Fair division of indivisible goods under risk - Sylvain Bouveret

IntroductionA toy-example :ExamplesA set of bottles of wine to share. . .Objects : bottles of wineAgents : wine amateursFair division of indivisible goods under risk3 / 19

IntroductionA toy-example :ExamplesA set of bottles of wine to share. . .Objects : bottles of wineAgents : wine amateursA more realistic example :A co-funded Earth-observing satellite to operate. . .Agents : the countries that have co-funded thesatelliteObjects : observation requests posted by theagentsFair division of indivisible goods under risk3 / 19

IntroductionCentralized allocationA classical way to solve the problem :Ask the agents to give a score (weight, utility. . .) w(o) to each object oConsider that they have additive preferences → u(π) = ∑ o∈π w(o)Find an allocation that maximizes min i∈A u(π(i)) (egalitarian solution[Rawls, 1971])The Santa-Claus problem [Bansal and Sviridenko, 2006]Bansal, N. and Sviridenko, M. (2006).The santa claus problem.In Proceedings of the thirty-eighth annual ACM symposium on Theory of computing, pages 31–40. ACM.Fair division of indivisible goods under risk4 / 19

IntroductionAdding uncertaintyNow, we might be unsure of the quality of the objects when they areallocated.Fair division of indivisible goods under risk5 / 19

IntroductionAdding uncertaintyNow, we might be unsure of the quality of the objects when they areallocated.The bottles can be tainted.The weather can be cloudy over the observed area.Fair division of indivisible goods under risk5 / 19

IntroductionAdding uncertaintyNow, we might be unsure of the quality of the objects when they areallocated.The bottles can be tainted.The weather can be cloudy over the observed area.If we have some probabilistic information on the quality of an object, howcan we take it into account in the allocation process ?Fair division of indivisible goods under risk5 / 19

IntroductionAdding uncertaintyNow, we might be unsure of the quality of the objects when they areallocated.The bottles can be tainted.The weather can be cloudy over the observed area.If we have some probabilistic information on the quality of an object, howcan we take it into account in the allocation process ?We assume that :each object can be in two possible states : good or bad (bad = utility 0)each object o has a probability p(o) to be goodthese probabilities are independentFair division of indivisible goods under risk5 / 19

The modelResource allocation under riskResource allocation problemA tuple (A,O,W,p) with :A = {1,..,n} a set of agentsO = {1,..,l} a set of objectsW ∈ M n,l (R + ) a matrix of weights (given by agents to objects)p ∈ [0,1] l the probability for each object to be in good state.Fair division of indivisible goods under risk6 / 19

The modelResource allocation under riskResource allocation problemA tuple (A,O,W,p) with :A = {1,..,n} a set of agentsO = {1,..,l} a set of objectsW ∈ M n,l (R + ) a matrix of weights (given by agents to objects)p ∈ [0,1] l the probability for each object to be in good state.Notations :S : the set of 2 l states of the worldgood(s) ⊆ O : the set of objects in good states in s ∈ Su i,s (π) the utility of agent i in s with allocation π.Fair division of indivisible goods under risk6 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4Risk :∀j, p j = 0.5Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4Risk :∀j, p j = 0.5Allocations :π = 〈{j 1, j 2}, {j 3}〉π ′ = 〈{j 1}, {j 2, j 3}〉Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :Risk :Allocations :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4∀j, p j = 0.5π = 〈{j 1, j 2}, {j 3}〉π ′ = 〈{j 1}, {j 2, j 3}〉Profiles :π −→π ′ −→[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4[0 0 0 0 5 5 5 50 4 1 5 0 4 1 5]]Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :Risk :Allocations :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4∀j, p j = 0.5π = 〈{j 1, j 2}, {j 3}〉π ′ = 〈{j 1}, {j 2, j 3}〉Profiles :π −→π ′ −→[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4[0 0 0 0 5 5 5 50 4 1 5 0 4 1 5]]Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :Risk :Allocations :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4∀j, p j = 0.5π = 〈{j 1, j 2}, {j 3}〉π ′ = 〈{j 1}, {j 2, j 3}〉Profiles :π −→π ′ −→[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4[0 0 0 0 5 5 5 50 4 1 5 0 4 1 5]]Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :Risk :Allocations :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4∀j, p j = 0.5π = 〈{j 1, j 2}, {j 3}〉π ′ = 〈{j 1}, {j 2, j 3}〉Profiles :π −→π ′ −→[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4[0 0 0 0 5 5 5 50 4 1 5 0 4 1 5]]Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :Risk :Allocations :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4∀j, p j = 0.5π = 〈{j 1, j 2}, {j 3}〉π ′ = 〈{j 1}, {j 2, j 3}〉Profiles :π −→π ′ −→[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4[0 0 0 0 5 5 5 50 4 1 5 0 4 1 5]]Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }Preferences :Risk :Allocations :j 1 j 2 j 3i 1 5 4 2i 2 4 1 4∀j, p j = 0.5π = 〈{j 1, j 2}, {j 3}〉π ′ = 〈{j 1}, {j 2, j 3}〉Profiles :π −→π ′ −→[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4[0 0 0 0 5 5 5 50 4 1 5 0 4 1 5]]Fair division of indivisible goods under risk7 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }π[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4][0 0 0 0 5 5 5 5π ′ 0 4 1 5 0 4 1 5]Fair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }π[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4]E−→[4.52][0 0 0 0 5 5 5 5π ′ 0 4 1 5 0 4 1 5]Fair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }π[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4]E−→[ ]4.52⏐↓ min[0 0 0 0 5 5 5 5π ′ 0 4 1 5 0 4 1 5]2Fair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }π[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4]E−→[ ]4.52⏐↓ min[0 0 0 0 5 5 5 5π ′ 0 4 1 5 0 4 1 5]E−→2[2.52.5]Fair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }π[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4]E−→[ ]4.52⏐↓ min[0 0 0 0 5 5 5 5π ′ 0 4 1 5 0 4 1 5]E−→2[ ]2.52.5⏐↓ min2.5Fair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }π[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4]E−→[ ]4.52⏐↓ min[0 0 0 0 5 5 5 5π ′ 0 4 1 5 0 4 1 5]E−→2[ ]2.52.5⏐↓ min2.5π ′ ≽ E,minπFair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }π[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4⏐↓ min[0 0 0 4 0 4 0 4[0 0 0 0 5 5 5 5π ′ 0 4 1 5 0 4 1 5]]]E−→E−→[ ]4.52⏐↓min2[ ]2.52.5⏐↓ min2.5π ′ ≽ E,minπFair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }π[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4⏐↓ min[0 0 0 4 0 4 0 4[0 0 0 0 5 5 5 5π ′ 0 4 1 5 0 4 1 5]E−→[ ]4.52⏐↓min]E−→ 1.5\2]E−→[ ]2.52.5⏐↓ min2.5π ′ ≽ E,minπFair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4⏐π↓ min[0 0 0 4 0 4 0 4[0 0 0 0 5 5 5 5π ′ 0 4 1 5 ⏐↓ 0 4 1 5min[0 0 0 0 0 4 1 5]E−→[ ]4.52⏐↓min]E−→ 1.5\2]]E−→[ ]2.52.5⏐↓min2.5π ′ ≽ E,minπFair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4⏐π↓ min[0 0 0 4 0 4 0 4[0 0 0 0 5 5 5 5π ′ 0 4 1 5 ⏐↓ 0 4 1 5min[0 0 0 0 0 4 1 5]E−→[ ]4.52⏐↓min]E−→ 1.5\2]E−→[ ]2.52.5⏐↓min]E−→ 1.25\2.5π ′ ≽ E,minπFair division of indivisible goods under risk8 / 19

The modelExample3 objects {j 1 ,j 2 ,j 3 }, 2 agents {i 1 ,i 2 }[0 0 4 4 5 5 9 90 4 0 4 0 4 0 4⏐π↓ min[0 0 0 4 0 4 0 4[0 0 0 0 5 5 5 5π ′ 0 4 1 5 ⏐↓ 0 4 1 5min[0 0 0 0 0 4 1 5]E−→[ ]4.52⏐↓min]E−→ 1.5\2]E−→[ ]2.52.5⏐↓min]E−→ 1.25\2.5π ′ ≽ E,minπ or π ≽ min,Eπ ′ !Fair division of indivisible goods under risk8 / 19

The modelTiming effect [Myerson, 1981]Fair division of indivisible goods under risk9 / 19

The modelTiming effect [Myerson, 1981]Ex-ante collective utility( ∑acu(π) = min Pr(s) · u i,s (π)i∈As∈S)Fair division of indivisible goods under risk9 / 19

The modelTiming effect [Myerson, 1981]Ex-ante collective utility( ∑acu(π) = min Pr(s) · u i,s (π)i∈As∈S)Ex-post collective utilitypcu(π) = ∑ ( )Pr(s) · min u i,s(π)i∈As∈SFair division of indivisible goods under risk9 / 19

The modelTiming effect [Myerson, 1981]Ex-ante collective utility( ∑acu(π) = min Pr(s) · u i,s (π)i∈As∈S)Ex-post collective utilitypcu(π) = ∑ ( )Pr(s) · min u i,s(π)i∈As∈SProposition∀π,acu(π) ≥ pcu(π)Fair division of indivisible goods under risk9 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = min ( ∑ Pr(s) · u i,s (π))i∈As∈SFair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = min ( ∑ Pr(s) · u i,s (π))i∈As∈SFair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = min ( ∑ Pr(s) ·i∈As∈S∑j∈π i ∩good(s)w ij )Fair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = min ( ∑ Pr(s) ·i∈As∈S∑j∈π i ∩good(s)w ij )Fair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = min ( ∑ Pr(s) ·i∈As∈S∑j∈π i ∩good(s)w ij )Fair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = min ( ∑ ∑Pr(s) · [j ∈ good(s)] · w ij )i∈Aj∈π is∈SFair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = min ( ∑ ∑Pr(s) · [j ∈ good(s)] · w ij )i∈Aj∈π is∈SFair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = mini∈A ( ∑ j∈π i∑Pr(s) · [j ∈ good(s)] ·w ij )s∈S} {{ }p jFair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = mini∈A ( ∑ j∈π i∑Pr(s) · [j ∈ good(s)] ·w ij )s∈S} {{ }p jThe ˜w ij = p j w ij can be pre-computed in linear timeFair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = mini∈A ( ∑ j∈π i∑Pr(s) · [j ∈ good(s)] ·w ij )s∈S} {{ }p jThe ˜w ij = p j w ij can be pre-computed in linear timeEx-ante collective utility∑acu(π) = mini∈A˜w ijj∈π iFair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationex-ante optimizationacu(π) = mini∈A ( ∑ j∈π i∑Pr(s) · [j ∈ good(s)] ·w ij )s∈S} {{ }p jThe ˜w ij = p j w ij can be pre-computed in linear timeEx-ante collective utilityRisk-free equivalent problem∑acu(π) = min˜w iji∈Aj∈π i(A,O,W,p) ⇐⇒ (A,O, ˜W)Fair division of indivisible goods under risk10 / 19

Ex-ante vs ex-post computationEx-post optimizationpcu(π) = ∑ ( )Pr(s) · min u i,s(π)i∈As∈SFair division of indivisible goods under risk11 / 19

Ex-ante vs ex-post computationEx-post optimizationpcu(π) = ∑ ( )Pr(s) · min u i,s(π)i∈As∈S. . . Even the computation of the ex-post utility of a given allocation isnot easy :we cannot pre-compute expected utilitieswe must enumerate all the states of the worldFair division of indivisible goods under risk11 / 19

Ex-ante vs ex-post computationEx-post optimizationpcu(π) = ∑ ( )Pr(s) · min u i,s(π)i∈As∈S. . . Even the computation of the ex-post utility of a given allocation isnot easy :we cannot pre-compute expected utilitieswe must enumerate all the states of the world→ Computation of ex-post utility suspected to be #P-complete.Fair division of indivisible goods under risk11 / 19

Ex-ante vs ex-post computationEx-post optimizationpcu(π) = ∑ ( )Pr(s) · min u i,s(π)i∈As∈S. . . Even the computation of the ex-post utility of a given allocation isnot easy :we cannot pre-compute expected utilitieswe must enumerate all the states of the world→ Computation of ex-post utility suspected to be #P-complete.Branch on objects (good / bad states) → possible heuristics : smallshares first.Fair division of indivisible goods under risk11 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3pcu(π) =j 1j 2Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3pcu(π) =Bj 1j 2p = 0.5Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3j 1BBGpcu(π) =+ 0.5 × min{?,0}j 2B G B Gp = 0.5Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3j 1BBGGpcu(π) =+ 0.5 × min{?,0}j 2B G B Gp = 0.5Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3j 1BBGBGpcu(π) =+ 0.5 × min{?,0}j 2B G B Gp = 0.25Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3j 1BBGBGpcu(π) =+ 0.5 × min{?,0}+ 0.125 × min{0,4}j 2B G B GBp = 0.125Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3j 1j 2BBGB G B GBBGGpcu(π) =+ 0.5 × min{?,0}+ 0.125 × min{0,4}+ 0.125 × min{4,4}p = 0.125Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3j 1j 2BBGB G B GBBGGBGpcu(π) =+ 0.5 × min{?,0}+ 0.125 × min{0,4}+ 0.125 × min{4,4}+ 0.125 × min{5,4}p = 0.125Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3j 1j 2BBGB G B GBBp = 0.125GGBGGpcu(π) =+ 0.5 × min{?,0}+ 0.125 × min{0,4}+ 0.125 × min{4,4}+ 0.125 × min{5,4}+ 0.125 × min{9,4}Fair division of indivisible goods under risk12 / 19

Ex-ante vs ex-post computationExample2 agents {i 1 ,i 2 }, 3 objects {j 1 ,j 2 ,j 3 }, π = 〈{j 1 ,j 2 },{j 3 }〉•j 3j 1j 2BBGB G B GBBp = 0.125GGBGGpcu(π) =+ 0.5 × min{?,0}+ 0.125 × min{0,4}+ 0.125 × min{4,4}+ 0.125 × min{5,4}+ 0.125 × min{9,4}= 1.5Fair division of indivisible goods under risk12 / 19

Ex-post optimization – Exact proceduresBranch and BoundVariables : objectsQuestion : to whom is it allocated ?Heuristics : give to the poorest agent the object she prefersPossible cuts : ex-ante collective utility acu of a virtual allocation whichgives to all agents all the still unallocated objectsThe ex-post utility is computed only at each leaf of the search treeFair division of indivisible goods under risk13 / 19

Ex-post optimization – Exact proceduresMixed utilityThe ex-post utility is computed only at each leaf of the search tree. . .Fair division of indivisible goods under risk14 / 19

Ex-post optimization – Exact proceduresMixed utilityThe ex-post utility is computed only at each leaf of the search tree. . .. . . but it is still very expansive. . .Fair division of indivisible goods under risk14 / 19

Ex-post optimization – Exact proceduresMixed utilityThe ex-post utility is computed only at each leaf of the search tree. . .. . . but it is still very expansive. . .Wouldn’t it be possible to define and use a better approximation thanacu ?Fair division of indivisible goods under risk14 / 19

Ex-post optimization – Exact proceduresMixed utilityThe ex-post utility is computed only at each leaf of the search tree. . .. . . but it is still very expansive. . .Wouldn’t it be possible to define and use a better approximation thanacu ?Idea of the mixed utility mcu(π,Ω) :a set Ω of objects which are computed ex-anteobjects from O \ Ω are still computed ex-postwe still use acu as an upper bound in the search treewe use mcu(π,Ω) at each leaf to avoid unnecessary ex-post computationsFair division of indivisible goods under risk14 / 19

Ex-post optimization – Exact proceduresSome resultsn l (a) (b) (c) (d)5 ≤ 9 100 100 100 1005 10 49 52 89 1005 11 1 1 10 525 ≥ 12 0 0 0 0n l (a) (b) (c) (d)7 ≤ 8 100 100 100 1007 9 27 47 100 1007 10 0 1 19 327 ≥ 11 0 0 0 0Figure: Number of instances solved in 30 seconds (over 100 instances)10010090(b)90tps calculs ex-post (% temps total)8070(a)6050(c)4030(d)20106 7 8 9 10 11tps calculs ex-post (% temps total)80(b)70(a)605040(c)302010(d)07 7.5 8 8.5 9 9.5 10objetsobjetsFigure: Percentage of the total time used for ex-post computations for 5 and7 agents (average over 100 instances)Fair division of indivisible goods under risk15 / 19

Ex-post optimization – Incomplete searchAn approached algorithmApproximate computation of pcu :with mixed collective utilitywith a Monte-Carlo procedureFair division of indivisible goods under risk16 / 19

Ex-post optimization – Incomplete searchAn approached algorithmApproximate computation of pcu :with mixed collective utilitywith a Monte-Carlo procedureCustomized greedy stochastic algorithm [Bresina, 1996] :"best" solutions storedexact evaluation of stored solutionsFair division of indivisible goods under risk16 / 19

Ex-post optimization – Incomplete searchSome results110.90.80.90.70.8ucp/ucp*0.60.50.4N=200ucp/ucp*0.70.6|Oep| = 80.3N=10000.5|Oep| = 40.20.1N=50000.400 20 40 60 80 100 120temps(s)(a) Monte Carlo approximation, fordifferent numbers of draws0.30 20 40 60 80 100 120temps(s)(b) Mixed utility based approximation, fordifferent sizes of ΩFigure: Evolution of the best solution (average over 100 instances with 5agents and 12 objects)Fair division of indivisible goods under risk17 / 19

ConclusionSummaryThe model : a Santa-Claus problem under risktwo possible states for each objecteach object is in good state with a given probabilitytwo possible egalitarian collective utility functions : ex-ante and ex-postEx-ante case can be reduced to risk-free allocationEx-post optimization :a (supposed) quite harder problema branch-and-bound algorithm with mixed utilitysome incomplete methodsFair division of indivisible goods under risk18 / 19

ConclusionFuture workOn this problem :Missing complexity resultHow to choose the objects in Ω for mixed utility ?Better algorithms ?Other problems :Matching problems (l ≤ n) with other CUFRelaxing probabilistic independence (Bayesian networks)More possible states for each objectFair division of indivisible goods under risk19 / 19

ConclusionReferencesBresina, J. L. (1996).Heuristic-Biased Stochastic Sampling.In Proceedings of the 13th AAAI Conference on Artificial Intelligence (AAAI-96), pages 271–278, Portland, OR.Myerson, R. B. (1981).Utilitarianism, egalitarianism, and the timing effect in social choice problems.Econometrica, 49(4) :883–897.Rawls, J. (1971).A Theory of Justice.Harvard University Press, Cambridge, Mass.Traduction française disponible aux éditions du Seuil.Fair division of indivisible goods under risk20 / 20