐
𝒃𝟑 𝒃𝟒
𝒃𝟐 5 2 3 𝒃𝟑 3 1 𝒃𝟒 2 3 4
𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒃𝟐 5 2 3 𝒃𝟑 3 1 𝒃𝟒 2 3 4
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 145
Ø Decisions are public, all agents receive the same outcome
Ø Each agent should receive their “dictator utility” multiplied by 1/𝑜
Ø Each agent would receive their proportional share if they were allowed to
change the outcome on a single issue
Ø The MNW outcome satisfies Prop1 + PO in the public decisions setting
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 146
Issue 1 Issue 2 𝒅𝟐
𝟐
𝒅𝟑
𝟐
𝒅𝟒
𝟐
𝒅𝟐
𝟑
𝒅𝟑
𝟑
𝒅𝟒
𝟑
𝒃𝟐 3 1 2 5 1 𝒃𝟑 2 2 1 3 4 1 𝒃𝟒 4 4 3 2 𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 3 1 2 5 1 𝒃𝟑 2 2 1 3 4 1 𝒃𝟒 4 4 3 2
Ø Each good can give a positive utility to multiple agents simultaneously
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 147
Issue 1 Issue 2 𝒅𝟐
𝟐
𝒅𝟑
𝟐
𝒅𝟒
𝟐
𝒅𝟐
𝟑
𝒅𝟑
𝟑
𝒅𝟒
𝟑
𝒃𝟐 3 1 2 5 1 𝒃𝟑 2 2 1 3 4 1 𝒃𝟒 4 4 3 2 𝒉𝟐 𝒉𝟑 𝒉𝟒 𝒉𝟓 𝒉𝟔 𝒉𝟕 𝒃𝟐 3 1 2 5 1 𝒃𝟑 2 2 1 3 4 1 𝒃𝟒 4 4 3 2
Ø Exactly one of {%, 3, 4} and exactly one of {C, i, j} must be chosen Ø Partition matroid constraint
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 148
Ø An allocation of public goods 𝐷 is in (𝜀, 𝛽)-core if for every subset of agents
𝑇 ⊆ 𝑂, there is no feasible allocation of public goods 𝐷D such that 𝑇 𝑜 ⋅ 𝑣! 𝐷D ≥ 1 + 𝜀 ⋅ 𝑣! 𝐷 + 𝛽 for all 𝑗 ∈ 𝑇, and at least one inequality is strict.
4
𝑣- 4 = 1
Ø (0,1)-core generalizes a guarantee very similar to Prop1
EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 149
Ø Public goods are ground set elements Ø Feasible allocations are basis of a matroid Ø Generalizes public decisions (thus goods allocation) and multiwinner voting
Ø For matroid constraints, a (0,2)-core allocation exists, and for constant 𝜗 > 0,
a (0,2 + 𝜗)-core allocation can be computed in polynomial time.
Ø Algorithm: Maximize smooth Nash welfare ∏!∈( 1 + 𝑣! 𝐷 Ø For 𝜗 > 0, (0,1 − 𝜗)-core allocations may not exist.
Open Problem: Does there always exist a (0,1)-core allocation?
AAAI 2020 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 150
Ø For “matching constraints” and constant 𝜀 ∈ (0,1], a (𝜀, 8 + 6/𝜀)-core
allocation can be computed in polynomial time.
Ø Algorithm: Maximize a slightly different smooth NW ∏!∈( 1 + 4/𝜀 + 𝑣! 𝐷 Ø For 𝜀 > 0 and 𝛽 < 1, a (𝜀, 𝛽)-core allocation may not exist. Ø Open problem: Does there always exist a (0,1)-core allocation?
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