Slide 1Slide 2Big pictureBig picture (some more)let’s start simpleWho’s the winner?Condorcet’s ParadoxSocial Choice/RankingTwo questionsE.g. 1E.g. 2E.g. 3what are some properties we’d like?NotationSome propertiesSlide 16Slide 17The case for |A| = 2The case for |A| = 3 (or more)The Borda systemCopeland’s MethodSocial Choice functions…What about social-choice functions?Slide 24Slide 25Slide 26Again, some simple cases…PluralityInstant-Runoff VotingWhat are some good social ranking and social choice functions for |A| >= 3?Slide 31Slide 32Gibbard-SatterthwaiteArrow’s Theorem Gibbard-Satterthwaite Theorem Two important results with very similar proofsSo what do we do?Mechanisms with moneySelfishnessAuctionsSlide 39PictureTry #1Try #2Try #3Truth-telling is a good strategy hereSlide 45Wanna play a game?You can get…Range VotingSlide 4915-251Great Theoretical Ideas in Computer ScienceSocial Choice, Votingand AuctionsLecture 17 (October 21, 2008)Big pictureSay you like pizza > hotdogs >> burgers.We ask you for your preferences, so that we can order food for the review session.I’ll choose the option that gets most 1st place votes.What should you tell me?What if you know: the rest of the class has 20 votes each for hotdogs and burgers, and 15 for pizza?Big picture (some more)People have private informationYou want to get that informationIf you ask them, will they tell you the truth?Well, they might lie, if telling a lie helps them.(“strategic behavior”)How do you elicit the truth?let’s start simpleWho’s the winner?Three candidates in townThree voters in town1st: a > b > c2nd: b > c > a3rd: c > a > bWho’s the winner?What’s the best ordering of the candidates?Condorcet’s ParadoxMarie Jean Antoine Nicolas de Caritat, Marquis de Condorcet1st: a > b > c2nd: b > c > a3rd: c > a > bGiven any potential winner (say a),a majority prefer another candidate (c) to this person.Social Choice/RankingA: set of alternatives (say, {a,b,c,d})L: set of all possible rankings or linear orderings of these alternativese.g., a > d > b > c, or b > c > d > aN people each with their rankingWant to combine these into one “social ranking”Two questionsGiven the rankings of the N individuals:Social Choice:Output the alternative that’s the “winner”i.e., output an element of ASocial Ranking:Output a ranking that “best captures” the rankings of the individuals.E.g. 1A = {a, b}In this case, L = { (a>b), (b>a)}Population:person 1: a>bperson 2: a>bperson 3: b>aperson 4: a>bSocial Choice: maybe use “plurailty” and output aSocial Ranking: (a>b)E.g. 2A = {a, b, c}L = { (a>b>c), (a>c>b),(b>c>a),(b>a>c), (c>a>b), (c>b>a)}Population:person 1: a>b>cperson 2: a>b>cperson 3: b>a>cperson 4: c>a>bSocial Choice: maybe use “plurailty” and output aSocial Ranking: maybe (a>b>c)E.g. 3A = {a, b, c}L = { (a>b>c), (a>c>b),(b>c>a),(b>a>c), (c>a>b), (c>b>a)}Population:person 1: a>b>cperson 2: b>c>aperson 3: c>a>bSocial Choice: For each output, majority of people prefer some other candidateSocial Ranking: ???what are some properties we’d like?NotationSocial ranking functionF: LN Ltakes (>1, >2, …, >N) >outputSocial choice functionG: LN Atakes (>1, >2, …, >N) aSome propertiesUnanimity:F is unanimous if when all individuals have a>b for some a,b in A, then the output satisfies a>b.Some propertiesIndependent of Irrelevant Alternatives:F is IIA if the relative ranking of a and b in the outcome depends only on the voters’ rankings of a and b.I.e., whenever all voters rank a and b the same,the output is the same, regardless of the otheralternatives.Some propertiesDictatorVoter j is a dictator in F if F(>1, >2, …, >N) = >jF is a dictatorship if there is some j that is a dictator in F.The case for |A| = 2Fact:If there are 2 alternatives, the IIA property is trivially satisfied.Facts:Note that plurality satisfies unanimity, and is not a dictatorship.The case for |A| = 3 (or more)Plurality does not immediately apply…The Borda systemCopeland’s MethodSocial Choice functions…What about social-choice functions?Remember a social choice function outputs a single choiceI.e., G: LN A, takes (>1, >2, …, >N) aSome propertiesUnanimity:G is unanimous if when all individuals have a at the top of their rankings, then G outputs a.Some propertiesMonotone:G is monotone if wheneverG(>1, >2, …, >j, …, >N) = aand G(>1, >2, …, >j’, …, >N) = a’then it must be the case that voter j moved a’ above a in his ranking.(I.e., G is “incentive-compatible”. It does not reward lying.)Some propertiesDictatorVoter j is a dictator in G if G(>1, >2, …, >N) = choice at top of >jG is a dictatorship if there is some j that is a dictator in G.Again, some simple cases…PluralityOutput the option at the top of most people’s rankings.Unanimity:Monotonicity:Instant-Runoff VotingRemove alternative with fewest first-place votes, and repeat.Unanimity:Monotonicity:What are some good social ranking and social choice functionsfor |A| >= 3?The case for |A| = 3 (or more)Theorem (Arrow)Any social ranking function with |A| = 3 or more that satisfies unanimity and IIA is a dictatorship.The case for |A| = 3 (or more)Theorem (Gibbard-Satterthwaite)Any social choice function with |A| = 3 or more that satisfies unanimity and monotonicityis a dictatorship.Gibbard-SatterthwaiteNote that we wanted to ask people for their (secret) preferences, and use that to picka social choice.We don’t want them to lie. (Hence we want thesocial choice function to be monotone.)But that is impossible. Arrow’s TheoremGibbard-Satterthwaite TheoremTwo important resultswithvery similar proofsSo what do we do?How to get around these impossibility results?Two solutions:1) Money…2) Change the representation…Mechanisms with moneyMeasure not just that a preferred to b, but also “by how much”…Each individual j (or player j) has a “valuation” for each alternative a in A. Denoted as vj(a)Also, each player values money the same.So, if we choose alternative a, and give $m to j, then j’s “utility” is vj(a) + mSelfishnessEach player acts to maximize her utility.AuctionsSuppose there is a single item a to be auctioned.Each player has value vj(a) (or just vj) for it.If item given to j, and j pays $p, thenutility(j) = vj – pandutility(j’) = 0 for all other players j’.AuctionsHowever, auctioneer does not know these private valuations.Auctioneer wants to give the item to the person who values it the most.
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