## Generalized scoring rules and the frequency of coalitional manipulability (2008)

### Cached

### Download Links

- [cs.duke.edu]
- [www.cs.duke.edu]
- [www.cs.duke.edu]
- [www.cs.duke.edu]
- [people.seas.harvard.edu]
- [www.cs.rpi.edu]
- [ftp.cs.duke.edu]
- DBLP

### Other Repositories/Bibliography

Venue: | In Proceedings of the Ninth ACM Conference on Electronic Commerce (EC |

Citations: | 58 - 18 self |

### BibTeX

@INPROCEEDINGS{Xia08generalizedscoring,

author = {Lirong Xia},

title = {Generalized scoring rules and the frequency of coalitional manipulability},

booktitle = {In Proceedings of the Ninth ACM Conference on Electronic Commerce (EC},

year = {2008}

}

### OpenURL

### Abstract

We introduce a class of voting rules called generalized scoring rules. Under such a rule, each vote generates a vector of k scores, and the outcome of the voting rule is based only on the sum of these vectors—more specifically, only on the order (in terms of score) of the sum’s components. This class is extremely general: we do not know of any commonly studied rule that is not a generalized scoring rule. We then study the coalitional manipulation problem for gener-alized scoring rules. We prove that under certain natural assump-), then tions, if the number of manipulators is O(n p) (for any p < 1 2 the probability that a random profile is manipulable is O(n p − 1 2), where n is the number of voters. We also prove that under another set of natural assumptions, if the number of manipulators is Ω(n p) (for any p> 1) and o(n), then the probability that a random pro-2 file is manipulable (to any possible winner under the voting rule) is 1 − O(e −Ω(n2p−1)). We also show that common voting rules satisfy these conditions (for the uniform distribution). These results generalize earlier results by Procaccia and Rosenschein as well as even earlier results on the probability of an election being tied.

### Citations

1393 |
Counterspeculation, auctions and competitive sealed tenders
- Vickrey
- 1961
(Show Context)
Citation Context ...(the rule does not simply always choose the most-preferred alternative of a single fixed voter). This is in sharp contrast to settings with quasilinear preferences, where, for example, VCG mechanisms =-=[20, 6, 14]-=- are strategy-proof. Although a manipulation is guaranteed to exist (for reasonable rules), in order for the manipulating agent to use it, she must also be able to find it. Recent research has studied... |

900 | Probability; Theory and Examples
- Durrett
- 1991
(Show Context)
Citation Context ...ard normal distribution N(0, 1). Let σ 2 = E((Y1− E(Y1)) 2 ), ρ = E(|Y1 − E(Y1)| 3 ). Then we have: Pr(|Y | < f(N)) =Pr(− E(Y1)N σ √ N − f(N) σ √ N < Y − E(Y1)N σ √ N Then by the Berry-Esséen theorem =-=[10]-=-, Pr(|Y | < f(N)) <Φ(− E(Y1)N σ √ N = � E(Y − 1 )N σ √ f(N) + N σ √ N − E(Y1 )N σ √ f(N) − N σ √ N < 2f(N) σ √ N f(N) + σ √ ) − Φ(−E(Y1)N N σ √ N × 1 √ 2π + Cρ σ 3√ N N(0,1)(x)dx + Cρ σ 3√ N < − E(Y1)... |

789 |
Multipart pricing of public goods
- Clarke
- 1971
(Show Context)
Citation Context ...(the rule does not simply always choose the most-preferred alternative of a single fixed voter). This is in sharp contrast to settings with quasilinear preferences, where, for example, VCG mechanisms =-=[20, 6, 14]-=- are strategy-proof. Although a manipulation is guaranteed to exist (for reasonable rules), in order for the manipulating agent to use it, she must also be able to find it. Recent research has studied... |

770 |
A Measure of Asymptotic Efficiency for Tests of a Hypothesis Based on the Sum of Observations
- Chernoff
- 1952
(Show Context)
Citation Context ... depend on n such that the minimal difference between any two components of f(P M c ) is at least dcn p , when n → ∞. (End of the proof of Claim 1.) ✷ The next lemma is known as Chernoff’s inequality =-=[5]-=-. Lemma 2 (Chernoff’s inequality) Let N ∈ N. Let Y1, . . . , YN be N i.i.d. random variables with variance σ 2 . Let Y = � N ζ=1 Yζ. For any 0 ≤ k ≤ 2 √ Nσ, Pr(|Y −E(Y )| ≥ k √ Nσ) ≤ 2e −k2 /4 . For a... |

636 |
Incentives in teams
- Groves
- 1973
(Show Context)
Citation Context ...(the rule does not simply always choose the most-preferred alternative of a single fixed voter). This is in sharp contrast to settings with quasilinear preferences, where, for example, VCG mechanisms =-=[20, 6, 14]-=- are strategy-proof. Although a manipulation is guaranteed to exist (for reasonable rules), in order for the manipulating agent to use it, she must also be able to find it. Recent research has studied... |

406 | Strategy-proofness and Arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions
- Satterthwaite
- 1975
(Show Context)
Citation Context ...at is, a voter can sometimes make herself better off by declaring her preferences insincerely. A rule that is not vulnerable to manipulation is called strategy-proof. The GibbardSatterthwaite theorem =-=[13, 18]-=- states that when there are three or more alternatives, there is no strategy-proof voting rule that satisfies non-imposition (for every alternative, there exist votes that would make that alternative ... |

287 |
Manipulation of Voting Schemes: A General Result
- Gibbard
- 1973
(Show Context)
Citation Context ...at is, a voter can sometimes make herself better off by declaring her preferences insincerely. A rule that is not vulnerable to manipulation is called strategy-proof. The GibbardSatterthwaite theorem =-=[13, 18]-=- states that when there are three or more alternatives, there is no strategy-proof voting rule that satisfies non-imposition (for every alternative, there exist votes that would make that alternative ... |

143 | Single transferable vote resists strategic voting
- Bartholdi, Orlin
- 1991
(Show Context)
Citation Context ...ion can be made computationally hard, thereby erecting a computational barrier against manipulation. A number of results have been obtained that show that finding a successful manipulation is NP-hard =-=[3, 2, 7, 11, 9, 15]-=-. Some of these results consider manipulation by an individual voter, whereas others consider the more general case of manipulation by a coalition of voters. However, all of these hardness results are... |

139 |
The computational difficulty of manipulating an election
- Bartholdi, Tovey, et al.
- 1989
(Show Context)
Citation Context ...ion can be made computationally hard, thereby erecting a computational barrier against manipulation. A number of results have been obtained that show that finding a successful manipulation is NP-hard =-=[3, 2, 7, 11, 9, 15]-=-. Some of these results consider manipulation by an individual voter, whereas others consider the more general case of manipulation by a coalition of voters. However, all of these hardness results are... |

124 |
When are elections with few candidates hard to manipulate
- Conitzer, Sandholm, et al.
(Show Context)
Citation Context ...ion can be made computationally hard, thereby erecting a computational barrier against manipulation. A number of results have been obtained that show that finding a successful manipulation is NP-hard =-=[3, 2, 7, 11, 9, 15]-=-. Some of these results consider manipulation by an individual voter, whereas others consider the more general case of manipulation by a coalition of voters. However, all of these hardness results are... |

109 | Universal voting protocol tweaks to make manipulation hard
- Conitzer, Sandholm
- 2003
(Show Context)
Citation Context |

92 | Junta distributions and the averagecase complexity of manipulating elections
- Procaccia, Rosenschein
- 2007
(Show Context)
Citation Context ...at can find a manipulation for most instances. Several recent results seem to suggest that indeed, in various senses, hard instances of the manipulation problem are the exception rather than the rule =-=[17, 8, 16, 21]-=-. The results in this paper add to the body of work that suggests that the manipulation problem is usually easy to solve. For a very large class of voting rules, we show that in most cases, as the num... |

77 | Nonexistence of voting rules that are usually hard to manipulate
- Conitzer, Sandholm
- 2006
(Show Context)
Citation Context ...at can find a manipulation for most instances. Several recent results seem to suggest that indeed, in various senses, hard instances of the manipulation problem are the exception rather than the rule =-=[17, 8, 16, 21]-=-. The results in this paper add to the body of work that suggests that the manipulation problem is usually easy to solve. For a very large class of voting rules, we show that in most cases, as the num... |

53 |
Dichotomy for voting systems
- Hemaspaandra, Hemaspaandra
(Show Context)
Citation Context |

51 | Hybrid voting protocols and hardness of manipulation
- Elkind, Lipmaa
- 2005
(Show Context)
Citation Context |

46 | Algorithms for the coalitional manipulation problem
- Zuckerman, Procaccia, et al.
- 2009
(Show Context)
Citation Context ...at can find a manipulation for most instances. Several recent results seem to suggest that indeed, in various senses, hard instances of the manipulation problem are the exception rather than the rule =-=[17, 8, 16, 21]-=-. The results in this paper add to the body of work that suggests that the manipulation problem is usually easy to solve. For a very large class of voting rules, we show that in most cases, as the num... |

24 | Average-case tractability of manipulation in voting via the fraction of manipulators
- Procaccia, Rosenschein
- 2007
(Show Context)
Citation Context |

15 |
How large should a coalition be to manipulate an election
- Slinko
- 2004
(Show Context)
Citation Context ...ns, when the number of manipulators is no more than a constant, the probability that manipulation can be done is O( 1 √ n ), where n is the number of voters, under any positional scoring rule. Slinko =-=[19]-=- showed that under a particular condition on the probability distribution, under any faithful positional scoring rule (that is, all the scores in the scoring vector are different) the ratio of the num... |

13 |
A Note on the Probability of a Tied Election
- Beck
- 1975
(Show Context)
Citation Context ...es a general version of various results on the probability that an election ends up (roughly) in a tie, that is, a single voter can change the winner; this probability is also called the voting power =-=[4, 12]-=-. Knowing this probability is also interesting from the perspective of a voter who is determining her incentive to vote. Again, all of the existing results consider only the much smaller class of posi... |

7 | The mathematics and statistics of voting power
- Gelman, Katz, et al.
- 2002
(Show Context)
Citation Context ...es a general version of various results on the probability that an election ends up (roughly) in a tie, that is, a single voter can change the winner; this probability is also called the voting power =-=[4, 12]-=-. Knowing this probability is also interesting from the perspective of a voter who is determining her incentive to vote. Again, all of the existing results consider only the much smaller class of posi... |