Direct algorithms for balanced two-person fair divisions of - - PowerPoint PPT Presentation

Direct algorithms for balanced two-person fair divisions of indivisible items: A computational study

Dagstuhl Fair Division Workshop 6 – 10 June 2016

• D. Marc Kilgour

Wilfrid Laurier University

Rudolf Vetschera

University of Vienna

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Agenda

Direct fair division algorithms Research questions Method: Computational study Results Outlook

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Discrete fair division problems

n players; typically, n = 2 A set S of indivisible items Ordinal preference information (ranking of items from best

to worst)

Players have different preferences for items Output: Allocation of items to players that has desirable

properties such as

− Envy-free: No player strictly prefers subset allocated to opponent − Maximin: Rank of least preferred item assigned to any player is

maximal

− Borda Maximin: Minimum Borda score of any player is maximal

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Fair division algorithms

All algorithms considered here create balanced allocations

(same number of items to both players)

Contested pile algorithms:

− First allocate “easy” items:

Each player receives most preferred unallocated item If both prefer same item, put it on Contested Pile Repeat until all items are allocated or on CP

− Allocate items from the contested pile using specific properties

(e.g. same ranking by both players)

Direct algorithms

− Input: Complete (strict) rankings of both players − Allocate items in one pass, without creating a Contested Pile

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Recent direct FD algorithms (for 2 players)

SA (sequential algorithm): (Brams, Kilgour & Klamler 2016a) SD (Singles-Doubles), ISD (Iterated Singles-Doubles): (Brams,

Kilgour & Klamler 2016b)

SD1, ISD1: Modifications of SD and ISD developed for this

study

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SA algorithm

Sequentially process the input rankings Allocate most preferred items to respective players if they

are different

If an item is contested, the player whose next unallocated

item has higher rank (i.e., is more preferred) receives this item as compensation, while the opponent receives the contested item. In case of ties, algorithm branches.

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SA algorithm

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Uncontested Contested, unique comp. Contested, branch Allocation 1: {1, 3, 5} {2, 4, 6} Allocation 2: {1, 3, 6} {2, 4, 5}

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SD / ISD ( SD1/ISD1) Algorithms

Maximin rank: Min x: every item is at worst xth for some player

Single item: Item with rank ≤ maximin rank for only one player

Phase 1: Allocate single items to player who prefers them Phase 2: Process remaining rankings top-down.

− If items are not contested, allocate to respective players − If item is contested, check if other player can be compensated with lower

ranked item so that resulting allocation is envy-free

− If this is possible, allocate accordingly (branch if possible for both) − If not possible, terminate indicating failure

ISD: Iteratively check for new single items after allocation of single

items

SD1/ISD1: If no envy-free allocation is possible, proceed with non

envy-free allocation (possibly branching)

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SD algorithm

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Uncontested Single items Allocation 1: {1, 3, 4} {2, 5, 6} Allocation 2: {1, 3, 5} {2, 4, 6} Contested, EF compensation

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ISD algorithm

1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Contested,

• nly one EF

Iterated Single items Allocation: {1, 3, 4} {2, 5, 6} Note: {1, 2, 3} {4, 5, 6} is not EF and therefore not generated

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Theoretical results

Property SA SD ISD Envy-free Yes, if exists Yes Yes Max-Min No guarantee Yes Yes Borda Max min No guarantee No guarantee No guarantee

Brams, Kilgour, Klamler 2016a, 2016b

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Research questions

Effects of ambiguity in algorithms:

− How many alternative allocations are created when an algorithm

branches?

− What problem characteristics influence branching and number of

allocations?

Effects of lack of guaranteed properties

− Do allocations usually have these properties anyway? − Do they have other desirable properties?

Effects of strategic play

− How often is truthful reporting a Nash equilibrium?

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Computational study

Complete enumeration of up to n=10 items W.l.g. items numbered according to ranking of player A Consider all permutations as possible rankings of player B For strategic play, all permutations are strategies

→ for 10 items, (10!)2 strategies need to be analyzed

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Numbers of allocations

1 2 4 0% 20% 40% 60% 80% 100%

SA SD ISD SD1 ISD1

1 2 3 4 6 8 0% 20% 40% 60% 80% 100%

SA SD ISD SD1 ISD1

1 2 3 4 5 6 8 ≥10 0% 20% 40% 60% 80% 100%

SA SD ISD SD1 ISD1

1 2 3 4 5 6 7 8 9 ≥10 0% 20% 40% 60% 80% 100%

SA SD ISD SD1 ISD1

4 Items 6 Items 8 Items 10 Items

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Percentage of problems with unique allocations

4 6 8 10 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

SA SD ISD SD1 ISD1

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Conflict vs. number of allocations

SA SD ISD SD1 ISD1 Conflict: rank correlation between rankings Problems with 10 items

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Comments on allocation counts

Branching does not occur that often (for small problems) In many cases, algorithms generate unique solution For algorithms that always generate solution, number of

allocations increases with conflict levels

Other algorithms do not generate any allocation for high

conflict situations

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Fraction envy-free among all generated allocations

4 6 8 10 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% SA SD1 ISD1

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Abilities of Algorithms to find Envy-Free Allocations

SA & SD SA & ISD SA & SD1 SA & ISD1 SD & ISD SD & SD1 SD & ISD1 ISD & SD1 ISD & ISD1 SD1 & ISD1 2 4 6 8 10 12 14 16 SA & SD SA & ISD SA & SD1 SA & ISD1 SD & ISD SD & SD1 SD & ISD1 ISD & SD1 ISD & ISD1 SD1 & ISD1 100 200 300 400 500 600 700 800 900 1000 SA & SD SA & ISD SA & SD1 SA & ISD1 SD & ISD SD & SD1 SD & ISD1 ISD & SD1 ISD & ISD1 SD1 & IS 20000 40000 60000 80000 100000 120000 SA & SD SA & ISD SA & SD1 SA & ISD1 SD & ISD SD & SD1 SD & ISD1 ISD & SD1 ISD & ISD1 SD1 & IS 1000000 2000000 3000000 4000000 5000000 6000000 7000000

4 items 8 items 10 items

(total # of EF allocations not computed)

6 items None Second First Both

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Comments on Envy freeness

Considerable overlap in the envy-free solutions generated All algorithms fail to find significant fraction of EF allocations

(except for n = 4)

SA finds fewer allocations than other algorithms

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How rare are fairness properties?

4 6 8 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% All properties MaxMin & MMBorda EF & MMBorda EF & MaxMin MMBorda MaxMin EF None

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Summary of properties

SA SD ISD SD1 ISD1

5 10 15 20 25 30 35 40 45

SA SD ISD SD1 ISD1

200 400 600 800 1000 1200 1400

SA SD ISD SD1 ISD1

10000 20000 30000 40000 50000 60000 70000 80000 90000

SA SD ISD SD1 ISD1

1000000 2000000 3000000 4000000 5000000 6000000 7000000 8000000 9000000

All

MM & MMB EF & MMB EF & MM

MMB MM EF None 4 6 10 8

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Fraction of all allocations generated that have all three properties

4 6 8 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% SA SD ISD SD1 ISD1

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Combinations of properties

Only SA can possibly generate non-maximin allocation

(theoretical property)

Maximin Borda is more rare than Maximin All algorithms are quite good at finding most allocations

having all desirable properties; SA is slightly weaker

SA generates fewer allocations that are simultaneously EF &

Maximin

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Nash Equilibrium

What are the strategy sets?

− Rankings for which algorithm in question generates an allocation

(provided that other player plays sincerely)

What does “improve” mean (if algorithm produces multiple

allocations)?

− Strategy leads to set of allocations that absolutely dominates

allocations generated by sincere strategy (i.e. worst outcome has better Borda score than best outcome under sincere)

Both players choose strategy = ranking of items input to algorithm Sincere strategy = truthful ranking Nash Equilibrium: No player improves by unilateral change of

• strategy. Do sincere strategies form a Nash Equilibrium?

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How often is sincere play a Nash equilibrium?

4 6 8 10 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100% SA SD ISD SD1 ISD1

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Comments on Manipulability

For all algorithms, vulnerability to strategic play increases

with number of items

For smaller problems, algorithms that always generate

allocations (SA, SD1, ISD1) are more vulnerable than algorithms that generate only EF allocations

For larger problems, this advantage vanishes

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Summary and outlook

All algorithms are quite efficient at finding the "needle in the

haystack": an allocation with many fairness properties

SA is slightly worse at finding allocations with many

desirable properties

Non-uniqueness of allocations seems to be a minor problem Due to complexity issues, only problems of limited size could

be tested

For larger problems, efficient ways of identifying incentives

to deviate from sincere play are needed

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Thank you for your attention!

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Number of allocations

nAlloc SA SD ISD SD1 ISD1 1 54.17% 92.86% 92.86% 54.17% 54.17% 2 37.50% 7.14% 7.14% 37.50% 37.50% 4 8.33% 0.00% 0.00% 8.33% 8.33% nAlloc SA SD ISD SD1 ISD1 1 56.81% 78.48% 80.53% 53.19% 54.58% 2 35.00% 21.11% 19.06% 32.36% 32.92% 3 0.56% 0.00% 0.00% 1.94% 0.00% 4 7.08% 0.41% 0.41% 10.83% 10.83% 6 0.00% 0.00% 0.00% 1.11% 1.11% 8 0.56% 0.00% 0.00% 0.56% 0.56% nAlloc SA SD ISD SD1 ISD1 1 58.60% 61.40% 67.02% 46.03% 50.24% 2 33.40% 34.81% 30.24% 36.53% 35.53% 3 0.93% 0.88% 0.00% 2.89% 0.00% 4 6.38% 2.87% 2.69% 10.85% 10.91% 5 0.01% 0.00% 0.00% 0.16% 0.00% 6 0.14% 0.03% 0.03% 2.22% 2.00% 8 0.52% 0.01% 0.01% 1.10% 1.10% >=10 0,02% 0,00% 0,00% 0,22% 0,22% SA SD ISD SD1 ISD1 1 60.01% 47.54% 55.54% 38.01% 44.41% 2 32.22% 41.89% 37.60% 40.17% 38.93% 3 1.15% 3.17% 0.11% 4.39% 0.09% 4 5.87% 6.71% 6.22% 12.17% 12.13% 5 0.02% 0.06% 0.00% 0.33% 0.00% 6 0.23% 0.45% 0.37% 2.84% 2.44% 7 0.00% 0.00% 0.00% 0.02% 0.00% 8 0.47% 0.17% 0.17% 1.46% 1.45% 9 0.00% 0.00% 0.00% 0.02% 0.00% >=10 0,03% 0,00% 0,00% 0,58% 0,55%