Ordinal welfarism ■ Ordinal welfarism pursues the welfarist program Voting and Social in those situations where cardinal measurement of individual welfare is either unfeasible, Choice unreliable or ethically untenable ■ In most real life elections voters are not asked to express more than an “ordinal” opinion of the names on the ballot Chapter 4 Moulin ■ If the outcome depends on intensity of voters’ feelings, a minority of fanatics will influence the outcome more than a quiet majority Ordinal welfarism Ordinal welfarism ■ The identification of welfare with preferences, ■ In the ordinal world collective decision making can only be defined if we specify the set A of feasible outcomes and of preferences with choice, is an intellectual (states of the world), and for each agent i a preference construction at the center of modern economic relation Ri on A. ■ The focus is on the distribution of decision power thinking ■ Two central models of social choice theory: a voting ■ Social choice theory adapts the welfarist program problem and a preference aggregation problem to the ordinalist approach ■ These are the most general microeconomic models of cdm because they make no restrictive assumptions ■ Individual welfare can no longer be separated neither on the set A of outcomes or on the admissible from the set A of outcomes to which it applies preference profile of the agents.
Condorcet versus Borda Where Condorcet and Borda agree ■ 21 voters and three ■ Plurality voting is the most widely used voting No. 6 7 8 candidates a,b,c method voters ■ Plurarily elects a yet b is ■ Each voter chooses one of the competing Top b c a more convincing candidates and the candidate with the largest compromise (a more support wins often below b) c b b ■ Condorcet and Borda argued that plurality voting ■ Borda tally: Score is seriously flawed because it reflects only the a=16,b=27,c=20 distribution of the “top” candidates and fails to ■ Condorcet winner b: Bot a a c take into account entire relation of voters bPc, bPa,cPa Where Borda and Condorcet Where Borda and Condorcet Disagree Disagree ■ The profile of 26 voters ■ Borda’s argument relies on No of 15 11 No of 15 11 scoring convention and three candidates voters voters ■ General family of scoring ■ Plurality winner “a” (also include Borda’s and plurality a b a b Condorcet winner) as special cases: ■ Borda winner is “b” – ■ Plurality: s1=1, sk=0 for all k eleven “minority” voters b c b c ■ Borda sk=p-k for k=1, … ,p dislike “a” more than ■ In this example depending on fifteen “majority” dislike “b” scores either a or b selected c a c a but never c (this flexibility contrasts Condorcet)
Condorcet against Scoring Method Condorcet against Scoring Method ■ 81 voters, 3 ■ b wins for any choice of 30 3 25 14 9 30 3 25 14 9 scores, s between [0,1] with candidates s=0 plurality, s=1/2 Borda ■ “b” is plurality and a a b b c a a b b c ■ c fares badly in both scoring Borda winner and Borda (c much more often between b and a when ■ Condorcet winner "a" b c a c a b c a c a b is first choice than between aPb by 42/29 and aPc a and b when a is first choice by 58/23 ■ a Condorcet winer and is c b c a b c b c a b unaffected by the position of a sure loser c score (b) = 39+30s >score (a)=33+34s > score (c) = 9+17s Top score = 1, bottom 0 and s middle Condorcet cycle The Reunion Paradox ■ Majority relation may Two disjoint groups (34 and n1 n2 n3 10 6 6 12 35 members each) who cycle a b b c vote for same candidates ■ n1+n2>n3=>aPb a c b Candidate “a” is majority b a c a ■ n1+n3>n2=>bPc winner among bottom c c a b group (right-handed) ■ n2+n3>n1=>cPa b a c 18 17 Among top group (left- ■ No Condorcet winner handed) we have a cycle a c ■ Proposed to break and removing weakest link c b a cycle at weakest link leads to “a” c a b b
Voting over Resource Allocation Voting over resource allocation ■ For political elections with a few candidates ■ Majority voting works well in a number of arbitrary preferences are a reasonable allocation problems but produces assumption systematic cycling in others ■ When the issue concerns allocation of ■ Scoring methods are hopelessly resources some important restrictions impractical when the set of A outcomes is come into play large (and typically modelled as an infinite set), also because of IIA property Voting over Time shares ex. 4.5 Single-Peaked Preferences ■ Example 2.6: Location of a Facility ■ Can choose any mixture (x1, … x5) where xi represents u y ( ) | y x | = − − time share and sum to one i i ■ Set N agents partitioned into five disjoint groups of one- F z ( ) minded fans ■ If one group has a majority (>n/2) then that station is a y y * 1 y y * 1 + + Condorcet winner and it is played all the time y y * F ( ) F y ( *) 1 F ( ) < ⇒ < = ⇒ − > 2 2 2 2 ■ If no group has an absolute majority then the majority relation is strongly cyclic. ■ Destructive competition: failure of the logic of private contracting (negative externalities)=> instability and unpredictability
Single-Peaked Preferences Single-Peaked Preferences ■ The coincidence of Condorcet and ■ Given an ordering of the set A, we write x<y Utilitarian optimum depends on particular when x on left of y assumption of common utility = distance ■ we say that z is “between” x and y if either x ≤ ■ However, median of distribution is a z ≤ y or y ≤ z ≤ x Condorcet winner (if not util optimum) for a ■ The preference relation Ri is single-peaked much larger domain of individual with peak xi if xi is the top outcome of Ri and preferences called single-peaked for all other outcomes x prefers any outcome preferences in between. Single-Peaked preferences and Single-Peaked preferences IIA ■ Definition of feasible set far away from A does not matter, e.g., [0,100] median 35
Condorcet method is strategy- Strategy proofness example proof ■ A voter has no incentive to lie strategically when reporting a peak of her preferences ■ Even if a group of voters join forces to jointly misrepresent their peaks, they cannot find a move from which they all benefit Proof: Strategy Proofness ■ Ultimate test of incentive-compatibility in mechanism design ■ Simple truth is always best move (whether or not I have information about other agents messages) ■ Two important examples of strategy-proof mechanisms: majority voting over single-peaked preferences and atomistic competitive equilibrium
Gibbard-Satterthwaite theorem ■ Any voting method defined for all rational preferences over a set A of three or more outcomes must fail the strategy proofness property: at some preference profile some agent will be able to “rig” the election to her advantage by reporting untruthfully ■ Technically equivalent to Arrow’s IT 25
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