Arrow’s Theorem zCompleteness - (Transitivity) If x beats y and y beats z in the social ordering then x beats z. zUniversal Domain - the rules must apply to all possible combinations of individual preference orderings. zPareto Optimality- x beats y whenever everyone individually prefers x to y. zIndependence of irrelevant alternatives - The collective preferences between two alternatives never depend on individual preferences regarding other alternatives. zNon-Dictatorship. No individual is so powerful that, for every pair of alternatives x and y, x beats y socially even if she prefers x to y while everyone else prefers y to x. CUPID
Implications zArrow showed that these 5 criteria are incompatible with any system of preference aggregation. zExample: Simple Majority Rule
Simple Voting Paradox zMajority Rule Violates Transitivity xSuppose we have a preference ordering as follows: zProblem: xCollective preferences are cyclic and every feasible alternative is unstable, such that: x>y, y>z; and z>x. xNo unique stable outcome: z x y
Significance zWe can't predict what will be the outcome under a majority rule setting. zThere is no true social welfare maximizing outcome zThen majority government cannot be modeled as maximizing anything, if any outcome is possible no matter how unrepresentative.
When Does Stability Occur? zBlack’s Theorem: yOne-dimensional issue space yIdeal (bliss) points: the one point on the line they prefer to all others yThe only stable point on the line is the median xmxm x6x6 x7x7 x3x3 x2x2 x1x1 x5x5 yX m is the Condorcet winner because it is the point that beats all others.
Median Voter Result zIn general, for any distribution of voters with single peakedness there will be a median voter. zIf we assume that x1) Individuals have single peaked preference in a one-dimensional space and x2) A proposal can be freely amended yThen the outcome is always the ideal point of the median voter.
Plott Conditions zIn more than one dimension, however, no single equilibrium point exists. zUnless… the Plott Conditions are met: yEach individual’s ideal point can be paired with another that is exactly on the opposite side of the Median X M. x5x5 x7x7 x6x6 x3x3 x2x2 x1x1 xMxM
Chaos Theory zDivide the Dollar Game xDistributive politics has no unique outcome zDevil Agenda Setter… things are getting worse: xIf no Condorcet winner exists, then majority rule voting can lead to anywhere in the policy space. x1x1 x3x3 x2x2
Strategic Voting zAssume individuals vote sophisticatedly: yA bill is proposed along with an amendment a = amended bill b = bill q = status quo yPreference are: ySequence: xAn amendment is voted on against the bill xThen which ever wins is pitted against the status quo.
Structure Induced Equilibrium zQuestion: Why do we observe so much stability? xGovernmental institutions are stable xPolicy change in incremental zAnswer: Institutions Create Stability xIf a policy space is divided among legislators xAnd these legislators have agenda control and special rights xThen outcomes will not be chaotic zProcedures induce Institutional Equilibrium
Committee System & Stability yCommittees are given monopoly power over a jurisdiction. yPolicy may be unrepresentative of median chamber, but it is stable.