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 x i ∈ { 1 , . . . , 100 } , i = 1 , 2 – Winner = Participant who is closer to 2 / 3 of the average of x 1 and x 2 . – Payoff = 2 / 3 of the average of x 1 and x 2 . • Question: What number will the participants select? • Answer: 0! Why???? 1

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? 2

Social Choice • Q: Are there any reasonable mechanisms for aggregating the opinions of many? • Social Choice Setup: (Kenneth Arrow, 1951) – Set of alternatives: X = { x 1 , . . . , x m } – 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 q i (list of preferences for all pairs of alternatives) • Social Choice Function: A function SC ( · ) of the form: Societal Preferences = SC ( Individuals’ Preferences ) or mathematically q N = SC ( q 1 , . . . , q n ) where q N 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. 3

Axiom #1: Domain and Range • Axiom #1 restricts attention to reasonable individual preferences ( q 1 , . . . , q n ) and rea- sonable societal preferences q N • Reasonable preferences = Preferences that can be express as rankings or ordered lists – Preferences ( q i ): 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 q i ) – 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 our social choice mechanism should satisfy for certain profiles), we want to make sure we restrict attention to reasonable preferences 4

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 = ( q 1 , . . . , q n ) 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 q N = SC ( q ) , then x ≻ N y in q ′ N = SC ( q ′ ) – If x ∼ N y in q N = SC ( q ) , then x ≻ N y or x ∼ N y in q ′ N = SC ( q ′ ) – If y ≻ N x in q N = 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: �� x x x y y � x �� � SC = y y y x x y and �� x x x x y � y �� � SC = y y y y x x 5

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 = ( q 1 , . . . , q n ) – Suppose for some pair of alternatives x and y , we have x ≻ i y for all i ∈ N – Then the societal choice q N = SC ( q 1 , . . . , q n ) should also satisfy x ≻ N y 6

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 q N = SC ( q ) , then x ≻ N y in q ′ N = SC ( q ′ ) – If x ∼ N y in q N = SC ( q ) , then x ∼ N y in q ′ N = SC ( q ′ ) – If y ≻ N x in q N = SC ( q ) , then y ≻ N x in q ′ N = SC ( q ′ ) • Ex: Axiom 4 seeks to avoid situations like the following:       x x x x y x  = SC y y y y x y      z z z z z z and       z x x x z y  = SC x z y z y x      y y z y x z 7

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 = ( q 1 , . . . , q n ) the following statements are true for any pair of alternatives x, y : – x ≻ N y in q N = SC ( q ) if and only if x ≻ j y in q j – x ∼ N y in q N = SC ( q ) if and only if x ∼ j y in q j – y ≻ N x in q N = SC ( q ) if and only if y ≻ j x in q j 8

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... 9