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Manipulation of Voting Manipulation of Voting Schemes: A General ResultSchemes: A General Result

By Allan GibbardBy Allan Gibbard

Presented byPresented byRishi KantRishi Kant

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RoadmapRoadmap

IntroductionIntroduction Definition of termsDefinition of terms 33 Brief overviewBrief overview 44 ImportanceImportance 1010

DiscussionDiscussion Definition of termsDefinition of terms 1313 PropertiesProperties 1414 Proof of statementProof of statement 1616

ConclusionConclusion

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Definition of termsDefinition of terms Voting scheme – a decision making system that Voting scheme – a decision making system that

depends solely on the preferences of depends solely on the preferences of participants, and leaves nothing to chanceparticipants, and leaves nothing to chance

Dictatorial – no matter what the other Dictatorial – no matter what the other participants’ preferences are, the outcome is participants’ preferences are, the outcome is always decided by the preference given by the always decided by the preference given by the dictatordictator

True preference – the player’s preference if he True preference – the player’s preference if he were the only participant / dictatorwere the only participant / dictator

Non-trivial voting scheme – a voting scheme in Non-trivial voting scheme – a voting scheme in which not every player has a dominant strategywhich not every player has a dominant strategy

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ProblemProblem

Can one design a voting scheme whose Can one design a voting scheme whose outcome is solely based on the true outcome is solely based on the true preference of each participant ?preference of each participant ?

Answer: Not unless the game is dictatorial Answer: Not unless the game is dictatorial or has less than 3 outcomesor has less than 3 outcomes

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Formal statementFormal statement

““Any non-dictatorial voting scheme with at Any non-dictatorial voting scheme with at least 3 possible outcomes is subject to least 3 possible outcomes is subject to individualindividual manipulationmanipulation””

Interpretation:Interpretation:

Given a voting scheme (and certain Given a voting scheme (and certain circumstances) it is possible for an circumstances) it is possible for an individual to force his desired outcome by individual to force his desired outcome by disguising his true preferencedisguising his true preference

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ExampleExample

4 contestants – w, x, y, z4 contestants – w, x, y, z3 voters – a, b, c3 voters – a, b, cEach voter ranks contestants (as i j k l) Each voter ranks contestants (as i j k l)

according to his/her preferenceaccording to his/her preference11stst gets 4 points, 2 gets 4 points, 2ndnd gets 3 … gets 3 …Whoever has most points winsWhoever has most points wins

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ExampleExample

Let the true preference of each voter be:Let the true preference of each voter be:

a => w x y za => w x y z

b => w x y zb => w x y z

c => x w y zc => x w y z

If every voter put down his/her trueIf every voter put down his/her true

preference then w would win [11 points]preference then w would win [11 points]

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ExampleExample

However, for the given situation c can forceHowever, for the given situation c can force

the winner to be x by pretending that histhe winner to be x by pretending that his

preference order is differentpreference order is different

a => w x y za => w x y z

b => w x y zb => w x y z

c => x w y z c => x w y z c => x y z w c => x y z w

x will now win with 10 pointsx will now win with 10 points

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NotesNotes

Point to note: c could influence the voting Point to note: c could influence the voting scheme only due to the given circumstancesscheme only due to the given circumstances If a and b had slightly different orderings e.g. If a and b had slightly different orderings e.g.

a => w y z x, then c would not be successfula => w y z x, then c would not be successful

Thus,Thus, subject to individual manipulation subject to individual manipulation means means that there is at least one scenario for which an that there is at least one scenario for which an individual can force the outcome that he wants individual can force the outcome that he wants => voting scheme is not totally tamper proof=> voting scheme is not totally tamper proof

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ImportanceImportance

No non-trivial decision making system that No non-trivial decision making system that depends on depends on informed self-interestinformed self-interest can can guarantee that the outcome was based on guarantee that the outcome was based on the true preferences of the participantsthe true preferences of the participants Informed self-interest => everyone knows Informed self-interest => everyone knows

everyone else’s true preference and will act in everyone else’s true preference and will act in their own best interesttheir own best interest

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ImportanceImportance

With respect to Mechanism design, this With respect to Mechanism design, this result deals with the question:result deals with the question:

““Would an agent reveal his/her true Would an agent reveal his/her true preference to the principal?”preference to the principal?”

The answer: Only for binary or dictatorial The answer: Only for binary or dictatorial choice schemes => only binary or dictatorial choice schemes => only binary or dictatorial choices are DOM-implementablechoices are DOM-implementable

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RoadmapRoadmap

IntroductionIntroduction Definition of termsDefinition of terms 33 Brief overviewBrief overview

44 ImportanceImportance 1010

DiscussionDiscussion Definition of termsDefinition of terms 1313 Important propertiesImportant properties 1414 Proof of statementProof of statement 1616

ConclusionConclusion

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Definition of termsDefinition of terms

Game form – Any decision making system Game form – Any decision making system in which the outcome depends upon the in which the outcome depends upon the individual actions (strategies) individual actions (strategies)

Dominant strategy – a strategy that gives Dominant strategy – a strategy that gives the best possible outcome to a player no the best possible outcome to a player no matter what strategies others choosematter what strategies others choose

Straightforward game – a game in which Straightforward game – a game in which everyone has a dominant strategyeveryone has a dominant strategy

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PropertiesProperties

Properties of game formsProperties of game formsGame forms leave nothing to chanceGame forms leave nothing to chancePlayers in game forms may or may not have Players in game forms may or may not have

“honest” strategies“honest” strategiesGame forms always have a single outcome – Game forms always have a single outcome –

there are no tiesthere are no tiesGame forms may be used to characterize any Game forms may be used to characterize any

non-chance decision making systemnon-chance decision making system

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PropertiesProperties

Properties of voting schemesProperties of voting schemesVoting schemes are a special case of game Voting schemes are a special case of game

forms in which the players’ preferences are forms in which the players’ preferences are their strategiestheir strategies

Every player in a voting schemes has a true Every player in a voting schemes has a true preference (honest strategy)preference (honest strategy)

Voting schemes do not have to be democratic Voting schemes do not have to be democratic or count all individuals alikeor count all individuals alike

Voting schemes must always have an Voting schemes must always have an outcome, even if the outcome is inactionoutcome, even if the outcome is inaction

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Intuitive proofIntuitive proof

1.1. Given a non-dictatorial voting scheme with more than 3 Given a non-dictatorial voting scheme with more than 3 outcomesoutcomes

2.2. Assume theorem: Every straightforward game form Assume theorem: Every straightforward game form with at least 3 possible outcomes is dictatorialwith at least 3 possible outcomes is dictatorial

3.3. Non-dictatorial => not straightforward => not every Non-dictatorial => not straightforward => not every player / agent has a dominant strategyplayer / agent has a dominant strategy

4.4. No dominant strategy => true preference cannot be No dominant strategy => true preference cannot be dominantdominant

5.5. True preference not dominant => possible for a True preference not dominant => possible for a different preference to give a better outcomedifferent preference to give a better outcome

6.6. Voting scheme cannot guarantee true preference for Voting scheme cannot guarantee true preference for all players and can thus be manipulatedall players and can thus be manipulated

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Formal approach usedFormal approach used

Proving theorem:Proving theorem:

““Every straightforward game form with at Every straightforward game form with at least 3 possible outcomes is dictatorialleast 3 possible outcomes is dictatorial””

is equivalent to proving theorem:is equivalent to proving theorem:

““Any non-dictatorial voting scheme with at Any non-dictatorial voting scheme with at least 3 possible outcomes is subject to least 3 possible outcomes is subject to individual manipulationindividual manipulation””

as shown by previous slideas shown by previous slide

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Formal approach usedFormal approach used

Proved by invoking Arrow Impossibility Proved by invoking Arrow Impossibility TheoremTheorem

Arrow Impossibility Theorem states:Arrow Impossibility Theorem states:““Every social welfare function violates at least Every social welfare function violates at least one of Arrow’s conditionsone of Arrow’s conditions””where Arrow’s conditions are:where Arrow’s conditions are:

1.1. ScopeScope2.2. UnanimityUnanimity3.3. Pair wise determinationPair wise determination4.4. Non-dictatorshipNon-dictatorship

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Formal approach usedFormal approach used

1.1. A social welfare function is generated from a A social welfare function is generated from a straightforward game form with 3+ outcomesstraightforward game form with 3+ outcomes

2.2. The social welfare function is shown to The social welfare function is shown to conform to the first 3 Arrow conditions – conform to the first 3 Arrow conditions – Scope, Unanimity, Pair wise determinationScope, Unanimity, Pair wise determination

3.3. Thus, the function must violate the non-Thus, the function must violate the non-dictatorial condition => it must be dictatorialdictatorial condition => it must be dictatorial

4.4. The dictator of the social welfare function is The dictator of the social welfare function is proven to be the dictator of the game formproven to be the dictator of the game form

5.5. Hence the theorem is provedHence the theorem is proved

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RoadmapRoadmap

IntroductionIntroduction Definition of termsDefinition of terms 33 Brief overviewBrief overview

44 ImportanceImportance 1010

DiscussionDiscussion Definition of termsDefinition of terms 1313 Important propertiesImportant properties 1414 Proof of statementProof of statement 1616

ConclusionConclusion

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ConclusionConclusion

Results proved in the paper:Results proved in the paper:1.1. ““Every straightforward game form with at Every straightforward game form with at

least 3 possible outcomes is dictatorialleast 3 possible outcomes is dictatorial””

2.2. ““Any non-dictatorial voting scheme with at Any non-dictatorial voting scheme with at least 3 possible outcomes is subject to least 3 possible outcomes is subject to individual manipulationindividual manipulation””

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ConclusionConclusion

Comments about the paper:Comments about the paper:The paper is written in a self-contained The paper is written in a self-contained

fashion i.e. one does not need to refer to fashion i.e. one does not need to refer to other sources to decipher the contentother sources to decipher the content

The paper is well-structuredThe paper is well-structuredThe paper leaves the rigorous math proof to The paper leaves the rigorous math proof to

the end making it easy to followthe end making it easy to followThe paper could elaborate on the implications The paper could elaborate on the implications

of the result a bit moreof the result a bit more

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Thank youThank you

EndEnd