Recent Advances in Fair Resource Allocation Rupert Freeman and - PowerPoint PPT Presentation

P ROPORTIONALITY : E VEN -P AZ • E VEN -P AZ • Input: Ø Interval [𝑦, 𝑧] , number of agents 𝑜 (assume a power of 2 for simplicity) • Recursive procedure: Ø If 𝑜 = 1 , give [𝑦, 𝑧] to the single agent. Ø Otherwise: o Each agent 𝑗 marks 𝑨 ! such that 𝑤 ! 𝑦, 𝑨 ! = 𝑤 ! 𝑨 ! , 𝑧 o 𝑨 ∗ = 𝑜 2 &' mark from the left. ⁄ o Recurse on [𝑦, 𝑨 ∗ ] with the left 𝑜/2 agents, and on [𝑨 ∗ , 𝑧] with the right 𝑜/2 agents. • Query complexity: Θ(𝑜 log 𝑜) EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 19

Complexity of Proportionality • Theorem [Edmonds and Pruhs, 2006]: Ø Any protocol returning a proportional allocation needs Ω(𝑜 log 𝑜) queries in the Robertson-Webb model. • Hence, E VEN -P AZ is provably (asymptotically) optimal! EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 20

Envy-Freeness EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 21

Envy-Freeness : Few Agents • 𝑜 = 2 agents : C UT - AND - CHOOSE (2 queries) • 𝑜 = 3 agents : S ELFRIDGE -C ONWAY (14 queries) Gets complex pretty quickly! EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 22

Envy-Freeness : Few Agents • [Brams and Taylor, 1995] Ø The first finite (but unbounded) protocol for any number of agents • [Aziz and Mackenzie, 2016a] Ø The first bounded protocol for 4 agents (at most 203 queries) • [Amanatidis et al., 2018] 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 23

Envy-Freeness • Theorem [Aziz and Mackenzie, 2016b] Ø There exists a bounded protocol for computing an envy-free allocation with 𝑜 agents, which requires 𝑃(𝑜 & """" ) queries Ø After 𝑃 𝑜 3&#4 queries, the protocol can output a partial allocation that is both proportional and envy-free • What about lower bounds? EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 24

Complexity of Envy-Freeness • Theorem [Procaccia, 2009] Any protocol for finding an envy-free allocation requires Ω(𝑜 6 ) queries. Open Problem Bridge the gap between 𝑃(𝑜 3 !!!! ) upper bound and Ω 𝑜 6 lower bound for envy-free cake-cutting • Theorem [Stromquist, 2008] There is no finite (even unbounded) protocol for finding a simple envy-free allocation for 𝑜 ≥ 3 agents. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 25

Equitability EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 26

Upper Bound: 𝑜 = 2 Agents • Existence Ø Suppose we cut the cake at 𝑦 to form pieces [0, 𝑦] and [𝑦, 1] Ø Let 𝑔 𝑦 = 𝑤 % 0, 𝑦 − 𝑤 3 𝑦, 1 o Note that 𝑔 0 = −1 , 𝑔 1 = 1 , and 𝑔 is continuous Ø By the intermediate value theorem: ∃𝑦 ∗ such that 𝑔 𝑦 ∗ = 0 Ø Allocation 𝐵 % = [0, 𝑦 ∗ ] and 𝐵 3 = [𝑦 ∗ , 1] is equitable • Theorem [Cechlárová and Pillárová, 2012] 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 27

Upper Bound: 𝑜 > 2 Agents • Theorem [Cechlárová and Pillárová, 2012] Ø This technique can be extended to 𝑜 agents to find an 𝜗 -equitable allocation in 𝑃 𝑜 ln ⁄ % 5 queries. • Theorem [Procaccia and Wang, 2017] Ø There exists a protocol for 𝑜 agents which finds an 𝜗 -equitable allocation in 𝑃 ⁄ % 5 ln ⁄ % 5 queries. Ø Intuition: o If 𝑜 ≤ ⁄ ( ) , use above protocol for finding an equitable 𝜗 -equitable allocation. o If 𝑜 > ⁄ ( ) , 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. ⁄ • While this is a “bad” allocation, it is 𝜗 -equitable. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 28

Lower Bound • Theorem [Procaccia and Wang, 2017] Any protocol for finding an 𝜗 -equitable allocation must require " # ⁄ OP Ω queries. " # ⁄ OP OP • Theorem [Procaccia and Wang, 2017] 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. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 29

Price of Fairness EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 30

Price of Fairness • Measures the worst-case loss in social welfare due to requirement of a fairness property 𝑌 • Social welfare of allocation 𝐵 is the sum of values of the agents Ø Denoted 𝑡𝑥 𝐵 = ∑ !∈( 𝑤 ! 𝐵 ! • Let ℱ denote the set of feasible allocations and ℱ 0 denote the set of feasible allocations satisfying property 𝑌 max S∈ℱ 𝑡𝑥(𝐵) 𝑄𝑝𝐺 0 = sup S∈ℱ $ 𝑡𝑥(𝐵) max Q " ,…,Q ! EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 31

Price of Fairness • Theorem [Caragiannis et al., 2009] For cake-cutting, the price of proportionality is Θ 𝑜 , and the price of equitability is Θ 𝑜 . • Theorem [Bertsimas et al., 2011] 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. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 32

Efficiency EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 33

Efficiency • Weak Pareto optimality (WPO) 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” • Pareto optimality (PO) 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 34

PO+EF+EQ: (Non-)Existence • Theorem [Barbanel and Brams, 2011] With two agents, there always exists 2 an allocation that is envy-free (thus proportional), equitable, and Pareto optimal. Ø Their algorithm has similarities to the more popular “adjusted winner” algorithm, which we will see later in the tutorial. 1 • With 𝑜 ≥ 3 agents, PO+EQ is impossible 1 2 0 1 - EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 35

What about PO+EF? • Competitive Equilibrium from Equal Incomes (CEEI) 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 • Theorem [Weller, 1985] 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 36

s-CEEI • Strong Competitive Equilibrium from Equal Incomes (s-CEEI) 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 Maximum bang-per-buck • Theorem [Segal-Halevi and Sziklai, 2018] For cake-cutting, an s-CEEI allocation always exists. Every s-CEEI allocation is envy-free and Pareto optimal. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 37

s-CEEI and Nash-Optimality • An allocation 𝐵 ∗ is called Nash-optimal if 𝐵 ∗ ∈ arg max " Π #∈% 𝑤 # 𝐵 # • Theorem [Segal-Halevi and Sziklai, 2018] For cake-cutting, the set of s-CEEI allocations is exactly the same as the set of Nash-optimal allocations. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 38

Nash-Optimality Example 𝑦 ∶ 1 − 𝑦 • Due to PO, suppose: 1.5 3 4 ] Ø Agent 1 gets 𝑦 fraction of [0, ⁄ 3 4 ] and Ø Agent 2 gets 1 − 𝑦 fraction of [0, ⁄ 3 4 , 1] all of [ ⁄ 1 Ø 𝑤 % 𝐵 % = 𝑦 (4$3;) 3 4 + ⁄ % 4 = c Ø 𝑤 3 𝐵 3 = 1 − x ⋅ ⁄ Allocated Allocated 4 to agent 1 to agent 2 (5W6X) M 5 ⇒ 𝑦 = ⁄ 5 < • Maximize 𝑦 ⋅ Ø Nash-optimal allocation: 1 3 2 3 0 1 - - ( + , 𝑤 ( 𝐵 ( = ⁄ , - o 𝐵 ( = 0, ⁄ ( + , 1 , 𝑤 + 𝐵 + = ⁄ ( + ⁄ o 𝐵 + = EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 39

Nash-Optimality = s-CEEI 𝑦 ∶ 1 − 𝑦 • Still must be PO, so like before 1.5 3 4 ] Ø Agent 1 buys 𝑦 fraction of [0, ⁄ 3 4 ] and Ø Agent 2 buys 1 − 𝑦 fraction of [0, ⁄ 3 4 , 1] all of [ ⁄ 1 Allocated Allocated • Prices: 𝑄 0, ⁄ 6 5 6 5 , 1 ⁄ = 𝑏, 𝑄 = 𝑐 to agent 1 to agent 2 Ø Spending: 𝑏 ⋅ 𝑦 = 1 , 𝑏 ⋅ 1 − 𝑦 + 𝑐 = 1 o Hence, 𝑏 + 𝑐 = 2 1 3 2 3 0 1 - - • Two cases: 𝑦 < 1 or 𝑦 = 1 Price = 𝑏 Price = 𝑐 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 40

Nash-Optimality = s-CEEI 𝑦 ∶ 1 − 𝑦 • 𝑦 < 1 1.5 Ø Agent 2 buys parts of both pieces Ø MBB: 1 1 3 2 3 ] ] 4 3 , ] 2 3 𝑐 = ⇒ 𝑏 = 2𝑐 ⇒ 𝑏, 𝑐 = ] 𝑏 Allocated Allocated to agent 1 to agent 2 4 C Ø Substituting in 𝑏 ⋅ 𝑦 = 1 , we get 𝑦 = ⁄ o Same as Nash-optimal solution 1 3 2 3 0 1 - - 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

Nash-Optimality = s-CEEI 𝑦 ∶ 1 − 𝑦 • 𝑦 = 1 1.5 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 1 3 2 3 ] ] Allocated Allocated 𝑐 ≥ ⇒ 𝑏 ≥ 2𝑐 𝑏 to agent 1 to agent 2 Ø Contradiction! Ø So there is no s-CEEI with 𝑦 = 1 1 3 2 3 0 1 - - 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

Strategyproofness EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 43

Strategyproofness (SP) • Direct-revelation mechanisms 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 ! • Strategyproofness (deterministic mechanisms) Ø A direct-revelation mechanism ℎ is called strategyproof if D ∶ 𝑤 ! ℎ ! 𝑤 % , … , 𝑤 & D , … , 𝑤 & ) ∀𝑤 % , … , 𝑤 & , ∀𝑗, ∀𝑤 ! ≥ 𝑤 ! (ℎ ! 𝑤 % , … , 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 44

Strategyproofness (SP) • Strategyproofness (randomized mechanisms) Ø Technically, referred to as “truthfulness-in-expectation” o When referring to SP for randomized mechanisms, we will refer to this concept Ø A randomized direct-revelation mechanism ℎ is called strategyproof if D ∶ 𝐹 𝑤 ! ℎ ! 𝑤 % , … , 𝑤 & D , … , 𝑤 & ∀𝑤 % , … , 𝑤 & , ∀𝑗, ∀𝑤 ! ≥ 𝐹 𝑤 ! ℎ ! 𝑤 % , … , 𝑤 ! Ø That is, no agent 𝑗 can achieve a higher expected value by misreporting her valuation, regardless of what the other agents report o Expectation is over the randomness of the mechanism EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 45

Deterministic SP Mechanisms • Theorem [Menon and Larson ’17, Bei et al. ‘17] 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 46

Deterministic SP Mechanisms • SP+PO is easy to achieve Ø E.g. serial dictatorship • SP+PO+EQ is impossible 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 47

Randomized SP Mechanisms • We want the mechanism always return an allocation satisfying a subset of {EQ,EF,PO}, and be SP in expected utilities • Recall: PO+EQ allocations may not exist 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 48

Randomized SP Mechanisms • Theorem [Mossel and Tamuz, 2010; Chen et al. 2013] There is a randomized SP mechanism that always returns an EF+EQ allocation. Ø Recall: In a perfect partition 𝐶 , 𝑤 ! 𝐶 E = ⁄ % & 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 49

= Impossibility Existential Summary = Possibility SP+PO+EF+EQ Rand SP+PO+EF SP+PO+EQ SP+EF+EQ PO+EF+EQ Det Rand Det Rand Rand Rand SP+PO SP+EF SP+EQ PO+EF PO+EQ EF+EQ Det Det Det Det Det Rand Rand Rand EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 50

Special Cases EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 51

Piecewise Constant/Uniform Valuations Piecewise constant density function 0 1 Piecewise uniform density function 0 1 Special case of piecewise constant EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 52

Possibilities • Theorem [Chen et al., 2013] 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. • Theorem [Aziz and Ye, 2014] 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 53

EF in Robertson-Webb • Theorem [Kurokawa et al., 2013] 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 ! 𝐵 1 = 𝑤 ! 𝐵 1 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 54

PO in Robertson-Webb • Non-wastefulness 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 • Theorem [Ianovski, 2012; Kurokawa et al., 2013] No finite protocol in the Robertson-Webb model can always produce a non-wasteful allocation, even for piecewise uniform valuations. • This is the reason we did not provide query complexity results when discussing PO EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 55

Burnt Cake Division EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 56

Model • Same as regular cake, except agents now have non-positive valuation for every piece of the cake Ø 𝑔 ! 𝑦 ≤ 0, ∀𝑦 ∈ [0,1] Ø Hence, 𝑤 ! 𝑌 ≤ 0, ∀𝑌 ∈ 𝒝 • Equitability and perfect partitions carry over from the goods case 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 57

Dividing a Burnt Cake • Theorem [Peterson and Su, 2009] 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? EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 58

(Homogeneous) Divisible Goods EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 59

Model • Agents: 𝑂 = {1, 2, … , 𝑜} • Resource: Set of divisible goods 𝑁 = 𝑕 2 , 𝑕 6 , … , 𝑕 b • Allocation 𝐵 = 𝐵 2 , … , 𝐵 3 Ø 𝐵 ! = 𝐵 !,1 1∈[H] 3 Ø ∀𝑗, 𝑘: 𝐵 !,1 ∈ [0,1] 2 Ø ∀𝑘: ∑ ! 𝐵 !,1 ≤ 1 1 • Assume additive valuations 𝑤 - 𝐵 - = ∑ 4 𝐵 -,4 𝑤 - (𝑕 4 ) 𝑕 ! 𝑕 " 𝑕 # • Special case of cake cutting (up to normalization) EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 60

𝑜 = 2: Adjusted Winner Procedure [Brams and Taylor 1996] • Input: Normalized valuation functions ⁄ • Order the goods by ratio 𝑤 2 (𝑕) 𝑤 6 (𝑕) . 𝒉 𝟐 𝒉 𝟑 𝒉 𝟒 𝒉 𝟓 𝒉 𝟔 𝒉 𝟕 20 30 30 10 5 5 𝒃 𝟐 15 10 15 20 10 15 10 30 𝒃 𝟑 ⁄ ⁄ 𝑤 ! (𝑕) 𝑤 # (𝑕) high 𝑤 ! (𝑕) 𝑤 # (𝑕) low • Divide the goods so that agent 1 receives goods 𝑕 2 , … , 𝑕 4W2 , agent 2 receives goods 𝑕 4.2 , … , 𝑕 b for some 𝑘 , and 𝑤 2 𝐵 2 = 𝑤 6 𝐵 6 Ø 𝑕 1 is divided between the agents, if necessary EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 61

𝑜 = 2: Adjusted Winner Procedure [Brams and Taylor 1996] • Theorem [Brams and Taylor 1996]: Ø The adjusted winner procedure is envy-free (and therefore proportional), equitable and Pareto optimal • Breaks down for 𝑜 > 2 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 𝒉 𝟐 𝒉 𝟑 Ø EF + PO: Can achieve with CEEI 𝒃 𝟐 1 0 𝒃 𝟑 1 0 𝒃 𝟒 0 1 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 62

CEEI • With a fixed set of items, the definition of s-CEEI (that we will now call just CEEI) becomes simpler. • Equilibrium price 𝑞 4 > 0 for each good 𝑕 4 Ø Assume for simplicity that ∀𝑘 ∃𝑗 with 𝑤 ! 𝑕 1 > 0 Q % (c & ) Q % (c ' ) • CE: If 𝐵 -,4 > 0 then ≥ for all 𝑙 d & d ' • EI: ∑ 4 𝑞 4 𝐵 -,4 = 1 for all 𝑗 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 63

Eisenberg-Gale convex program • Can compute a CEEI allocation as the solution to the Eisenberg-Gale [1959] convex program: max h log 𝑣 ! 𝑡. 𝑢. !∈$ ∀𝑗: 𝑣 ! ≤ h 𝐵 !," 𝑤 ! 𝑕 " . ! ∈/ ∀𝑘: h 𝐵 !," ≤ 1 !∈$ ∀𝑗, 𝑘: 𝐵 !," ≥ 0 • Theorem [Orlin 2010, Végh 2012]: 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 64

Strategyproofness • CEEI solution is fair and efficient but not strategyproof. Ø It is strategyproof in the large (SP-L) [Azevedo and Budish 2018] though • Theorem [Han et al. 2011]: 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 . o Social welfare = ∑ !∈$ 𝑤 ! (𝐵 ! ) • Theorem [Cole et al. 2013]: 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 65

SP + Prop + EF • SP + Prop + EF is trivial! Just allocate everyone an equal fraction of each good. Ø What if we also want PO? • Theorem [Schummer 1996]: Ø It is impossible to achieve SP + Prop + PO. Ø SP + PO: Serial dictatorship. • SP + Prop + EF can also be achieved non-trivially [Freeman et al. 2019] 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) EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 66

Allocating Divisible Goods + Bads EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 67

Model • Agents: 𝑂 = {1, 2, … , 𝑜} • Resources: Set of divisible “items” 𝑁 = 𝑝 2 , 𝑝 6 , … , 𝑝 b • Allocation 𝐵 = 𝐵 2 , … , 𝐵 3 Ø 𝐵 ! = 𝐵 !,1 1∈[H] Ø ∀𝑗, 𝑘: 𝐵 !,1 ∈ [0,1] Ø ∀𝑘: ∑ ! 𝐵 !,1 ≤ 1 • Assume additive valuations: 𝑤 - 𝐵 - = ∑ 4 𝐵 -,4 𝑤 - (𝑝 4 ) Ø However, 𝑤 ! (𝑝 1 ) can be positive, zero, or negative • We’ll refer to s-CEEI simply as CEEI in this case EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 68

Achieving EF+PO • Theorem [Bogomolnaia et al. 2017] 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: 1. If it is possible to give a positive utility to each agent (who can receive a positive utility), then maximizing the Nash welfare gives the unique CEEI utility profile. 2. Else, if the all-zero utility profile is feasible and Pareto optimal, then it is the unique CEEI utility profile. 3. Else, there can be exponentially many CEEI utility profiles, which give non-positive utility to 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 69

Not Covered • Nash equilibria of cake-cutting • Optimal cake-cutting Ø Algorithms for maximizing social welfare subject to fairness constraints • Number of cuts and moving knives protocols Ø Possibility and impossibility results for 𝑜 − 1 cuts • Multidimensional cakes • Randomized or strategyproof Robertson-Webb protocols • Non-additive valuations • … EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 70

Indivisible Goods EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 71

Model • A gents: 𝑂 = {1, 2, … , 𝑜} • Resource: Set of indivisible goods 𝑁 = 𝑕 2 , 𝑕 6 , … , 𝑕 b • Allocation 𝐵 = 𝐵 2 , … , 𝐵 3 ∈ Π 3 𝑁′ is a partition of 𝑁′ for some 𝑁 s ⊆ 𝑁 . • Each agent 𝑗 has a valuation 𝑤 - ∶ 2 t → ℝ . 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 72

Valuation Functions • Additive: ∀𝑌, 𝑍 with 𝑌 ∩ 𝑍 = ∅: 𝑤 - 𝑌 ∪ 𝑍 = 𝑤 - 𝑌 + 𝑤 - 𝑍 Ø Equivalently: 𝑤 ! 𝑌 = ∑ U∈V 𝑤 ! (𝑕) Most results for additive Ø Value for a good independent of other goods received valuations unless stated otherwise • Submodular: ∀𝑌, 𝑍 ∶ 𝑤 - 𝑌 ∪ 𝑍 + 𝑤 - (𝑌 ∩ 𝑍) ≤ 𝑤 - 𝑌 + 𝑤 - 𝑍 Ø Equivalently: ∀𝑌, 𝑍 with 𝑌 ⊆ 𝑍: 𝑤 ! 𝑌 ∪ 𝑕 − 𝑤 ! 𝑌 ≥ 𝑤 ! 𝑍 ∪ 𝑕 − 𝑤 ! (𝑍) • Subadditive: ∀𝑌, 𝑍 with 𝑌 ∩ 𝑍 = ∅: 𝑤 - 𝑌 ∪ 𝑍 ≤ 𝑤 - 𝑌 + 𝑤 - 𝑍 • Submodular and subadditive definitions capture the idea of diminishing returns. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 73

Need new guarantees! EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 74

Envy-Freeness up to One Good EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 75

Envy-Freeness up to One Good (EF1) [Lipton et al 2004, Budish 2011] • An allocation is envy-free up to one good (EF1) if, for all agents 𝑗, 𝑘 , there exists a good 𝑕 ∈ 𝐵 4 for which 𝑤 - 𝐵 - ≥ 𝑤 - (𝐵 4 ∖ 𝑕 ) • “Agent 𝑗 may envy agent 𝑘 , but the envy can be eliminated by removing a single good from 𝑘 ’s bundle.” Ø Note: We don’t consider 𝐵 1 = ∅ a violation of EF1. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 76

Round Robin Algorithm • Fix an ordering of the agents 𝜏 . • In round 𝑙 mod 𝑜 , agent 𝜏 u selects their most preferred remaining good. • Theorem: Round robin satisfies EF1. Phase 1 Phase 2 Animation Credit: Ariel Procaccia EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 77

Algorithm for Achieving EF1 • Envy graph: Edge from 𝑗 to 𝑘 if 𝑗 envies 𝑘 • Greedy algorithm [Lipton et al. 2004] 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. o Can prove this loop terminates in a polynomial number of steps • Removing the most recently added good from an agent’s bundle removes envy towards them. • Neither this algorithm nor round robin is Pareto optimal. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 78

EF1 with Goods and Bads [Aziz et al. 2019] • An allocation is envy-free up to one item (EF1) if, for all agents 𝑗, 𝑘 , there exists an item 𝑝 ∈ 𝐵 - ∪ 𝐵 4 for which 𝑤 - 𝐵 - ∖ 𝑝 ≥ 𝑤 - (𝐵 4 ∖ 𝑝 ) • Round robin fails EF1 𝒑 𝟐 𝒑 𝟑 𝒑 𝟒 𝒑 𝟓 2 1 -4 -4 𝒃 𝟐 2 -3 -4 -4 𝒃 𝟑 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 79

Double Round Robin • Let 𝑃 W = {𝑝 ∈ 𝑃: ∀𝑗 ∈ 𝑂, 𝑤 - 𝑝 ≤ 0} denote all unanimous bads and 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 . • Double round robin: 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 80

Double Round Robin • Theorem [Aziz et al. 2019]: 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) 𝑃 # 𝑃 $ 𝒑 𝟐 𝒑 𝟑 𝒑 𝟒 𝒑 𝟓 2 1 -4 -4 𝒃 𝟐 2 -3 -4 -4 𝒃 𝟑 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 81

Maximum Nash Welfare • Maximum Nash Welfare (MNW): Select the allocation that maximizes the geometric mean of agent utilities (more on this later). 2/3 𝐵 = arg max z 𝑤 - 𝐵 - - Ø This is just Nash-optimality from earlier • What if ∏ - 𝑤 - 𝐵 - = 0 for all allocations? Ø Find an allocation that maximizes |{𝑤 ! 𝐵 ! > 0}| , and subject to that maximizes %/& ˆ 𝑤 ! 𝐵 ! !:6 # X # YZ EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 82

EF1 + PO • Theorem [Caragiannis et al. 2016]: 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. ∗ = arg max ∗ ) Ø EF1: Let 𝑕 ! U∈X # 𝑤 ! (𝑕) . Not-too-hard proof shows 𝑤 1 (𝐵 1 ) ≥ 𝑤 1 (𝐵 ! ∖ 𝑕 ! for all 𝑘 . 𝒉 𝟐 𝒉 𝟑 𝒉 𝟒 𝒉 𝟓 𝒉 𝟔 𝒉 𝟕 2 1 3 0 1 2 𝒃 𝟐 10 1 1 1 2 5 𝒃 𝟑 3 1 3 0 5 2 𝒃 𝟒 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 83

Computing EF1 + PO • The MNW allocation is strongly NP-hard to compute (reduction from X3C). Ø Actually, it’s APX-hard [Lee 2017]. • Special case: Binary valuations 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 84

Computing EF1 + PO • Theorem [Barman et al. 2018]: 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(𝐵 E ∖ 𝑕 1 ) Ø Need different entitlements because CEEI might not exist with indivisibilities o Two agents, one item… EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 85

Computing EF1 + PO 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 86

EF1 + PO for Bads • Theorem [Aziz et al. 2019]: 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? EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 87

Proportionality up to One Good EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 88

Proportionality up to One Good (Prop1) [Conitzer et al. 2017] • An allocation is proportional up to one good (Prop1) if, for every agent 𝑗 , there exists a good 𝑕 for which ≥ 𝑤 - 𝑁 𝑤 - 𝐵 - ∪ 𝑕 𝑜 𝒉 𝟐 𝒉 𝟑 𝒉 𝟒 1 3 3 𝒃 𝟐 1 3 3 𝒃 𝟑 = 4 ≥ 7 2 = 𝑤 - (𝑁) 𝑤 2 𝐵 2 ∪ 𝑕 6 𝑜 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 89

Prop1 + PO • Any algorithm that satisfies EF1 + PO is also Prop1 + PO. Ø MNW Ø Barman et al. [2018] algorithm • Theorem [Barman and Krishnamurthy 2019]: Ø An allocation satisfying Prop1 + PO can be computed in strongly polynomial time. • Allocation is a careful rounding of the fractional CEEI allocation. Ø In contrast, there exist instances in which no rounding of the fractional CEEI allocation will give EF1 [Caragiannis et al., 2016]. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 90

Envy-Freeness up to the Least Valued Good EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 91

Envy-Freeness up to the Least Valued Good [Caragiannis et al. 2016] 𝒉 𝟐 𝒉 𝟑 𝒉 𝟒 10 5 5 𝒃 𝟐 10 𝒃 𝟑 𝜗 𝜗 • An allocation is envy-free up to the least valued good (EFX) if, for all agents 𝑗, 𝑘 , and every 𝑕 ∈ 𝐵 4 with 𝑤 - 𝑕 > 0 , 𝑤 - 𝐵 - ≥ 𝑤 - 𝐵 4 ∖ 𝑕 . EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 92

Leximin Allocation • Leximin allocation: Ø First, maximize the minimum utility any agent receives. Subject to this, maximize the second-minimum utility. Then the third-minimum utility, etc. 𝒉 𝟐 𝒉 𝟑 𝒉 𝟒 𝒉 𝟓 𝒉 𝟔 𝒉 𝟕 2 1 3 0 1 2 𝒃 𝟐 10 1 1 1 2 5 𝒃 𝟑 3 1 3 0 5 2 𝒃 𝟒 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 93

𝒉 𝟐 𝒉 𝟑 𝒉 𝟒 𝒉 𝟓 Satisfying EFX 𝒃 𝟐 4 1 2 2 𝒃 𝟑 4 1 2 2 𝒃 𝟒 4 1 2 2 • Theorem [Plaut and Roughgarden, 2018]: Ø The Leximin allocation satisfies EFX + PO for agents with (general) identical valuations. • Theorem [Plaut and Roughgarden, 2018]: 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? EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 94

Satisfying EFX • What about partial allocations satisfying EFX? Ø Easy! We can just throw all goods away and take the empty allocation. • Theorem [Caragiannis et al. 2019]: 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 95

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 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 96

Maximin Share EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 97

Maximin Share [Budish 2011] • “If I partition the goods into 𝑜 bundles and receive an adversarially chosen bundle, how much utility can I guarantee myself?” u (𝑇) = • Define 𝑁𝑁𝑇 - (v " ,…,v ' )∈w ' (x) min max 2y4yu 𝑤 - (𝑄 4 ) 3 (𝑁) • MMS allocation: One for which 𝑤 - 𝐵 - ≥ 𝑁𝑁𝑇 - Q % (t) 3 (𝑁) ≤ • Note that 𝑁𝑁𝑇 - 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 98

Maximin Share [Budish 2011] 𝒉 𝟐 𝒉 𝟑 𝒉 𝟒 𝒉 𝟓 𝒉 𝟔 𝒉 𝟕 2 1 3 0 1 2 𝒃 𝟐 10 1 1 1 2 5 𝒃 𝟑 3 1 3 0 5 2 𝒃 𝟒 3 (𝑁) = min 3, 3, 3 = 3 𝑁𝑁𝑇 2 3 (𝑁) = min 10, 5, 5 = 5 𝑁𝑁𝑇 6 3 (𝑁) = min 4, 5, 5 = 4 𝑁𝑁𝑇 5 EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 99

Achieving Maximin Allocations • Theorem [Procaccia and Wang 2014]: Ø There exist instances for which no allocation satisfies MMS. • Instead, consider approximations. & (𝑁) Ø c-MMS: allocation for which 𝑤 ! 𝐵 ! ≥ 𝑑 ⋅ 𝑁𝑁𝑇 ! E (𝑁) for some 𝑙 > 𝑜 Ø Guarantee 𝑤 ! 𝐵 ! ≥ 𝑁𝑁𝑇 ! • Theorem [Budish 2011]: &#% (𝑁) for Ø There always exists an allocation that satisfies 𝑤 ! 𝐵 ! ≥ 𝑁𝑁𝑇 ! every agent 𝑗. EC'19, AAAI'20 and AAMAS'20 Tutorial on Recent Advances in Fair Resource Allocation – Rupert Freeman and Nisarg Shah 100