Game Theory: Lecture #2 Outline: Information exchange Beauty - - PDF document

Game Theory: Lecture #2

Outline:

• Information exchange
• Beauty contest
• Social choice

Information-Based System

• Last lecture: Engineering for social systems possess unique set of challenges (e.g., inef-

ficiencies emergent behavior, information exchange, etc.)

• Example: Keynesian beauty contest (1936)
• Problem setup:

– Information: 100 pictures published in local newspaper – Goal: Obtain public opinion of most beautiful individual

• Mechanisms:

– M1: Request readers to send in opinions? – M2: Pay readers fixed amount to send in opinions? – M3: Pay readers to send in opinions, where the payment amount depends on the quality of their opinions?

• Issues:

– M1: No incentive to report – M2: No incentive to report truthfully – M3: Incentive to report according to perceived societal preferences

• Q: Are there any reasonable mechanisms available to attain information? M3 seems

reasonable... What are the issues?

• Example: Mathematical formulation of beauty contest

– Two participant each select a number xi ∈ {1, . . . , 100}, i = 1, 2 – Winner = Participant who is closer to 2/3 of the average of x1 and x2. – Payoff = 2/3 of the average of x1 and x2.

• Question: What number will the participants select?
• Answer: 0! Why????

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Voting Paradox

• Q: Are there any reasonable mechanisms for aggregating the opinions of many?
• Example: Voting paradox (Marquis de Condorcet, 1785)

– Fixed monetary budget can go to one cause: Health, Security, or Education – Voters: Left Party: 3 members Middle Party: 4 members Right Party: 5 members – Preferences: Left (3) Middle (4) Right (5) health education security security health education education security health

• Mechanism #1: Single vote, majority rules

– Result: Security – Issue: More prefer Health (7) to Security (5)

• Mechanism #2: Pairwise voting

– Result: Health (7) ≻ Security (5), Security (8) ≻ Education (4) – Social preferences: Health ≻ Security ≻ Education – Issue: More prefer Education (9) to Health (3)

• Mechanism #3: Group voting, strength = population size

– Issue: Would anyone favor this mechanism?

• Mechanism #4: Choice of strongest party (dictatorship)

– Issues?

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Social Choice

• Q: Are there any reasonable mechanisms for aggregating the opinions of many?
• Social Choice Setup: (Kenneth Arrow, 1951)

– Set of alternatives: X = {x1, . . . , xm} – Set of individuals: N = {1, . . . , n} – Preferences: For each individual i ∈ N and pair of alternatives x, x′ ∈ X, then exactly one of the following is satisfied: – x ≻i x′ (i prefers x to x′) – x′ ≻i x (i prefers x′ to x) – x ∼i x′ (i views x and x′ as equivalent) – Express preferences of individual i as qi (list of preferences for all pairs of alternatives)

• Social Choice Function: A function SC(·) of the form:

Societal Preferences = SC(Individuals’ Preferences)

• r mathematically

qN = SC(q1, . . . , qn) where qN encodes a list of preferences for all pairs of alternatives in X.

• Note: Social choice function returns a list of preferences for all pairs of alternatives in
• X. Why is this more desirable than purely top choice?
• Direction:

– Shift focus from case-by-base analysis – Focus on underlying properties mechanism should possess – Temporarily ignore strategic component (i.e., individuals will report truthful prefer- ences)

• We will refer to the properties that our social choice mechanism should satisfy as Axioms.

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Axiom #1: Domain and Range

• Axiom #1 restricts attention to reasonable individual preferences (q1, . . . , qn) and rea-

sonable societal preferences qN

• Reasonable preferences = Preferences that can be express as rankings or ordered lists

– Preferences (qi): For each individual i ∈ N and pair of alternatives x, x′ ∈ X, then exactly one of the following is satisfied: – x ≻i x′ (i prefers x to x′) – x′ ≻i x (i prefers x′ to x) – x ∼i x′ (i views x and x′ as equivalent) – Additional requirements: For all i ∈ N – x ∼i x for all x ∈ A – Completeness: (all pair of alternatives accounted for in qi) – Transitivity: x ≻i x′, x′ ≻i x′′ ⇒ x ≻i x′′ x ≻i x′, x′ ∼i x′′ ⇒ x ≻i x′′ . . . – Summary: Reasonable implies that preferences can be expressed by a list/order

• Ex: Majority rules does not satisfy Axiom 1. Why?
• Reasonable preferences exclude the following phenomena

– Do you prefer Chicken or Steak? Chicken – Do you Steak or Fish? Steak – Do you Fish or Chicken? Fish

• When discussing reasonable properties of a social choice mechanism (i.e., properties that
• ur social choice mechanism should satisfy for certain profiles), we want to make sure

we restrict attention to reasonable preferences

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Axiom #2: Positive Association

• Axiom #2 seeks to ensure that if a given alternative gains in popularity, then this alter-

native should only improve in the resulting societal ranking.

• Formal definition of Axiom #2: Positive Association

– Consider two preference profiles q = (q1, . . . , qn) and q′ = (q′

1, . . . , q′ n)

– Suppose for some pair of alternatives x and y, the preference profiles q and q′ satisfy the following for all i ∈ N: – If x ≻i y in q, then x ≻i y in q′ – If x ∼i y in q, then either x ≻i y or x ∼i y in q′ – If y ≻i x in q, then either x ≻i y, x ∼i y, or y ≻i x in q′ – Then one of following must be true: – If x ≻N y in qN = SC(q), then x ≻N y in q′

N = SC(q′)

– If x ∼N y in qN = SC(q), then x ≻N y or x ∼N y in q′

N = SC(q′)

– If y ≻N x in qN = SC(q), then x ≻N y or x ∼N y or y ≻N x in q′

N = SC(q′)

• Ex: Axiom 2 seeks to avoid situations like the following:

SC x x x y y y y y x x

• =

x y

• and

SC x x x x y y y y y x

• =

y x

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Axiom #3: Unanimous Decision

• This is the most straightforward of all the axioms
• If all voters prefer x to y, then the societal choice should prefer x to y.
• Formal definition of Axiom #3: Unanimous Decision

– Consider any preference profiles q = (q1, . . . , qn) – Suppose for some pair of alternatives x and y, we have x ≻i y for all i ∈ N – Then the societal choice qN = SC(q1, . . . , qn) should also satisfy x ≻N y

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Axiom #4: Independence of irrelevant alternative

• Axiom #4 seeks to ensure that the presence of an irrelevant alternative, i.e., an alternative

z that does not change the relative preference between a pair of alternative x, y, does not change the societal preference

• Formal definition of Axiom #4: Independence of irrelevant alternative

– Let q and q′ be any two preference profiles – Suppose the preference between x and y is the same for each individual in the preference profiles q and q′, i.e., – If x ≻i y in q, then x ≻i y in q′ – If x ∼i y in q, then x ∼i y in q′ – If y ≻i x in q, then y ≻i x in q′ – Then the societal preference between x and y should also be the same, i.e., – If x ≻N y in qN = SC(q), then x ≻N y in q′

N = SC(q′)

– If x ∼N y in qN = SC(q), then x ∼N y in q′

N = SC(q′)

– If y ≻N x in qN = SC(q), then y ≻N x in q′

N = SC(q′)

• Ex: Axiom 4 seeks to avoid situations like the following:

SC     x x x x y y y y y x z z z z z     =   x y z   and SC     z x x x z x z y z y y y z y x     =   y x z  

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Axiom #5: Non-Dictatorship

• Axiom #5 seeks to ensure that there is no dictator. That is, there is no single individual

that dictates the societal choice for any individual preference profiles

• Formal definition of Axiom #5: Non-Dictatorship

– Consider any society with at least 3 individuals – There does not exist an individual {j} such that for any preference profile q = (q1, . . . , qn) the following statements are true for any pair of alternatives x, y: – x ≻N y in qN = SC(q) if and only if x ≻j y in qj – x ∼N y in qN = SC(q) if and only if x ∼j y in qj – y ≻N x in qN = SC(q) if and only if y ≻j x in qj

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Social Choice

• Goal: Derive social choice function SC(·) that satisfies

Reasonable Social Choice = SC(Reasonable Individuals’ Preferences)

• Q: What is a reasonable social choice function?
• Definition: A reasonable social choice function SC(·) must satisfy Axioms 1-5.

– Axiom 1: Domain and Range of f – Axiom 2: Positive Association – Axiom 3: Unanimous Decision – Axiom 4: Independence of irrelevant alternative – Axiom 5: Non-dictatorship

• Q: Does there exist a social choice function SC(·) that satisfies Axioms 1-5? No!
• Theorem (Arrow, 1951): If any social choice function SC(·) satisfies Axioms 1-4,

then the social choice function necessarily does not satisfy Axiom 5.

• Proof - Next Lecture...

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