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Social Choice Theory, Lecture Notes - Computer Science, Study notes of Computers and Information technologies

Prof. Yiling Chen, Computer Science, Social Choice Theory, Pairwise Elections, Harvard, Lecture Notes

Typology: Study notes

2010/2011

Uploaded on 10/27/2011

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Download Social Choice Theory, Lecture Notes - Computer Science and more Study notes Computers and Information technologies in PDF only on Docsity! Social Choice Theory Yiling Chen October 27, 2008 CS286r Fall’08 Social Choice Theory 1 Introduction Social choice: preference aggregation Our settings I A set of agents have preferences over a set of alternatives I Taking preferences of all agents, the mechanism outputs a social preference over the set of alternatives or output a single winner I Hope to satisfy some desired properties Voting protocols are examples of social choice mechanisms Readings: SLB 9.1 – 9.4 CS286r Fall’08 Social Choice Theory 2 Pairwise Elections       2 prefer Obama to McCain 2 prefer McCain to Hillary 2 prefer Obama to Hillary   CS286r Fall’08 Social Choice Theory 5 More Voting Protocols Pairwise elimination I Pair candidates with a schedule I The candidate who is preferred by a minority of voters is deleted I Repeat until only one candidate is left Slater I The overall ordering that is inconsistent with as few pairwise elections as possible is selected. I NP-hard Kemeney I The overall ordering that is inconsistent with as few votes on pairs of candidates as possible. I NP-hard ... and many other voting rules What is the perfect voting protocol? CS286r Fall’08 Social Choice Theory 6 Condorcet Condition A candidate is a Condorcet winner if it wins all its pairwise elections. A voting protocol satisfies the Condorcet condition, if the Condorcet winner, if exists, must be elected by the protocol. Condorcet winner may not exist. Many voting protocols do not satisfy the Condorcet condition. CS286r Fall’08 Social Choice Theory 7 Voting Paradox: Sensitivity to A Losing Candidate 35 agents: a  c  b 33 agents: b  a  c 32 agents: c  b  a Which alternative is the winner under plurality voting? Which alternative is the winner under Borda voting? What happens if c drops off? CS286r Fall’08 Social Choice Theory 10 Notations N: a set of individuals, |N| = n A: a set of alternatives, |A| = m i : agent i ’s preference over A (e.g. ai i a3 i a5) L: the set of total orders, ∈ L Ln: the set of preference profiles, [] ∈ Ln A social welfare function is a function W : Ln → L W : the preference ordering selected by W A social choice function is a function C : Ln → A CS286r Fall’08 Social Choice Theory 11 Social Welfare Function: Pareto Efficiency A social welfare function W is Pareto efficient if for any a1, a2 ∈ A, ∀a1 i a2 implies that a1 W a2. It means that when all agents agree on the ordering of two alternatives, the social welfare function must select the ordering. CS286r Fall’08 Social Choice Theory 12 Arrow’s Impossibility Results (1951) If |A| ≥ 3, any social welfare function W can not simultaneously satisfy I Pareto efficiency I Independence of irrelevant alternatives I Nondictatorship Most influential result in social choice theory Read the proof Maybe asking for a complete ordering is too much? Let’s consider social choice functions. CS286r Fall’08 Social Choice Theory 15 Social Choice Function: Weak Pareto Efficiency A social choice function C is weakly Pareto efficient if for any preference profile [] ∈ Ln, if there exist a pair of alternatives a1 and a2 such that ∀i ∈ N , a1 i a2, then C () 6= a2. It means that a dominated alternative can not be selected. Weak Pareto efficiency implies unanimity: If a1 is the top choice for all agents, we must have C [] = a1. Pareto efficient rules satisfy week Pareto efficiency. But the reverse is not true. CS286r Fall’08 Social Choice Theory 16 Social Choice Function: Strong Monotonicity A social choice function C is strongly monotonic, if for any preference profile [] withC [] = a, then for any other preference profile [′] with the property that ∀i ∈ N, ∀a′ ∈ A, a ′i a′ if a i a′, it must be that C [′] = a. Strong monotonicity means that if I The current winner is a I We change the preference profile in the way such that for if alternative a′ ranks below a previously it is still below a in the new preference Then, a is the winner for the new preference profile. An example with STV 9 agents: a  b  c 12 agents: a  b  c 9 agents: b  c  a ⇒6 agents: b  c  a 7 agents: c  a  b 7 agents: c  a  b None of our rules satisfy strong monotonicity CS286r Fall’08 Social Choice Theory 17 Social Choice Function: Manipulability A social choice function is manipulable if some voter can be better off by lying about his preference An example with plurality voting 1 agent: a  b  c 2 agents: b  c  a 2 agents: c  b  a CS286r Fall’08 Social Choice Theory 20 Social Choice Function: Onto A social choice function C is onto if for each a ∈ A there is a preference profile [] ∈ Ln such that C ([]) = a. Onto means that every alternative can be a winner under some preference profile. CS286r Fall’08 Social Choice Theory 21 Gibbard-Satterthwaite’s Impossibility Results (1973, 1975) If |A| ≥ 3, any social choice function can not simultaneously satisfy I Nonmanipulable I Onto I Nondictatorship What’s possible? CS286r Fall’08 Social Choice Theory 22
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