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# Social choice

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Social Theory introduction up to Arrow's and Gibbard–Satterthwaite's Theorems. Have fun.

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### Social choice

1. 1. Social Choice Theory Enrico Franchi efranchi@ce.unipr.it
2. 2. Group Decisions}  Agents are required to choose among a set of outcomes Ω = {ω1 , ω2 ,…}}  Agents can choose one outcome in Ω}  Agents can express a preference of outcomes}  Let Π ( Ω ) the set of preference orderings of outcomes}  We also write ω 1 i ω 2 to express that agent i prefers ω1 to ω2 2 Enrico Franchi (efranchi@ce.unipr.it)
3. 3. Social Welfare}  A social welfare function takes the voter preferences and produces a social preference order: f : Π (Ω) → Π (Ω) N or in the slightly simplified form: f : Π (Ω) → Ω N}  We write ω 1 * ω 2 to express that the first outcome ranked above the second in the social outcome 3 Enrico Franchi (efranchi@ce.unipr.it)
4. 4. Plurality}  Simplest voting procedure: used to select a single outcome (candidate)}  Everyone submits his preference order, we count how many times each candidate was ranked first}  Easy to implement and to understand}  If the outcomes are just 2, it is called simple majority voting}  If they are more than two, problems arise 4 Enrico Franchi (efranchi@ce.unipr.it)
5. 5. Voting in the UK}  Three main parties: Voters Labour Party (left-wing) }  }  Liberal Democrats (center- Conser vative Labour left) Party Party }  Conservative Party (right- 44% 44% wing)}  Left-wing voter: ω L  ω D  ω C}  Center voter:ω D  ω L  ω C}  Right-wing voter:ω C  ω D  ω L Liberal Democ}  Tactical Voting rats}  Strategic Manipulation 12% 5 Enrico Franchi (efranchi@ce.unipr.it)
6. 6. Condorcet’s Paradox}  Consider this election: Ω = {ω 1 ,ω 2 ,ω 3 } Ag = {1,2,3} ω 1 1 ω 2 1 ω 3 ω 3 2 ω 1 2 ω 2 ω 2 3 ω 3 3 ω 1}  No matters the outcome we choose: two thirds of the electors will be unhappy 6 Enrico Franchi (efranchi@ce.unipr.it)
7. 7. Sequential Majority}  Series of pair-wise elections, the winner will go on to the next election}  An agenda is the strategy we choose to order the elections (linear, binary tree)}  An outcome is a possible winner if there is some agenda which would make that outcome the overall winner}  An outcome is a Condorcet winner if it is the overall winner for every possible agenda}  Can we choose the agenda to choose a winner? 7 Enrico Franchi (efranchi@ce.unipr.it)
8. 8. Borda Count and Slater Ranking}  Borda Count }  We have K outcomes }  Each time an outcome is in the j-th position for some agent, we increment its counter by K-j }  We order the outcomes according to their counter }  Good for single candidates}  Slater ranking }  Tries to be as close to the majority graph as possible }  Unfortunately, is NP-hard 8 Enrico Franchi (efranchi@ce.unipr.it)
9. 9. Properties}  Pareto condition: if every agent ranks ωi above ωj, then ω i * ω j }  Plurality, Borda}  Condorcet winner: if an outcome is a Condorcet winner, then it should be ranked first }  Sequential majority elections}  Independence of Irrelevant Alternatives (IIA): social ranking of two outcomes should only be affected by the way that they are ranked in their preference orders }  Almost no protocol satisfies IIA 9 Enrico Franchi (efranchi@ce.unipr.it)
10. 10. Properties}  Dictatorship: a social welfare function f is a dictatorship if for some voter j we have that: f (ω1 ,…, ωN ) = ω j}  Unrestricted Domain:   for any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. }  E.g., not random, always answers, does not “loop” 10 Enrico Franchi (efranchi@ce.unipr.it)
11. 11. Arrow’s Theorem}  There is no voting procedure for elections with more than two outcomes that satisfies }  Non-dictatorship }  Unrestricted Domain }  Pareto }  Independence of Irrelevant Alternatives 11 Enrico Franchi (efranchi@ce.unipr.it)
12. 12. Gibbard-Satterthwaite’s Theorem}  Sometimes voters “lie” in order to obtain a better outcome}  Is it possible to devise a voting procedure that is not subject to such manipulation?}  Manipulation (i prefers ωi): f (ω 1 ,…,ω i …,ω n ) i f (ω 1 ,…,ω i …,ω n )}  The only procedure that cannot be manipulated and satisfies the Pareto condition is dictatorship 12 Enrico Franchi (efranchi@ce.unipr.it)
13. 13. Complexity and Manipulation}  Even if all procedures can be manipulated, can we devise procedures which are hard to manipulate?}  Hard means “difficult to compute” in an algorithmic sense, e.g., NP-Hard procedures}  These procedures are easy (polynomial) to compute?}  Second-order Copeland may be “difficult” to manipulate }  In theory it is NP-Hard }  However, it is only a worst case complexity 13 Enrico Franchi (efranchi@ce.unipr.it)
14. 14. References1.  Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations; Yoav Shoham and Kevin Leyton- Brown; Cambridge Press2.  Game Theory: Analysis of Conflict; Roger B. Myerson; Harvard Press3.  An Introduction to Multi-Agent Systems; Michael Wooldridge; Wiley Press 14 Enrico Franchi (efranchi@ce.unipr.it)