Econ 452Voting

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Voting rules: (Dixit and Skeath, ch 14)

Recall parkland provision decision:

Assume- n=10; - total cost of proposed parkland=38;- if provided, each pays equal share = 3.8- there are two groups of individuals in

society: 8 have 1 1 = , and remaining 2have 2 16. =

So: efficient to provide public good, sincesum of benefits (40) > total cost (38).

Suppose held a referendum on provision;would parkland be provided (with this financingscheme)?

Depends.on voting scheme used.

Possible schemes?

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General: classifiy vote aggregation methodsby number of options or candidates at any time:three types:

A. Binary: two alternatives - if only two, aggregate votes by majority

rule;

- if more than two: sequence of pairwisevotes, winner by majority rule;

- types?1. Condorcet method: round robin;

overall winner is the one who defeatsall others in pairwise contests.

2. successive elimination (variousindices)

B. Plurative methods: simultaneouslyconsider multiple options

1. plurality rule: - each voter has one vote; - candidate with most votes wins;- plurality may be less than majority

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2. Borda count: - voters rank order all options

- assign points to each, based on own ranking

- if 3 alternatives, then most preferred gets 3 points, least preferred gets 1

- sum points across voters - highest wins

3. Approval voting: - vote for any, and all, of which

approve;- winner(s): highest number of votes

C Mixed methods- multistage- combine binary and plurative

Voting paradoxes:

1. Condorcet Paradox: intransitivity of social ordering- best known (among economists)- may be no winner - that is, no alternative

which is successful against all others:

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Example: suppose three individuals(1,2,3) choosing one of three options(A,B,C); preferences are

- Ind'l 1: A;B;C- Ind'l 2: B;C;A- Ind'l 3: C;A;BResults? A wins in (A,B); B wins in (B,C);

C wins in (A,C)- social ordering: A;B;C;A

2. Reversal Paradox: - from Borda count: when slate of

candidates changes after votes, and newvote held

- need at least 4 candidates (A,B,C,D)- Suppose 7 voters

1 2 3 4 5 6 7A D A B D D BB A B C A A CC B C D B B DD C D A C C A

- With all four options, winner is A:A: 2*4 + 3*3 + 2*1=19 B: 2*4 + 2*3 + 3*1=15C: 2*3 + 2*2 + 3*1=13 D: 3*4 + 2*2 + 2*1=18

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Now: suppose discovered that C not validalternative - vote again, with A,B,D on ballot(same voters):

1 2 3 4 5 6 7A D A B D D BB A B D A A DD B D A B B A

Now, winner is D: A: 2*3 + 3*2 + 2*1=14B: 2*3 + 2*2 + 3*1=13D: 3*3 + 2*2 + 2*1=15

(eliminated an irrelevant alternative?)

3. Agenda Paradox:- binary elections - winner goes on, loser

doesn't- matters who meets whom first.

4. Different methods give different outcomes

Example: 100 voters, in 3 distinct groups:40: A B C; ;25: B C A; ;35: C B A; ;

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Who wins election?

a) plurality: A

b) Borda count: each voter assignsnumbers 1,2,3 to candidates - 3indicating "most preferred"

A obtains 40x3 + 60x1 = 180B " 25x3 + 75x2 = 225C " 35x3 + 25x2 + 40x1 = 195Hence B wins.

c) Majority run-off: use first round toeliminate one alternative (lowestnumber of votes); second round pairstop two from first round. Here, C wins.

Is there a "best", reasonable voting rule?

No.

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Arrow's "impossibility theorem"

Six criteria for aggregation of preferences1. complete ranking2. transitive ranking3. Pareto property: unanimous

preferences within populationreflected in social ranking

4. ranking not independent ofpreferences of individuals in soc'y

5. ranking not dictatorial - not thereflection of the preferences of oneindividual

6. independent of irrelevant alternatives

Theorem? no such ranking exists.

How to choose between flawedmechanisms? One criterion is manipulability

- how easy is it to affect outcome bystrategic voting - not voting in accord withown preferences, to produce a result whichis more in accord with preferences.

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Consider a voting game in which 3 players(denoted 1, 2, and 3) are deciding amongthree alternatives (A, B, and C). AlternativeB is the "status quo" and alternatives A andC are "challengers". In the first stage,players choose which of the two challengersshould be considered; they do this bycasting votes for either A or C, with themajority choice being the winner andabstentions not allowed. In the secondstage, players vote between the status quo(B) and the winner of the first stage, withmajority rule again determining the winner.The players care only about the alternativethat is finally selected. The payoffs are

1 2 3( ) ( ) ( ) 2;u A u B u C= = = 1 2 3( ) ( ) ( ) 0;u B u C u A= = = 1 2 3( ) ( ) ( ) 1.u C u A u B= = =

Suppose that at each stage each player votesfor the alternative they most prefer as the finaloutcome.

a) What would the outcome be? Do thesestrategies constitute a Nash equilibrium?