• Markets are a discovery process that use prices to aggregate dispersed knowledge about scarcity, preferences, and opportunities regarding resources

• Individual decisions maximize individual preferences within constraints

• Politics might be considered a discovery process that uses votes to aggregate dispersed knowledge about individual preferences into a single group choice

• “Social choice theory” studies how to aggregate individual preferences into a consistent group preference to reach a collective decision for a group

• Collective choice aims to maximize “group preferences” within constraints

• In practice: analysis of alternative voting rules

$$\left[\begin{array}{c}A\\ B\\ C\end{array}\right],\left[\begin{array}{c}B\\ A\\ C\end{array}\right],\cdots ,\left[\begin{array}{c}C\\ B\\ A\end{array}\right]⟹\left(\begin{array}{c}A\\ C\\ B\end{array}\right)$$

• Voting rule: Broccoli vs. Carrots; then Winner vs. Apples

• Voting rule: Broccoli vs. Carrots; then Winner vs. Apples
  1. Broccoli: 2 vs. Carrots: 1

• Voting rule: Broccoli vs. Carrots; then Winner vs. Apples

  1. Broccoli: 2 vs. Carrots: 1
  2. Broccoli: 1 vs. Apples: 2

• Result: Apples win

• Voting rule: Broccoli vs. Carrots; then Winner vs. Apples

• Ben likes Apples the least

• He recognizes that under this voting rule, Apples will win

• Voting rule: Broccoli vs. Carrots; then Winner vs. Apples

• Ben likes Apples the least

• He recognizes that under this voting rule, Apples will win

• Suppose instead, in the first round, he votes for Carrots instead of Broccoli (even though he prefers Broccoli)

• Voting rule: Broccoli vs. Carrots; then Winner vs. Apples

  • 1: Broccoli: 1 vs. Carrots 2
  • 2: Carrots: 2 vs. Apples 1

• In effect, a vote for Carrots against his preferences in the first round ensures Carrots win the second round

• This is strategic voting: voting against one's true preferences to change the (often a later-round) outcome

Mr. Putin [is surprisingly popular] with ordinary Russians, most of whom preferred the stability that he brought to the more democratic chaos of Boris Yeltsin." - The Economist (June 9, 2012) Review of Masha Gessen, 2012, The Man Without a Face: The Unlikely Rise of Vladimir Putin

"[T]he most popular topic in thinking today is trying to understand how systems that are not Western, not liberal, not liberal democracies, and perhaps not even democracies, can nevertheless make their nations successful." Source

• Example: Consider a committee of three members (A, B, C)

• Vote is on how much to spend on budget to host a party

• Height is level of utility for each voter

• Each voter has single-peaked preferences
  • Ideal point of how much to spend (peak)
  • Utility decreases with distance (in each direction) away from ideal point

• Suppose any voter is allowed to make a proposal, e.g.

  • A will propose a budget of $50
  • B will propose a budget of $100
  • C will propose a budget of $300

• The question is, what will happen?

• Consider pairwise voting between alternatives...

• Suppose two proposals are put forth: $50 and $300

• Voters vote for proposal that is closer to their ideal point:

  • $50: A and B
  • $300: C
  • $50 wins

• Suppose two proposals are put forth: $50 and $100

• Voters vote for proposal that is closer to their ideal point:

  • $50: A
  • $100: B and C
  • $100 wins

• Suppose two proposals are put forth: $100 and $300

• Voters vote for proposal that is closer to their ideal point:

  • $100: A and B
  • $300: C
  • $100 wins

• $100, if it ever gets proposed, is a Condorcet winner, it will defeat any alternative

  • $100 $\succ$ $50
  • $100 $\succ$ $300

• This is because it is the median, it has enough supporters of alternatives on either side of it

  • Each side would rather support the median than the platform on the opposite side

• B is the “median voter” who has the median preference

• Median Voter Theorem (MVT): if preferences are single-peaked along a single issue dimension, the median preference will always beat any alternative in a pairwise vote

  • It is a Condorcet winner

• Suppose C goes off the deep end and proposes to spend $1,000 on the party

• What happens to the outcome?

• Suppose C goes off the deep end and proposes to spend $1,000 on the party

• What happens to the outcome? Nothing!

• Politics is resistant to changes at the margin, or at the fringes!

  • Only if the median moves will the outcome change

• Now consider a Presidential election

  • many voters, each with own ideal preference

• Aggregated together along a single dimension

  • e.g. "left" vs. "right"; "low tax rates" to "high tax rates", etc.

• Median Voter Theorem implies the median preference (M) will determine the outcome

• Median Voter Theorem implies the median preference (M) will determine the outcome

• Note the median need not be exactly in the middle, or median can shift

• Imagine two candidates, A and B in an election, who randomly start somewhere on the spectrum

• Imagine two candidates, A and B in an election, who randomly start somewhere on the spectrum

• Voters vote for the candidates closest to them on spectrum

  • B is closer to median, gets more votes
  • A is more extreme, gets fewer votes

• If A moves closer to the median (A'), gains more votes (at B's) expense

• The closer to the median (M) a candidate gets, the more likely they are to win

• Imagine a third candidate, C on the spectrum

• Imagine a third candidate, C on the spectrum

• Again, voters vote for who is closest to them

  • Splits the vote of candidate that is closest to C (i.e. A)

• Implication: Third parties cannot win, and may harm party that they are closest to on issues

• Imagine an election of where to move Tennessee's capital

• Voter preferences in table

• Memphis wins, with 42% of the vote
  • Even though 58% of voters preferred Memphis the least!

• Memphis (42%) and Nashville (26%) win first round

• Second round:

  • Memphis: 42%
  • Nashville: 58%

• Nashville wins

• Memphis (42%) and Nashville (26%) win first round

• Second round:

  • Memphis: 42%
  • Nashville: 58%

• Nashville wins