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Mechanism Design without Money

Lecture 13

Condorcet winner

• Voting rules are trying to generalize the concept of “majority” in different ways

• Condorcet proposed another way, which relies on pairwise elections

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Condorcet winner

• Recall that a beats b in pairwise elections, if more voters prefer a over b

• a is a Condorcet winner if a beats any other candidate in a pairwise election.

3Does it define a new voting

rule?

The Condorcet Paradox

Voter 1 Voter 2

Voter 3

c

b

a

a

c

b

b

a

c

4

a

bc

The Condorcet property

• As we saw, there might not be a Condorcet winner in some case

• However, if there is one, then it is unique

• We say that a voting rule has the Condorcet property, if the Condorcet winner is always selected (when there is one)

Voting rules: Borda• Given m candidates

– ith ranked candidate score m-i

– Candidate with greatest sum of scores wins

• Example– 42 votes: A>B>C>D– 26 votes: B>C>D>A– 15 votes: C>D>B>A– 17 votes: D>C>B>A

– B wins

Jean Charles de Borda, 1733-1799

• If the Borda rule is used, then a will win – a has 8 points, while b only has 7

Manipulationabcd

abcd

bacd

• But if voter 3 lies about his preferences…– Now a only has 6 points, and b wins!

• What would happen if we used Plurality?

Manipulationbdca

Manipulationabcd

abcd

Manipulation (2)

• For each voting rule we saw, there is a voting profile, in which one voter can manipulate the outcome by lying.

• Does it matter?

• Are there rules which are immune to such strategic behavior? NO (at least not good rules)

It depends…

Manipulation (3)

Definition: a voting rule f is strategy-proof (SP), if no (single) voter can ever benefit from lying about his preferences. Formally:

Claim: If |A|=2 (i.e. there are two candidates), then Plurality is Strategy-Proof– In this case all the rules we saw are also SP

)()',(,', ,|V| RR fRRfRVi iiii RR

The Gibbard-Satterthwaite Theorem

Definition: A voting rule f is dictatorial if there is an individual (the dictator) whose most preferred candidate is always chosen by f. formally:

Definition: A voting rule f is onto if it is possible for any of the candidates to win (given the right preference profile):

)1()(,, ||i

V RfVi RR R

afAa V )(,, || RR R

The Gibbard-Satterthwaite Theorem

The Gibbard-Satterthwaite theorem: If there are at least three candidates, any voting rule that is strategy-proof and onto is dictatorial.

– This is a powerful negative result

)1()(,, ||i

V RfVi RR RafAa V )(,, || RR R

)()',(,', ,|V| RR fRRfRVi iiii RR

(if |A|≥3

)

(no dictator)(onto)

(SP)

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Manipulating Borda

• Suppose you want to add k manipulators for Borda who would promote p

• Clearly, p should be ranked first in all of them• If we could rank no one else, we would be

happy– But we have to give points to other candidates

• So the goal is to give points to bad candidates who won’t win

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How hard can that be?

• Manipulating Borda with just two extra voters is NPC.– Reduction from a cousin of subset sum

• But we know that subset sum (and family) are not really hard..

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Greedy algorithm for k manipulators

• We want to promote p• Each manipulator puts p on top• Problem – we need to give points to other

candidates..– Set the manipulators one by one– At each manipulator be greedy

• Thm: If exists a manipulation with k-1 manipulators this will succeed with k manipulators

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Probability that a manipulation exists

• What is the probability that a manipulation by a single voter exists?– Pluarality: requires a tie on top, probability is

about 1/n1/2

– Other systems – inverse polynomial– Done by proving that condorcet’s cycles exist with

decent probability– Nathan Keller (math) works on these questions

Computational hardness as a barrier to manipulation

• A (successful) manipulation is a way of misreporting one’s preferences that leads to a better result for oneself

• Gibbard-Satterthwaite only tells us that for some instances, successful manipulations exist

• It does not say that these manipulations are always easy to find

• Do voting rules exist for which manipulations are computationally hard to find?

A formal computational problem • The simplest version of the manipulation problem:• CONSTRUCTIVE-MANIPULATION: – We are given a voting rule r, the (unweighted) votes of the other

voters, and an alternative p. – We are asked if we can cast our (single) vote to make p win.

• E.g., for the Borda rule:– Voter 1 votes A > B > C– Voter 2 votes B > A > C– Voter 3 votes C > A > B

• Borda scores are now: A: 4, B: 3, C: 2• Can we make B win?• Answer: YES. Vote B > C > A (Borda scores: A: 4, B: 5, C: 3)

“Tweaking” voting rules

• It would be nice to be able to tweak rules:– Change the rule slightly so that• Hardness of manipulation is increased (significantly)• Many of the original rule’s properties still hold

• It would also be nice to have a single, universal tweak for all (or many) rules

• One such tweak: add a preround [Conitzer & Sandholm IJCAI 03]

Adding a preround [Conitzer & Sandholm IJCAI-03]

• A preround proceeds as follows:– Pair the alternatives– Each alternative faces its opponent in a pairwise

election– The winners proceed to the original rule

• Makes many rules hard to manipulate

Preround example (with Borda)

Voter 1: A>B>C>D>E>FVoter 2: D>E>F>A>B>CVoter 3: F>D>B>E>C>A

A gets 2 pointsF gets 3 points

D gets 4 points and wins!

Voter 1: A>D>FVoter 2: D>F>AVoter 3: F>D>A

A vs B: A ranked higher by 1,2C vs F: F ranked higher by 2,3D vs E: D ranked higher by all

Match A with BMatch C with FMatch D with E

STEP 1:A. Collect votes and

B. Match alternatives (no order required)

STEP 2:Determine winners of preround

STEP 3:Infer votes on remaining

alternatives

STEP 4:Execute original rule

(Borda)

Matching first, or vote collection first?

• Match, then collect“A vs C,

B vs D”.“D > C > B > A”

“A vs C,B vs D”.

“A vs C,B vs D”.

“A > C > D > B”

•Collect, then match (randomly)

Could also interleave…• Elicitor alternates between:

– (Randomly) announcing part of the matching– Eliciting part of each voter’s vote

“A vs F” “C > D” “B vs E” “A > E”

…

“A vsF”

…

“A vs F”

How hard is manipulation when a preround is added?

• Manipulation hardness differs depending on the order/interleaving of preround matching and vote collection:

• Theorem. NP-hard if preround matching is done first• Theorem. #P-hard if vote collection is done first• Theorem. PSPACE-hard if the two are interleaved (for a

complicated interleaving protocol)• In each case, the tweak introduces the hardness for any rule

satisfying certain sufficient conditions– All of Plurality, Borda, Maximin, satisfy the conditions in all cases,

so they are hard to manipulate with the preround

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Bribery

• We are allowed to change the votes of a set of voters.

• Stronger than manipulation – we are deleting bad voters and replacing them with good ones

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Reducing Energy Consumption

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The Voting Process

a3 > a4 > a1 > a2

a4 > a2 > a1 > a3

a2 > a3 > a1 > a4

a1 > a2 > a3 > a4

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The Voting Process

a3 > a4 > a1 > a2

a4 > a2 > a1 > a3

a2 > a3 > a1 > a4

a1 > a2 > a3 > a4

a1 > a2

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The Voting Process

a3 > a4 > a1 > a2

a4 > a2 > a1 > a3

a2 > a3 > a1 > a4

a1 > a2 > a3 > a4

a1 > a2

a3 > a1 > a4 > a2?

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Persuasion Problems

• Given:– A set of alternatives– A set of voters with their preferences– A preferences list of the sender

• Is there a “good” set of suggestions?

• K-Persuasion: send at most k suggestions

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Add Safety Requirement

• What if not all the voters accept the suggestions?

• Safe-Persuasion– Is there a “good” and safe set of suggestions?

• r-Safe-persuasion: send at most r suggestions

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Persuasion ≠ Manipulation or Bribery

• In coalitional manipulation and bribery– The manipulators always obey their suggestions– There is no requirement that they will benefit from it– How the manipulators attain full knowledge?

• In persuasion– Voters can accept or decline the sender’s suggestions– Send suggestion only to voters that will benefit from

it, and we add safety requirement– The sender is the election organizer

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Some Complexity Results

Persuasion K-Persuasion Safe-Persuasion

K-Safe-Persuasion

Plurality P P P PVeto P P P PK-Approval P NP-complete NP-hard NP-hardBucklin P NP-complete NP-hard NP-hardBorda NP-complete NP-complete NP-hard NP-hard

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Some Intuition

• In Plurality– The score of the current winner cannot be decreased

• K-Approval and Bucklin– Persuasion is in P from UCM– K-Persuasion is hard from X3C

• In Borda– Augmenting the hardness proof of [Betzler et al.,

2011]

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Hardness of is safe for k approval

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Hardness for k approval

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Test

• Four out of five• Mix of easy and moderate questions – no one is

supposed to get a bad grade, not trivial to get a 100• You need to know the slides, and need to be able to

solve simple exercises (e.g. find a Nash equilibrium of a game)

• If there are several sections, scoring is not identical• Test from last year’s course is online

– Not exactly the same material, but same style of questions