Direct algorithms for balanced two-person fair divisions of indivisible items: A computational study D. Marc Kilgour Wilfrid Laurier University Rudolf Vetschera University of Vienna Dagstuhl Fair Division Workshop 6 – 10 June 2016
Agenda � Direct fair division algorithms � Research questions � Method: Computational study � Results � Outlook 2
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 3
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 4
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 5
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. 6
SA algorithm Contested, Contested, Uncontested unique comp. branch 1 1 2 2 3 3 4 4 5 5 6 6 2 3 4 1 5 6 Allocation 1: {1, 3, 5} {2, 4, 6} Allocation 2: {1, 3, 6} {2, 4, 5} 7
SD / ISD ( SD1/ISD1) Algorithms � Maximin rank : Min x : every item is at worst x th 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) 8
SD algorithm Contested, EF Uncontested Single items compensation 1 1 2 2 3 3 4 4 5 5 6 6 2 4 5 1 6 3 Allocation 1: {1, 3, 4} {2, 5, 6} Allocation 2: {1, 3, 5} {2, 4, 6} 9
ISD algorithm Iterated Single Contested, items only one EF 1 1 2 2 3 3 4 4 5 5 6 6 2 4 5 1 6 3 Allocation: {1, 3, 4} {2, 5, 6} Note: {1, 2, 3} {4, 5, 6} is not EF and therefore not generated 10
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 11
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? 12
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 13
Numbers of allocations 4 Items 6 Items 100% 100% SA SA 80% 80% SD SD 60% 60% ISD ISD SD1 SD1 40% 40% ISD1 ISD1 20% 20% 0% 0% 1 2 4 1 2 3 4 6 8 8 Items 10 Items 100% 100% SA SA 80% 80% SD SD ISD 60% 60% ISD SD1 SD1 40% 40% ISD1 ISD1 20% 20% 0% 0% 1 2 3 4 5 6 8 ≥10 1 2 3 4 5 6 7 8 14 9 ≥10
Percentage of problems with unique allocations 100% SA 90% SD ISD 80% SD1 70% ISD1 60% 50% 40% 30% 20% 10% 0% 4 6 8 10 15
Conflict vs. number of allocations SA SD ISD ISD1 SD1 Problems with 10 items Conflict: rank correlation between rankings 16
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 17
Fraction envy-free among all generated allocations 100% 90% SA SD1 80% ISD1 70% 60% 50% 40% 30% 20% 10% 0% 4 6 8 10 18
Abilities of Algorithms to find Envy-Free Allocations 4 items 6 items 16 1000 900 None 14 800 12 700 10 600 Second 500 8 400 6 300 200 4 First 100 2 0 0 SA & ISD SA & ISD1 SD & SD1 ISD & SD1 SD1 & ISD1 SA & SD SA & SD1 SD & ISD SD & ISD1 ISD & ISD1 SA & ISD SA & ISD1 SD & SD1 ISD & SD1 SD1 & ISD1 Both SA & SD SA & SD1 SD & ISD SD & ISD1 ISD & ISD1 10 items 8 items (total # of EF allocations not computed) 120000 7000000 100000 6000000 80000 5000000 4000000 60000 3000000 40000 2000000 20000 1000000 0 0 SA & ISD SA & ISD1 SD & SD1 ISD & SD1 SD1 & IS 19 SA & ISD SA & ISD1 SD & SD1 ISD & SD1 SD1 & IS SA & SD SA & SD1 SD & ISD SD & ISD1 ISD & ISD1 SA & SD SA & SD1 SD & ISD SD & ISD1 ISD & ISD1
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 20
How rare are fairness properties? 100% 90% 80% All properties 70% MaxMin & MMBorda 60% EF & MMBorda EF & MaxMin 50% MMBorda 40% MaxMin EF 30% None 20% 10% 0% 4 6 8 21
Summary of properties 45 1400 All 40 1200 35 4 6 MM & 1000 30 MMB 800 25 20 600 EF & 15 MMB 400 10 200 5 EF & 0 0 MM SA SD ISD SD1 ISD1 SA SD ISD SD1 ISD1 90000 9000000 MMB 10 80000 8000000 8 70000 7000000 60000 MM 6000000 50000 5000000 40000 4000000 EF 30000 3000000 20000 2000000 10000 None 1000000 0 0 SA SD ISD SD1 ISD1 SA SD ISD SD1 ISD1 22
Fraction of all allocations generated that have all three properties 100% 90% 80% 70% SA 60% SD 50% ISD SD1 40% ISD1 30% 20% 10% 0% 4 6 8 23
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 24
Nash Equilibrium 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? � 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) 25
How often is sincere play a Nash equilibrium? 100% 90% 80% SA SD 70% ISD 60% SD1 ISD1 50% 40% 30% 20% 10% 0% 4 6 8 10 26
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 27
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 28
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