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Cambridge University Press 978-1-107-06043-2 - Handbook of Computational Social Choice Edited by Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang and Ariel D. Procaccia Index More information

Index

adjacency matrix, 60 bloc voting, 207 Adjusted Winner, 305 blocking coalition, 360 afﬁne equivalence, 28 blocking pair, 337 agenda, 79, 454 Borda in judgment aggregation, 402 score, asymmetric, 28 agenda control, 456, 457, 460 score, symmetric, 28 for balanced binary trees, 459, 460 voting rule (Borda count), 28, 97, 147, 154, 163, #Agenda Control Problem, 462 164 agenda-implementability, 80 Borda, Jean-Charles de, 3 alternative vote, 37 bracket, 454, 465, 472, 473 amendment procedure, 78 Brams-Taylor algorithm, 321 anonymity, 31 Bribery, 161, 163 in judgment aggregation, 404 $Bribery, 161, 163, 164 anonymous game, 374 bribery, 159, 473 antiplurality voting rule, 37, 147 approximability of, 158, 166 apportionment, 15 in combinatorial domains, 166 appreciation of friends game, 373 in judgment aggregation, 167, 425 approval voting rule, 53, 147, 152, 154, 163 in path-disruption games, 167 sincere-strategy preference-based, 147, 154 bribery problem, 161 approximate manipulation, 140 priced, 159 approximation algorithm, 118, 387 with discrete price functions, 161 Archimedean property, 40, 152 with $discrete price functions, 161 argumentation, 423 with swap-bribery price functions, 161 Arrovian social choice, 261 shift, 164 Arrow’s Theorem, 6 BTT conditions, 132 in judgment aggregation, 419 Bucklin voting, 148, 154, 163 automated reasoning, 14, 425 simpliﬁed, 148 AV, see approval voting rule budgeted social choice, 209 aversion to enemies game, 373 bye, 459 axiomatic method, 30, 427 C1 social choice function, 38, 57, 60 Baldwin voting rule, 37 C2 social choice function, 38, 85 Banks set, 68 C3 social choice function, 39 Banzhaf index, 382 cake division problem, 265 computational properties, 386 campaign management, 159, 168 belief merging, 206, 421 with truncated ballots, 168 binary aggregation, 422 Campbell-Kelly Theorem, 35 binomial arborescence, 463 candidate cloning, 158, 168 bipartisan set, 66 candidate decloning, 168 of a weighted tournament, 101 Carroll, Lewis, see Dodgson, Charles Lutwidge Black’s rule, 97 CCAC, 150, 154 Black’s Theorem, 51 CCAUC, 150, 154

529

© in this web service Cambridge University Press www.cambridge.org Cambridge University Press 978-1-107-06043-2 - Handbook of Computational Social Choice Edited by Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang and Ariel D. Procaccia Index More information

530 index

CCAV, 151, 154 majority, 175 CCDC, 150, 154 strong unanimity, 174 CCDV, 151, 154 transitivity, 175 CCPC-TE, 155, 154 unanimity, 174 CCPC-TP, 155, 154 conservative extension, 81 CCPV-TE, 155, 154 consistency, 40, 276 CCPV-TP, 155, 154 of a judgment set, 402 CCRPC-TE, 155, 154 of an aggregator, 405 CCRPC-TP, 155, 154 constructive manipulation, 473 Chamberlin and Courant, 208 Constructive-Control-by-Adding-an- choice set, 60, 367 Unlimited-Number-of-Candidates, 150 claims problem, 265 Constructive-Control-by-Adding- classical fair division problem, 264 Candidates, 150 closure under complementation, 402 Constructive-Control-by-Adding-Voters, 151 closure under propositional variables, 412 Constructive-Control-by-Deleting- coalition, 402 Candidates, 150 coalition structure, 357 Constructive-Control-by-Deleting-Voters, 151 coalitional manipulation, 473 Constructive-Control-by-Partition-of- coarsening of a tournament solution, 61 Candidates, 155 Coleman index, 382 Constructive-Control-by-Partition-of- collaborative ﬁltering, 452 Voters, 155 collective combinatorial optimization, 16 Constructive-Control-by-Runoff-Partition- combinatorial domain, 198, 200 of-Candidates, 155 connection with judgment aggregation, 423 Contagion Lemma, 6 committee elections, 197, 198, 207 contractual Nash stability (CNS), 360 communication complexity, 235 contractually individual stability (CIS), 361 compact preference representation language, 210 control, 149 compilation complexity, 143, 247 approximability of, 158 complement constructive, 149 of a coalition (C), 402 by adding an unlimited number of candidates, of a propositional formula (∼ϕ), 402 150, 152, 153 complement-freeness by adding candidates, 149, 150, 152, 153 of a judgment set, 402 by adding voters, 149, 152, 153 of an aggregator, 405 by deleting candidates, 149, 150, 152, 153 complete vs. incomplete information, 138 by deleting voters, 149, 152, 153 completeness by partition of candidates, 154, 155 of a judgment set, 402 by partition of voters, 154, 155 of an aggregator, 405 by runoff partition of candidates, 154, 155 completion (extension) principle, 204 destructive, 154 complex competition format, 474 by deleting candidates, 156 complexity theory, 17 experimental study of, 158 component of a tournament, 60 in combinatorial domains, 158 composition property, 277 in judgment aggregation, 159, 425 composition-consistency, 62 in sequential elections, 159 computational complexity of manipulation, 131 online computer-assisted theorem proving, 14 in sequential elections, 159 conditional preference table, 212 control problem, 149 conditional preferential independence, 211 counting variant of a, 167 conditionally lexicographic preference, 213 converse consistency, 276 Condorcet cooperative game, 13, 358, 378 consistency, 35, 61, 148, 453, 455 cooperative vs. noncooperative game theory, 144 domain (DCondorcet), 35 Copeland voting rule, 28, 63, 148, 153, 154, 163, 464 extension, 35, 453, 454 cooperative game, with non-transferable utility, 358 generative model, 471 core, 360 in judgment aggregation, 403 computational properties, 383 jury theorem, 184 deﬁnition, 380 loser, 35, 63 core stability, 360 Marie Jean Antoine Nicolas de Caritat, Marquis de, covering relation, 67 4, 24 deep, 82 non-losers, 63 McKelvey, 82 paradox, 4, 34, 57 CP-net, 211 principle, 25 crowdsourcing, 17, 424 voting rule, 152, 154 cumulative voting, 207 winner, 34, 58, 103, 148, 454, 464 winning set, 210 DCAC, 154 consensus class, 174 DCAUC, 154 Condorcet, 175 DCAV, 154

index 531

DCDC, 154 electoral control, see control DCDC, 156 electorate DCDV, 154 single-crossing, 158 DCPC-TE, 154 nearly, 158 DCPC-TE, 158 Elevator Lemma, 276 DCPC-TP, 154 envy DCPC-TP, 158 cycle, 301 DCPV-TE, 154 maximal, 300 DCPV-TP, 154 possible / necessary, 303 DCRPC-TE, 154 ratio, 302 DCRPC-TE, 158 envy-freeness, 295, 313 DCRPC-TP, 154 equal treatment of equals, 269, 278 DCRPC-TP, 158 equal-division lower bound, 271 Debord’s Theorem, 86 equal-division Walrasian rule, 270 decomposition of a tournament, 60 equitability, 313 Deegan-Packel index, 382 essential set (ES), 82, 101 descending demand procedure, 308 even-number-negatable agenda, 417 destructive manipulation, 473 Even-Paz algorithm, 314 dichotomous preferences, 55 expected quality of the winner, 472 dichotomy result Extension-Bribery, 168 on control, 158 on manipulation, 158 f -equivalence, 248 on possible winners, 158, 164 fair allocation, 261 dictatorship, 31 fair division with single-peaked preferences, 264 in judgment aggregation, 415 fallback voting, 148, 154, 163 discursive dilemma, 400 false-name-proofness, 144 distance, Hamming, 205 Field Expansion Lemma, 6 distance, 175 Fishburn’s classification, 38 discrete, 176 Fisher market, 327 edge reversal, 177 fixed electorate, 29 footrule, 176 fooling set, 236 Kendall tau, 176 fractional hypergraph matching, 375 Sertel, 177 frequency of manipulability, 141 swap, 176 full P -agenda control, 457, 458 vote insertion, 177 full agenda control, 457 votewise, 180 fully proportional representation, 208 distance rationalizability, 104 distance-based aggregation Gale–Shapley algorithm, 334, 338, 339 in judgment aggregation, 410, 425 game-theoretic models of manipulation, 142 distortion, 250 Gamson’s hedonic game, 374 distributed fair division, 308 general agenda control, 457 districting, 15 geometric reward mechanism, 443 doctrinal paradox, 400 GETCHA, 71, 82 Dodgson approximation, 118 Gibbard-Satterthwaite Theorem, 46, 76, 128 Dodgson score, 104 quantitative versions, 141 Dodgson voting rule, 45, 103, 125 GOCHA, 83 Dodgson winner problem, 109, 110 group activity selection game, 374 Dodgson, Charles Lutwidge, 4, 103 group classification, 16 DodgsonScore problem, 111 Group Contraction Lemma, 6 domain, 264 group recommendation, 16 domain restrictions, 29 Groupthink, 439 in judgment aggregation, 407 dominance relation, 59 Hamming distance, 205 double-elimination tournament, 474 between two judgment sets, 403 duality, 282 Hare voting rule, 37 Dubins-Spanier algorithm, 314 hedonic coalition formation game, additively Duggan-Schwartz Theorem, 50 separable, 364 dummy player, 381 hedonic coalition formation game, fractional, 357, dynamic programming, 116 374 dynamic social choice, 17 hedonic coalition formation game, with dynamics, 143 B-preferences, 365 hedonic coalition formation game, with ε-monotonic probability matrix, 460 W-preferences, 366 edit distance, 104 hedonic coalition net, 362 effort, 472 heuristic algorithm egalitarian-equivalence, 272 frequently self-knowingly correct, 113 election fraud detection, 159 highish outdegree, 467, 468

532 index

homogeneity, 122 Llull, Ramon, 3 Hospitals/Residents problem (hr), 336, 337 lobbying problem, 166 Hospitals/Residents problem with Couples (hrc), logic, 401, 425 344 lower bound on welfare, 271 Hospitals/Residents problem with Ties (hrt), 343 LP-tree, 213 House Allocation problem (ha), 346, 349 Housing Market (hm), 347, 349 majority cycle, 34 hybrid voting rules, 138 majority graph, 456 hypercube, 205, 211 majority margin, 85 majority relation, 58 idempotency of a tournament solution, 61 majority rule, 185 IIA, see independence of irrelevant alternatives in judgment aggregation, 403, 409, 414 immediate acceptance rule, 270 Mallows model, 227 impartial culture, 226 Manipulation, 163, 164 implementation by agenda, 456 manipulation imposition, 32 approximability of, 158 imputation, 380 in judgment aggregation, 413 inapproximability online of Dodgson score/ranking, 124 in sequential elections, 159 of StrongYoung score, 126 manipulation as a computational problem independence basic variant, 132 in judgment aggregation, 404 coalitional manipulation, 134 independence of irrelevant alternatives, 5 coalitional weighted manipulation, 135 individual rationality, 359 margin-of-victory problem, 167 Individually Rational Lists of Coalitions (IRLC), marginal contribution net, 362 362 marginal preference, 200 individual stability (IS), 360 Markov set, 65 instant run-off voting (IRV) voting rule, 37 marriage game, 374 integer programming, 19 match, 453, 454 irresolute voting rules, 131 matching, 267, 337, 346, 353 issues, 200 greedy maximum, 353 iterative voting, 14 individually rational, 347, 350 maximum Pareto optimal, 351 judgment aggregation, 159 pairwise stable, 345 judgment aggregation axioms Pareto optimal, 346, 349 anonymity, 404 popular, 352, 353 complement-freeness, 405 rank-maximal, 353 completeness, 405 setwise stable, 345 consistency, 405 stable, 337, 340, 344, 345 independence, 404 strict core, 347, 349 monotonicity, 404 strongly stable, 343 neutrality, 404 super-stable, 343 nondictatoriality, 415 weak core, 347 systematicity, 405 weakly stable, 343 unanimity, 404 matching representation of a seeding, 465 judgment aggregation rule, 403 maximal lottery, 66 judgment set, 402 maximin voting rule, 148, 154, 163 computation, 97 k-approval voting rule, 37, 147, 163 maximin share, 297 k-veto voting rule, 147, 163 maximum likelihood estimator, 187 Kemeny’s voting rule, 86, 45, 112 for ranking (MLERIV), 189 approximation algorithms, 90 for winner (MLEWIV), 193 computational hardness, 88 maximum regret, 231 in judgment aggregation, 410 maxmin fairness, 295 parameterized algorithms, 91 May’s Theorem, 34 practical algorithms, 93 McGarvey’s Theorem, 58 Kendall’s tau distance, 44, 87 mechanism, 340, 347 Kendall-Wei method, 64 core, 349 king, 464 Core from Fixed Endowments (CFE), 349 knockout tournament, 454 Core from Random Endowments (CRE), 349 balanced, 459, 473 Random Serial Dictatorship (RSD), 348 Roth-Vande Vate, 342 linear programming, 19 Serial Dictatorship (SD), 348 linear programming formulation strategyproof, 340, 341, 350, 351 of Dodgson score, 120 mechanism design, 13, 129 link prediction, 452 median order, 87

index 533

median property, 414 ownership Min-Cut recommendation system, 440 collective, 263 minimal TC-retentive set, 74 individual, 263 minimal consistent enlargement, 282 mixed, 263 minimal covering set semi-collective, 263 of a weighted tournament, 102 ownership data, 263 minimal covering set, 72 minimal extending set, 72 P -agenda control, 457, 459, 460 minimally inconsistent set, 403 PageRank, 430 minimax approval voting, 205 pair-negatable agenda, 419 minimax regret, 231 pairwise comparison, 225, 235, 453 misrepresentation, 208, 252 pairwise majority monotonic probability matrix, 459, 472 >μ,27 monotonicity, 122 pairwise majority rule (PMR), 34 down, 42 paradoxes of power, 388, 389 half-way, 42 parameterized complexity in judgment aggregation, 404 of Dodgson score, 116 Maskin, 42 of StrongYoung elections, 115 of a tournament solution, 61 Pareto-optimality, 5, 32, 61, 68, 78, 295, 359, 455, one-way, 42 456 strong, 42 parity rule, 418 weak, 41 partial preference, 225 Monroe scheme, 208 participation, 42, 455 Muller-Satterthwaite Theorem, 47 party afﬁliation game, 369 multi-level marketing, 441 peak amount, 265 multiagent systems, 423 perfect partition, 359 multiple election paradoxes, 201, 221 picking sequences, 307 multiple elimination tournament, 473 pivotal player, 382 multiple referenda, 197, 198 Pliny the Younger, 3 multiwinner elections, 197, 207 PLS (polynomial local search), 368 plurality with run-off voting rule, 37 Nanson’s voting rule, 37 plurality voting rule, 27, 147, 152, 154, 161, 163 computation, 97 polynomial hierarchy, 108 Nash stability, 360 polynomial-time many-one reduction, 149 necessary winner, 228 population monotonicity, 275 net preference (NetP (a>b)), 27 positional scoring rule, 147 network analysis, 376 pure, 158 neutrality, 31, 455 positive responsiveness, 33 in judgment aggregation, 404 possible and necessary winner problems, 144 no-domination of equal division, 271 possible winner, 228 no-envy, 269 Possible-Winner, 164 for groups, 272 potential function, 365 in trades, 269 PPAD, 375 noise model preference agenda, 403 Mallows, 186, 187 preference elicitation, 239 rankings, 187 preference query, 235 winners, 193 preference ranking, 26 nontrivial agenda, 402 Preference Reﬁnement Algorithm, 371 nonunique-winner model, 160 preferences, 264 norm, 180 additive, see modular null object, 266 compact representation, 289 lifting, 287 O-legal, 216 modular, 287 object, 267 monotonic, 294 desirable, 266 responsive, 287 undesirable, 266 separable, 199, 200, 364 object allocation single-peaked, 51, 112, 130, 141, 158, 166, 238, priority-augmented, 267 265 object-allocation problem, 266 value-restricted, 52 objects-and-money allocation problem, 266 premise-based rule, 411 odd-max-clique, 108 price function, 159 opportunity, 272 discrete, 160 optimal seeding, 472 $discrete, 160 order preservation in awards, 278 family of price functions, 160 orthogonal decomposition into cyclic and cocyclic swap-bribery, 160 components, 39 price of fairness, 296, 325

534 index

principal ultrafilter, 416 relational requirement, 273 priority resoluteness, 29 absolute, 263 resource data, 263 relative, 263 resource monotonicity, 274 priority data, 263 Robertson-Webb model, 316 priority order, 267 robust winner, 246 probability distribution, 471 robustness principle, 276 probability matrix, 457 roommate game, 374 probability of winning, 455 round-robin tournament, 473, 474 probably approximately correct (PAC) learning, Rural Hospitals Theorem, 339 254 product of tournaments, 60 safe strategic voting, 144 profile, 26 safety of the agenda, 419 in judgment aggregation, 402 SCF, see social choice function profile completion, 226 School Choice problem (sc), 337, 341, profile of weak rankings, 26 350 proper reward mechanism, 443 Schulze’s voting rule, 98, 148, 154, 163 proportionality, 313 computation, 98 pseudodistance, 175 Schwartz set, 83 Public Good index, 382 Schwartz’s conjecture, 74 punctual requirement, 271 scoring rule, 37, 177, 189, 193 with full coverage, 271 characterization, 194 compound, 40 quota, 378 continuity of, 40 quota rule, 408 scoring vector, 147 Seed Algorithm, 467 Random Walk recommendation system, seeding, 454, 455 439 seeding efficiency, 474 randomized social choice, 14, 130 selection systems, 451 range voting rule, 148, 154, 163 Selfridge-Conway algorithm, 315 normalized, 148, 154 Sen’s Possibility Theorem, 52 ranked pairs voting rule, 98, 148, 163 sequential majority comparison voting rule (SMC), computation, 99 36, 454, 460 ranking scoring rule, 45 sequential voting protocol, 216 ranking system, 430 set extension, 77, 365 ranking system axioms Fishburn’s extension, 77 Collapsing, 433 Kelly’s extension, 77 Isomorphism, 432 Shapley value, 381 Proxy, 434 characterization, 382 Self Edge, 433 computational properties, 385, 386 Vote by Committee, 433 Shapley-Shubik voting index, 382 rationalizable choice function, 245 Shift-Bribery, 164 recommendation system, 436 simple agenda, 78, 414 recommendation system axioms simple game, 378, 390 Independence of Irrelevant Stuff, 437 simple ranking scoring function, 187, 193 Neighborhood Consensus, 437 simplified median property, 420 No Groupthink, 440 Simpson’s voting rule, see maximin voting rule Positive Response, 436 simultaneous voting, 200 Scale Invariance, 438 sincere voting, 79 Symmetry, 436 single transferable vote, 37, 148, 163 Transitivity, 438 single-elimination tournament, 454, 465 Trust Propagation, 438 single-peakedness, see preferences, single-peaked referal reward axioms approximation, 15 Additivity, 444 single-valuedness, 268 Budget, 443 skew-adjacency matrix, 60 Child Dependence, 444 Slater rule, 64 Depth Level Dependence, 444 in judgment aggregation, 411 Local Split Proof, 448 SMC-tree, 460, 473 Monotonicity, 448 Smith set, 71, 82 Split Proof, 448 Smith-consistent, 454 SubTree, 442 SNTV, 207 Unbounded Reward, 443 social choice function, 29 referral forest, 442 social distance games, 374 referral tree, 441 social preference function, 44 refinement of a tournament solution, 61 social welfare, 225, 250, 323 reinforcement, 40, 455 social welfare function, 5, 29

index 535

solidarity, 273 ultraﬁlter, 415 sophisticated voting, 79 unanimity SP-AV, see approval voting, sincere-strategy in judgment aggregation, 404 preference-based unavailable candidate model, 245 split proof reward mechanism, 448 uncovered set, 67, 464 Splitting Lemma, 6 of a weighted tournament, 102 stability, 279, 355 undercut procedure, 306 stability of a tournament solution, 61 uniform quota rule, 408 Stable Marriage problem (sm), 337 uniform rule, 270 Stable Marriage problem with Incomplete lists (smi), unique-winner model, 112, 160 337 upper bound on welfare, 271 standard seeding, 472 utilitarian optimality, 295 strategic sequential voting, 220 utility function, cardinal, 250 strategyproofness, 127, 355 Fishburn-strategyproofness, 77 value density function, 312 Kelly-strategyproofness, 77 piecewise constant, 312 of tournament solutions, 76 piecewise uniform, 312 strict core, 360 variable electorate, 29 strict strong Nash stability (SSNS), 360 vector weighted voting games, 393 strong individual stability (SIS), 360 veto, 147, 163 strong superset property, 61 veto player, 383 StrongYoung voting rule, 105 voter weight vs. voter power, 388 StrongYoung winner problem, 112 voting equilibrium, 256 STV, see single transferable vote voting network, 435 subtournament, 59 voting protocol, 236 summary of a tournament, 60 voting rule, 147 Summing Contributions property, 445 Borda, voting rule (Borda count), 191, 192 superking, 466, 471 Condorcet-consistent, 148 Superseed Algorithm, 466 continuity, 152 Support-Bribery, 166 distance rationalizable, 178 Swap-Bribery, 161, 164 immune to a control type, 151 SWF, see social welfare function Kemeny’s voting rule, 186, 192 sybil attacks, 446 resistant to a control type, 152 local split, 447 susceptible to a control type, 151 split, 446 Tideman’s voting rule, 191 systematicity, 405 vulnerable to a control type, 152 voting situation, 27 p 2 , 107, 112, 125 voting tree, 454, 472, 473 TE, see ties eliminate tentative acceptance rule, 270 weak composition-consistency, 62 Tideman’s voting rule, 123 weak Condorcet winner, 103 ties eliminate, 155 weak tournament, 81 ties promote, 155 WeakDodgson voting rule, 104 Top Covering Algorithm, 370 WeakDodgson winner problem, 112 top cycle, 36, 71, 455, 457 weight, 378 top responsive preferences, 367 total weight of a coalition, 379 Top Trading Cycles (TTC) algorithm, 348, 349 weight vs. power, see voter weight vs. voter power top-k voting (query), 243 weighted tournament, 28, 85 totally blocked agenda, 417 weighted voting games, 377, 379 tournament, 28, 59 Weighted-Bribery, 161, 163, 164 tournament equilibrium set, 74 Weighted-$Bribery, 161 tournament ﬁxing problem, 463, 464 Weighted-Manipulation, 164 tournament graph, 456 welfare dominance under preference replacement, tournament solution, 60 275 TP, see ties promote winning coalition, 405 trade robustness, 391 worst-case vs. typical-case hardness, 139 transferable utility cooperative game, 358 trivial tournament solution, 62 yes/no voting systems, 378, 390 trust based recommendations, 435 Young score, 104 trust systems, 452 Young winner problem, 112 truth-biased voters, 143 Young’s voting rule, 103, 104