Mechanism design without money Voting and ranking What will 2020 - PowerPoint PPT Presentation

Algorithmic game theory Ruben Hoeksma January 6, 2020 Mechanism design without money Voting and ranking

What will 2020 bring (for this course) This week : Voting and ranking Next week : No lectures 20 & 21 Jan: Student presentations 27 & 28 Jan: Final lectures: recap and exam preparation 24 Feb : Exams

The voting and ranking problems Candidates: A: Albert Aalderink B: Birgit Becker C: Camila Cortes Voters have a preference over the candidates e.g.,: C ≻ i B ≻ i A . How can we find: ◮ a single winner (voting) ◮ a complete ranking (ranking) from the preferences?

Formal Definition (Voting) Given a preference profile ( ≻ 1 , . . . , ≻ n ) for n agents on m candidates, Γ, produce a single winner W ∈ Γ. Definition (Ranking) Given a preference profile ( ≻ 1 , . . . , ≻ n ) for n agents on m candidates produce a society ranking >

Example Two candidates: A: Albert Aalderink B: Birgit Becker Voting rule: Voters vote for one person and the person who gets most wins. Cool properties: ◮ There is no better outcome (for any reasonable definition of better) ◮ The identity of the voters does not matter ◮ There is no incentive to strategize.

Plurality voting Definition (Plurality voting) Every voter votes for one candidate. The candidate with highest number of votes wins. Example 35%: A ≻ B ≻ C 25%: B ≻ A ≻ C 40%: C ≻ A ≻ B

(Instant-)Runoff voting Definition (Runoff voting) Every voter votes for one candidate. The candidate with the least votes is eliminated. Repeat until one candidate has 50% of votes. Definition (Instant-Runoff voting) Every voter makes a complete ranking over the candidates. Run a runoff vote using the voters rankings. Example 35%: A ≻ B ≻ C 25%: B ≻ A ≻ C 40%: C ≻ A ≻ B

(Instant-)Runoff voting Definition (Runoff voting) Every voter votes for one candidate. The candidate with the least votes is eliminated. Repeat until one candidate has 50% of votes. Definition (Instant-Runoff voting) Every voter makes a complete ranking over the candidates. Run a runoff vote using the voters rankings. Example 30%: A 60%: A 45%: B → 40%: C 25%: C

(Instant-)Runoff voting Definition (Runoff voting) Every voter votes for one candidate. The candidate with the least votes is eliminated. Repeat until one candidate has 50% of votes. Definition (Instant-Runoff voting) Every voter makes a complete ranking over the candidates. Run a runoff vote using the voters rankings. Example 30%: A ≻ B ≻ C 45%: B ≻ C ≻ A 25%: C ≻ A ≻ B

Dictatorship Definition (Dictatorship) Pick one voter. The candidate that that voter prefers wins. Positive property: No incentive to misreport preferences.

Other properties Anonymity: The voters are anonymous, i.e., if two (or more) voters switch their votes, the outcome remains the same. Monotonicity: If one voter moves candidate A up in their preferences and everything else remains the same, A does not get a worse ranking. Definition (Condorcet winner / loser) Consider two candidates, the one who is preferred by more voters gets a point. Do this for every candidate pair. A candidate with m − 1 points is Condercet winner. A candidate with 0 points is Condorcet loser. Condorcet winner/loser criterion: If there is a Condorcet winner (loser), then they are the winner (a loser) of the election.

Condorcet winner/loser Example 35%: A ≻ B ≻ C 25%: B ≻ A ≻ C 40%: C ≻ A ≻ B Implication Plurality voting does not satisfy the Condorcet winner criterion and does not satisfy the Condorcet loser criterion

Positional voting Definition (Positional voting) Assign a number a i to each position i . Candidates get a i points for each voter that has them on position i of their preference list. The Candidate with the highest total number of points wins. Example: Plurality voting is a 1 = 1, a i = 0, for all i ≥ 2. Definition (Borda count) Borda count voting is a positional voting rule with a 1 = m , a 2 = m − 1,. . . , a m = 1 35: A ≻ B ≻ C 25: B ≻ A ≻ C 40: C ≻ A ≻ B

Mechanism design without money Arrow’s impossibility theorem for ranking rules

Strategic vulnerability Definition (Ranking) Given a preference profile ( ≻ 1 , . . . , ≻ n ) for n agents on m candidates Γ, produce a society ranking > . Definition (Strategic vulnerability) A ranking rule is strategically vulnerable if there are an agent i , a preference profile ( ≻ 1 , . . . , ≻ n ), an alternative preference report ≻ i , and two candidates A and B such that A ≻ i B and B > A but A > ′ B , where > is the social ranking under ( ≻ 1 , . . . , ≻ n ) and > ′ is the social ranking under ( ≻ 1 , . . . , ≻ i − 1 , ≻ ′ i , ≻ i +1 , . . . , ≻ n ). Informal The ranking cannot improve for a particular player from that player lying about their preferences.

Independence of irrelevant alternatives (IIA) Definition (Independent of irrelevant alternatives IIA) A ranking rule is independent of irrelevant alternatives (IIA) if for the ranking of candidates A and B only the relative ranking of those two candidates matters. I.e., if > is the ranking for ( ≻ 1 , . . . , ≻ n ) and > ′ is the ranking for ( ≻ ′ 1 , . . . , ≻ ′ n ), and A ≻ i B iff i B , then A > B iff A > ′ B . A ≻ ′ Lemma A ranking rule that is not IIA is strategically vulnerable.

Arrow’s impossibility theorem Definition (Unanimity) A ranking is unanimous, when, if all agents agree on the relative rank of two candidates A and B , then the ranking also agrees. I.e., if A ≻ i B for all i , then A > B . Lemma (Arrow’s theorem) A ranking rule for three or more candidates fulfills unanimity and IIA only if it is a dictatorship. That is, there is some agent i such that the ranking is equal to the preference i.

Arrow’s impossibility theorem - proof Definition (Polarizing candidate) A candidate B is polarizing with respect to a preference profile is each agent ranks B first or last. Lemma Consider a ranking rule that fulfills unanimity and IIA. If there is a polarizing candidate B in the strategy profile, then the ranking rule ranks B highest or lowest.

Arrow’s impossibility theorem - proof Definition ( B -pivotal agent) Given a candidate B , we call an agent i B -pivotal if there is a preference profile ( ≻ 1 , . . . , ≻ n ) and an alternative preference ≻ ′ i such that B is polarizing and ranked lowest under ( ≻ 1 , . . . , ≻ n ) and polarizing and ranked highest under ( ≻ 1 , . . . , ≻ i − 1 , ≻ ′ i , ≻ i +1 , . . . , ≻ n ). Lemma Consider a ranking rule that fulfills unanimity and IIA. For every candidate B, there is at least one B-pivotal agent. Lemma Consider a ranking rule that fulfills unanimity and IIA, any candidate B, and a B-pivotal agent i. Then, i is a dictator on Γ \ B. I.e., for A , C � = B, we have A > C iff A ≻ i C.

Impossibility theorem for voting - Gibbard-Satterthwaite Definition (Strategy-proofness) A voting rule is strategy-proof if for all preference profiles ( ≻ 1 , . . . , ≻ n ), all agents i , and candidates A and B the following holds. If A ≻ i B and B wins under ( ≻ 1 , . . . , ≻ n ), then A does not win under any false report ≻ ′ i of agent i . Theorem (Gibbard-Satterthwaite) If a voting rule for three or more candidates is onto (that is, every candidate can be elected) and strategy-proof, then it is a dictatorship. That is, there is some agent i such that always agent i’s most preferred candidate wins.

Single-peaked preferences Suppose all candidates are on the real line [0 , 1] and all voters i have a preference p i ∈ [0 , 1] such that, if B > A > p i or B < A < p i , we have A ≻ i B . m a p i 0 1 � x i is not strategy-proof. With x i the reported peak of agent i . The average a = 1 n The median m , the ⌈ n 2 ⌉ -th x i if n is odd and the average of the n 2 -th and n 2 + 1-st value if n is even is strategy-proof.