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 ai to each position i. Candidates get ai points for each voter that has them on position i of their preference list. The Candidate with the highest total number

• f points wins.

Example: Plurality voting is a1 = 1, ai = 0, for all i ≥ 2.

Definition (Borda count)

Borda count voting is a positional voting rule with a1 = m, a2 = m − 1,. . . , am = 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

A ≻′

i B, then A > B iff A >′ B.

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

• r 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 pi ∈ [0, 1] such that, if B > A > pi or B < A < pi, we have A ≻i B.

1 pi a m

With xi the reported peak of agent i. The average a = 1

n

xi is not strategy-proof.

The median m, the ⌈ n

2⌉-th xi 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.