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Game Theory

Game Theory. Section 3: Social Choice. Agenda. Motivating questions and modus operandi Key terms May’s Theorem Arrow’s Theorem The Condorcet winner Condorcet Procedure: Voting Scheme Cycling Single-peaked preferences Median Voter Theorem. Motivating Questions.

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Game Theory

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  1. Game Theory Section 3: Social Choice

  2. Agenda • Motivating questions and modus operandi • Key terms • May’s Theorem • Arrow’s Theorem • The Condorcet winner • Condorcet Procedure: Voting Scheme • Cycling • Single-peaked preferences • Median Voter Theorem

  3. Motivating Questions To Consider the Preferences of a Society • What are reasonable/desirable characteristics of the social (aggregate) preference? • What are the reasonable/desirable characteristics of individual preferences? • How to aggregate individual preferences? • Which schemes are possible, and which are impossible, and what are the real-world implications? • Voting, democracy, institutional design • Under what circumstances will it be rational to reveal information (or preferences) honestly? • When won’t it be rational to reveal info in this way?

  4. Modus Operandi (1) encourage honest information-sharing, and(2) use that information in a reasonable way? Are there systems which • First: Define the setting • Consider a case of perfect information • Define basic rules for allowable preferences • Second: Now, consider rules that are still admissible • Identify new allowable rules with desirable properties • Third: Decide whether these rules provide incentive for honest revelation of preferences Remember: We work with ordinal preferences (including ties)

  5. Key Terms • Social Welfare Functional (SWF) • Societal decision rule that creates a social ranking by aggregating individual rank orders • “Desireable” properties of the SWF • Paretian Property: unanimity rules • Symmetry among Agents: voting can be anonymous without changing the result • Neutrality among Alternatives: given 2 options, labels given to the possible alternatives don’t matter • Positive Responsiveness: if the social ranking is indifferent between two alternatives, and one individual changes preference in favor of one alternative, the social preference must change to favor that alternative

  6. Key Terms Additional “Desirable” Properties of a SWF • Transitivity: • If x is preferred to y, AND • If y is preferred to z, • Then x must be preferred to z • Independence of Irrelevant Alternatives (IIA) • Social preference for A vs. B is independent of individual’s preferences for all alternatives except A vs. B • Changes in individual preferences that don’t change preferences for A vs. B do not change social preference for A vs. B • The social preference for A vs. B is NOT affected by changes in individual i’s preference for A vs. CorD vs. E • IIA is also known as the pairwise independence property

  7. May’s Theorem: Step I Define the Setting: Consider the case of complete information, anddefine basic rules for allowable preferences Given • N individuals • Revealing preferences over two outcomes • i’s preferences:pi = (x>y)or(x>y)or(x~y) • SWF(p1, p2 , …, pN) = (x>y)or(x>y)or(x~y) • Allowing a social preference that complies with • Symmetry among agents (anonymity) • Neutrality among alternatives • Positive responsiveness

  8. May’s Theorem: Step II Consider the rules that are still admissible • Symmetry among agents • The # of people with given preferences matters, not who the people are • Neutrality among alternatives (2 alternatives) • None of the alternatives is favored by the SWF • The support for alternative A required to get A >s B is identical to the support for B necessary for B >s A • Positive responsiveness • Departures from a tie must break a tie So, what is still admissible?

  9. May’s Theorem: Step III May’s Thm. A social welfare functionalF(p1, p2, … , pN)corresponds to majority voting iff it satisfies:(1) symmetry among agents; (2) neutrality among alternatives; and (3) positive responsiveness In majority rule, you should vote for what you want. Either (1) no effect (2) break a tie (3) tie But a vote for what you don’t want can hurt you. Everything thus far has been in a two-alternative world Now, decide whether these rules provide incentive for honest revelation of preferences

  10. Arrow’s Theorem:Enter the Third Alternative Define the Setting: Consider the case of complete information, anddefine basic rules for admissible preferences Given • N individuals • Revealing preferences over three outcomes • i’s preferences:pi = (x>y~z), for example • SWF(p1, p2 , …, pN) = (x~z<y) for example • Allowing a social preference that complies with somewhat weaker properties than last time • Transitivity • Unanimity • Independence of Irrelevant Alternatives

  11. Arrow’s Theorem: Step II Consider the rules that are still allowable • Transitivity • Social preferences must satisfy A > B, B > C, A > C • Unanimity rules • IIA (pairwise independence): • Only myA vs. Bis relevant to society’sA vs. B • For itsA vs. B, society only needs myA vs. B Condorcet Preferences Alternative Preferences Is social order in Condorcet world going to match the social order in the Alternative world? B vs. C? A vs. B? A vs. C?

  12. Arrow’s Theorem: Step II Consider the rules that are still admissible • Transitivity • Social preferences must satisfy A > B, B > C, A > C • Unanimity rules • IIA (pairwise independence): • Only myA vs. Bis relevant to society’sA vs. B • For itsA vs. B, society only needs myA vs. B Condorcet Preferences Alternative Preferences Borda Count allowed?Majority rule allowed?Arbitrary SWF allowed? Can we reject Condorcet preferences?Is a dictatorship the only allowable SWF?

  13. Arrow’s Theorem: Step III With 3 or more alternatives, the only social welfare functional F(α1, α2, … , αN)satisfying(1) unanimity; (2) transitivity; and (3) independence of irrelevant alternativesand no restriction on the domain of preferences is a dictatorship, i.e., a social choice matching the individual preferences of a particular person (the dictator) regardless of others’ preferences Arrow’s Thm. Now, decide whether these rules provide incentive for honest revelation of preferences If you’re the dictator, you should vote your preference. If you’re not, you don’t matter…Else, 1, 2, and/or 3 don’t apply. If it’s IIA, be strategic! Consider the “irrelevant” alternatives!

  14. The Condorcet Winner • So, three or more alternatives spells trouble • Run-off rules are a coping mechanism • Because with 2 alternatives, you can use majority rule and get honest voting & stable outcomes • But Arrow’s Theorem throws a wrench in it • Can’t always be honest voting in elimination stages • One voting scheme: Condorcet procedure, which asks • Can one option win a majority against each of the others? • If so, that option is known as the Condorcet Winner

  15. Condorcet Winner Example Category = No. of voters = Most Preferred Least Preferred What constitutes a majority? Is ‘a’ the Condorcet Winner? Is ‘e’ the Condorcet Winner? Do > than ½ voters actually have the CW as their 1st choice? This example was taken from Shepsle, Bonchek “Analyzing Politics”, which took it from Malkevitch 1990.

  16. Cycling: The “Money Pump” No Condorcet Winner Raiffa called these “money pump” preferences; if we’re at any one policy, we could conceivably hold a new vote to change to any other policy, and just go round and round. Most Preferred Least Preferred

  17. Single-Peaked Preferences (S.P.P) A restriction on preferences that they are monotonically decreasing away from an ideal point Obviates Condorcet’s cycling problem Illustration: 3 options & strict preferences WLOG A > B > C C > B > AC > A > B B > C > A B > A > CA > C > B Can be ruled out by S.P.P. • Assume a society: {(A,B,C), (A,B,C), (B,C,A), (C,B,A), (B,A,C)} Utility Is S.P.P violated? Is there a Condorcet winner? Why? Policy A B C

  18. Median Voter Theorem With single-peaked preferences and an odd number of voters, the median of the voters’ ideal points is the Condorcet winner. a1 m a2 Who is the median voter? Why is the MV the CW?

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