Condorcet Consistent Bundling with Social Choice Shreyas Sekar, Sujoy Sikdar, Lirong Xia Optimal Bundle Selectionβ¦ Journals in a Library Algorithmica Limited space Software π J A I R ECONOMETRICA Many journals to choose from β¦ πβͺπ Users have preferences over the journals Examples: Committees, TV channel packages, insurance policies, β¦ β’ Linear orders on the journals: Software Optimal Bundle Selection: Preferences β» β» β»β― Motivating Question How to select bundles of size k to satisfy majority of users, who have preferences over individual items? Condorcet Criterion Copeland Maximin/Minimax This Work Ranked Pairs Schulze The Condorcet Criterion Candidate who defeats every other candidate by pair-wise majority, must be the winner. Software Voting rule Software > > Software > > > Software > Relaxing Condorcet: Different notions of winners β’ Copeland: most pairwise victories Condorcet winners may not always exist: how to select? β’ Maximin: maximize the minimum margin of victory Relaxing Condorcet via Tournament Graphs β’ Copeland: most pairwise victories β’ Maximin: maximize the minimum margin of victory Depend only on (weighted) tournament graph Tournament graph π Weighted Tournament graph π 1 2 Software 2 2 β’ Copeland: maximum outgoing edges in π β’ Maximin: maximize the minimum weight of any outgoing edge in π β’ Other Condorcet Consistent Rules: Ranked Pairs, Schulze, β¦ Condorcet criterion for Bundles Combinatorial Voting Care about every item Compare all bundles Our Work Proportional Representation Care about favorite item Compare bundle to items outside the bundle [Gehrlein, 1985] Condorcet Winning Every item in the bundle, defeats every item outside Bundle Another Angle on Gehrleinβs notion Strong Every item in the bundle, defeats every item outside β‘ Gehrlein Stability Defeats every neighboring bundle Local dominance: β’ Compare to neighboring bundles Weighted majority graphs for π-size bundles Software Software Software Local (Weighted) Majority graph of π-bundles π (π) (π (π) ) β’ Condorcet criterion: Bundle with outgoing edges to every neighbor β’ Local Condorcet winner [Xia et al. β08, Conitzer et al. β11] Generic Template for Condorcet Consistent Bundling Condorcet winning bundles may not always exist Single winner Condorcet rule depending on π (π) Bundling Condorcet rule depending on π (π) (π (π) ) Copeland(k) bundle: maximum outgoing edges in π (π) Maximin (k) bundle: maximizes the minimum weight of any outgoing edge in π (π) JAIR Example (N=100 users, k=3) Software 35 55 80 Software JAIR 90 Software 80 JAIR Software 70 Copeland winning bundle (score = 2 bundles defeated) JAIR Maximin Winning Bundle (score = 35) 55 (k) Copeland Mechanism β’ Copeland winner iff it maximizes pairwise defeats of items not in the bundle β’ Maximum size cut in π Theorem: Compute optimal Copeland (k) bundle efficiently (no ties case): οSelect π items with largest out-degree in π (k) Maximin Mechanism β’ Computing optimal bundle of size π is NP-hard β’ Relax size constraint β’ Maximin Score of Bundle B Largest β β₯ 0 s.t, at least β users prefer B to any neighbor Bβ. Size-Relaxing Maximin-Approx. Theorem Compute bundle of size β€ 2π that is as good as Opt.bundle of size k in terms of maximin score Why? β’ Seller may have some flexibility in deciding the size. β’ Cost borne by a benevolent seller, not buyer. β’ Buyers pay for π, and get a larger bundle that is at least as good as the best bundle of size π. Example: Government run library, tele-education package, β¦ Copeland πΌ - Dealing with ties β’ πΌ controls importance of tie J A I R β’πΌ=1 J A I R β’ πΌ = 0.5 J A I R β’πΌ=0 ECONOMETRICA Algorithmica COMSOC ECONOMETRICA ECONOMETRICA π Copeland (πΌ) β’ Copeland π (πΌ) score is ππ’π‘ππππππ + πΌ β π‘πππ in π (π) β’ Winner computation hard for πΌ = 0,1 β’ OPT for πΌ = 0.5 using greedy algorithm β’ Pick k items w/ highest score. β’ 0.5-approx. using MAX-DICUT β’ min(2Ξ±, 1/2Ξ±)-approx. using greedy algorithm Ranked Pairs and Schulze for Bundles π (π) Ranked Pairs (π) β’ Sort edges of π by dec. weight β’ Select edge if it does no cycle β’ Winner has no incoming edges π (π) Schulze (π) β’ ππ (π₯, π¦) β min weight of π₯ β π¦ path in π β’ πβ (π₯, π¦) = max ππ (π₯, π¦) π β’ Winner aβ : πβ πβ , π₯ β₯ πβ π₯, πβ Results Poly-time algorithm for winning Ranked Pairs (π) and Schulze (π) bundles
© Copyright 2026 Paperzz